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[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.", "problem_activity": "Narrative\nIn this activity, students identify right triangles. They also explain why certain given shapes are not right triangles. As they determine whether a triangle is a right triangle, students may use the corner of an index card or a piece of paper, use a protractor, or trace an angle in a triangle to compare with an angle elsewhere that they know to be a right angle. In doing so, students practice choosing tools strategically (MP5).\nRequired Materials\nMaterials to Gather\nIndex cards\nPatty paper\nProtractors\nLaunch\nGroups of 2\nProvide access to index cards, patty paper, and protractors.\nActivity\n5 minutes: independent work time\n2 minutes: partner discussion\nMonitor for students who use tools to check angles and those who estimate them visually. Ask the latter: “How might we check that the angles that look like right angles are\n\\(90^\\circ\\)\n?”\nStudent Facing\nIdentify all shapes that are right triangles. For each right triangle, mark the right angle with a small square.\n[IMG]\nExplain why the other shapes are not right triangles.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAdvancing Student Thinking\nStudents may not identify triangles C or D because of the orientation.\nConsider asking:\n“How can we show that a triangle has a right angle?”\n“Where else do you see triangles that may have perpendicular sides?”\n“What tools could you use to see if each triangle has any right angles?”\nActivity Synthesis\nInvite students to share their responses and how they found the right triangles in the set.\n“Were there any triangles that appeared to be right triangles but are not, or don’t appear to be right triangles but are?” (Triangle H looks like one but isn’t.)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.", "problem_activity": "Activity\nIn this warm-up, students considers what happens if an object is launched up in the air unaffected by gravity. The work here serves two purposes. It reminds students that an object that travels at a constant speed can be described with a linear function. It also familiarizes students with a projectile context used in the next activity, in which students will investigate a quadratic function that more realistically models the movement of a projectile—with gravity in play.\nStudents who use a spreadsheet to complete the table practice choosing tools strategically (MP5).\nLaunch\nAsk a student to read the opening paragraph of the activity aloud. To help students visualize the situation described, consider sketching a picture of a cannon pointing straight up, 10 feet above ground. Ask students to consider what a speed of 406 feet per second means in more concrete terms. How fast is that?\nStudents may be more familiar with miles per hour. Tell students that the speed of 406 feet per second is about 277 miles per hour.\nConsider arranging students in groups of 2 so they can divide up the calculations needed to complete the table. Provide access to calculators, if requested.\nStudent Facing\nA cannon is 10 feet off the ground. It launches a cannonball straight up with a velocity of 406 feet per second.\nImagine that there is no gravity and that the cannonball continues to travel upward with the same velocity.\nComplete the table with the heights of the cannonball at different times.\n| seconds | | | | | | |\n|------------------------------|---|---|---|---|---|---|\n| distance above ground (feet) | | | | | | |\nWrite an equation to model the distance in feet,\n\\(d\\)\n, of the ball\n\\(t\\)\nseconds after it was fired from the cannon if there was no gravity.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nAsk students how the values in the table are changing and what equation would describe the height of the cannonball if there were no gravity. Even without graphing, students should notice that the height of the cannonball over time is a linear function given the repeated addition of 406 feet every time\n\\(t\\)\nincreases by 1.\nTell students that, in the next activity, they will look at some actual heights of the cannonball.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.", "problem_activity": "Task\nHow many rectangles are in this picture?\n[IMG]\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.", "problem_activity": "Stage 4: Whole Number and Fraction Factors\nRequired Preparation\nMaterials to Gather\nColored pencils, crayons, or markers\nNumber cubes\nPaper clips\nMaterials to Copy\nBlackline Masters\nRectangle Rumble Stage 4 Spinner\nRectangle Rumble Stage 4 Grid\n\nNarrative\nStudents generate factors with a number cube and a spinner with fractions with denominators of 3, 6, or 12. Students use a\n\\(24 \\times 24\\)\ngrid that represents 16 square units.\nAdditional Information\nEach group of students need a paper clip, a number cube, and two different color writing utensils.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.", "problem_activity": "Problem 1\nA small box measures 5 in. across, 8 in. deep, and 6 in. tall. A book that measures 5\n$$\\times$$\n8\n$$\\times$$\n1 in. fits perfectly into the bottom of the box.\n[IMG]\na. What is the volume of one book? How many books will fit into the box? What is the volume of all of the books that will fit into the box?\nb. What is the volume of the box? How is this related to the volume of one book?\nc. A medium box measures 12 inches tall. A different book with dimensions 8\n$$\\times$$\n10\n$$\\times$$\n1 in. fits perfectly into the bottom of the medium box. What is the volume of the medium box?\nProblem 2\nThe same rectangular prism is positioned in two different ways, as shown below. For each prism, use the rectangular face that the prism is “sitting on” as the base of the prism.\n[IMG]\nFor each prism, find the area of the base and multiply by the height. Show that this is the same volume as when all three dimensions are multiplied together.\nProblem 3\nA rectangular prism has a base area of\n$${4{3\\over8}}$$\ninches\n$${^2}$$\n, a width of\n$$1 \\frac{1}{4}$$\ninches, and a height of\n$${6 {1\\over2}}$$\ninches. Kirby writes the equation\n$$V=4\\frac{3}{8}\\times1\\frac{1}{4}\\times6\\frac{1}{2}$$\nto represent the volume of the prism.\nDo you agree with Kirby's equation? Explain and find the volume of the prism.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.", "problem_activity": "Optional activity\nIn this activity, students begin by practicing their understanding of proportions and and probabilities by examining the data set they have available. In the fourth problem, students obtain a sample from the population using tools they choose (MP5)and examine the sample they selected to compare it to the expected proportions and probabilities calculated in the first 3 problems.\nThe problems are intended for students to use their own data set to answer. Although they are kept in pairs for the entire lesson, this activity should be done individually.\nLaunch\nKeep students in the same groups of 2. Give students 5–7 minutes of quiet work time followed by a whole-class discussion.\nIf possible, allow students to use their chosen method of random sampling to obtain a sample of 10 for this activity. Have items such as paper clips, scissors, 10-sided polyhedra, and other materials available for student use. The blackline master for the first activity in this lesson contains accurate spinners that could be used to select a random sample.\nAction and Expression: Internalize Executive Functions.\nChunk this task into more manageable parts. After students have solved the first 2-3 problems, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far as well as any questions they have before continuing.\nSupports accessibility for: Organization; Attention\nStudent Facing\nContinue working with the data set your teacher gave you in the previous activity. The data marked with a star all came from students at Springfield Middle School.\nWhen you select the first value for your random sample, what is the probability that it will be a value that came from a student at Springfield Middle School?\nWhat proportion of your entire sample would you expect to be from Springfield Middle School?\nIf you take a random sample of size 10, how many scores would you expect to be from Springfield Middle School?\nSelect a random sample of size 10.\nDid your random sample have the expected number of scores from Springfield Middle School?\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nThe purpose of this discussion is to connect the ideas of probability and random sampling from the unit.\nConsider these questions for discussion:\n“How is selecting a sample at random connected to probability?” (A random sample should give each value an equal chance of being chosen. Therefore, each value has a\n\\(\\frac{1}{100}\\)\nprobability of being chosen.)\n“How could we simulate the probability of getting at least 2 values in the sample of 10 from Springfield Middle School?” (Since 20% of the values come from Springfield Middle School, we could put 10 blocks in a bag with 2 colored red to represent Springfield Middle School. Draw a block from the bag, and if it is red, it represents a score from Springfield Middle School; replace the block and repeat. Get a sample of 10 and see if the sample has at least 2 red blocks. Repeat this process many times and use the fraction of times there are at least 2 red blocks as an estimate for the probability that a random sample will have at least 2 scores from Springfield Middle School.)\nRepresenting, Speaking: MLR7 Compare and Connect.\nInvite students to create a visual display of their random sample of size 10 and response to the question: “Did your random sample have the expected number of scores from Springfield Middle School?” Invite students to investigate each other’s work and compare their responses. Listen for the language students use to describe a random sample and assign a probability of each value being chosen. This will help students connect the ideas of probability and random sampling through discussion.\nDesign Principle(s): Optimize output (for representation); Cultivate conversation\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Understand a fraction as a number on the number line; represent fractions on a number line diagram.\nGrade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.", "problem_activity": "Narrative\nThe purpose of this What Do You Know About _____ is to invite students to share what they know about and how they can represent the number\n\\(\\frac{1}{8}\\)\n.\nLaunch\nDisplay the number.\n“What do you know about\n\\(\\frac{1}{8}\\)\n?”\n1 minute: quiet think time\nActivity\nRecord responses.\n“How could we represent the number\n\\(\\frac{1}{8}\\)\n?”\nStudent Facing\nWhat do you know about\n\\(\\frac{1}{8}\\)\n?\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\n“What connections do you see between different answers?”\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Use place value understanding to round multi-digit whole numbers to any place.\nGrade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.", "problem_activity": "Problem 1\na. What do you notice? What do you wonder?\n[IMG]\nb. Determine the value of the point on each number line in Part (a) above.\nProblem 2\na. Label each point with the value it represents.\n[IMG]\n[IMG]\n[IMG]\nb. Locate and label each number on the number line. Label the intervals with the numbers they represent if it helps.\n60\n[IMG]\n897\n[IMG]\n364\n[IMG]\nProblem 3\na. Plot 55 on a number line from 50 to 60.\nb. Plot 55 on a number line from 0 to 100.\nc. Plot 55 on a number line from 0 to 1,000.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Explain and use the relationship between the sine and cosine of complementary angles.", "problem_activity": "Narrative\nThe purpose of this activity is to introduce students to a new unit, the hundred. Students used connecting cubes to make tens in grade 1 and used tens and ones to count to and represent numbers within 120.\nIn an earlier unit, students were introduced to base-ten blocks and used base-ten diagrams to represent sums and differences within 100. Students build on this understanding as they use blocks to represent a starting number (96) and add ones until they reach a total value of 100. As students discover they have 10 tens, monitor for language they use to describe the total value of the blocks and the connections they make to their previous work with ones, tens, and three-digit numbers (MP7).\nEngagement: Provide Access by Recruiting Interest.\nGive the students a context to connect to their own lives. Tell the students that the base-ten blocks represent gum. The ones are single sticks of gum and the tens are packs of gum.\nSupports accessibility for: Conceptual Processing, Attention\nRequired Materials\nMaterials to Gather\nBase-ten blocks\nLaunch\nGroups of 2\nGive students access to base-ten blocks.\nDisplay photo of 8 tens and 16 ones.\n“Andre was using base-ten blocks to represent numbers. How many do you see? How do you see them?” (96. I see 8 tens and 16 ones. I see it as 9 tens and 6 ones.)\n30 seconds: quiet think time\n1 minute: partner discussion\nShare responses.\nActivity\n“Work with a partner to follow along and see what Andre discovered as he adds more blocks.”\n10 minutes: partner work time\nMonitor for students who explain that the value is 100 by:\ncounting by 1 from 80 or by 1 from 96\ncounting by 10\nStudent Facing\nHow many do you see? How do you see them?\n[IMG]\nAndre added more blocks.\n[IMG]\nWhat is the value of Andre’s blocks now?\nHow many tens and ones are in this number?\nAndre made the same number with the fewest amount of blocks possible. Draw a base-ten diagram to show what Andre’s number looks like now. Use your base-ten blocks to help.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAdvancing Student Thinking\nIf students represent 100 with a diagram that includes tens and ones, consider asking:\n“Can you explain how your diagram represents Andre’s total value using the fewest number of blocks possible?”\n“Is there a way to represent the same amount without any ones?”\nActivity Synthesis\nInvite previously identified students to share how they know the value of Andre’s blocks is 100.\n“If Andre used the fewest number of blocks, how many tens did he use?” (He had 10 tens.)\n“What if Andre wanted to use the greatest amount of blocks to show this number? What would he do?” (He’d decompose all the tens to use only ones. He’d use 100 ones.)\n“In this activity, you showed you know that 10 ones is the same as 1 ten. Just like 10 ones is the same as 1 ten, 10 tens is the same as 1 hundred.”\n“A hundred is a unit that is made up of 10 tens.”\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Prove the Laws of Sines and Cosines and use them to solve problems.", "problem_activity": "Warm-up\nStudents examine statements deciding in each case whether the statement is always true, sometimes true, or never true. Since figures are similar if one is the result of reflecting, rotating, translating, and dilating the other, and figures are congruent only as a result of the rigid transformations, congruent figures are similar but not vice versa. Though students have not studied how angles behave under dilations, students have had multiple opportunities to see that dilations influence the size of a shape but not its angles. They also know from work in grade 7 that scaled copies have corresponding angles with the same measure, and dilations have been characterized as a process that makes scaled copies.\nLaunch\nDisplay all 3 statements. Ask students to decide whether each of the statements is always, sometimes, or never true and give a signal when they have reasoning to support their decision.\nStudent Facing\nChoose whether each of the statements is true in\nall\ncases, in\nsome\ncases, or in\nno\ncases.\nIf two figures are congruent, then they are similar.\nIf two figures are similar, then they are congruent.\nIf an angle is dilated with the center of dilation at its vertex, the angle measure may change.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nDiscuss each statement one at a time with this structure:\nPoll the class on their answer choice and display the answers.\nIf everyone agrees on one answer, ask a few students to share their reasoning, recording it for all to see.\nIf there is disagreement on an answer, ask students with opposing answers to explain their reasoning to come to an agreement on an answer.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.", "problem_activity": "Task\nMaterials\nEach pair of students needs: * One worksheet * Two markers of different colors * One pair of dice\nAction\nStudent A rolls the two dice, finds the sum, and traces the number on the worksheet which corresponds to the answer with his/her marker. Student A then passes the dice to Student B who rolls both the dice, finds the sum and traces the correct number on the worksheet with his/her marker. Play continues this way until one of the numbers “wins” (i.e. all of the numbers of that quantity have been traced).\n[IMG]\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Display numerical data in plots on a number line, including dot plots, histograms, and box plots.", "problem_activity": "A city planner for a neighborhood of Boston is trying to determine if a new park should be built in the neighborhood. She goes around the neighborhood, knocking on doors, asking, “How many children under the age of 7 live in this household?” The data she collected from 28 households is shown in the frequency table below.\n| ** # of children under 7 years old in the household ** | ** Frequency ** |\n|---------------------------------------------------------------------------|------------------------------------|\n| 0 | 9 |\n| 1 | 6 |\n| 2 | 8 |\n| 3 | 4 |\n| 4 | 0 |\n| 5 | 1 |\na. Create a dot plot to represent the data. Describe your dot plot in words.\nb. What number would you use to represent the center of the data? Explain why you chose that number.\nc. Based on the data, what would you recommend to the city planner about the new park? Does this neighborhood seem like a good place to put the new park? Explain why or why not using the data.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Distinguish between correlation and causation.", "problem_activity": "Given the following data set, what statistical questions could you ask?\n[IMG]\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.", "problem_activity": "Narrative\nThe purpose of this activity is for students to find the value of quotients where the divisor is less than 1 and where the dividend is large enough that drawing a complete diagram is cumbersome. Instead, students are encouraged to use a diagram to find\nhow many divisor sized groups are in 1 whole and then multiply the dividend by that number to find the value of the quotient. When students use the value of\n\\(1 \\div 0.2\\)\nand multiplication to find the value of\n\\(12 \\div 0.2\\)\nthey are using regularity in reasoning (MP8).\nThis activity uses\nMLR1 Stronger and Clearer Each Time.\nAdvances: Conversing.\nAction and Expression: Internalize Executive Functions.\nInvite students to plan a strategy, including the tools they will use, for finding the value of the expression. If time allows, invite students to share their plan with a partner before they begin.\nSupports accessibility for: Conceptual Processing, Memory\nRequired Materials\nMaterials to Copy\nSmall Grids\nLaunch\nGroups of 2\nGive students access to blackline masters with hundredths grids.\nActivity\n3–5 minutes: independent work time\nMonitor for students who use the following strategies for problem 1:\nreason about how many divisor-sized groups are in 1 whole, possibly with a diagram, and then multiply by the dividend\nuse multiplication equations to find the value of the division expressions\nStudent Facing\nFind the value of the expression. Use a diagram if it is helpful.\n\\(12 \\div 0.2\\)\n[IMG]\nThis is the diagram and explanation Tyler used to justify why\n\\(12 \\div 0.2 = 60\\)\n.\n[IMG]\n[IMG]\nExplain how the expression\n\\(12 \\times (1 \\div 0.2)\\)\nrelates to Tyler's reasoning.\nFind the value of each expression.\n\\(14 \\div 0.5\\)\n\\(5 \\div 0.25\\)\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAdvancing Student Thinking\nIf a student is not sure how to explain Tyler’s reasoning in problem 2, ask them to read the expression to you and record the words they use. Then ask them how those words relate to the words Tyler used.\nActivity Synthesis\nAsk previously identified students to share their solutions and strategies for evaluating the expression in the first problem.\n“How was multiplication useful to find the value of the expression in the first problem?” (I didn't have to draw 12 wholes because I knew there were five 0.2s in each one so that would be\n\\(12 \\times 5\\)\ntotal.)\nMLR1 Stronger and Clearer Each Time\n“Share your response to problem 2 with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”\n3-5 minutes: structured partner discussion\nRepeat with 2 different partners.\n“Revise your initial draft based on the feedback you got from your partners.”\n2-3 minutes: independent work time\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.", "problem_activity": "Problem 3\nPre-unit\nThe value of the 6 in 618,204 has how many times the value as the 6 in 563? Explain or show your reasoning.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Order three objects by length; compare the lengths of two objects indirectly by using a third object.", "problem_activity": "Narrative\nThe purpose of this warm-up is for students to compare lengths of objects and notice when they are longer, shorter, or equal to each other in length. While students may notice and wonder many things about these images, comparing the length is an important discussion point.\nLaunch\nGroups of 2\nDisplay the image.\n“What do you notice? What do you wonder?”\n1 minute: quiet think time\nActivity\n“Discuss your thinking with your partner.”\n1 minute: partner discussion\nShare and record responses.\nStudent Facing\nWhat do you notice?\nWhat do you wonder?\n[IMG]\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\n“How can you describe the length of the pencil?” (The pencil is longer than the yellow cubes. The pencil is the same length as the purple cubes.)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.", "problem_activity": "Task\nOn the map below, $\\frac14$ inch represents one mile. Candler, Canton, and Oteen are three cities on the map.\n[IMG]\nIf the distance between the real towns of Candler and Canton is 9 miles, how far apart are Candler and Canton on the map?\nIf Candler and Oteen are $3\\frac12$ inches apart on the map, what is the actual distance between Candler and Oteen in miles?\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Analyze and solve pairs of simultaneous linear equations.", "problem_activity": "Problem 1\nSolve the system of linear equations using substitution. Describe how you can solve any system using substitution.\n$${x+2y=8}$$\n$${-3x+2y=16}$$\nProblem 2\nSolve the two systems below using substitution. Compare and contrast the two solutions. What do they each mean?\n| ** System A ** | ** System B ** |\n|-----------------------------------------------------|-----------------------------------------------------|\n| $${y=2(-x+3)}$$ $${2x+y=4}$$ | $${5x+20y=15}$$ $${x+4y=3}$$ |\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?", "problem_activity": "Activity\nIn this activity, students examine equations with one variable to determine the number of solutions. In the associated Algebra lesson, students will find the number of solutions for a system of equations. In the discussion after this activity, students are asked to look for ways they can connect the number of solutions to the equation without solving.\nLaunch\nArrange students in groups of 2.\nDemonstrate that there are three options for the number of values that make a linear equation true.\nOne. For example,\n\\(x+1 = 3\\)\nis only true when\n\\(x = 2\\)\nand not true for any other value of\n\\(x\\)\n.\nZero. For example,\n\\(x + 1 = x\\)\nis not true for any value of\n\\(x\\)\n.\nInfinite. For example\n\\(x + 1 = x + 1\\)\nis true for any value of\n\\(x\\)\n.\nTell students that they do not need to solve any of the equations, although rewriting the equations in other forms may help students recognize the number of solutions.\nStudent Facing\nHow many values of\n\\(x\\)\nmake each equation true?\n\\(3x + 1 = 10\\)\n\\(2x + 12 = 2x + 10 + 2\\)\n\\(2x = x + 2\\)\n\\(3(x+4) = 3x + 4\\)\n\\(\\frac{2x+6}{2} = x + 6\\)\n\\(0 = 0\\)\n\\(x + 3x - 4 = 7(x - \\frac{4}{7})\\)\n\\(0 = 6\\)\nWith your partner, discuss what you notice about the equations based on the number of solutions they have.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nThe purpose of the discussion is to recognize ways to determine the number of solutions from an equation without solving. Select students to share their answers as well as what they noticed during their discussions.\nTo continue the discussion, consider asking:\n\"What do you notice about equations that have one solution?\" (On each side of the equation, there are different number of\n\\(x\\)\nvariables.)\n\"Is there a way to know whether there will be infinite or no solutions based on the original equation?\" (When each side of the equation has the same number of\n\\(x\\)\nvariables and the constants are also equal, there will be infinite solutions. When each side of the equation has the same number of\n\\(x\\)\nvariables and the constants are not equal, there will be no solution.)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.", "problem_activity": "Narrative\nThe purpose of this activity is for students to try the algorithms they saw earlier in the lesson. The important thing is that they combine hundreds and hundreds, tens and tens, and ones and ones, which should be a familiar idea from grade 2. The synthesis provides an opportunity to show a different way of recording newly composed tens and hundreds when compositions are required, which will be discussed in more detail in subsequent lessons. Provide access to base-ten blocks for students to use to support their reasoning about the algorithms, in case requested.\nStudents analyze and improve a given explanation of how to find a sum, filling in details and using more precise language to explain the calculation more fully (MP3, MP6).\nThis activity uses\nMLR3 Clarify, Critique, Correct\n.\nAdvances: reading, writing, representing\nEngagement: Develop Effort and Persistence.\nChunk this task into more manageable parts. Check in with students to provide feedback and encouragement after each chunk.\nSupports accessibility for: Organization, Social-Emotional Functioning\nRequired Materials\nMaterials to Gather\nBase-ten blocks\nLaunch\nGroups of 2\nGive students access to base-ten blocks.\n“Now you are going to have a chance to try the algorithms that Lin and Han used in the last activity. Take a minute to think about which algorithm you want to use for each problem.”\n1 minute: quiet think time\nActivity\n“Work with your partner to try an algorithm to find the value of each sum.”\n5 minutes: partner work time\nMonitor for students who use Lin’s algorithm and Han's algorithm on the last problem.\nStudent Facing\nTry using an algorithm to find the value of each sum. Show your thinking. Organize it so it can be followed by others.\n\\(475 + 231\\)\n\\(136 + 389\\)\n\\(670 + 257\\)\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nSelect previously identified students to share their work on the last expression.\nMLR3 Clarify, Critique, Correct\nDisplay the following partially correct answer and explanation:\n[IMG]\nI added the ones, the tens, and then hundreds.\nRead the explanation aloud.\n“What is unclear?”\n1 minute: quiet think time\n2 minute: partner discussion\n“With your partner, work together to write a revised explanation.”\nDisplay and review the following criteria:\nExplanation for each step\nWords such as: first, next, then\n3–5 minute: partner work time\nSelect 1–2 groups to share their revised explanation with the class. Record responses as students share. (Sample explanation: First, I stacked the numbers vertically. Then, I added the ones to get 7 and recorded a 7 below the ones place. Next, I added 70 and 50 and got 120. I recorded 20 below the 7 and 100 below the 20. I added 600 and 200 to get 800 and recorded that below the 100. Then I added and wrote 927 as the answer.)\n“What is the same and different about the explanations?” (Both explanations say that they added the ones, tens, and hundreds, but the revised explanation gives more detail about how to record each step.)\n“How is this way of recording this work the same or different from Han’s method in the first activity?” (Han would record 120 in the second row where we record the tens. It was 12 tens, but that’s the same as 1 hundred and 2 tens. We could record 100 on one line and 20 on the next line.)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.", "problem_activity": "Activity\nIn this activity, students use a large data set to compare the size of students with different dominant hands. Students must select a variable that best represents the size of the students and compare the two groups. By selecting the relevant variable to analyze, students are engaging in aspects of mathematical modeling (MP4). Additionally, students must choose the appropriate tools (MP5) to analyze a large data set such as this.\nLaunch\nArrange students in groups of 2. Distribute a copy of the blackline master to each group.\nRepresentation: Internalize Comprehension.\nBegin the activity with concrete or familiar contexts. Engage in a classroom discussion drawing attention to the relationship between height, foot length, and arm span of right- and left-handed students in the class. This will help students make connections between the large data set and specific students in the class.\nSupports accessibility for: Conceptual processing; Memory\nStudent Facing\nIs there a connection between a student's dominant hand and their size? Use the table of information to compare the size of students with different dominant hands.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nSelect students to share their analysis of the data.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.", "problem_activity": "Activity\nThe purpose of this activity is for students to compare two different proportional relationships represented in different ways using the skills they have worked on over the past three lessons. Working in groups, students compare the relationships, responding to questions about their rate of change, which rate of change is higher, and one other situation-based question. Groups make a visual display for their problem set to explain each of their responses and convince others of their accuracy.\nIdentify groups using a variety of representations to share during the Activity Synthesis.\nLaunch\nRemind students that in previous lessons they identified representations of and created representations for a single proportional relationship. In this activity, they will consider representations of two\ndifferent\nproportional relationships and make comparisons between them.\nArrange students in groups of 2–3. Assign to each group (or ask groups to choose) one of the three question sets. Tell groups that they will make a visual display for their responses to the questions. The display should clearly demonstrate their reasoning and use multiple representations in order to be convincing. Let them know that there will be a gallery walk when they finish for the rest of the class to inspect their solutions’ accuracy.\nIf time allows, ask groups to complete all three problems and make a visual display for just one.\nAction and Expression: Internalize Executive Functions.\nTo support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, present one question at a time.\nSupports accessibility for: Memory; Organization\nStudent Facing\nElena babysits her neighbor’s children. Her earnings are given by the equation\n\\(y=8.40x\\)\n, where\n\\(x\\)\nrepresents the number of hours she worked and\n\\(y\\)\nrepresents the amount of money she earned.\nJada earns\n$\n7 per hour mowing her neighbors’ lawns.\nWho makes more money after working 12 hours? How much more do they make? Explain your reasoning by creating a graph or a table.\nWhat is the rate of change for each situation and what does it mean?\nUsing your graph or table, determine how long it would take each person to earn\n$\n150.\nClare and Han have summer jobs stuffing envelopes for two different companies.\nHan earns\n$\n15 for every 300 ​​​​​envelopes he finishes.\nClare’s earnings can be seen in the table.\n| number of envelopes | money in dollars |\n|-------------------------------------------------|----------------------------------------------|\n| 400 | 40 |\n| 900 | 90 |\nBy creating a graph, show how much money each person makes after stuffing 1,500 envelopes.\nWhat is the rate of change for each situation and what does it mean?\nUsing your graph, determine how much more money one person makes relative to the other after stuffing 1,500 envelopes. Explain or show your reasoning.\nTyler plans to start a lemonade stand and is trying to perfect his recipe for lemonade. He wants to make sure the recipe doesn’t use too much lemonade mix (lemon juice and sugar) but still tastes good.\nLemonade Recipe 1 is given by the equation\n\\(y=4x\\)\nwhere\n\\(x\\)\nrepresents the amount of lemonade mix in cups and\n\\(y\\)\nrepresents the amount of water in cups.\nLemonade Recipe 2 is given in the table.\n| lemonade mix (cups) | water (cups) |\n|---------------------|--------------|\n| 10 | 50 |\n| 13 | 65 |\n| 21 | 105 |\nIf Tyler had 16 cups of lemonade mix, how many cups of water would he need for each recipe? Explain your reasoning by creating a graph or a table.\nWhat is the rate of change for each situation and what does it mean?\nTyler has a 5-gallon jug (which holds 80 cups) to use for his lemonade stand and 16 cups of lemonade mix. Which lemonade recipe should he use? Explain or show your reasoning.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nStudent Facing\nAre you ready for more?\nHan and Clare are still stuffing envelopes. Han can stuff 20 envelopes in a minute, and Clare can stuff 10 envelopes in a minute. They start working together on a pile of 1,000 envelopes.\nHow long does it take them to finish the pile?\nWho earns more money?\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Extension Student Response.\nAnticipated Misconceptions\nSome students may confuse the values for the rate of change of a situation. For example, Lemonade Recipe 1's equation,\n\\(y=4x\\)\n, shows that the rate of change is 4 cups of water per cup of lemonade mix. Students may switch these values and think that the rate of change is 4 cups lemonade mix per cup of water. Ask students who do this to explain where in the original representations they see the rate of change. Students may need to list a few values or sketch a graph in order to see their mix-up between the two quantities.\nActivity Synthesis\nBegin with a gallery walk for students to see how other groups answered the same set of questions they did and how students answered questions about the other two contexts.\nInvite groups to share the strategies they used with the various representations. Consider asking groups the following questions:\n“What representations did you choose to answer the questions? Why did you pick them?”\n“What representation did you not use? Why?”\n“How did you decide what scale to use when you made your graph?”\n“Now that you have seen the work of other groups, is there anything about your display you would change if you could?”\nRepresenting and Conversing: MLR7 Compare and Connect.\nDuring the gallery walk, invite students to discuss “what is the same and what is different?” about the representations on the posters and then share with the whole class. Look for opportunities to highlight representations that helped students answer the questions and decide which scales to use for the graph. This will help students make connections and describe the usefulness of each type of representation.\nDesign Principle(s): Optimize output\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.", "problem_activity": "Stage 2: Subtract from 20\nRequired Preparation\nMaterials to Gather\nNumber cards 0–10\nMaterials to Copy\nBlackline Masters\nHow Close? Stage 2 Recording Sheet\n\nNarrative\nBefore playing, students remove the cards that show the number 10 and set them aside.\nEach student picks 4 cards and chooses 2 or 3 to subtract from 20 to get close to 0. The student whose difference is closest to 0 wins a point for the round. Students pick new cards so that they have 4 cards in their hand and then start the next round.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.", "problem_activity": "Optional activity\nIn this activity, students investigate the average rates of change of the three loan options from an earlier activity. Calculating average rates of change over the same interval of time but at different points in the lending period (early and then later) further reinforces the dramatic change that can result from compounding over time.\nLaunch\nRecall the previous work students have done on the three loans (A, B, C) and tell students that they are going to examine how these loans change over different time intervals.\nStudent Facing\nThe functions\n\\(a\\)\n,\n\\(b\\)\n, and\n\\(c\\)\nrepresent the amount owed (in dollars) for Loans A, B, and C respectively: the input for the functions is\n\\(t\\)\n, the number of years without payments.\nFor each loan, find the average rate of change per year between:\nthe start of the loan and the end of the second year\nthe end of the tenth year and the end of the twelfth year\nHow do the average rates of change for the three loans compare in each of the two-year intervals?\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nSelect students to share their calculations for the average rates of change for each loan. Invite students to share their observations about how the different average rates of change compare. If not mentioned by students, ask them to quantify how the average rate of change increased between the two intervals for the different loans, such as by calculating how many times larger the value for the second interval is when compared the first.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)", "problem_activity": "Task\nLucy has measuring cups of sizes 1 cup, $\\frac{1}{2}$ cup, $\\frac{1}{3}$ cup, and $\\frac{1}{4}$ cup. She is trying to measure out $\\frac{1}{6}$ of a cup of water and says, ''If I fill up the the $\\frac{1}{2}$ cup and then pour that into the $\\frac{1}{3}$ cup until it is full, there will be $\\frac{1}{6}$ of a cup of water left.''\nIs Lucy's method to measure $\\frac{1}{6}$ of a cup of water correct? Explain.\nLucy wonders what other amounts she can measure. Is it possible for her to measure out $\\frac{1}{12}$ of a cup? Explain.\nWhat other amounts of water can Lucy measure?\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.", "problem_activity": "Lily and Ravi are visiting the Redwood Forest in California. They want to measure the height of one of the trees, but they can’t climb it. However, they are able to measure the shadow of the tree. They determine the shadow of the tree is\n$$583{1\\over3}$$\nfeet long.\nIf Ravi is 6 feet tall and they determine his shadow is 10 feet long, find the height of the redwood tree.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.", "problem_activity": "Activity\nIn this activity, students mix different numbers of batches of a color recipe to obtain a certain shade of green. They observe how multiple batches of the same recipe produce the same shade of green as a single batch, which suggests that the ratios of blue to yellow for the two situations are equivalent. They also come up with a ratio that is not equivalent to produce a mixture that is a different shade of green.\nAs students make the mixtures, ensure that they measure accurately so they will get accurate outcomes. As students work, note the different diagrams students use to represent their recipes. Select a few examples that could be highlighted in discussion later.\nDigital\nLaunch\nArrange students in groups of 2–4. (Smaller groups are better, but group size might depend on available equipment.) Each group needs a beaker of blue water and one of yellow water, one graduated cylinder, a permanent marker, a craft stick, and 3 opaque white cups (either styrofoam, white paper, or with a white plastic interior).\nFor classes using the print-only version: Show students the blue and yellow water. Demonstrate how to pour from the beakers to the graduated cylinder to measure and mix 5 ml of blue water with 15 ml of yellow water. Demonstrate how to get an accurate reading on the graduated cylinder by working on a level surface and by reading the measurement at eye level. Tell students they will experiment with different mixtures of green water and observe the resulting shades.\nFor classes using the digital version: Display the dynamic color mixing cylinders for all to see. Tell students, “The computer mixes colors when you add colored water to each cylinder. You can add increments of 1 or 5. You can’t remove water (once it’s mixed, it’s mixed), but you can start over. The computer mixes yellow and blue.” Ask students a few familiarization questions before they start working on the activity:\nWhat happens when you mix yellow and blue? (A shade of green is formed.)\nWhat happens if you add more blue than yellow? (Darker green, blue green, etc.)\nIf necessary, demonstrate how it works by adding some yellows and blues to both the left and the right cylinder. Show how the “Reset” button lets you start over.\nRepresentation: Internalize Comprehension.\nProvide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.\nSupports accessibility for: Language; Conceptual processing\nStudent Facing\nIn the left cylinder, mix 5 ml of blue and 15 ml of yellow. This is a\nsingle batch\nof green.\nSuppose you have one batch of green but want to make more. Which of the following would produce the same shade of green?\nIf you're unsure, try creating the mixture in the right cylinder. Start with the amounts in a single batch (5 ml of blue and 15 ml of yellow) and . . .\nadd 20 ml of blue and 20 ml of yellow\ndouble the amount of blue and the amount of yellow\ntriple the amount of blue and the amount of yellow\nmix a single batch with a double batch\ndouble the amount of blue and triple the amount of yellow\nFor one of the mixtures that produces the same shade, write down the number of ml of blue and yellow used in the mixture.\nFor the same mixture that produces the same shade, draw a diagram of the mixture. Make sure your diagram shows the number of milliliters of blue, yellow, and the number of batches.\nSomeone was trying to make the same shade as the original single batch, but started by adding 20 ml of blue and 20 ml of yellow. How can they add more but still create the same shade of green?\nInvent a recipe for a\nbluer\nshade of green. Write down the amounts of yellow and blue that you used, and draw a diagram. Explain how you know it will be bluer than the original single batch of green before testing it out.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nPrint\nLaunch\nArrange students in groups of 2–4. (Smaller groups are better, but group size might depend on available equipment.) Each group needs a beaker of blue water and one of yellow water, one graduated cylinder, a permanent marker, a craft stick, and 3 opaque white cups (either styrofoam, white paper, or with a white plastic interior).\nFor classes using the print-only version: Show students the blue and yellow water. Demonstrate how to pour from the beakers to the graduated cylinder to measure and mix 5 ml of blue water with 15 ml of yellow water. Demonstrate how to get an accurate reading on the graduated cylinder by working on a level surface and by reading the measurement at eye level. Tell students they will experiment with different mixtures of green water and observe the resulting shades.\nFor classes using the digital version: Display the dynamic color mixing cylinders for all to see. Tell students, “The computer mixes colors when you add colored water to each cylinder. You can add increments of 1 or 5. You can’t remove water (once it’s mixed, it’s mixed), but you can start over. The computer mixes yellow and blue.” Ask students a few familiarization questions before they start working on the activity:\nWhat happens when you mix yellow and blue? (A shade of green is formed.)\nWhat happens if you add more blue than yellow? (Darker green, blue green, etc.)\nIf necessary, demonstrate how it works by adding some yellows and blues to both the left and the right cylinder. Show how the “Reset” button lets you start over.\nRepresentation: Internalize Comprehension.\nProvide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.\nSupports accessibility for: Language; Conceptual processing\nStudent Facing\nYour teacher mixed milliliters of blue water and milliliters of yellow water in the ratio\n\\(5:15\\)\n.\nDoubling the original recipe:\nDraw a diagram to represent the amount of each color that you will combine to double your teacher’s recipe.\nUse a marker to label an empty cup with the ratio of blue water to yellow water in this double batch.\nPredict whether these amounts of blue and yellow will make the same shade of green as your teacher’s mixture. Next, check your prediction by measuring those amounts and mixing them in the cup.\nIs the ratio in your mixture equivalent to the ratio in your teacher’s mixture? Explain your reasoning.\nTripling the original recipe:\nDraw a diagram to represent triple your teacher’s recipe.\nLabel an empty cup with the ratio of blue water to yellow water.\nPredict whether these amounts will make the same shade of green. Next, check your prediction by mixing those amounts.\nIs the ratio in your new mixture equivalent to the ratio in your teacher’s mixture? Explain your reasoning.\nNext, invent your own recipe for a\nbluer\nshade of green water.\nDraw a diagram to represent the amount of each color you will combine.\nLabel the final empty cup with the ratio of blue water to yellow water in this recipe.\nTest your recipe by mixing a batch in the cup. Does the mixture yield a bluer shade of green?\nIs the ratio you used in this recipe equivalent to the ratio in your teacher’s mixture? Explain your reasoning.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nStudent Facing\nAre you ready for more?\nSomeone has made a shade of green by using 17 ml of blue and 13 ml of yellow. They are sure it cannot be turned into the original shade of green by adding more blue or yellow. Either explain how more can be added to create the original green shade, or explain why this is impossible.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Extension Student Response.\nAnticipated Misconceptions\nIf any students come up with an incorrect recipe, consider letting this play out. A different shade of green shows that the ratio of blue to yellow in their mixture is not equivalent to the ratio of blue to yellow in the other mixtures. Rescuing the incorrect mixture to display during discussion may lead to meaningful conversations about what equivalent ratios mean.\nActivity Synthesis\nAfter each group has completed the task, have the students rotate through each group’s workspace to observe the mixtures and diagrams. As they circulate, pose some guiding questions. (For students using the digital version, these questions refer to the mixtures on their computers.)\nAre each group’s results for the first two mixtures the same shade of green?\nAre the ratios representing the double batch, the triple batch, and your new mixture all equivalent to each other? How do you know?\nWhat are some different ways groups drew diagrams to represent the ratios?\nHighlight the idea that a ratio is equivalent to another if the two ratios describe different numbers of batches of the same recipe.\nConversing: MLR8 Discussion Supports.\nAssign one member from each group to stay behind to answer questions from students visiting from other groups. Provide visitors with question prompts such as, \"These look like the same shade, how can we be sure the ratios are equivalent?\", \"If you want a smaller amount with the same shade, what can you do?\" or \"What is the ratio for your new mixture?\"\nDesign Principle(s): Cultivate conversation\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as “right rectangular prism.”)", "problem_activity": "Narrative\nThe purpose of this activity is for students to analyze a collection of connecting cubes that is arranged in towers of 10. Students analyze another student’s thinking about a representation of 48 cubes in towers of 10. When students explain that they disagree with Noah because a ten must include 10 ones, they show their understanding of a ten and the foundations of the base-ten system (MP3, MP7).\nMLR8 Discussion Supports.\nSynthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain why they agree or disagree with Noah. Invite groups to rehearse what they will say when they share with the whole class.\nAdvances: Speaking, Conversing, Representing\nRequired Materials\nMaterials to Gather\nConnecting cubes in towers of 10 and singles\nLaunch\nGroups of 2\nGive students access to connecting cubes in towers of 10 and singles.\n“Noah counted a collection of connecting cubes. He says there are 50 cubes. Do you agree or disagree? Explain how you know. You will have a chance to think about it on your own and talk to your partner about Noah’s thinking before you write your response.”\nActivity\n1 minute: quiet think time\n“Share your thinking with your partner.”\n2 minutes: partner discussion\n“Explain why you agree or disagree with Noah. Write the word ‘agree’ or ‘disagree’ in the first blank. Then write why you agree or disagree.”\n3 minutes: independent work time\nStudent Facing\nNoah organized his collection of connecting cubes.\n[IMG]\nHe counts and says there are 50 cubes.\nDo you agree or disagree?\nExplain how you know:\nI ____________________ with Noah because\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAdvancing Student Thinking\nIf students attempt to check Noah's count by counting all the cubes by one, consider asking:\n“How are you checking to see if there are 50 cubes?”\n“How could you use what you know about groups of ten to check Noah’s thinking?”\nActivity Synthesis\nInvite students to share whether they agree or disagree, and their reasons.\nIf needed, ask “How do you know there are not 50 cubes?” (50 is 5 tens. There are only 4 tens here. The last tower isn’t 10. There are 4 tens and 8 ones.)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.", "problem_activity": "Stage 2: Multiple Rectangles\nRequired Preparation\nMaterials to Gather\nFolders\nGrid paper\nInch tiles\nMaterials to Copy\nBlackline Masters\nCan You Build It Stage 2 Directions\nNumber Cards (0-10)\n\nNarrative\nBefore playing, students remove the cards that show 6 or higher and set them aside.\nStudents flip two number cards to get a number of tiles and then each partner tries to create as many rectangles as possible for that area. If both students each found all the rectangles, they get one point. A student gets two points for any rectangle they built that their partner did not. The player who gets the most points after eight rounds is the winner. Students may choose to draw their rectangle on grid paper, rather than use inch tiles.\nThis center stage is the first time Number Cards 0–10 are used in Grade 4, so they are provided as a blackline master. Students will continue to use these throughout the year. Consider copying them on cardstock or laminating them and keeping them organized to be used repeatedly.\nAdditional Information\nEach group of 2 needs at least 120 inch tiles and a set of number cards.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.", "problem_activity": "Narrative\nThe purpose of this activity is for students to compare three-digit numbers based on their understanding of place value. They are invited to explain or show their thinking in any way that makes sense to them. A number line is provided. Students may revise their thinking after locating the numbers on the number line, or may choose to draw diagrams to represent their thinking. During the activity synthesis, methods based on comparing the value of digits by place are highlighted.\nFor the last problem, students persevere in problem solving as there are many ways to make most of inequalities true but students will need to think strategically in order to fill out all of them. In particular, 810 can be used in the first, second, or fourth inequality but it needs to be used in the fourth because it is the\nonly\nnumber on the list that is larger than 793.\nRepresentation: Develop Language and Symbols.\nRepresent the problem in multiple ways to support understanding of the situation. For example, have one group member place the numbers on a number line to verify which number is greater. Another option is to have one group member make the numbers using base-ten blocks to prove which is greater.\nSupports accessibility for: Conceptual Processing, Organization\nLaunch\nGroups of 2\n“In the warm-up, you saw that different forms of writing a number can help you think about the value of each digit.”\nWrite\n\\(564\\phantom{3} \\boxed{\\phantom{33}}\\phantom{3}504\\)\non the board.\n“What symbol would make this expression true? Explain.” (>, because they both have 500, but the first number has 6 tens or 60 and the second number has 0 tens.)\nActivity\n“Today you will be comparing three-digit numbers by looking at place value.”\n“If it helps, you can use base-ten diagrams or expanded form to help you think about place value.”\n“Try it on your own and then compare with your partner.”\n5 minutes: independent work time\n5 minutes: partner work time\nMonitor for students who compare the numbers without drawing base-ten diagrams or using a number line.\nStudent Facing\nCompare the numbers.\n\\(>\\)\n,\n\\(=\\)\n, or\n\\(<\\)\n521\n\\(\\boxed{\\phantom{33}}\\)\n523\nExplain or show your thinking. If it helps, use a diagram or number line.\n\\(>\\)\n,\n\\(=\\)\n, or\n\\(<\\)\n889\n\\(\\boxed{\\phantom{33}}\\)\n878\nExplain or show your thinking. If it helps, use a diagram or number line.\nPlace the numbers in the blanks to make each comparison true. Use each number only once. Use base-ten diagrams or a number line if it helps.\n810\n529\n752\n495\n\\(\\underline{\\hspace{1 cm}} > 519\\)\n\\(687 < \\underline{\\hspace{1 cm}}\\)\n\\(\\underline{\\hspace{1 cm}} < 501\\)\n\\(\\underline{\\hspace{1 cm}} > 793\\)\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAdvancing Student Thinking\nIf students write comparison statements that are not true, consider asking:\n“Could you read each statement aloud?”\n“How did you know your statement is true?”\n“How could you use the base-ten diagram or number line to help you show whether your statement is true or false?”\nActivity Synthesis\nDisplay\n\\(564 > 504\\)\n.\nDisplay\n\\(500 + 60 + 4 > 500 + 4\\)\n.\n“We decided that this was a true statement. How does the expanded form of these numbers help justify our thinking?” (We can see the value of each place, so we can compare each digit.)\nInvite previously selected students to share how they compared numbers without drawing diagrams or using the number line.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”", "problem_activity": "Optional activity\nIn this activity, students explore percentages that describe parts of a whole. They find both\n\\(B\\)\nand\n\\(C\\)\n, where\n\\(A\\%\\)\nof\n\\(B\\)\nis\n\\(C\\)\n, in the context of available and consumed water on a hike.\nStudents who use a double number line may notice that the value of\n\\(B\\)\nis the same in both questions, so the same double number line can be used to solve both parts of the problem. To solve the second question, however, the diagram needs to be partitioned with more tick marks.\nAs in the previous task, students may solve using other strategies, including by simply multiplying or dividing, i.e.,\n\\((1.5) \\boldcdot 2 = 3.0\\)\nand\n\\(\\frac{80}{100} \\boldcdot (3.0) = 2.4\\)\n. Encourage them to also explain their reasoning with a double number line, table, or tape diagram. Monitor for at least one student using each of these representations.\nLaunch\nGive students quiet think time to complete the activity and then time to share their explanation with a partner. Follow with a whole-class discussion.\nStudent Facing\nDuring the first part of a hike, Andre drank 1.5 liters of the water he brought.\nIf this is 50% of the water he brought, how much water did he bring?\nIf he drank 80% of his water on his entire hike, how much did he drink?\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nStudent Facing\nAre you ready for more?\nDecide if each scenario is possible.\nAndre plans to bring his dog on his next hike, along with 150% as much water as he brought on this hike.\nAndre plans to drink 150% of the water he brought on his hike.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Extension Student Response.\nAnticipated Misconceptions\nAfter students create a double number line diagram with tick marks at 50% and 100%, some may struggle to know how to fit 80% in between. Encourage them to draw and label tick marks at 10% increments or work with a table instead. Some students may think that the second question is asking for the amount of water Andre drank on the second part of the hike. Clarify that it is asking for his total water consumption on the entire hike.\nActivity Synthesis\nSelect 1–2 students who used a double number line, a table of equivalent ratios, and a tape diagram to share their strategies. As students explain, illustrate and display those representations for all to see.\nAsk students how they knew what 100% means in the context. At this point it is not necessary for students to formally conceptualize the two ways percentages are used (to describe parts of whole, and to describe comparative relationships). Drawing their attention to concrete and contextualized examples of both, however, serves to build this understanding intuitively.\nReading, Writing, Speaking: MLR3 Clarify, Critique, Correct.\nBefore students share their answers for the second question, present an incorrect answer and explanation. For example, “If Andre drank 80% of his water on his entire hike, then he drank 1.2 liters of water because 0.8 times 1.5 liters is 1.2 liters.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss in partners, listen for students who identify and clarify the assumption in the statement. For example, the author assumed that Andre brought 1.5 liters for his entire hike; however, the problem states that Andre drank 1.5 liters of the water he brought. This will remind students to carefully identify the quantity they are finding a percentage of.\nDesign Principle(s): Optimize output (for explanation); Maximize meta-awareness\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.", "problem_activity": "Task\nFor each pair of figures, decide whether these figures are the same size and same shape. Explain your reasoning.\n[IMG]\n[IMG]\n[IMG]\nWhat does it mean for two figures to be the same size and same shape?\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.\nGrade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.", "problem_activity": "Launch\nDisplay one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.\nStudent Facing\nEvaluate mentally:\n\\(1 - \\frac12\\)\n\\(1 - \\frac{1}{10}\\)\n\\(1-\\frac{3}{10}\\)\n\\(1-\\frac{5}{17}\\)\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nAsk students to share their strategies for each problem. Record and display their responses for all to see. To focus the discussion for the rest of the lesson, consider asking:\nTo involve more students in the conversation, consider asking:\n“Who can restate\n\\(\\underline{\\hspace{.5in}}\\)\n’s reasoning in a different way?”\n“Did anyone have the same strategy but would explain it differently?”\n“Did anyone solve the problem in a different way?”\n“Does anyone want to add on to\n\\(\\underline{\\hspace{.5in}}\\)\n’s strategy?”\n“Do you agree or disagree? Why?”\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.", "problem_activity": "Stage 3: 20 Cubes\nRequired Preparation\nMaterials to Gather\nConnecting cubes\nMaterials to Copy\nBlackline Masters\nWhat's Behind My Back Stage 3 Recording Sheet\n\nNarrative\nStudents work with 20 cubes, organized into two towers of 10 cubes. One partner snaps the towers and puts one part behind their back and shows the other part to their partner. Their partner figures out how many cubes are behind their back. Students record an addition equation with a blank to represent the missing cubes.\nAdditional Information\nEach group of 2 needs 20 connecting cubes.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.", "problem_activity": "Narrative\nThe purpose of this activity is for students to learn stage 2 of the Estimate and Measure center. Students estimate and measure the length of objects. This stage includes centimeters, inches, and feet. Students should only measure in centimeters at this point in the unit. In this activity, students also find the difference between their estimate and their measurement for extra practice.\nRequired Materials\nMaterials to Gather\nMetersticks\nObjects of various lengths\nRulers (centimeters)\nMaterials to Copy\nEstimate and Measure Stage 2 Recording Sheet\nRequired Preparation\nGather or identify objects of various lengths (pencils, markers, books, glue, scissors, shoe, tape dispenser, side of desk, length of bulletin board).\nLaunch\nGroups of 2\nGive students access to centimeter rulers and meter sticks.\nGive each student a recording sheet.\n“We are going to learn a new center called Estimate and Measure. Today we are going to estimate and measure the length of different objects.”\n“First you and your partner will decide on an object in the room to measure. Then you will decide what tool you will use to measure the object.”\nChoose an object to measure and hold it up for students to see.\n“What tool would you use to measure this object?”\n“Before measuring the object, you and your partner will both estimate the length of the object using the unit you chose. Your estimates do not have to be the same. Record your estimate.”\nDemonstrate estimating the length of your object and recording it on the recording sheet.\n“Now work with your partner to measure the object and record the actual measurement.”\nDemonstrate measuring the object, thinking aloud as you line up the endpoints and use the number on the tool to determine the length.\nRecord the actual measurement on the recording sheet.\n“If you have time: Once you have estimated and measured the object, find the difference between your estimate and the actual length.”\nDemonstrate finding the difference.\n“Put your object back and choose another object to measure.”\nActivity\n10 minutes: partner work time\nActivity Synthesis\nDisplay a row of the recording sheet with this information:\nObject: notebook\nEstimate: 20 cm\nActual Measurement: 31 cm\n“How can we find the difference?” (We can add on to 20 until we get to 31. We can subtract 20 from 31.)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.", "problem_activity": "[IMG]\nDetermine the polygon described by the given system of inequalities. Explain your reasoning.\nFind the perimeter. Round your answer to the nearest tenth.\nFind the area in simplest radical form.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.", "problem_activity": "Optional activity\nEarlier in this lesson, students found the area of regions around circles and the area of fractions of circles in separate problems. In this activity, students combine these two strategies to find the area of a complex real-world object. Students engage in MP2 as they decide how to decompose the running track into measurable pieces and how to use the given information about the dimensions of the track to calculate areas.\nLaunch\nArrange students in groups of 2. Give students 3–4 minutes of partner work time followed by small-group and whole-class discussions.\nAction and Expression: Internalize Executive Functions.\nProvide students with a graphic organizer to organize the information provided in the problem and to structure their problem-solving strategy. The graphic organizer should include the prompts: “What do I need to find out?”, “What do I need to do?”, “How I solved the problem.”, and “How I know my answer is correct.”\nSupports accessibility for: Language; Organization\nStudent Facing\nThe field inside a running track is made up of a rectangle 84.39 m long and 73 m wide, together with a half-circle at each end. The running lanes are 9.76 m wide all the way around.\n[IMG]\nWhat is the area of the running track that goes around the field? Explain or show your reasoning.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAnticipated Misconceptions\nSome students may think they can calculate the area of the running track by multiplying half of the perimeter times the radius, as if the shape were just a circle. Prompt them to see that they need to break the overall shape into rectangular and circular pieces.\nActivity Synthesis\nPair the groups of 2 to create groups of 4. Have students compare answers and explain their reasoning until they reach an agreement.\nAs a whole group, discuss how finding the area of the track in this activity was similar or different to solving the two area problems in the previous activity. Make sure to highlight these points:\nLike the first problem in the previous activity, finding the area of the track can be done by finding the area of the larger shape (the track and the field inside) and then taking away the area of the field inside.\nLike the second problem in the previous activity, there are semicircles whose area can be found by composing them to make a full circle or by taking half the area of the corresponding full circle.\nRepresenting, Speaking, Listening: MLR7 Compare and Connect.\nAsk students to prepare a visual display that shows how they calculated the area of the running track that goes around the field, and look for students with different strategies for calculating the area of the curved parts of the track. As students investigate each other’s work, ask students to share what worked well in a particular approach. During this discussion, listen for and amplify any comments that clarify that the curved parts of the running track are half circles with a smaller circle missing from the inside. Then encourage students to make connections between the quantities representing areas and the shapes in the diagram. Ask questions such as, “What does the quantity 6,719.561 represent in the diagram?” and “What does the quantity 4,183.265 represent in the diagram?” During this discussion, listen for and amplify language students use to reason that these quantities represent the areas of larger and smaller circles, respectively. This will foster students’ meta-awareness and support constructive conversations as they compare strategies for finding the area of a complex real-world object.\nDesign Principle(s): Cultivate conversation; Maximize meta-awareness\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.", "problem_activity": "Activity\nIn this activity students build off the specific calculations from the previous lesson to generalize the perimeter of a polygon inscribed in a circle of radius 1. The relatively unstructured presentation of this activity is purposeful. Students work with their groups to determine what information they need, how to calculate in the specific cases, and how they can express those repeated procedures in a generalized formula (MP8). Monitor for groups who have generalized any part of the process.\nDigital\nLaunch\nEncourage students to refer to the examples from the previous lesson as they work to generalize.\nAction and Expression: Internalize Executive Functions.\nChunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Look for students who are struggling to begin and review examples from the previous lesson. Record their thinking on a display and keep the work visible as students continue to work.\nSupports accessibility for: Organization; Attention\nStudent Facing\nThe applet shows a regular\n\\(n\\)\n-sided polygon inscribed in a circle.\nCome up with a general formula for the perimeter of the polygon in terms of\n\\(n\\)\n. Explain or show your reasoning.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nPrint\nLaunch\nEncourage students to refer to the examples from the previous lesson as they work to generalize.\nAction and Expression: Internalize Executive Functions.\nChunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Look for students who are struggling to begin and review examples from the previous lesson. Record their thinking on a display and keep the work visible as students continue to work.\nSupports accessibility for: Organization; Attention\nStudent Facing\nHere is one part of a regular\n\\(n\\)\n-sided polygon inscribed in a circle of radius 1.\nCome up with a general formula for the perimeter of the polygon in terms of\n\\(n\\)\n. Explain or show your reasoning.\n[IMG]\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAnticipated Misconceptions\nIf students are struggling invite them to go back to the problems from the previous lesson to generalize the process. (Draw in the altitude. Find the measure of the central angle. Find the length of the opposite leg.) Suggest students generalize each step before trying to write a single formula.\nActivity Synthesis\nInvite previously selected groups to share one step at a time:\ngeneralize the angle measure\ngeneralize the segment length\nextend to the whole perimeter\nInvite another student to summarize by explaining where each piece of\n\\(P=2n \\boldcdot \\sin \\left( \\frac{360}{2n} \\right)\\)\nappears in the diagram.\nRepresenting, Conversing: MLR7 Compare and Connect.\nUse this routine to prepare students for the whole-class discussion. At the appropriate time, invite students to create a visual display showing their general formula for the perimeter of the polygon in terms of\n\\(n\\)\nand their reasoning. Allow students time to quietly circulate and analyze the formulas and reasoning in at least 2 other displays in the room. Give students quiet think time to consider what is the same and what is different. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations that highlight what information each group used, any calculations they completed, and how they expressed those procedures in a generalized formula.\nDesign Principle(s): Optimize output (for generalization); Cultivate conversation\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Interpret multiplication as scaling (resizing), by:\n- Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.\n- Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.", "problem_activity": "Narrative\nThe purpose of this activity is for students to draw two diagrams that represent a unit fraction of a unit fraction. Students work with the same unit fractions in both diagrams. The directions were intentionally written to encourage students to partition a unit square in different ways. Students initially partition the square into thirds in the first problem and into fourths in the second problem. Students may complete the diagrams in a way that makes sense to them. During the synthesis, students will connect both of the diagrams to the expressions\n\\(\\frac {1}{4} \\times \\frac {1}{3}\\)\nand\n\\(\\frac {1}{3} \\times \\frac {1}{4}\\)\n.\nLaunch\nGroups of 2\nActivity\n2 minutes: Independent work time\n10 minutes: partner work time\nMonitor for students who:\ndraw diagrams like those in the student responses\nrecognize that both diagrams have the same amount shaded\ncan explain the different ways they partitioned each diagram\nStudent Facing\nShow\n\\(\\frac{1}{3}\\)\nof the square.\nShade\n\\(\\frac{1}{4}\\)\nof\n\\(\\frac{1}{3}\\)\nof the square.\nHow much of the whole square is shaded?\n[IMG]\nShow\n\\(\\frac{1}{4}\\)\nof the square.\nShade\n\\(\\frac{1}{3}\\)\nof\n\\(\\frac{1}{4}\\)\nof the square.\nHow much of the whole square is shaded?\n[IMG]\nHow are the diagrams the same and how are they different?\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAdvancing Student Thinking\nIf students do not identify how much of the whole square is shaded, ask: “How can you adapt your diagram to show how many pieces would be in the whole square?”\nActivity Synthesis\nAsk previously selected students to share their responses in the order given.\n“How are the diagrams the same?” (They both have\n\\(\\frac {1}{4}\\)\nand\n\\(\\frac {1}{3}\\)\nin them. They both have\n\\(\\frac {1}{12}\\)\nof the whole square shaded.)\n“How are the diagrams different?” (In one diagram, I started by showing\n\\(\\frac {1}{3}\\)\nof the square and then shaded in\n\\(\\frac {1}{4}\\)\nof\n\\(\\frac {1}{3}\\)\n. In the other diagram, I started by showing\n\\(\\frac {1}{4}\\)\nof the square and then shaded in\n\\(\\frac {1}{3}\\)\nof\n\\(\\frac {1}{4}\\)\n.)\nDisplay:\n\\(\\frac{1}{12}\\)\n,\n\\(\\frac{1}{4}\\times\\frac{1}{3}\\)\n,\n\\(\\frac{1}{3}\\times\\frac{1}{4}\\)\n“How do the diagrams represent these expressions?”\nAdapt diagrams as needed in order to show all three expressions.\n“How do we know that both squares have the same amount shaded?” (They both have\n\\(\\frac {1}{12}\\)\nshaded.)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.", "problem_activity": "Circle\n$$A$$\nis shown below.\n[IMG]\nDraw two diameters of the circle.\nIdentify the shape defined by the end points of the two diameters.\nExplain why this shape is always the result.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (This standard is limited to problems posed with whole numbers and having whole number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).)", "problem_activity": "Problem 1\na. Your teacher will read you a series of situations. Explain what new information you get with each one.\nVersion 1: Mrs. Powell buys a few boxes with some binders in each box. She plans to give out the binders to some students.\nVersion 2: Mrs. Powell buys a few boxes with some binders in each box. She plans to give out the same number of binders to each student.\nVersion 3: Mrs. Powell buys a few boxes with some binders in each box. She plans to give out 3 binders to each student.\nVersion 4: Mrs. Powell buys a few boxes with the same number of binders in each box. She plans to give out 3 binders to each student.\nVersion 5: Mrs. Powell buys a few boxes with 6 binders in each box. She plans to give out 3 binders to each student.\nVersion 6: Mrs. Powell buys 4 boxes with 6 binders in each box. She plans to give out 3 binders to each student.\nb. What question do you think this problem is going to ask you to solve? Write the question, then find its answer.\nProblem 2\nTen children equally share 40 almonds. How many almonds will 3 of those children get?\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.", "problem_activity": "Task\nImagine you are a ninja that can slice solid objects straight through. You have a solid cube in front of you. You are curious about what 2-dimensional shapes are formed when you slice the cube. For example, if you make a slice through the center of the cube that is parallel to one of the faces, the cross section is a square:\n[IMG]\n[IMG]\nFor each of the following slices, (i) describe using precise mathematical language the shape of the cross section. (ii) draw a diagram showing the cross section of the cube.\n[IMG]\na. A slice containing edge AC and edge EG b. A slice containing the vertices C, B, and G. C. A slice containing the vertex A, the midpoint of edge EG, and the midpoint of edge FG.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.", "problem_activity": "Warm-up\nThe purpose of this warm-up is for students to reason about square roots by estimating the value of each expression. The values given as choices are close in range to encourage students to use the square roots they know to help them estimate ones they do not. These understandings will be helpful for students in upcoming activities where they will be applying the Pythagorean Theorem.\nWhile four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem.\nLaunch\nDisplay one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all previous problems displayed throughout the talk. Follow with a whole-class discussion.\nStudent Facing\nWhich estimate is closest to the actual value of the expression? Explain your reasoning.\n\\(\\sqrt{24}\\)\n4\n4.5\n5\n\\(\\sqrt7\\)\n2\n2.5\n3\n\\(\\sqrt{42}\\)\n6\n6.5\n7\n\\(\\sqrt{10}\\)\n+\n\\(\\sqrt{97}\\)\n13\n13.25\n13.5\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nAsk students to share their strategies for each problem. Record and display their responses for all to see. After each student shares, ask if their chosen estimate is more or less than the actual value of each expression.\nSpeaking: MLR8 Discussion Supports.\n: Display sentence frames to support students when they explain their strategy. For example, \"First, I _____ because . . .\" or \"I noticed _____ so I . . . .\" Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.\nDesign Principle(s): Optimize output (for explanation)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.", "problem_activity": "Task\nRosa ran $\\frac13$ of the way from her home to school. She ran $\\frac14$ mile. How far is it between her home and school?\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.", "problem_activity": "Problem 1\nPoint\n$$Q$$\nis located at\n$$(-3,2)$$\n. It is reflected over at least one axis and is now located at\n$$(-3,-2)$$\n. Describe the reflection that took place.\nProblem 2\nRiley reflects point\n$$L$$\n, located at\n$$(5,4)$$\n, over the\n$$y$$\n-axis. They determine its new location is at\n$$(4,-5)$$\n. Did Riley correctly identify the coordinates of the reflected point? Explain your reasoning.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Prove the Laws of Sines and Cosines and use them to solve problems.", "problem_activity": "Warm-up\nThe mathematical purpose of this activity is for students to describe the relationship between variables using mathematical terminology, such as strong or weak relationship and positive or negative relationship. These terms were defined in the previous lesson. Students must reason abstractly and quantitatively (MP2) to determine the type of relationship.\nLaunch\nArrange students in groups of 2. Tell students there are many possible answers for the questions. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.\nTo help students understand some of the context, explain that, for many modern cars, it is recommended that the oil be changed every 5,000 miles driven or every 5 months.\nStudent Facing\nDescribe the strength and sign of the relationship you expect for each pair of variables. Explain your reasoning.\nUsed car price and original sale price of the car.\nUsed car price and number of cup holders in the car.\nUsed car price and number of oil changes the car has had.\nUsed car price and number of miles the car has been driven.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nThe purpose of this discussion is for students to discuss the strength of relationships in preparation for having students distinguish between\ncausal relationships\nand statistical relationships.\nEncourage the use of the terms strong or weak relationship and positive or negative relationship in the discussion.\nThe discussion should focus on the reasoning for the last 2 problems and how they are similar and different. Here are some questions for discussion.\n“How is the relationship between the car price and number of oil changes similar to the relationship between the car price and number of miles driven?” (They both have a strong, negative relationship.)\n“How is the relationship between the car price and number of oil changes different from the relationship between the car price and number of miles driven?” (The number of miles the car has been driven seems more directly related to the price than the number of oil changes. It probably has a stronger negative relationship than the number of oil changes.)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.", "problem_activity": "Problem 1\nWrite each rational number in the form\n$${a\\over{b}}$$\n, where\n$$a$$\nand\n$$b$$\nare integers, and\n$$b\\neq0$$\n.\na.\n$${0.3333}$$\nb.\n$${-5.05}$$\nc.\n$${\\sqrt{64}}$$\ne.\n$${\\sqrt{4\\over25}}$$\nf.\n$${\\sqrt{1.21}}$$\nProblem 2\nA student wrote two repeating decimals as fractions using the strategy shown below.\n| Given the decimal $$0.\\overline{12}$$ Let $$x=0.\\overline{12}$$ Then $$100x=12.\\overline{12}$$ |https://d37su8u1z9ynie.cloudfront.net/uploads/2019/02/06/math8_u7_l5_ap2a.PNG| |https://d37su8u1z9ynie.cloudfront.net/uploads/2019/02/06/math8_u7_l5_ap2a.PNG| image:: https://d37su8u1z9ynie.cloudfront.net/uploads/2019/02/06/math8_u7_l5_ap2a.PNG | Given the decimal $$0.\\overline{123}$$ Let $$x=0.\\overline{123}$$ Then $$1000x=123.\\overline{123}$$ |https://d37su8u1z9ynie.cloudfront.net/uploads/2019/02/06/math8_u7_l5_ap2b.PNG| |https://d37su8u1z9ynie.cloudfront.net/uploads/2019/02/06/math8_u7_l5_ap2b.PNG| image:: https://d37su8u1z9ynie.cloudfront.net/uploads/2019/02/06/math8_u7_l5_ap2b.PNG |\n|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\na. Describe the student’s strategy in your own words.\nb. Use the same strategy to write the two decimals below as fractions.\n$${0.\\overline{54}}$$\n$${0.1\\overline{23}}$$\nProblem 3\nWhat is the sum of\n$${0.1}$$\nand\n$${0.\\overline3}$$\nwritten as a fraction?\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as “right rectangular prism.”)", "problem_activity": "Narrative\nThe purpose of this card sort is for students to connect words and phrases to visual representations of partitioned shapes. Students begin by sorting the cards in a way that makes sense to them, and then are invited to sort the shapes based on the language that can be used to describe them. In the activity synthesis, the phrases “two of the halves” and “four of the fourths” are introduced to describe shapes where all pieces are shaded in.\nKeep cards D and N available to use during the lesson synthesis.\nAs students sort the shape cards they look for structure, namely the number of pieces making up each shape, whether or not the pieces are equal, and the number of those pieces that are shaded (MP7).\nMLR2 Collect and Display.\nCirculate, listen for and collect the language students use as they describe the shapes. On a visible display, record words and phrases such as: “shaded,” “half,” “fourth,” “quarter,” “equal,” “not equal,” “whole,” “shape.” Invite students to borrow language from the display as needed, and update it throughout the lesson.\nAdvances: Conversing, Speaking\nRequired Materials\nMaterials to Copy\nShaded Parts Cards\nRequired Preparation\nCreate a set of Shaded Parts Cards for each group of 2.\nLaunch\nGroups of 2\nGive each group a set of cards.\n“Sort your cards in a way that makes sense to you. Then explain how you sorted them.”\n3 minutes: partner work time\nShare responses.\nActivity\n“Now, sort the cards into the categories shown on your paper. Think about what words you can use to describe the shape on each card. Write the letter of each card in the space for it's category.”\n5 minutes: partner work time\nStudent Facing\nSort the cards into categories in a way that makes sense to you. Explain how you sorted the cards.\n[IMG]\nSort your cards into these categories.\nWrite the letter of each card in the space for its category.\nA fourth or quarter is shaded\n[IMG]\nA half is shaded\nThe whole shape is shaded\n[IMG]\nNot equal pieces\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nDisplay card J.\n“How can we describe this shape?” (It is a square. It is cut into halves. The whole shape is shaded.)\n“When a shape is split into halves and they are both shaded, we say the whole shape is shaded or two of the halves are shaded.”\nRepeat with card E.\n“When a shape is split into fourths and they are all shaded, we say the whole shape is shaded or four of the fourths are shaded.”\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.", "problem_activity": "Activity\nIn this activity, students again apply what they learned about quadratic expressions and their graphs. This time, they solve a contextual problem. Students consider how to change the parameters of a quadratic expression that models the movement of a digitally animated object—a peanut—such that it travels in a parabolic path and meets certain restrictions (MP4).\nProviding time to notice and wonder will elicit the idea that this equation shows the horizontal and vertical movement of the peanut, which will be useful when students adjust the equation in a later activity. While students may notice and wonder many things about the graphs and equation, connecting the vertex of the graph to the parameters in the equation are the important discussion points. This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).\nFor the second question, there are many possible values of\n\\(h\\)\nand\n\\(k\\)\nthat students could use. As students work, notice those who choose to:\nShift the starting position of the peanut closer to the wall (translating the graph horizontally to the right).\nStay in the same starting position but shift the vertex to a point closer to the wall.\nShift both the peanut and the vertex of the parabolic path.\nIf students choose to use graphing technology to help them experiment with expressions or verify their predictions, they practice choosing tools strategically (MP5).\nDigital\nLaunch\nYou may wish to use the applet during the synthesis to demonstrate student equations. There is a copy of the applet with an option to show the path of the peanut.\nRepresenting, Conversing: MLR7 Compare and Connect.\nAs students share their values of\n\\(h\\)\nand\n\\(k\\)\nwith the class, call students’ attention to the different ways the output is represented graphically within the context of the situation. Take a close look at the equation to distinguish what the\n\\(h\\)\nand\n\\(k\\)\nrepresent in each situation. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.\nDesign Principle(s): Maximize meta-awareness\nRepresentation: Internalize Comprehension.\nProvide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content.\nSupports accessibility for: Language; Conceptual processing\nStudent Facing\nMai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to show the path of the main character, an animated peanut, jumping over a wall.\nMai uses the equation\n\\(y = \\text-0.1(x-h)^2+k\\)\nto represent the path of the jump.\n\\(y\\)\nrepresents the height of the peanut as a function of the horizontal distance it travels\n\\(x\\)\n. On the screen, the base of the wall is located at\n\\((22, 0)\\)\n, with the top of the wall at\n\\((22,4.5)\\)\n.\nThe dashed curve in the picture shows the graph of one equation Mai tried, where the peanut fails to make it over the wall.\n[IMG]\nWhat are the values of\n\\(k\\)\nand\n\\(h\\)\nin this equation?\nThis applet is programmed to use Mai’s equation. Experiment with the applet to find an equation to get the peanut over the wall, and keep it on the screen. Be prepared to explain your reasoning.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nPrint\nLaunch\nAsk students to read the first couple of paragraphs in the statement. Then, display this graph for all to see. Tell students that when Mai used the equation\n\\(y = \\text -0.1(x-16)^2+7\\)\n, this is the path of the peanut’s jump.\n[IMG]\nAsk students what they notice and wonder about the equation and the graph. Make sure students see the connection between the parameters in the equation and the vertex of the graph before they proceed with the activity.\nProvide continued access to graphing technology, in case students choose to use it.\nRepresenting, Conversing: MLR7 Compare and Connect.\nAs students share their values of\n\\(h\\)\nand\n\\(k\\)\nwith the class, call students’ attention to the different ways the output is represented graphically within the context of the situation. Take a close look at the equation to distinguish what the\n\\(h\\)\nand\n\\(k\\)\nrepresent in each situation. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.\nDesign Principle(s): Maximize meta-awareness\nRepresentation: Internalize Comprehension.\nProvide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content.\nSupports accessibility for: Language; Conceptual processing\nStudent Facing\nMai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to model the path of the main character, an animated peanut, jumping over a wall.\n[IMG]\nMai uses the equation\n\\(y = \\text-0.1(x-h)^2+k\\)\nto represent the path of the jump.\n\\(y\\)\nrepresents the height of the peanut as a function of the horizontal distance it travels,\n\\(x\\)\n.\nOn the screen, the base of the wall is located at\n\\((22, 0)\\)\n, with the top of the wall at\n\\((22,4.5)\\)\n. The dashed curve in the picture shows the graph of 1 equation Mai tried, where the peanut fails to make it over the wall.\n[IMG]\nWhat are the values of\n\\(h\\)\nand\n\\(k\\)\nin this equation?\nStarting with Mai’s equation, choose values for\n\\(h\\)\nand\n\\(k\\)\nthat will guarantee the peanut stays on the screen but also makes it over the wall. Be prepared to explain your reasoning.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAnticipated Misconceptions\nSome students may randomly choose values for\n\\(h\\)\nand\n\\(k\\)\nuntil they find a combination whose graph meets the requirements. Encourage them to reason about the problem more systematically, by considering what they learned about the vertex form.\nActivity Synthesis\nSelect students with contrasting strategies to share their equations and explanations for the second question. To verify that students’ proposed equations meet the animation criteria, consider graphing students’ proposed equations for all to see. Highlight the three strategies mentioned in the Activity Narrative.\nPoint out one distinction between the graphs in this activity and previous graphs that represent projectile motions. In earlier cases, the output represented the height of an object as a function of\ntime\n, so the graph of the function and physical trajectory of the object may be very different (for example, the object may be going straight up and then fall straight down, but the graph shows a parabola). Here, they see a model where the height of an object is a function of\nhorizontal distance\n, so the path of the object in a projectile motion does resemble the graph representing the function.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Use place value understanding to round whole numbers to the nearest 10 or 100.", "problem_activity": "Problem 1\nThe following picture graph represents how much time Jasmine spends exercising during the week.\n[IMG]\nHow many minutes did Jasmine spend exercising during the week?\nProblem 2\nThis bar graph shows how long, in minutes, four people spent reading over the weekend.\n[IMG]\nHow many fewer minutes did Fred and Tomoko spend reading than Simon and Rosario?\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.", "problem_activity": "Warm-up\nThis Math Talk encourages students to think about the fact that subtracting a number is equivalent to adding the opposite of that number (that is,\n\\(100-5=100+\\text-5\\)\n) and to rely on the structure of the expressions on each side of the equal sign to mentally solve problems. The understandings elicited here will be helpful later in the lesson when students reason that\n\\((x-1)(x-4)\\)\nis equivalent to\n\\((x+\\text-1)(x+\\text-4)\\)\n, which in turn helps them expand the expression using diagrams and the distributive property.\nSome students may reason about the value of\n\\(n\\)\nby reasoning. For example, to find\n\\(n\\)\nin\n\\(40-8=40+n\\)\n, they see that the left side of the equation is 32 and reason that the number being added to 40 must be negative, and that it must be -8. Others may carry out the steps for solving equations, for example, by subtracting 40 from both sides, or by adding 8 to both sides and then subtracting 48 from both sides.\nAny of these strategies are fine. The key point is to recognize that the number being added on one side is the opposite of the value being subtracted from the other side. If after a couple of questions students show an understanding of this, and if time is limited, it is not essential for students to complete all questions.\nAs students solve these equations mentally, they practice looking for and making use of structure (MP7).\nLaunch\nDisplay one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.\nRepresentation: Internalize Comprehension.\nTo support working memory, provide students with sticky notes or mini whiteboards.\nSupports accessibility for: Memory; Organization\nStudent Facing\nSolve each equation mentally.\n\\(40-8=40+n\\)\n\\(25+\\text-100=25-n\\)\n\\(3-\\frac12=3+n\\)\n\\(72-n=72+6\\)\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nAsk students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:\n“Who can restate\n\\(\\underline{\\hspace{.5in}}\\)\n’s reasoning in a different way?”\n“Did anyone have the same strategy but would explain it differently?”\n“Did anyone solve the problem in a different way?”\n“Does anyone want to add on to\n\\(\\underline{\\hspace{.5in}}\\)\n’s strategy?”\n“Do you agree or disagree? Why?”\nIf not already made clear in students’ explanations, highlight that subtracting a number is gives the same outcome as adding the opposite of that number.\nTell students that remembering that subtraction can be thought of in terms of addition can help us rewrite quadratic expressions such as\n\\((x-5)(x+2)\\)\nor\n\\((x-9)(x-3)\\)\n, where one or both factors are differences.\nSpeaking: MLR8 Discussion Supports.\nDisplay sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.\nDesign Principle(s): Optimize output (for explanation)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.\nGrade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.", "problem_activity": "Problem 1\nMs. Roll is making a mural using square tiles. She has 96 tiles and wants the mural to be 8 tiles tall.\na. How many tiles long will Ms. Roll’s mural be?\nb. Write one or more equations to show how you solved this problem.\nProblem 2\nAyana starts to draw an area model to solve\n$$79\\div3$$\n, which is shown below.\n[IMG]\na. Why did Ayana make a rectangle with an area of 60 and a length of 20?\nb. How much of the dividend still needs to be divided by 3? How do you know?\nc. Complete the area model by extending the one Ayana started above.\nd. Write the quotient and remainder for\n$$79\\div3$$\n. Where are those values represented in the area model above?\ne. Use multiplication to check your answer.\nProblem 3\na. Solve each of the following word problems. Show or explain your work.\nThe price of a video game is $52. A video game costs 2 times as much as a jigsaw puzzle. What is the price of the jigsaw puzzle?\nMs. Needham squeezed 85 ounces of orange juice. She wants to pour it into glasses that hold 6 ounces each. How many glasses will Ms. Needham need to hold all the juice?\nb. Did you use an area model to represent either or both problems above? If not, could you have? Why or why not?\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.", "problem_activity": "Optional activity\nIn this optional activity, students revisit the representation of a multiplication chart, which may be familiar from previous grades; however, in this activity, the multiplication chart is extended to include negative numbers. Students identify and continue patterns (MP8) to complete the chart and see that it fits the patterns in the chart for the product of two negative numbers to be a positive number.\nThe blackline master has a multiplication chart that also includes the factors 1.5, -1.5, 3.2, and -3.2, so that students can see how the patterns extend to rational numbers that are not integers. Encourage students to complete the rows and columns for the integers first and then come back to 1.5, -1.5, 3.2, and -3.2 later. Directions are included on the blackline master for a way that students can fold their papers to hide the non-integers while they fill in the integers. If you want students to do this, it would be good to demonstrate and walk them through the process of folding their paper.\nDigital\nLaunch\nArrange students in groups of 3. If desired, distribute 1 copy of the blackline master to every student and instruct students to ignore the chart printed in their books or devices. (Also if desired, instruct students to fold their papers according to the directions on the top and right sides of the chart, so that the decimal rows and columns are temporarily hidden.) Give students 30 seconds of quiet think time. Have them share what patterns they notice about the numbers that are already filled in. Give the groups 5 minutes of work time followed by whole-class discussion.\nIf students have access to the digital materials, students can use the applet to complete the chart. The applet helps students focus on fewer of the numbers and patterns at a time, similar to the purpose of folding the blackline master. Also, the applet gives students immediate feedback on whether their answers are correct which helps them test their theories about ramifications of multiplying by a negative number.\nRepresentation: Internalize Comprehension.\nActivate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.\nSupports accessibility for: Memory; Conceptual processing\nStudent Facing\nLook at the patterns along the rows and columns and continue those patterns to complete the table. When you have filled in all the boxes you can see, click on the \"More Boxes\" button.\nWhat does this tell you about multiplication by a negative?\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nPrint\nLaunch\nArrange students in groups of 3. If desired, distribute 1 copy of the blackline master to every student and instruct students to ignore the chart printed in their books or devices. (Also if desired, instruct students to fold their papers according to the directions on the top and right sides of the chart, so that the decimal rows and columns are temporarily hidden.) Give students 30 seconds of quiet think time. Have them share what patterns they notice about the numbers that are already filled in. Give the groups 5 minutes of work time followed by whole-class discussion.\nIf students have access to the digital materials, students can use the applet to complete the chart. The applet helps students focus on fewer of the numbers and patterns at a time, similar to the purpose of folding the blackline master. Also, the applet gives students immediate feedback on whether their answers are correct which helps them test their theories about ramifications of multiplying by a negative number.\nRepresentation: Internalize Comprehension.\nActivate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.\nSupports accessibility for: Memory; Conceptual processing\nStudent Facing\nComplete the\nshaded\nboxes in the multiplication square.\n[IMG]\nLook at the patterns along the rows and columns. Continue those patterns into the unshaded boxes.\nComplete the whole table.\nWhat does this tell you about multiplication with negative numbers?\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAnticipated Misconceptions\nSome students may need a reminder of how a mutliplication chart works: the factors are listed at the end of the rows and columns, and their products go in the boxes.\nActivity Synthesis\nThe most important takeaway is that it makes sense for the product of two negative numbers to be a positive number, whether or not the numbers are integers. This fits in with the patterns in the extended multiplication chart. Those patterns depend on the distributive property. For example, the reason the numbers in the top row go up by 5s is that\n\\(5(n+1) = 5n + 5\\)\n. So when students extend the pattern to negative numbers, they are extending the distributive property.\nDisplay a complete chart for all to see, and ask students to explain the ways in which the chart shows that the product of a negative and a negative is a positive. The general argument involves assuming that a pattern observed in a row or column will continue on the other side of 0.\n[IMG]\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.", "problem_activity": "Narrative\nThe purpose of this activity is for students to learn stage 2 of the Greatest of Them All center. Students use digit cards to create the greatest possible number. As each student draws a card, they choose where to write it on the recording sheet. Once a digit is placed, it can’t be moved. Students compare their numbers using\n\\(<\\)\n,\n\\(>\\)\n, or\n\\(=\\)\n. The player with the greater number in each round gets a point.\nStudents should remove cards that show 10 from their deck.\nMLR2 Collect and Display.\nSynthesis: Direct attention to the words collected and displayed on the anchor chart from the previous lessons. Invite students to borrow place value language from the display as needed.\nAdvances: Conversing, Reading\nRequired Materials\nMaterials to Gather\nNumber cards 0–10\nMaterials to Copy\nGreatest of Them All Stage 2 Recording Sheet\nLaunch\nGroups of 2\nGive each group a set of number cards and each student a recording sheet.\n“Now you will be playing the Greatest of Them All center with your partner.”\n“You will try to make the greatest three-digit number you can.”\nDisplay number cards and recording sheet.\nDemonstrate picking a card.\n“If I pick a (2), I need to decide whether I want to put it in the hundreds, tens, or ones place to make the largest three-digit number.”\n“Where do you think I should put it?” (I think it should go in the ones place because it is a low number. In the hundreds place, it would only be 200.)\n30 seconds: quiet think time\nShare responses.\n“At the same time, my partner is picking cards and building a number, too.”\n“Take turns picking a card and writing each digit in a space.”\n“Read your comparison aloud to your partner.”\nActivity\n“Now play a few rounds with your partner.”\n15 minutes: partner work time\nActivity Synthesis\nSelect a group to share a comparison statement. For example:\n\\(654 > 349 \\)\nand\n\\(349 < 654\\)\n.\n“I noticed that partners had different comparison statements for the same numbers. How can they both be true?”\n“If I draw an 8, where should I choose to place it and why?” (I would put it in the hundreds place since it’s almost the highest number I could\ndraw\n. I might not get 9. With 800, I have a good chance for my number to be larger than my partner’s.)\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”", "problem_activity": "Problem 1\nConsider the statement: A negative number raised to any whole number power will be negative.\nIs this statement always, sometimes, or never true? Justify your response.\nProblem 2\nConsider the statement: An odd number raised to an odd power will be odd.\nIs this statement always, sometimes, or never true? Justify your response.\nProblem 3\nWhat is the last digit of\n$${7^{2023}}$$\n? Explain.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?", "problem_activity": "Problem 1\nWhich scatter plot shows a positive nonlinear relationship? Explain how you know.\n[IMG]\nProblem 2\nSketch a scatter plot in the coordinate plane below so that the following statements are true about your scatter plot:\nThere is a negative linear relationship between the variables.\nThere is a cluster of data.\nThere are no outliers in the data.\n[IMG]\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.", "problem_activity": "Activity\nThis activity prompts students to graph equations to solve problems and think about appropriate values of their variables. The context is the thickness and the area of a sheet of paper that is repeatedly folded.\nFor both equations, only whole numbers make sense as inputs. Students consider why this makes sense in this context. (This is in contrast, for example, to the price of producing solar energy in the earlier activity. While the data given is on an annual basis, it made sense to “connect the dots” because there is a price for producing solar energy at any given time.) Without instruction about how to create a discrete graph with their graphing technology, students are likely to produce a continuous graph. This is a good opportunity to discuss how to interpret a model for a particular context (MP4).\nLook for the different ways students use to find out how many folds are needed to reach a certain thickness. Identify students who used a table, a graph, or an equation so they can share later.\nLaunch\nGive each student a sheet of paper. Ask them to fold it in half, and in half again, as many times as they can. (To save paper, you might have students do this in pairs.) Once they have folded it as many times as they can, ask a few students to share how many times they folded it, and estimate the thickness of the folded paper. Tell students that they are going to investigate the relationship between the number of folds and the thickness of the paper.\nProvide access to graphing technology. This activity requires students to be mindful about the scale or graphing window. The activities that specifically target that skill are optional, so this gives students a chance to practice scaling the axes.\nReading, Writing, Speaking: MLR3 Clarify, Critique, Correct.\nPresent an incorrect answer and explanation for the question “How many folds does it take before the folded sheet of paper is more than 1 mm thick?” Consider using this ambiguous response: “It takes 4.5 folds, because I looked on the graph to check the points.” Ask students to identify the error, critique the reasoning, and write an improved explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe the point using both the graph and a created table. This helps students evaluate, and improve on, the written mathematical arguments of others, as they clarify their understanding of discrete graphs.\nDesign Principle(s): Optimize output (for explanation); Maximize meta-awareness\nStudent Facing\nThe thickness\n\\(t\\)\nin millimeters of a folded sheet of paper after it is folded\n\\(n\\)\ntimes is given by the equation\n\\(t = (0.05) \\boldcdot 2^n\\)\n.\nWhat does the number\n\\(0.05\\)\nrepresent in the equation?\nUse graphing technology to graph the equation\n\\(t = (0.05) \\boldcdot 2^n\\)\n.\nHow many folds does it take before the folded sheet of paper is more than 1 mm thick? How many folds before it is more than 1 cm thick? Explain how you know.\nThe area of a sheet of paper is 93.5 square inches.\nFind the remaining, visible area of the sheet of paper after it is folded in half once, twice, and three times.\nWrite an equation expressing the visible area\n\\(a\\)\nof the sheet of paper in terms of the number of times it has been folded\n\\(n\\)\n.\nUse graphing technology to graph the equation.\nIn this context, can\n\\(n\\)\ntake negative values? Explain your reasoning.\nCan\n\\(a\\)\ntake negative values? Explain your reasoning.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nStudent Facing\nAre you ready for more?\nUsing the model in this task, how many folds would be needed to get 1 meter in thickness? 1 kilometer in thickness?\nDo some research: what is the current world record for the number of times humans were able to fold a sheet of paper?\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Extension Student Response.\nAnticipated Misconceptions\nIf students have trouble organizing their data points, suggest that they use a table.\nActivity Synthesis\nFocus the discussion on:\nHow students identified the growth factor, given an equation or a description.\nHow students determined how many folds are needed to reach a certain thickness. Invite previously identified students to share their strategies.\nThe meaningful values of the variables in the equations:\nHighlight that a discrete graph is more appropriate in both situations: only the whole-number values make sense for the number of folds because it is not possible to fold the paper a fractional number of times.\nPractically speaking, a large number of folds, such as 36, is not possible.\nDemonstrate, using graphing technology, how to graph the equations and invite students to share how they chose an appropriate graphing window. If students just typed an equation, most likely the tool plotted a continuous curve. This is an opportunity to talk about interpreting the continuous graph in this context: really, only whole-number values for\n\\(n\\)\nare meaningful. Time permitting, you might demonstrate how to graph only whole number values of\n\\(n\\)\nusing your graphing technology of choice. Typically, this is done by using the technology to generate a table based on a rule. Here is an example:\n[IMG]\nTime permitting, consider asking students if the equations\n\\(t = (0.05)\\boldcdot 2^n \\)\nand\n\\(a = (93.5) \\boldcdot \\left(\\frac{1}{2}\\right)^n \\)\ncould be plotted on the same set of axes. Even though the variable\n\\(n\\)\n(the number of folds) is the same for the two equations, the thickness\n\\(t\\)\nand the area\n\\(a\\)\nare mesaured in different units and so cannot be plotted on the same axes.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.", "problem_activity": "Solve. Show or explain your work.\na.\n$$\\begin{align} 7,396 \\\\ {–\\ 2,145}& \\\\ \\hline \\end{align}$$\nb.\n$$87,625-5,031=$$\n______\nc.\n$$\\begin{array}{r} &19,850\\\\ -\\!\\!\\!\\!\\!\\!&15,761\\\\ \\hline \\end{array}$$\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.", "problem_activity": "Problem 1\nQuiana measures the width of her doorway entrance using a measuring tape. She measures the width as 33 inches. She later finds a better tape measure that measures to the nearest half of an inch. After re-measuring, she determines that the actual measure of the doorway is 32 ½ inches.\na. What is the amount of error from using the first measuring tape?\nb. What is the error as a percentage of the actual width?\nProblem 2\nPaul and Kimberly both estimate the weight of their dogs before bringing them to the vet. The table below shows each of their estimates along with the actual weights of the dogs from the vet’s office.\n| | ** Estimate (lbs.) ** | ** Actual (lbs.) ** |\n|----------------|------------------------------------------|----------------------------------------|\n| Paul's dog | 10 | 11.5 |\n| Kimberly's dog | 84 | 82.5 |\nPaul notices that both his and Kimberly’s estimates were only 1.5 pounds off from the actual weights. Paul says that means his estimate was as good as Kimberly’s estimate.\nDo you agree with Paul? Find the percent error for each person’s guess of their dog’s weight, and use it in your answer.\nProblem 3\nIn science class, Justine measured a rock sample and found it to be 15 grams. Her teacher told her that the scale is off and the actual weight is 0.3 grams more than her measurement.\nWhat is the percent error? Give your answer to the nearest percent.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.", "problem_activity": "Warm-up\nThis warm-up prompts students to compare four lines. It invites students to explain their reasoning and hold mathematical conversations, and allows you to hear how they use terminology and talk about lines and their properties. To allow all students to access the activity, each figure has one obvious reason it does not belong. Encourage students to move past the obvious reasons (e.g., line\n\\(t\\)\nhas a different color) and find reasons based on geometric properties (e.g., a slope triangle of line\n\\(u\\)\nis not similar to the slope triangles of the other three lines).\nSo far, we have only considered lines with positive slopes. The purpose of this warm-up is to suggest similarities (same vertical and horizontal lengths of slope triangles) and differences (since they are not parallel, there is something fundamentally different going on here) between lines whose slopes have the same absolute value but opposite signs.\nAs students share their responses, listen for important ideas and terminology that will be helpful in the work of this lesson. Students may:\nIdentify lines that have the same slope.\nIdentify points where lines intersect.\nDistinguish between lines with positive and negative values for slope and be able to articulate the difference clearly.\nLaunch\nArrange students in groups of 2–4 and provide access to geometry toolkits. Display the image of the four lines for all to see. Ask students to indicate when they have noticed one line that does not belong and can explain why. Give students 2 minutes of quiet think time and then time to share their thinking with their group. After everyone has conferred in groups, ask each group to offer at least one reason a particular line doesn’t belong.\nStudent Facing\nWhich line doesn’t belong?\n[IMG]\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nAfter students have conferred in groups, invite each group to share one reason why a particular line might not belong. Record and display the responses for all to see. After each response, ask the rest of the class if they agree or disagree. Since there is no single correct answer to the question asking which shape does not belong, attend to students’ explanations and ensure the reasons given are correct.\nDuring the discussion, prompt students to use mathematical terminology (parallel, intersect, slope) correctly. Also, press students on unsubstantiated claims. For example, a student may claim that\n\\(u\\)\ndoes not belong because it has a different slope. Ask how they know for sure that its slope is different from the other lines. Demonstrate drawing a slope triangle and computing slope.\nBased on the work done up to this point in the unit, students are likely to assume that the slope of\n\\(v\\)\nis\n\\(\\frac13\\)\n. In the discussion, solicit the idea that there is something fundamentally different about line\n\\(v\\)\ncompared to the others. You could use informal language like “uphill” and “downhill,” or “tilt direction.” The expressions positive and negative slope do not need to be introduced at this time.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.", "problem_activity": "Stage 3: Find Shapes\nRequired Preparation\nMaterials to Gather\nPicture books\nMaterials to Copy\nBlackline Masters\nPicture Books Stage 3 Recording Sheet\n\nNarrative\nStudents look through picture books and notice and describe shapes they see in the pictures.\nVariation:\nStudents may record the shapes they see with drawings or words.\nAdditional Information\nEach group of 2–4 needs at least one picture book that shows a variety of shapes throughout the book.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "yes" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Compute (using technology) and interpret the correlation coefficient of a linear fit.", "problem_activity": "Warm-up\nThe mathematical purpose of this activity is for students to reason about the area under a normal curve using what they know about composition and decomposition of areas.\nIf students are using a table of values with\n\\(z\\)\n-scores for this lesson, this work will be essential to finding the desired areas in many cases.\nLaunch\nRemind students that the phrase “the area under the curve” means the area between the\n\\(x\\)\n-axis and the curve.\nStudent Facing\nThe images show a normal curve with mean of 40 and standard deviation of 2.\nThe area under the curve to the left of 39 is 0.3085.\n[IMG]\nThe area under the curve to the left of 43 is 0.9332.\n[IMG]\nSince it is a normal curve, we know that the total area under the curve is 1. Use the given areas to find the areas in question. Explain your reasoning for each.\nFind the area under the curve to the right of 39.\nFind the area under the curve between 39 and 43.\nFind the area under the curve to the left of 40.\nFind the area under the curve between 39 and 40.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nThe purpose of this discussion is to make sure students can use the given areas to compose or decompose them to find additional areas. For each question, ask a student to explain how they reasoned about their answer. Highlight places where students used the symmetry of the distribution or cases where they could add or subtract known areas to find new areas.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.", "problem_activity": "Activity\nThe purpose of this activity is to use the new skills of multiplying and dividing rational numbers to represent and solve problems in a new context (MP4). In this activity students are using using multiplication and division of negatives and working with a proportional relationship with a negative constant of proportionality. Students should use what they know about proportional relationships to help them (for example; points lie on a straight line, line passes through zero, there is a point\n\\((1, k)\\)\nthat lies on the line).\nDigital\nLaunch\nRemind students that we can model positions below the surface with negative values, so drilling 30 feet down is represented with -30 feet.\nAsk students what they remember about proportional relationships. Examples:\nOften represented with an equation in the form\n\\(y=kx\\)\n.\nThe constant of proportionality, often called\n\\(k\\)\n, is the change in\n\\(y\\)\nfor a change by 1 in\n\\(x\\)\n.\nA graph representing a proportional relationship is a line through (0,0) and\n\\((1,k)\\)\nArrange students in groups of 3 during the discussion.\nRepresentation: Internalize Comprehension.\nProvide appropriate reading accommodations and supports to ensure students have access to written directions, word problems and other text-based content.\nSupports accessibility for: Language; Conceptual processing\nStudent Facing\nA water well drilling rig has dug to a height of -60 feet after one full day of continuous use.\nAssuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?\nIf the rig has been running constantly and is currently at a height of -147.5 feet, for how long has the rig been running?\nUse the coordinate grid to show the drill’s progress.\nAt this rate, how many hours will it take until the drill reaches -250 feet?\n[IMG]\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nPrint\nLaunch\nRemind students that we can model positions below the surface with negative values, so drilling 30 feet down is represented with -30 feet.\nAsk students what they remember about proportional relationships. Examples:\nOften represented with an equation in the form\n\\(y=kx\\)\n.\nThe constant of proportionality, often called\n\\(k\\)\n, is the change in\n\\(y\\)\nfor a change by 1 in\n\\(x\\)\n.\nA graph representing a proportional relationship is a line through (0,0) and\n\\((1,k)\\)\nArrange students in groups of 3 during the discussion.\nRepresentation: Internalize Comprehension.\nProvide appropriate reading accommodations and supports to ensure students have access to written directions, word problems and other text-based content.\nSupports accessibility for: Language; Conceptual processing\nStudent Facing\nA water well drilling rig has dug to a height of -60 feet after one full day of continuous use.\nAssuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?\nIf the rig has been running constantly and is currently at a height of -147.5 feet, for how long has the rig been running?\nUse the coordinate grid to show the drill’s progress.\n[IMG]\nAt this rate, how many hours will it take until the drill reaches -250 feet?\n[IMG]\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAnticipated Misconceptions\nSome students may struggle to get the length of time out of the phrase \"one full day of continuous use,\" wondering whether the drill was running at night. Let them know the drill has been going for 24 hours.\nActivity Synthesis\nHave students to share their solutions with each other in groups of three and work to come to agreement.\nTo wrap up, emphasize that because we can now complete any calculation with any rational number, we can extend constants of proportionality\n\\(k\\)\nto include negative values. Because the operations are now well-defined, we can answer questions using such a constant.\nSpeaking: MLR8 Discussion Supports.\nUse this routine to support whole-class discussion. Invite select groups to share their solution and reasoning with the class. After each group presents, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they heard to a partner, before selecting one or two students to share with the class. Ask the original group if this accurately reflects their thinking. This will provide more students with an opportunity to describe what they have learned about proportional relationships with a negative constant of proportionality.\nDesign Principle(s): Support sense-making\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.", "problem_activity": "Warm-up\nBy now students have written many division equations based on verbal descriptions of situations. This warm-up prompts them to go in the other direction: to interpret a division expression and write a fitting question the expression could help answer. Then, they trade descriptions with a partner and reason about the value of the same expression in someone else’s story.\nAs students write their descriptions, look out for scenarios that are unclear and ask their authors clarifying questions. Select 2–3 students whose descriptions clearly match the expression to share later.\nStudents may struggle to answer their partner’s question because the descriptions are confusing or do not match the given expression. Encourage them to ask the questions they have about the description, but still try to find the answer to the given expression.\nLaunch\nArrange students in groups of 2. Explain to students the two halves of this activity: to write a situation and a question that can be represented by the equation\n\\(12 \\div \\frac23 = ?\\)\n, and to trade papers with their partner and find the answer to their partner’s question.\nGive 2 minutes of quiet think time for the first question, another minute for the second question, and follow with a whole-class discussion.\nStudent Facing\nThink of a situation with a question that can be represented by the equation\n\\(12 \\div \\frac23 = \\,?\\)\nDescribe the situation and the question.\nTrade descriptions with your partner, and answer your partner’s question.\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nAnticipated Misconceptions\nSome students may struggle writing a context. Ask them to reflect on the work they have done and what the division expression means for them, or suggest that they write related multiplication expressions to help them understand the division expression.\nActivity Synthesis\nAsk selected students to share their stories with the whole group. After each student shares, have another student explain how the story matches the division equation. Afterward, ask these and other students for the value of the given expression, and write and display the completed division equation for all to see.\nIf not already mentioned by students, highlight that the answer is the same for all scenarios because they are all based on the same division expression.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]

[ { "role": "user", "prompt_template": "You are a math expert reviewing K-12 curriculum. Does this problem or activity enable students to completely learn the following concept or skill? Answer 'yes' if it does, and 'no' if it does not. Answer no if the problem helps students understand part but not all of a concept or skill. \n\nProblem/activity: \n{problem_activity}\n\nConcept/skill: \n{standard_description}", "standard_description": "Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...", "problem_activity": "Narrative\nIn this activity, students learn how to use a protractor. They align a protractor to the vertex and a ray of an angle so that its measurement can be read. The given angles are oriented in different ways, drawing students’ attention to the two sets of scales on a protractor. Students need to consider which set of numbers to pay attention to and think about a possible explanation for when or how to use each scale. Moreover, one of the scales is only marked in increments of 10 degrees so if students use this scale they need to reason carefully about the precise angle measure (MP6).\nThis activity uses\nMLR1 Stronger and Clearer Each Time. Advances: reading, writing.\nAction and Expression: Internalize Executive Functions.\nInvite students to estimate the size of the angle before finding each precise measurement. Offer the sentence frame: “This angle will be greater than _____ and less than _____. It will be closer to _____.”\nSupports accessibility for: Conceptual Processing, Visual-Spatial Processing, Attention\nLaunch\nGroups of 2–4\nActivity\n5 minutes: independent work time\n2–3 minutes: group discussion\nMonitor for students who find each measurement by:\nlooking for the scale with a ray on\n\\(0^\\circ\\)\nand counting up on that scale\nsubtracting the two numbers (on the same scale) that the rays pass through\nMLR1 Stronger and Clearer Each Time\n“Share your response to the last problem with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”\n3–5 minutes: partner discussion\nRepeat with 2–3 different partners.\n“Revise your initial draft based on the feedback you got from your partners.”\n2–3 minutes: independent work time\nStudent Facing\nHere are four angles whose sizes you may have estimated earlier. A protractor has been placed over each angle. Measure the size of each angle in degrees.\n[IMG]\n[IMG]\n[IMG]\n[IMG]\nElena and Kiran are measuring an angle with a protractor. Elena says the angle is\n\\(80^\\circ\\)\n. Kiran says it shows\n\\(100^\\circ\\)\n. Why might they end up with different measurements? Which one is correct? Explain your reasoning.\n[IMG]\nStudent Response\nTeachers with a valid work email address can\nclick here to register or sign in\nfor free access to Student Response.\nActivity Synthesis\nSelect students who used different strategies to share their responses and reasoning.\nDisplay the image of Elena and Kiran’s angle. If no students mentioned that\n\\(80^\\circ\\)\nis not possible because the angle clearly appears greater than a right angle (\n\\(90^\\circ\\)\n), consider asking how Elena can use what she knows about right angles to think about her measurement.\n", "response_template": null, "aligns": null }, { "role": "assistant", "prompt_template": null, "standard_description": null, "problem_activity": null, "response_template": "{aligns}", "aligns": "no" } ]