6th to 8th Grade - Gateway 2

The materials reviewed for Experience Math Grades 6 through 8 meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

Conceptual understanding is developed throughout the course using a three-part lesson structure. Each lesson has a Mind’s On Activity, an Action Task, and a Consolidate Activity. During teacher guidance in the Action task, the "And the Point Is" notes clarify the specific conceptual goal behind each activity. Students have individual opportunities to demonstrate conceptual understanding through performance tasks, wonder tasks, and your turn problems. Conceptual understanding is assessed through math routines such as Math Congress and Gallery Walks, Exit Tickets, and Self/Peer Assessment. Examples include:

• Grade 6, Topic 3: Ratio, Lesson 2, Action Task, Questions 1-4, students develop conceptual understanding of equivalent ratios. Action Task states, “Daniela’s class has 24 students. 1. At the beginning of September, 16 students were 11 years old and 8 students were 12 years old. a. Use a model to show the ages of students in Daniela’s class. b. Write a ratio that compares the number of 11-year-olds and 12-year-olds. c. Write an equivalent ratio for the ratio in Question 1(b). Use your model to explain how you know the ratio is equivalent.” Problems 2–4 are structured the same, with part c asking students to explain how their models show an equivalent ratio. Teacher Guidance, Using the Action Task states, “ Students could record their responses on the reproducible page, Equivalent Ratios. Students might use counters of different colors or shapes to model the 11- and 12-year-olds. You could encourage some students to use tape diagrams. Keep in mind that it might be easier for students to share their work in the Consolidate part of the lesson if everyone uses the same color or shape counter to represent each age group. Note that the sample answers provided use red counters for 11-year-olds and blue counters for 12-year-olds. You might choose to do Question 1 as a class if you feel students need the support. Specifically, you might use a model with arrays of counters to find and show equivalent ratios. Conversation Starters: Could the counters be arranged in an array? How about in a long line? What ratio do you see if you only look at the top line in a tape diagram? What if you look at the top two lines? As the children turn 12, how do the ratios change?” (6.RP.3)

• Grade 7, Topic 11: Collecting Data, Lesson 1, Exit Ticket, students demonstrate conceptual understanding of designing statistical surveys. The Exit Ticket states, “Suppose you were collecting data about pizza preferences of teenagers by using a sample. How might the wording of the question affect the data you collect? How might who you ask affect the data you collect?” (7.SP.1)

• Grade 8, Topic 5: Proportional Relationships and Slope, Lesson 2, Your Turn Questions, students develop conceptual understanding of slope as they use similar triangles to understand why the slope is the same between any two points on a non-horizontal line in the coordinate plane. Question 5 states, “Describe the transformations that show that Triangle C and Triangle D are similar.” A graph of a proportional relationship is shown. The hypotenuse of Triangle C lies on the line with vertices (0, 0), (2, 0), and (2, 5), and the hypotenuse of Triangle D also lies on the line with vertices (4, 10), (8, 10), and (8, 20). Question 6 states, “Explain why you can use any two points on a line to determine its slope.” (8.EE.6)

The materials reviewed for Experience Math Grades 6 through 8 meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters. 

The materials include multiple pathways for fluency development, such as Number Talks and Data Talks, which target fluency and flexibility in thinking about numbers and operations while allowing students to use their own problem-solving strategies. Students engage in practice through Your Turn activities that provide procedural practice opportunities. Fluency is embedded within the three-part lesson structure, Minds On, Action Task, and Consolidate, and is assessed within the program’s assessment system, which includes specific skill and concept questions that help teachers monitor and support students’ procedural fluency development. Examples include:

• Grade 6, Topic 11: Using Algebra, Lesson 2, Exit Ticket, students demonstrate procedural skill and fluency as they evaluate algebraic expressions involving whole-number exponents. Exit Ticket states, “What is the value of 3n24+2m when n = 12 and m = 4.” (6.EE.1)

• Grade 7, Topic 6: Geometry, Lesson 3, Additional Practice, students demonstrate procedural skill and fluency as they use facts about complementary and supplementary angles. Question 2 states, “What is the measure of the angle that is complementary to a 55\degree angle? A. 35\degree B. 55\degree C. 145\degree D. 350\degree” Question 3 states, “What is the measure of the angle that is supplementary to a 35\degree angle? A. 35\degree B. 55\degree C. 145\degree D. 350\degree” (7.G.5)

Grade 8, Topic 9: Linear Equations and Systems, Lesson 5, Your Turn Questions, students develop procedural skill and fluency in solving systems of equations algebraically and through graphing. Question 7 states, “Use the system of equations below to answer parts (a) and (b). 4.5x - 2.6y = 12.3, 1.8x + 0.4y = 5.1 a. Solve the system of equations algebraically. B. Graph the equations to see if your solution from part a) is the intersection point of the lines.” Question 8 states, “Solve this system of equations using substitution. 4\frac{1}{2}s-3t=15\frac{1}{2}, -2s+t=3\frac{1}{2}.” Question 9 states, “Solve this system of equations by combining equations and eliminating a variable. 4\frac{1}{2}s-3t=15\frac{1}{2}, t=3\frac{1}{2}+2s.” Teacher Guidance, Your Turn: Questions, And the Point Is … states, “Questions 7, 8, and 9 suggest methods for solving, but you can allow students to use other methods if you wish. Question 8 encourages students to be precise when substituting an expression involving one variable for the other variable.” (8.EE.8b)

The materials reviewed for Experience Math Grades 6 through 8 meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters.

Multiple routine and non-routine applications of mathematics are included throughout the grade level, with single- and multi-step application problems embedded within the three-part lesson structure, Minds On, Action Task, and Consolidate. Students engage in Wonder Tasks, Making Connections Tasks, and Brain Benders. Wonder Tasks are three-act, problem-based activities in which students formulate questions about situations and communicate their thinking. Making Connections Tasks integrate concepts from multiple domains and provide opportunities for students to apply mathematical skills and concepts while solving complex, multi-part problems. Brain Benders present problem-solving opportunities in real-world contexts. Examples include:

• Grade 6, Games and Activities, Making Connections Task, Fish Tank, students apply their understanding of percent and geometry to design a fish tank that meets certain specifications. Students are given an image of a rectangular prism labeled “Volume = 3,600 in³.” Making Connections Task states, “Use the following information for Questions 1−4. The pet store has a fish tank, but the fish are getting too big for the tank. The store owner wants to build a new tank that is 75% larger.” Questions 1-4 ask students “Use the following information for Questions 1−4. The pet store has a fish tank, but the fish are getting too big for the tank. The store owner wants to build a new tank that is 75% larger. 1. What is the volume of the new tank? Explain your thinking. 2. What are some possible dimensions of the new tank? 3. If 1 gallon of water is 231 cubic inches ($$1 gal = 231 in.^4$$), what is the capacity of the new tank? Round your answer to the nearest gallon. 4. A hose fills the new tank with water at a rate of 6 gallons per minute. How long will it take to fill the tank? Explain. 4. A hose fills the new tank with water at a rate of 6 gallons per minute. How long will it take to fill the tank? Explain.” Teacher Guidance, Making Connections Task: Percent and Geometry, And the Point Is … states, “Math instruction is often domain-specific, but there are many opportunities for students to complete tasks that touch on concepts from two (or more) domains. Making Connections Tasks will help students build important connections between mathematical concepts from different domains. They are not intended to be used for evaluation. This task integrates the Ratios and Proportions domain and the Geometry domain. It will be helpful for students to have completed the Percents topic and Rate topic before they do the task. In addition, students will need to recall their previous work with Volume in Grade 5. At the start of the task, you may show students an object in the shape of a rectangular prism (e.g., a tissue box, an eraser, etc.). Have students discuss a method for calculating the volume of the rectangular prism in pairs, then as a class. If students do not recall the formulas for volume of a rectangular prism, share the formulas: V=l\times w\times h and V=b\times h. How You Could Handle: These Success Criteria for the Making Connections Task should be shared with students when the task is assigned: I made a fish tank that is 75% larger than the original size and gave the possible dimensions of the tank. I show my thinking and explain what I did to create the new fish tank.” (6.RP.3)

• Grade 7, Topic 9: Algebra, Lesson 3, Action Task, students apply their understanding as they use the skill of solving equations, along with the conceptual understanding that an equation must remain balanced when applying this knowledge to solve a real-world problem. Question 1 states, “Malcolm made the following purchase at a bookstore. a. Write an equation to model the situation. b. Use a pan balance to model and solve the equation you wrote in part (a). Explain your steps.” A receipt of sale shows “1 Large book 2, 5 Small book ?, Total 23.00.” Action Task states, “In questions 2 and 3, use a method other than a pan balance to solve the equation. 2. 17 = 3x + 9 3. 8x-4\frac{1}{2}=31\frac{1}{2} 4. Why might a pan balance not be easy to use to solve the equations in Questions 2 and 3?” (7.EE.3)

• Grade 8, Topic 8: The Pythagorean Theorem, Planning and Resources, Topic Task, Making Connections Task, Designing a Hiking Trail, students apply their understanding of the Pythagorean Theorem as they analyze and make recommendations regarding hiking trails. Designing a Hiking Trail states, “Hiking trails in California’s many state and national parks are informally classified as easy, moderate, difficult, or challenging. These classifications are based on factors such as trail length, change in elevation during the hike, and trail conditions. Trail designers often limit the steepness of trails to reduce erosion and lessen rapid water runoff. 1. a. Use the Pythagorean Theorem to find the distance to the summit along the steep trail. A hiker must choose between a steep trail or a longer, winding trail to reach the summit of a hill with an elevation of 375 feet. The horizontal distance (run) from the trail’s start to directly below the summit is 900 feet. The vertical elevation (rise) of the summit is 375 feet. b. Using the longer, winding path, the trail to the summit is a distance of 1,328 feet. How much longer is the winding trail than the steep trail? c. Using graph paper, draw a model of the steep trail. Include a label showing which measurement is the steep trail. 2. Using your steep trail model, what is the slope of the trail, written as a percentage? (Hint: Use slope = rise/run to determine the trail’s slope.) 3. a. If you were designing a new trail to the summit of the hill in item 1b, what would be the steepest allowable slope of the new trail, based on the half rule? One principle in trail design is the “half rule,” which states that the slope (or grade) of a trail should be no more than half the slope of the hill. 4. Research techniques that trail designers use. Then suggest ideas to reduce the steepness of the trail and help prevent erosion.” (8.G.7)

The materials reviewed for Experience Math Grades 6 through 8 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade as reflected by the standards.

Multiple aspects of rigor are engaged simultaneously across the materials to develop students' mathematical understanding of individual lessons. Each topic within the curriculum supports a variety of instructional approaches that incorporate conceptual understanding, procedural fluency, and application in a balanced way. 

For example:

• Grade 6, Topic 3: Ratio, Lesson 4, Your Turn Questions, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they solve an application problem involving ratios. Students are asked to represent the ratio in different ways and explain their thinking. Question 3 states, “The ratio of the distance from the sun to Mars compared to the distance from the sun to Jupiter is about 15: 20. If Mars is about 140 million miles from the sun, about how many million miles away is Jupiter from the sun? Use both a ratio table and a graph to show your thinking.” (6.RP.3)

• Grade 7, Topic 8: Operations with Rational Numbers, Lesson 3, Minds On Activity, students demonstrate conceptual understanding and application as they solve an addition and subtraction problem involving rational numbers. Students apply their understanding by creating addition and subtraction equations to represent temperature changes and by interpreting these equations within the context of the problem. Minds On Activity states, “One January, the average temperature in a town in Alaska changed -1.4\degreeF by from the second week to the third week. 1. What might the two average temperatures have been? 2. What addition equation involving the temperatures would be true? 3. What subtraction equation involving the temperatures would be true?” Students are given a number line with integers labeled from -10 to 10. (7.NS.1)

• Grade 8, Topic 1: Rigid Motions, Lesson 5, Action Task, students demonstrate conceptual understanding and application as they explore the relationship between transformations and congruence. Question 1 states, “Use Triangle ABC. a. Show the image Triangle A’” B’” C’” of Triangle ABC if you perform the following transformations in a row: Translate 2 units to the right and 3 units up to get A’B’C’. Reflect in the line of reflection to get A’B’C’. Rotate 90\degree clockwise around vertex A’. b. How do you know Triangle A’”B’”C’” is congruent to Triangle ABC?” A grid is shown with right triangle ABC, where segment AC measures 5 units and segment BC measures 2 units. A line of reflection is shown in the center of the grid. Question 2 states, “Use Triangles ABC and DEF. a. Use a combination of translations, rotations, and/or reflections to get from Triangle ABC to Triangle DEF. Show all of your steps. b. Are the two triangles congruent? How do you know? c. What other combinations of steps might you use to move from Triangle ABC to Triangle DEF? Show your steps.” A grid is shown with congruent equilateral triangles, including triangle ABC, where the short side BC is horizontal, and triangle DEF, which is rotated and translated so that the short side EF is horizontal. Question 3 states, “Create a polygon on a grid. Create a second congruent polygon. It should have exactly the same side lengths and angle measurements. Show what transformations you need to make to get from the first polygon to the second polygon.” (8.G.2)

The materials reviewed for Experience Math Grades 6 through 8 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP1 throughout the year. MP1 is found in the three-part lesson structure as students make sense of problems during Minds On discussions, persevere through open-ended Action Tasks, and reflect on strategies in the Consolidate portion of lessons. Essential Understandings connect new learning to big ideas, helping students analyze problem information and plan solution pathways, while Parallel Tasks provide differentiated entry points that allow all learners to engage in productive struggle. Students regularly interpret conditions in real-world problems, compare quantities, and choose tools to solve problems, while teacher supports such as prompts, embedded coaching videos, and “Probing and Extending” questions guide students in evaluating their solutions and asking “Does this make sense?” 

Across the grades, students engage in open-ended tasks that support key components of MP1, including sense-making, strategy development, and perseverance. These tasks prompt them to make sense of mathematical situations, develop and revise strategies, and persist through challenges. They are encouraged to analyze problems by engaging with the information and questions presented, using strategies that make sense to them, monitoring and evaluating their progress, determining whether their answers are reasonable, reflecting on and revising their approaches, and increasingly devising strategies independently. 

Examples Include: 

• Grade 6, Topic 9: Dividing Fractions, Lesson 2, Student Experience Book, students interpret given information to solve a word problem using fraction division to find a unit rate. After solving the problem with the provided fractions, they demonstrate understanding by choosing two different fractions and repeating the problem. Teacher Experience Book, Action, Teacher Guidance, states, “Students think of similar problems and repeat this task multiple times.” And the Point is… states, “The Action Task is an open task so that many fractions can be tried by many students, which allows for more data that can then be generalized. Students choose different pairs of fractions to work with, but are given some constraints for these reasons: The second fraction should be greater than \frac{1}{2} to ensure that students move away from using unit fractions. However, there is no stipulation that the second fraction should be less than 1 or greater than 1 since it really does not matter. There did not have to be a stipulation about the first fraction either, but often students respond better to open questions when there is at least one constraint.” Student Experience Book, Action Task, “In Questions 1-4, consider the following: Martina crocheted ---- of a scarf in --- hours. 1. Choose a fractions less than \frac{1}{2} for the first blank and fraction greater than \frac{1}{2} for the second blank. 2. How much of a scarf, or how many scarves, could Martina crochet in 1 hour? 3. What could you multiply the first fraction by to the answer you got? 4. Choose two different fractions, and repeat the problem several times.”

• Grade 7, Topic 1: Probability of Single Events, Lesson 5, Student Experience Book, students research everyday situations that involve predicting and making decisions based on probability, and then create a poster showing their situations and how probability is used in each. Teacher Experience Book, Action, Teacher Guidance states, “Students need to come up with ideas of where probability is used in everyday life and make the connection that they will help the poster reader understand how probability is used there.” Student Experience Book, Action Task, “1. Do some research to learn about everyday situations where probability is used. 2. Create a poster to show three or four of these situations, with an example of how probability is used in each situation to help predict and make decisions. Be specific in each example.” 

• Grade 8, Topic 12: Volume, Lesson 2, Student Experience Book, students interpret the context of comparing the volumes of a cone and a cylinder, use the given formulas, and determine possible dimensions that satisfy the condition of the cone’s volume being about 2 ft^3 more than the cylinder’s. They explain their reasoning by generating at least two possibilities. Students evaluate whether their answers make sense by checking their calculations and reasoning about the relationship between the cone’s and cylinder’s volumes. Teacher Experience Guide, Action, Teacher Guidance states, “students problem solve to figure out possible cones to meet required conditions pertaining to their volumes.” Student Experience Book Action Task states, “The volume of a cone is about 2 ft^3 more than the volume of a cylinder. What could be the dimensions of each be in inches? Include at least two possibilities.” An image shows a cylinder and a cone with the two formulas that represent each. Teacher Experience Guide, Using the Action Task states, “You might begin by asking students how they know that even though 1 ft = 12 in., 1 cubic foot is not 12 cubic inches. You might review why it is 12\times12\times12 in^3.” Teacher Guidance, Conversation Starters states, “How would the cone’s and cylinder’s volumes compare if they had the same height and the same radius? Will the cone’s or cylinder’s dimensions increase? Do you want to keep that measurement in cubic feet or do you want to change it? Can you just cut the height and radius in half?”

The materials reviewed for Experience Math Grades 6 through 8 meet expectations for supporting the intentional development of MP2: Reason abstractly and quantitatively, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP2 throughout the year. MP2 is found in the three-part lesson structure as students move between concrete, pictorial, and abstract representations during Minds On discussions, reason about quantities and relationships in open-ended Action Tasks, and justify strategies in the Consolidate portion of lessons. Essential Understandings connect new learning to big ideas, supporting students in decontextualizing and contextualizing problems, while Parallel Tasks provide differentiated entry points that allow all learners to represent quantities flexibly. Students regularly create and interpret representations, consider units, and explain the meaning of quantities, while teacher supports such as prompts, embedded coaching videos, and “Probing and Extending” questions guide students in deepening their reasoning and making connections between mathematical ideas 

Across the grades, students participate in tasks that support key components of MP2, including reasoning with quantities, representing situations symbolically, and interpreting the meaning of numbers and symbols in context. These tasks encourage students to consider the units involved in a problem, analyze the relationships between quantities, and connect real-world scenarios to mathematical representations. Teachers are guided to support this development by modeling the use of mathematical notation, asking clarifying and probing questions, and facilitating conversations that help students make connections between multiple representations. 

Examples Include: 

• Grade 6, Topic 3: Ratio, Lesson 4, Student Experience Book, students generate ratios based on measurements of their individual height, shoe length, shoe width, and arm length, and represent equivalent ratios about a hypothetical giant using a variety of strategies. Student Experience Book, Action Task states, “3. Measure your arm length and compare it to your height. About how long would the arms be of the person big enough to fit the giant shoes? Explain your thinking.” Teacher Experience Guide, Action states, “Students relate the context to the numerical values they choose to use in their calculations.” Teacher Experience Guide, Using the Action Task states, “Although students are asked to measure their own shoe length and shoe width, they don’t need to measure both to figure out the height—they have a choice. This will be discussed in the Consolidate. Also, when figuring out arm length, students have a choice of using ratios relating arm length to height, arm length to shoe length, or arm length to shoe width. Students should think about whether an estimate is sufficient for this problem and why. If a student chooses to be more precise, that is fine, but the Consolidate Questions will raise the issue of why being precise may not make sense given this context. Encourage students to use ratio tables, graphs, and possibly diagrams to write equivalent ratios relevant for the questions posed.”

• Grade 7, Topic 11: Collecting Data, Lesson 3, Student Experience Book, students create many random samples from a set of data to infer something about a bigger population. Student Experience Book, Action Task states, “Here is some data gathered from 60 seventh graders about how many hours of homework they usually do per week.” Students see numbers from 2 to 12 in a chart. 1. Randomly select 10 different samples of 8 students. Tell how you decided on each sample. 2. How similar or different are what the samples tell you? 3. Based on what you collected, what do you predict as a typical number of hours of homework for seventh grade students? 4. Suppose the actual number of hours of homework is two more than your prediction. What is the percent error of your prediction?” Teacher Experience Guide states, “Students make decisions about reasonable inferences based on a set of varied data.” Teacher Experience Guide, And the Point is… states, “This Action Task gives students a chance to see how similar different random samples might or might not be. shows students how to informally use data from many samples to make inferences about a larger population.” Teacher Experience Guide, Using the Action Task states, “Question 4 asks students to calculate percent error. You can help students to understand that percent error is the accuracy of a measured or estimated value compared to an actual value. For example, if you predict that the mean of the scores in a classroom on a math test is an 88 and the mean is actually a 92, the percent error of your prediction is calculated.” Teacher Experience Guide, Conversation Starters include, “What tool might you use to collect a random sample of 10 pieces of data? What will you do with the 10 values? What will you do if the samples are really different values?”

• Grade 8, Topic 2: Powers and Roots, Lesson 4, Student Experience Book, students work with square roots, estimating their value and describing their properties. Student Experience Book, Action Task states, “1. Complete the following. a. Choose two of these conjectures to investigate. 1. If you double a number, you double its square root. 2. The square roots of even perfect squares are even, and the square roots of odd perfect squares are odd. 3. If numbers are 100 apart, their square roots are about 5 to 10 apart. 4. The square roots of consecutive numbers could be as little as 0.001 apart. b. Decide whether each conjecture is true or not true, and explain why.” Teacher Experience Guide, Action states, “Students use their reasoning skills to decide which conjecture makes sense to investigate.” Teacher Experience Guide, And the Point Is… states, “Part of the intention of this Action Task is to provide practice with square roots, but part of the intention is to allow students to see the properties of square roots (for instance, how square roots of consecutive numbers get increasingly close together as numbers increase, and how square roots and their squares are not proportional — the square 36 and its double, 72, are not in the same ratio as their square roots 6 and 8.48).” Teacher Experience Guide, Using the Action Task states, “Remind students that a conjecture is a suggestion about what might be true. Emphasize that they need to verify each conjecture with examples combined with reasoning and try to explain why it is true or why it is false.” Teacher Experience Guide, Conversation Starters include, “Do you think you should try only big numbers or big and little ones? How many numbers do you think you should try? How far apart are the square roots of 4 and 5? How about the square roots of 1,004 and 1,005?”

The materials reviewed for Experience Math Grades 6 through 8 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP3 throughout the year. MP3 is found in the three-part lesson structure, students share and justify their reasoning during Minds On discussions, construct arguments and solve open-ended problems in Action Tasks, and critique the reasoning of others in the Consolidate portion of lessons. Open questions provide multiple entry points and create opportunities for students to explain their thinking and respond to peers, supporting the development of mathematical communication skills. Structured routines such as Math Congress and Gallery Walks provide spaces for students to examine, compare, and critique mathematical strategies. Students regularly explain why their reasoning is mathematically valid and analyze the reasoning of others, while teacher supports such as prompts, embedded coaching videos, and “Probing and Extending” questions guide students in deepening their arguments and making connections between mathematical ideas.

Across the grades, students participate in tasks that support key components of MP3. These include constructing mathematical arguments, analyzing errors in sample student work, and explaining or justifying their thinking orally or in writing using concrete models, drawings, numbers, or actions. Students are encouraged to listen to or read the arguments of others, evaluate their reasoning, and ask clarifying questions to strengthen or improve the argument. They also have opportunities to make and test conjectures as they solve problems. Teachers are guided to support this development by providing opportunities for students to engage in mathematical discourse, set clear expectations for explanation and justification, and compare different strategies or solutions. Teachers are prompted to ask clarifying and probing questions, support students in presenting their solutions as arguments, and facilitate discussions that help students reflect on and refine their reasoning.

Examples Include: 

• Grade 6, Topic 2: Factors and Multiples, Lesson 1, Student Experience Book, students find multiples of a whole number and explain how one number is a multiple of another number. Student Experience Book, Action Task states, “A group of friends compete in a gaming tournament. Aidan earns 8 times as many tokens as Ben. Kendra earns 8 times as many tokens as Delia. 1. Do you know who has more tokens, Aidan or Kendra? Explain your thinking. 2. Suppose Aidan has more tokens than Kendra. a. How many more tokens might Aidan have? How many tokens would Ben and Delia have? Think of more than one possibility. Explain your choices. b. Is there a difference between Aidan’s and Kendra’s amounts that is not possible? Explain your answer.” Teacher Experience Guide, Action states, “Students construct viable arguments when they figure out who has more tokens and how they know, and when they decide which differences in amounts are not possible.” Teacher Experience Guide, And the Point Is states, “This Action Task calls for students to create multiples of 8 and 6 and focuses on how far apart multiples of 8 or multiples of 6 can be. The numbers 8 and 6 were selected to give students practice with more challenging multiples than those of smaller numbers such as 2, 3, 4, or 5. This Action Task reinforces the idea that multiples of a number can be large or small. Students will encounter this idea in Question 1.” Teacher Experience Guide, Connecting Ideas and Experiences states, “MLR8 Discussion Supports To encourage students to share their own unique perspectives, you can do the following: Ask students to think about why there are coins for 1¢, 5¢, 10¢, and 25¢, but not coins for other amounts, like 24¢, 55¢, and 60¢. Discuss ideas with the class. If time is limited, have students complete Questions 1 and 2.” Teacher Experience Guide, Conversation Starters states, “How many tokens could Ben and Delia have? Could Aidan or Kendra have an odd number of tokens? Is there a limit to how many tokens Ben could have?”

• Grade 7, Topic 1: Probability of Single Events, Lesson 2, Student Experience Book, students use a fraction to represent an experimental probability. They also determine and explain if the probabilities change when the results of other experiments are combined. Student Experience Book, Action Task states, “1. Put the 10 birthdate slips in a bag. Pull one birthdate out of the bag and record the date you chose. Put the birthdate back in the bag, mix up the slips of paper, and pull out another date. Repeat the above until you have chosen 10 dates. 2. Based on your experiment, what fraction describes the probability of each outcome? a. The first digit is 4. b. The fifth digit is 2. c. The second digit is 4. d. The eighth digit is 7. e. The fourth digit is 7. 3. Choose 10 more dates from the bag, one at a time. Combine your results with those in Question 1. 4. Based on your experiment, what fraction describes the probability of each outcome? a. The first digit is 4. b. The fifth digit is 2. c. The second digit is 4. d. The eighth digit is 7. e. The fourth digit is 7. f. Display the results from parts (a) to (e) on a probability line. 5. Did your probabilities change? Explain.” Teacher Experience Guide, Action states, “Students explain whether probabilities changed or not when they combined results from different experiments.” Teacher Experience Guide, And the Point Is states, “Some students may think a probability has changed if it goes from \frac{0}{10} to \frac{0}{20} or from \frac{5}{10} to \frac{10}{20} since some of the numbers involved are different. Reinforce that we are now talking about the size of a fraction. Remind students that \frac{0}{20}=\frac{10}{20} and \frac{10}{20}=\frac{5}{10}. Students will begin to see that the randomness of probability experiments will often, but not always, lead to changes in experimental probability when more trials are carried out. They will explore this idea further in the Consolidate discussion. However, if events are certain or impossible, their probabilities will not change.” Teacher Experience Guide, Using the Action Task states, “Provide each pair with a paper bag and a copy of the reproducible page Ten Birthdates. Have students cut apart the slips of paper and put all 10 slips in the bag. If time is limited, students could share their results in Question 1 with another pair rather than completing Question 3 on their own.” Teacher Experience Guide, Conversation Starters states, “Why do you think it’s important to mix up the slips again? How will you keep track of all the outcomes? Why do you think it makes sense that some of the probabilities change?”

• Grade 8, Topic 11: Data, Lesson 1, Student Experience Book, students create relative frequency charts and use them in a real-world situation. Then they argue why the charts they created show an association between variables. Student Experience Book, Action Task states, “1. A restaurant did a survey of people who ordered chicken or fish dishes, beef dishes, or none of those at dinner. They also kept track of whether people ordered dessert or not. These are their results for a particular week. a. Create a relative frequency table based on the data above. b. Does there seem to be an association between the type of main course ordered and whether people order dessert? 3. Create a two-way table about restaurant orders that you think does not show much association between two variables. Explain your thinking.” Students see a table with Chicken or Fish, Beef, No Chicken/Fish/Beef, and Totals across the top. The left side is labeled Dessert, No Dessert, and Totals. The results of the survey are displayed in the table under the appropriate headings. Teacher Experience Guide, Action states, “Students argue about why their created tables do or do not show an association between variables.”Teacher Experience Guide, And the Point is… states, “This Action Task provides the opportunity for students to explore several tables that are already provided to make decisions about whether variables might be associated.”

The materials reviewed for Experience Math Grades 6 through  8 meet expectations for supporting the intentional development of MP4: Model with mathematics, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP4 throughout the year. MP4 is found in the three-part lesson structure as students identify real-world problems and create mathematical models during Minds On discussions, apply and test models in Action Tasks, and refine their models in the Consolidate portion of lessons. Wonder Tasks, Making Connections Tasks, and Brain Benders provide authentic contexts where students select and use mathematical tools and representations to solve problems. Students regularly build and interpret models with physical and virtual manipulatives, while teacher supports such as embedded coaching videos, “And the Point Is” sections, and “Probing and Extending” questions guide students in making connections between models and mathematical ideas.

Across the grades, students participate in tasks that support the intentional development of MP4, including putting problems or situations in their own words and identifying important information, using the math they know to solve problems and everyday situations, modeling the situation with appropriate representations and strategies, describing how their model relates to the problem situation, and checking whether their answer makes sense, revising the model when necessary.

Examples Include: 

• Grade 6, Topic 9: Dividing Fractions, Lesson 3, Student Experience Book, students solve problems involving the division of fractions from everyday situations. They plan a strategy to find a solution using a model. Student Experience Book, Action Task states, “Solve three of these problems. Use pictures or models to show your thinking for at least one of the problems. For each problem, explain why dividing makes sense. Problems 1 and 3 have blanks. You will choose the number for the blank, but you must choose a fraction or mixed number. 1. The area of a field is 20\frac{1}{2} square yards. The length of the field is \square yards. The perimeter of the field is greater than 30 yards. How wide is the field? 2. Malcolm ran 5\frac{1}{2} miles today, which is \frac{11}{12} of the distance that he ran yesterday. a. How far did Malcolm run yesterday? b. What fraction of today’s distance is yesterday’s distance? 3. Novia created a green mosaic in 🗋 hours. It took her 3\frac{1}{2} times as long to make a blue mosaic. How much of the blue mosaic did Novia complete in 1 hour?” Teacher Experience Guide, Action states, “Students can use a model of their choosing to help them visualize the problems in this task.” Teacher Experience Guide, In This Task states, “Students choose and solve three problems involving the division of fractions from a selection of problems derived from everyday situations. They explain why division is used to solve them.” Teacher Experience Guide, And the Point Is states, “This Action Task varies in level of difficulty to meet the needs of as many students as possible. Questions 3 and 5 are probably the most challenging problems. There is extraneous information in Question 4 since we want students to encounter such problems and attend to only using the information they need.” Teacher Experience Guide, Using the Action Task states, “You could decide to lead students to appropriate problems or you could let them choose for themselves. Letting students choose the numbers in a couple of the problems also makes them more accessible.” Teacher Experience Guide, Conversation Starters include, “Did you need information about the perimeter? Why or why not?” “Will the quotient be greater than or less than 1? Why do you think that?” and “Is division the only operation you used? Explain.”

• Grade 7, Topic 2: Adding and Subtracting Integers, Lesson 3, Student Experience Book, students solve problems involving integer subtraction in the context of temperature change. They plan a strategy to find a solution using a model. Student Experience Book, Action Task states, “1. Answer these questions about temperature changes. a. Choose a temperature for each blank from these numbers. At least one number must be negative. -20, -14, -8, -7, -6, -1, +7, +10, +12, +15. The temperature changes from ___\degreeC to ___\degreeC on Tuesday. b. Use a model to determine the integer that describes how much the temperature changed and in which direction. c. Write an addition equation and a subtraction equation to describe what happened. d. Look at the subtraction equation you wrote in part (c). Explain how using additive inverses can allow you to write two equations that mean the same thing. 2. Repeat Question 1 for three other pairs of numbers. Make sure you use a different model at least once.” Teacher Experience Guide, Action states, “Students model a temperature change situation in various ways, including using a number line or counters, drawing representations, and writing an equation.” Teacher Experience Guide, In This Task states, “Students choose two temperatures (one of which must be a negative value), use counters or number lines to model the difference between them, and write related addition and subtraction sentences.” Teacher Guide, And the Point Is states, “This Action Task has students use subtraction as the inverse of addition, asks students to use counters and number lines to gain practice with both, asks students to choose values in order to allow for a richer discussion later about how subtraction works, and has students write both addition and subtraction equations to make the connection between the two.” Teacher Experience Guide, Connecting Ideas and Experiences states, “Ask students what is the coldest temperature they have ever experienced? What is the hottest temperature they have ever experienced?” Teacher Experience Guide, Conversation Starters include, “If the temperature changed from −1° to −6°, would you describe that as (−1) − (−6) or (−6) − (−1)?” “When you subtract two integers to show the temperature change, when would the difference be positive?” and “Is it possible to subtract two negative integers and get a negative integer?”

• Grade 8, Topic 5: Proportional Relationships and Slope, Lesson 2, Student Experience Book, students graph to model proportional relationships and how the slope relates to the equation associated with the line. They plan a strategy to find a solution using a model. Student Experience Book, Action Task states, “1. a. Tony’s recipe for one batch of pasta sauce uses three cloves of garlic. Create a table of values and graph the line showing the proportional relationship between the number of cloves of garlic and the number of batches. b. Choose two points on your line where the x-coordinates are 3 apart. c. Using your graph, determine the rise and run between the two points, then calculate the slope of the line. d. Now choose two points on the line where the x-coordinates are 9 apart. e. Using your graph, show the rise and run between those two points and determine the slope of the line.” Teacher Experience Guide, Action states, “Students use graphs to model proportional relationships.” Teacher Experience Guide, In This Task states, “Students connect pairs of points on the same line where the x-coordinates are either 3 apart or 9 apart, draw triangles that show rise and run, and observe that the triangles are similar.” Teacher Experience Guide, And the Point Is states, “This Action Task suggests a number of proportional relationships to make a generalization more comfortable, uses very different unit rates in the relationships, again to make a generalization more comfortable, uses simple factors like tripling to make the task more accessible, makes it possible for students to really see the similar triangles that can be used to show that any two points along the line can be used determine the slope.” Teacher Experience Guide, Conversation Starters include, “If your table of values, what does each coordinate stand for? If you used an equation, what would it be? Can you be sure the triangles are similar before you measure?” Teacher Experience Guide, After the Action Task states, “The focus of this task is on proportional relationships and helping students see why the slope of a line doesn’t change no matter what two points on the line are used to determine it. After the task, discuss with students that not all linear relationships are proportional ones, however every line can be thought of as a proportional relationship where the y-value is increased or decreased by a constant amount. Ask students to create a table of values and a graph for the line y = 5x. Help them see why the slope is 5. Then, create a table of values and a graph for the line y = 5x + 4. Help them see that it is the same sort of line, but every y-value has been increased by 4. Point out that the slope didn’t change, so the slant of the line looks the same. By adding 4 to y = 5x, the change was moving the line up four units. Point out that since the proportional relationship y = 5x goes through (0, 0), you can look at where the line y = 5x + 4 crosses the y-axis to see how much all the y-values have increased. Explain that the point where the line crosses the y-axis is called the y-intercept.”

The materials reviewed for Experience Math Grades 6 through 8 meet expectations for supporting the intentional development of MP5: Use appropriate tools strategically, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP5 throughout the year. MP5 is found in the three-part lesson structure as students access a range of physical and digital mathematical tools and learn to select and use them strategically based on context. Classroom manipulative kits from hand2mind are aligned with Brainingcamp virtual manipulatives, allowing students to move between concrete and digital representations. During Action Tasks, students choose tools such as base ten blocks, fraction tiles, or geometric instruments to support problem solving and reflect on their effectiveness. Embedded teacher supports, including Dr. Marian Small’s coaching videos, the Teacher Assistance Panel, “And the Point Is” sections, and “Probing and Extending” questions, guide teachers in helping students evaluate when tools are helpful and how they connect to mathematical ideas.

Across the grades, students participate in tasks that support key components of MP5. These include selecting and using appropriate tools and strategies to explore mathematical ideas, solve problems, and communicate their thinking. Students are encouraged to consider the advantages and limitations of various tools such as manipulatives, drawings, measuring devices, and digital technologies, and to choose tools that best support their understanding and reasoning. They have opportunities to use tools flexibly for investigation, calculation, representation, and sense-making. Teachers are guided to support this development by making a variety of tools available, modeling their effective use, and encouraging students to make strategic decisions about when and how to use tools. Teacher materials prompt opportunities for student choice in tool selection, promote discussion around tool effectiveness, and support the comparison of multiple tools or representations. Teachers are also encouraged to highlight how different tools can yield different insights and to help students reflect on their tool choices as part of the problem-solving process.

Examples Include: 

• Grade 6, Topic 15: Surface Area and Volume, Lesson 2, Student Experience Book, students construct nets that will fold into prisms or pyramids of their choice, and they analyze and describe the nets. Student Experience Book, Action Task states, “You are designing boxes shaped like prisms or pyramids. You must be able to flatten the boxes for storage. 1. Choose two prism shapes and two pyramid shapes. Build nets of these objects. 2. Take a photo of each net, or trace around the shapes and sketch each net. Describe what shapes are in each net and explain why each net makes sense.” Student Experience Book, Action Task also states, “2. Choose one of your prism nets and one of your pyramid nets. Rearrange the pieces to make other correct nets for the same objects. 3. Choose one of your prism nets and one of your pyramid nets. Rearrange the pieces to make other correct nets for the same objects.” Teacher Experience Guide, Action states, “Students use 2-D shapes connected appropriately to create nets.” Teacher Experience Guide, Using the Action Task states, “Students will see that each 3-D object has many possible nets. Provide 3-D construction shapes or dynamic geometry software. Provide, as well, models of prisms and pyramids in case students want to study them more carefully or use them to help construct nets. Ensure students have copies of the reproducible page Net Faces available.” Teacher Experience Guide, And the Point Is… states, “This Action Task provides an opportunity for students to figure out how to create a net more independently, without a model.” Teacher Experience Guide, Conversation Starters states, “What sorts of faces will the net for this box have? How many? Will you use any triangles in this net? Any rectangles? Could the rectangles be different sizes in this net or not?”

• Grade 7, Topic 12: Probability of More Than One Event, Lesson 1, Student Experience Book, students use models to determine the theoretical probability of outcomes of two independent events. Student Experience Book, Action Task states, “1. You roll the number cube and draw a colored cube without looking. What is the probability that each situation could occur? Use a model to show your thinking. a. You roll an even number and draw a blue cube. b. You roll an odd number and draw the green cube. c. You roll either 3 or 5 and draw a red cube or a blue cube. d. You roll 6 and draw the green cube. 2. What combination of colors and rolls could have each of these probabilities? a. \frac{2}{5} b. 0.5. c.\frac{9}{30}.” Teacher Experience Guide, Action states, “Students use charts and area diagrams to help them solve probability problems.” Teacher Experience Guide, And the Point Is… states, “There is a combination of figuring out the outcome and choosing outcomes to achieve certain probabilities. The latter is done second so that it is supported by earlier calculations.” Teacher Experience Guide, Using the Action Task states, “Provide each pair with a number cube and a container that has two red cubes, two blue cubes, and one green cube. Provide centimeter grid paper for making charts or encourage students to sketch them.” Teacher Experience Guide, Conversation Starters states, “Should the probability of rolling a 2 and drawing a green cube be the same as the probability of rolling a 2 and drawing a blue cube? Why or why not? How do you know the probability of rolling a 6 and drawing a blue cube is less than the probability of just rolling a 6? Do you think that the probability rolling an even number and drawing a blue cube should be more or less than \frac{1}{2}? Why?”

• Grade 8, Topic 1: Rigid Motions, Lesson 1, Student Experience Book, students perform translations on a coordinate grid and draw conclusions about what happens to each vertex of a translated shape. Student Experience Book, Action Task states, “1. Draw a triangle on a coordinate grid. Position one vertex at (–1, –3). Position the other vertices at intersecting lines on the grid. Choose a letter to name each of the vertices. Perform the following translations with the triangle you drew. For each translation, explain what happens to the coordinates of each vertex. a. The vertex at (-1,-3) moves to (8,10). b. Any vertex moves to (-7,-10). c. Any vertex moves to (-3,7). 3. Compare the angle measures in the pre-image to the images. What do you notice? b. Compare the side lengths of the pre-image and the images. What do you notice?” Teacher Experience Guide, Action states, “ Students use coordinate grids, transparencies or tracing paper, protractors, and rulers to help them draw conclusions about the effects of translations. Alternatively, they could use geometry software to explore translations.” Teacher Experience Guide, In This Task states, “Students perform translations and generalize about what happens to measures of line segment lengths, angles, and the parallelism of lines.” Teacher Experience Guide, And the Point Is… states, “This Action Task is designed to help students realize that once they know how the coordinates change for one vertex of a shape after a translation, they know how the coordinates change for every point on the shape.” Teacher Experience Guide, Using the Action Task states, “Make sure students have opportunities to use both geometry software as well as coordinate grids, tracing paper or transparencies, rulers and protractors to perform the translations and measure angles and line segment lengths.” Teacher Experience Guide, Conversation Starters states, “Why does the image end up in the quadrant it does? Does the image look like the pre-image? In what ways? Do you think you will end up with a trapezoid?”

The materials reviewed for Experience Math Grades 6 through 8 meet expectations for supporting the intentional development of MP6: Attend to precision, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP6 throughout the year. MP6 is found in the three-part lesson structure as students develop precision in communicating their mathematical thinking and justifying their solution approaches. Action Tasks provide opportunities for students to explain their reasoning and use mathematical language accurately. The Student Experience Book includes journal prompts that support reflection and articulation of ideas with clarity. During the Consolidate phase, students share strategies and refine explanations, while the teacher supports. Dr. Marian Small’s coaching videos, the Teacher Assistance Panel, “And the Point Is” sections, and “Probing and Extending” questions guide them to use precise vocabulary, attend to details, and connect strategies to mathematical ideas.

Across the grades, students engage in tasks that support key components of MP6. These include formulating clear explanations, using grade-level appropriate vocabulary and conventions, applying definitions and symbols accurately, calculating efficiently, and specifying units of measure. Students are also expected to label tables, graphs, and other representations appropriately, and to use precise language and notation when presenting mathematical ideas. Teachers support this development by modeling accurate mathematical language, ensuring students understand and apply precise definitions, and providing feedback on usage. Lessons include opportunities for students to clarify the meaning of symbols, evaluate the accuracy of their own and others’ work, and refine their communication. Teachers are encouraged to facilitate discussions and structure tasks that promote attention to detail in both reasoning and representation.

Examples Include: 

• Grade 6, Topic 2: Factors and Multiples, Lesson 3, Student Experience Book, students figure out the prime factorizations of several composite numbers. Student Experience Book, Action Task states, “1. Tony challenges his friend to write large numbers as the product of as many whole numbers greater than 1 as possible. He starts with 144 and writes 2\times72 but sees he can go further and writes it as 2\times2\times36. Can he go further still? Explain your reasoning. Write each number in Questions 2–6 as the product of as many factors in one expression as you can. Do not include 1. 2. 60 3. 156 4. 2,304 5. 2,310 6. 10,000. Use the numbers in Questions 2–6 to answer the questions below. 7. Which number has the greatest factor? What is the factor? 8. Which number has the most factors? How many factors does it have? 9. Which number has the greatest number of identical factors? What is the factor? 10. Which number includes the most factors that are different from one another? How many different factors does it include? What are they? 11. What do you notice about your answers to Questions 2–6?”. Teacher Experience Guide, Action states, “Students factor each number and continue factoring until they have listed all of the prime factors correctly.” Teacher Experience Guide, Conversation Starters states, “What are you looking for when deciding if a factor is prime or not? Do you think if a number is bigger, it should have more prime factors? Do you think the first factoring you choose makes a difference in the final result?” Teacher Experience Guide, After the Action Task states, “Inform students that the process they engaged in is called writing the prime factorization, which is factoring a number down until all the factors left are prime numbers. You can show students the strategy of creating factor trees to write prime factorizations. Tell students that to factor a number into prime numbers, begin by factoring the number into two known factors. Ask students to think about the properties of the number when choosing the factors (e.g., Is it even? Is it a multiple of 5?). Then, to create the tree, they break each factor down into its factors until there are only prime numbers left. Display the factor tree for 60 from Question 2 in the Action Task.” 

• Grade 7, Topic 7: Multiplying and Dividing Integers, Lesson 5, Student Experience Book, students write and evaluate integer expressions involving several operations, including multiplication, division, and exponents. Student Experience Book, Action Task states, “1. Create an expression. Use four or all five of these numbers in any order. Insert three or more operation symbols and a set of parentheses but only where they make a difference. Use at least one multiplication symbol and one division symbol. Repeat twice more to create three different expressions. 2. Evaluate each of your expressions. Describe how you determined each value. 3. For each expression, explain how you know your set of parentheses made a difference. 4. Choose one of your expressions from Question 1 and insert an exponent of 2. Evaluate the expression now.” Teacher Experience Guide, Action states, “Students carefully apply the rules for orders of operations.” Teacher Experience Guide, Conversation Starters states, “Would parentheses make a difference if you started with 2\times5+(–30)? Why might you use the division between 30 and 5? What would you do to get a big value?”

• Grade 8, Topic 12: Volume, Lesson 1, Student Experience Book, students create cylinders and prisms and compare their volumes. Student Experience Book, Action Task states, “1. Create your own cylinder. a. Choose a height for your cylinder, and choose a radius for its base. Choose a height that is less than or equal to 10 inches and choose a radius that is less than or equal to 5 inches. b. Describe a prism that has the same height you chose in part (a) and that has a base area that is close to the circular base area in part (a). c. What is the volume of your prism? d. How do you think the volume of your prism should relate to the volume of your cylinder?” Teacher Experience Guide, Action states, “Students are careful with the units they use and how they are combined.” Teacher Experience Guide, Using the Action Task states, “You may need to remind students of the formulas for the area of a circle: A=\pi r^2. …” Teacher Experience Guide, Conversation Starters states, “How can you make sure the base area of your prism is close to the area of the base of the cylinder? How do you figure out the volume of the prism?” Teacher Experience Guide, After the Action Task states, “Help students see that a cylinder is like a prism with a base that has a lot of sides. For example, … You could write the following: Volume of cylinder = Area of base height of cylinder or V=\pi r^2\times h Or V = \pi r^2h.”

The materials reviewed for Experience Math Grades 6 through 8 meet expectations for supporting the intentional development of MP7: Look for and make use of structure, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP7 throughout the year. MP7 is found in the three-part lesson structure as students look for and make use of structure to identify patterns and mathematical relationships. Action Tasks provide opportunities for students to recognize regularities in repeated calculations, geometric configurations, and algebraic expressions that support problem solving. The Student Experience Book highlights Essential Understandings that help students connect new ideas to broader mathematical structures. During the Consolidate phase, students share strategies and explain how patterns or structures informed their reasoning. Teacher supports, including Dr. Marian Small’s coaching videos, the Teacher Assistance Panel, “And the Point Is” sections, and “Probing and Extending” questions, guide students to deepen their reasoning and make connections between mathematical ideas.

Across the grades, students engage in tasks that support key components of MP7. These include looking for and describing patterns, identifying structure within mathematical representations, and decomposing complex problems into simpler, more manageable parts. Students are encouraged to analyze problems for underlying structure and to consider multiple solution strategies. They make generalizations based on repeated reasoning and use those generalizations to solve problems efficiently. Teachers support this development by selecting tasks that highlight mathematical structure and by prompting students to attend to and describe patterns they notice. Lessons provide opportunities for students to compare approaches, justify their reasoning, and reflect on how structure helps deepen their understanding. Teachers are encouraged to facilitate discussions and structure lessons in ways that promote the recognition and use of patterns and structure in problem-solving.

Examples Include: 

• Grade 6, Topic 6: Area, Lesson 1, Student Experience Book, students cut paper parallelograms into pieces and rearrange the pieces to relate the area of a parallelogram to the area of a rectangle. Student Experience Book, Action Task states, “1. Use your copy of Areas of Parallelograms. Figure out the area of each parallelogram without using a grid. You may cut the parallelograms and rearrange the pieces. 2. Prove that your areas are correct by using a centimeter grid or ruler. 3. Think about the measurements of a parallelogram. a. Which measurements of a parallelogram matter most when you figure out the area? b. Which measurements of a parallelogram do you not really need to know when you figure out its area? 4. What rule could help you figure out the area of a parallelogram?” Teacher Experience Guide, Action states, “Students decompose a parallelogram and rearrange the parts to form a rectangle to make sense of the area formula.” Teacher Experience Guide, And the Point Is… states, “This Action Task … allows students to explore the base and height of parallelograms. The parallelograms used were chosen deliberately because they all have either the same base and height or an interchanged base and height. One shape was turned so that students could think about which part of the shape to call the base and which part to call the height … provides opportunities for students to discuss which measurements matter and which do not. This discussion is important as it will help students understand why the formula does not include all four side lengths of a parallelogram.” Teacher Experience Guide, Conversation Starters states, “Would the area change if you cut the parallelogram differently? Would the area change if the parallelogram were taller? Which operation(s) will your rule involve? Multiplication, addition, subtraction or division?” Teacher Experience Guide, After the Action Task states, “Most students will likely cut off the triangle from one end of a parallelogram and put it onto the other end, but some might form two trapezoids and rearrange them into a rectangle.”

• Grade 7, Topic 9: Algebra, Lesson 2, Student Experience Book, students use variable shape sets to factor and expand algebraic expressions to make equivalent expressions. Student Experience Book, Action Task states, “1. Use the variable shape set to make a design that models the expression 2c + 4d. Then, rearrange the design to show 2(c + 2d) instead. 2. Make a design to show 3c + 6e + 3d. Then, rearrange the design to show 3(c + 2e + d) instead. For each expression in Questions 3–7, suggest an equivalent expression. Explain why you think your suggestion is equivalent. 3. 5c - 15e + 25d. 4. 4(x - 2y).” Teacher Experience Guide, Action states, “Students rearrange pattern blocks to make sense of equivalent expressions.” Teacher Experience Guide, And the Point Is… states, “This Action Task uses visual models to help students make sense of generalizations that involve factoring and expanding … extends examples in Questions 3 to 7 to allow students to generalize to symbolic factoring and expanding. This also presents an opening to show how these generalizations can be helpful in computational situations … does not introduce the words factor and expand. This allows students to make their own choices first about how to write equivalent expressions. These terms will be introduced in the Consolidate discussion.” Teacher Experience Guide, Conversation Starters states, “What does 2(c + 2d)mean? What do you notice about all the coefficients? What would be another way of saying x+\frac{1}{2}x?”

• Grade 8, Topic 2: Powers and Roots, Lesson 1, Student Experience Book, students represent whole numbers, fractions, and negative decimals as the product of powers, including at least one positive integer exponent and one negative integer exponent. Student Experience Book, Action Task states, “Write each number as a product of powers including at least one power with a positive integer exponent and one with a negative integer exponent. List a few possibilities for each. Choices include: 7^1 , 10^2, 5^{-2}, 12^{-1}, (-5)^{5}, 10^{-3}, 25^{2}. 1. 150 2. \frac{3}{4} 3. –6.25”. Teacher Experience Guide, Action states, “Students write the same number several different ways.” Teacher Experience Guide, And the Point Is… states, “This Action Task uses numbers that are varied, including fractions less than 1 and negative numbers, to help students see that using negative exponents might make sense in any of those situations … includes a negative number so that students might observe that a negative number must be raised to a negative power for the result to be negative … includes –6.25 as one of the numbers since 625 is a power of both 5 and 25.” Teacher Experience Guide, Using the Action Task states, “Point out that writing a number as a product of powers could start with writing a product of numbers, then writing each number as a power. For example: 25=125\times\frac{1}{5}, or 5^{3}\times5^{-1}. 300=3\times100, or 3^{1}\times10^{2}2. 300=900\dive3, or 900\times\frac{1}{3}, or 30^{2}\times3^{-1}.” Teacher Experience Guide, Conversation Starters states, “Is 150 a multiple of any perfect squares? How might that help? Could your power be a power of a fraction? Could using a multiple of the number help? How do you know that you need to use an odd power of a negative somewhere?”

The materials reviewed for Experience Math Grades 6 through  8 meet expectations for supporting the intentional development of MP8: Look for and express regularity in repeated reasoning, in connection with grade-level content standards, as expected by the mathematical practice standards.

Students across the 6-8 grade band engage with MP8 throughout the year. MP8 is found in the three-part lesson structure as students look for and express regularity in repeated reasoning. Action Tasks provide opportunities for students to recognize patterns in calculations and mathematical structures, then apply these observations to develop general methods and efficient solution strategies. The Student Experience Book highlights Essential Understandings that connect new learning to broader mathematical concepts. During the Consolidate phase, students reflect on their processes and explain generalizable strategies through sharing routines such as Math Congress and Student Sharing. Teacher supports, including Dr. Marian Small’s coaching videos, the Teacher Assistance Panel, “And the Point Is” sections, and “Probing and Extending” questions, guide students to analyze repetitions, explain why patterns occur, and evaluate the reasonableness of their results.

Across the grades, students engage in tasks that support key components of MP8. These include noticing and using repeated reasoning to make sense of problems, recognizing patterns, and developing efficient, mathematically sound strategies. Students are encouraged to create, describe, and explain general methods, formulas, or processes based on patterns they identify. Lessons provide opportunities for students to evaluate the reasonableness of their answers and refine their approaches through discussion and reflection. Teachers support this development by structuring tasks that highlight repeated reasoning, prompting students to make generalizations, and guiding them to build conceptual understanding, distinct from relying on memorized tricks. Instructional guidance also encourages teachers to model and elicit strategies that build toward formal algorithms or representations through consistent reasoning.

Examples Include: 

• Grade 6, Topic 3: Ratio, Lesson 2, Student Experience Book, students model ratio problems with counters, write ratios to describe the situations, and write equivalent ratios. Student Experience Book, Action Task states, “Daniela’s class has 24 students. 1. At the beginning of September, 16 students were 11 years old and 8 students were 12 years old. a. Use a model to show the ages of students in Daniela’s class. b. Write a ratio that compares the number of 11-year-olds and 12-year-olds. c. Write an equivalent ratio for the ratio in Question 1(b). Use your model to explain how you know the ratio is equivalent. 2. By October 1, two of the 11-year-olds had turned 12. a. How many 11-year-olds and 12-year-olds are there now? Make a new model to show the relationship of the students’ ages. b. Write a ratio that compares the number of 11-year-olds and 12-year-olds now. c. Write an equivalent ratio for the ratio in Part 2(b). Use your model to explain how you know the ratio is equivalent. 3. By November 1, two more 11-year-olds had turned 12. a. How many 11-year-olds and 12-year-olds are there now? Make a new model of the students’ ages. b. Write a ratio that compares the number of 11-year-olds and 12-year-olds now. c. Write an equivalent ratio for the ratio in Part 3(b). Use your model to explain how you know the ratio is equivalent. 4. By December 1, two more 11-year-olds had turned 12. a. How many 11-year-olds and 12-year-olds are there now? Make a new model of the students. b. Write a ratio that compares the number of 11-year-olds and 12-year-olds now. c. Write an equivalent ratio for the ratio in Part 4(b). Use your model to explain how you know the ratio is equivalent”. Teacher Experience Guide, Action states, “Students see that they can multiply the terms of a ratio by any whole number to form an equivalent ratio.” Teacher Experience Guide, In This Task states, “Students model ratio problems with counters, write ratios to describe the situations, and write equivalent ratios.” Teacher Experience Guide, And the Point Is… states, “This Action Task uses the numbers 24, 16, and 8 to make modeling equivalent ratios more straightforward … uses models to make it easier for students to explain how they know that two ratios are equivalent … includes work with both part-to-part and part-to-whole comparisons … may lead to some students noticing that equivalent ratios work the same way as equivalent fractions—the terms are multiplied or divided by the same amount because that retains the same multiplicative relationship, or comparison.” Teacher Experience Guide, Conversation Starters states, “Could the counters be arranged in an array? How about in a long line? What ratio do you see if you only look at the top line in a tape diagram? What if you look at the top two lines? As the children turn 12, how do the ratios change?”

• Grade 7, Topic 8: Operations with Rational Numbers, Lesson 1, Student Experience Book, students write negative mixed numbers as rational numbers in decimal form. Student Experience Book, Action Task states, “1. Claire walked 2\frac{2}{} of a mile to the west of her apartment building. a. Choose one of these values for the denominator in the mixed number above: 3 5 11 12 15. How would you describe that distance to the west using a decimal? b. Repeat part (a) two more times using different values for the denominator. 2. What is the same and what is different about your decimal values in Question 1? 3. Write each of your mixed numbers from Question 1 as integer multiples of a unit fraction. 4. Suppose Claire walked 5\frac{3}{} miles east of her home and wrote that distance as a decimal. a. Predict two or three values for the denominator that would end up with decimal tenths or hundredths or thousandths. Test your predictions. b. Predict two or three values for the denominator that would end up with repeating decimals. Test your predictions.” Teacher Experience Guide, Action states, “Students begin to recognize when a fraction’s decimal representation is likely to terminate and when it’s likely to repeat. They notice patterns in the long division and can see when calculations are repeated.” Teacher Experience Guide, In This Task states, “Students write negative mixed numbers as rational numbers in decimal form.” Teacher Experience Guide, And the Point Is states, “In this Action Task, students … relate negative mixed numbers to rational numbers in decimal form … represent a given fraction as a multiple of a unit fraction … notice that changing a fraction to a decimal seems to result in either a ‘regular’ (terminating) decimal or a repeating decimal.” Teacher Experience Guide, Conversation Starters states, “What would have happened if one of the choices had been a denominator of 10? What is a unit fraction? Why will the multiple be negative? How did the fact that the numerator was 3 affect your predictions?”

1. Grade 8, Topic 10: Angle Relationships, Lesson 1, Student Experience Book, students investigate the relationships among the angles that are formed when a transversal crosses parallel and non-parallel lines. Student Experience Book, Action Task states, “1. Use lined paper. Choose two of the parallel lines on the paper and darken them. Then draw another line to form a 50\degree angle. 2. Measure the other seven angles where the transversal crosses the two parallel lines. What do you notice? 3. Repeat Steps 1 and 2 but change the angle from 50\degree to a different acute angle. Then repeat the steps again but change it to an obtuse angle. 4. What do you notice about the eight angles that are formed when a transversal crosses two parallel lines? 5. Repeat Steps 1 and 2, except draw two lines that are not parallel, then draw a line across them. What is the same about this drawing and the previous drawings with parallel lines? What is different?” Teacher Experience Guide, Action states, “Students notice what is consistent and what is not when they repeatedly cut parallel and non-parallel lines with a transversal.” Teacher Experience Guide, In This Task … states, “Students use lined paper to help them create parallel lines, then investigate the relationships among the angles that are formed when another line crosses both parallel lines.” Teacher Experience Guide, And the Point Is… states, “This Action Task encourages students to notice that no matter what angle is selected for one of the angles where a parallel line meets a transversal, all the other angles are either that size or 180\degree minus that size … includes both acute and obtuse angles to show that the type of angle in a particular spot does not change the overall relationships … also includes a transversal that crosses lines that are not parallel to show that the angle relationships are different in that case. Students will learn that checking these angle relationships is a way to test for parallelism.” Teacher Experience Guide, Conversation Starters states, “What makes you think those lines are parallel? Are the angles all the same measurement or only some of them? How many different angle sizes are you getting?”