6th to 8th Grade - Gateway 2

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 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. The curriculum is structured to systematically build students' conceptual understanding by introducing problems that allow for multiple approaches, which are then guided toward a more formal understanding.

As stated in the Program Guide, Curriculum, page 27 of the PD Library, “Lessons develop students’ conceptual understanding by inviting them into familiar or accessible contexts and asking them for their own ideas before presenting more formal mathematics.” This approach is reflected throughout the curriculum, where students engage with problems and questions that promote conceptual development across the grade level.

The Math of the Unit sections provide a clear explanation of how conceptual understanding is developed throughout each unit. Lessons present various representations and contexts, offering students opportunities to choose strategies that align with their understanding. Daily independent practice problems give students the chance to independently demonstrate their conceptual understanding.

Teacher guidance in each lesson is designed to facilitate discussions that connect representations and concepts, further supporting students’ understanding. These components align with the goal of providing students opportunities to develop and independently demonstrate their conceptual understanding across the grade level.

Examples include:

• Grade 6, Unit 2: Introducing Ratios, Lesson 4, Screen 7, Balancing Act, students develop conceptual understanding as they learn the concept of a ratio and use ratio language to describe the relationship between two quantities. The materials state, “The table shows some ratios of limes to lychees that balance the scale. Dyani says 22 limes will balance with 55 lychees. Will the 22:55 ratio balance?” A table is provided with the headings Limes and Lychees. The Limes column contains 2 and 20. The Lychees column contains 5 and 50. (6.RP.3)

• Grade 7, Unit 5: Operations With Positive and Negative Numbers, Lesson 2, Screen 6, Decide and Defend, students develop conceptual understanding as they use a vertical number line to apply and extend their previous knowledge of addition and subtraction to add and subtract rational numbers. The materials state, “Marc and Naoki are trying to evaluate 3-(-2). Marc says, This is like adding 2 anchors, so the submarine goes down to 1. Naoki claims, This is like removing 2 anchors, so the submarine goes up to 5. Whose thinking is correct?” Choices include: Marc (1), Naoki (5), Both, or Neither. An image of a submarine in water, with a vertical number line next to it, is provided. The submarine is placed at 3 on the number line, with a point labeled Naoki at 5 and a point labeled Marc at 1. (7.NS.1)

• Grade 8, Unit 3: Proportional and Linear Relationships, Sub-Unit Quiz, Screen 5, Problem 3, students demonstrate conceptual understanding as they interpret diagrams or graphs that represent proportional relationships in context. Problem 3 states, “The graph shows the cost vs. weight relationship for two types of rice that can be bought at a bulk food store. Select all of the true statements. The cost as a unit rate for conventional rice is 1.50 per pound. The cost vs. weight relationships for both organic and conventional rice are proportional relationships. Conventional rice costs twice as much per pound as organic rice. It costs 8 to buy 12 pounds of conventional rice. The relationship between cost, c, and weight, w, for organic rice can be represented by the equation c=3w.” Two lines are provided showing both organic rice and conventional rice graphed comparing the cost (dollars) to weight (lb). (8.EE.5 & 8.EE.6)

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 8 meet expectations for providing intentional opportunities for students to develop procedural skills and fluency, especially where called for in specific content standards or clusters. The curriculum is designed to systematically build procedural fluency by offering multiple opportunities for students to practice and apply skills in a variety of contexts.

As noted in the Amplify Desmos Math PD Library, Getting Started, Grades 6-A1, Program Guide, Curriculum, page 23, "Procedural fluency is embedded throughout activities and in daily lesson practice." The curriculum includes structures that support procedural fluency development, such as Repeated Challenges, where students engage with a series of problems on the same topic, and Challenge Creators, which encourage students to create and solve challenges based on learned concepts. These activities provide repeated practice of procedural skills and help students connect conceptual understanding to fluency, reinforcing mastery.

The Math of the Unit sections explain how procedural skills and fluency are developed across each unit. Lessons incorporate Warm-Ups and Instructional Routines that regularly support procedural fluency in whole-group settings. Lesson Activities align with grade-level standards, further reinforcing procedural skills and fluency.

Students independently reinforce procedural fluency through daily practice problems, including spiral reviews, available both digitally and in print. Fluency is supported through collaborative learning structures, practice days before quizzes and assessments, and personalized, adaptive exercises focusing on fact families, mental calculations, and number sense for whole numbers and fractions. Fluency cards from earlier grades are also included to support fluency standards from prior levels, ensuring the continued development of foundational skills.

Examples include:

• Grade 6, Unit 5: Decimal Arithmetic, Lesson 5, Screen 9, Repeated Challenges, students demonstrate procedural skill and fluency as they multiply multi-digit decimals using the standard algorithm. The materials state, “There are many challenge problems on this screen!.” Teachers instruct students “to solve as many problems as they have time for.” Some expressions include: (0.04)\\sdot(0.6), (0.03)\\sdot(0.02), (0.2)\\sdot(0.007). (6.NS.3)

• Grade 7, Unit 6: Expressions, Equations, and Inequalities, Lesson 7, Lesson Practice, Screen 5, students demonstrate procedural skill and fluency as they solve equations that involve positive and negative numbers in the form of px+q=r or p(x+q)=r. The materials state, “Solve each equation. Problem 4. -4x=-28. Problem 5. -4(x+1)=-28. Problem 6. x-(-7)=-1. Problem 7. -3x+7=-1.” (7.EE.4a) 

• Grade 8, Unit 4: Linear Equations and Linear Systems, Lesson 4, Screen 2, Activity 1 Launch, students develop procedural skill and fluency as they solve a linear equation in one variable. Activity 1 states, “Sadia and Amir started solving the same equation. Discuss: What was the first step each person took?” Teacher Moves states, “Launch: Consider asking: ‘How is the structure of this equation different from equations we’ve seen before?’ Responses vary. This equation has two terms in parentheses multiplied by a factor. ‘Did both Sadia’s and Amir’s first steps keep the equation balanced? How do you know?’ Discuss the distributive property if it does not come up naturally.” (8.EE.7)

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 8 meet expectations for providing intentional opportunities for students to develop and apply mathematical concepts and skills in both routine and non-routine applications. The curriculum is structured to build students' abilities by introducing problems that allow for multiple approaches and guide students toward problem-solving strategies.

As stated in the Amplify Desmos Math PD Library, Getting Started, Grades 6-A1, Program Guide, Curriculum, page 23, “Students also have opportunities to apply what they have learned to new mathematical or real-world contexts. Concepts are often introduced in context and most units end by inviting students to apply their learning…” This approach is reflected throughout the curriculum, where students engage with both routine and non-routine problems that promote mathematical application in real-world contexts and abstract scenarios.

The Math of the Unit sections provide clear explanations of how mathematical application is developed throughout each unit. Students are given opportunities to apply their learning to new contexts. Routine and non-routine application problems appear throughout the materials, particularly in the final lessons of each unit, which invite students to apply their learning to real-world scenarios.

The materials support students' procedural fluency through routine problems, including multi-step exercises, and encourage strategic thinking with non-routine problems. In Challenge Creators, students design problems for themselves and their classmates, incorporating both routine and non-routine applications. The Make My Challenge activity also allows students to create and solve challenges, reinforcing both types of applications.

Throughout the curriculum, students are provided with opportunities to independently demonstrate their ability to apply mathematical concepts and skills, both routinely and non-routinely, across the grade level.

Examples include:

• Grade 6, Unit 3: Unit Rates and Percentages, Lesson 10, Screen 12, Show What You Know, students engage in a routine application problem as they solve problems involving percentages. The materials state, “Callen bought new sneakers for $60. Miko bought sneakers that cost 80% of that price. How much did Miko pay for his sneakers?” (6.RP.3)

• Grade 7, Unit 4: Proportional Relationships and Percentages, Lesson 11, Screen 6, Sheets of Stickers, students independently engage in a non-routine application problem as they compute unit rates with ratios of fractions. The materials state,  “Alex is thinking about buying stickers by the sheet. Four sheets cost $14. How much would 11\\frac{1}{2} sheets cost?” (7.RP.A)

• Grade 8, Unit 5, Functions and Volume, Lesson 6, Screen 3, Tyler on the Slide, students engage in a non-routine application problem sketch a graph of a function based on a qualitative situation. The materials state, “Sketch a graph representing the relationship between Tyler's waist height and time.” Students are provided with a video clip showing Tyler moving around at the playground and an interactive graph for them to sketch their representation. (8.F.5)

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 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 topics or units. Each unit within the curriculum supports a variety of instructional approaches that incorporate conceptual understanding, procedural fluency, and application in a balanced way.

The Math of the Unit sections outline how the three aspects of rigor are balanced throughout each unit, indicating which lessons address each aspect independently and which combine them. The Math that Matters Most pages for each sub-unit detail how strategies, skills, and language related to the topic are developed, ensuring that rigor is present in both isolated and integrated forms. The Lesson Overview pages for each lesson describe how the rigor within the lesson connects to prior and future learning, reinforcing the coherence and balance of rigor across the grade level.

Examples include:

• Grade 6, Unit 4: Dividing Fractions, Lesson 3, Screen 10, Synthesis, students demonstrate conceptual understanding and apply it while interpreting and computing quotients of fractions. The materials state, “How can you use an equation or a diagram to figure out how many \\frac{1}{2}-cup scoops you need to make 6 cups?” A diagram is included which shows four cards one with 6\\div\\frac{1}{2}=?, \\frac{1}{2}\\times?=6, a picture of six measuring cups with a line through the half and a tape diagram with 12 equal parts each labeled \\frac{1}{2} and the entire diagram measures 6. (6.NS.1)

• Grade 7, Unit 2: Introducing Proportional Relationships, Lesson 9, Screen 7, Lesson Synthesis, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they distinguish between proportional and non-proportional relationships, write equations that represent each type, and explain the reasoning behind their choices. The materials state, “Write two equations: one that represents a proportional relationship and one that does not. Describe how you know whether an equation represents a proportional relationship.” (7.RP.2)

• Grade 8, Unit 3: Proportional and Linear Relationships, Lesson 7, Screen 2-3, Gather Data & Double the Cups, Double the Height?, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they make connections between proportional relationships and linear equations). The materials state, “Let’s look at different stacks of cups. Drag the movable point to adjust the number of cups. Record at least two data points. Sylvia found that a stack of 5 cups has a height of 15 centimeters. She thinks that a stack of 10 cups will have a height of 30 centimeters. Is she correct? Yes or No.” Students are given a stack of three cups, which measures 12.2 cm. (8.EE.B)

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 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. It is explicitly identified for teachers in the Unit Teacher Edition. The Math of the Unit, Focus identifies the MP standards used and describes how students will engage with the standards. The Math Practices are also identified with a description of how the practice standard is developed in Focus and Coherence, Today’s Goal’s for each lesson with development identified within Warm-Ups and Activities. Course Overview, Grade Overview lists math practices correlated to each lesson. Additionally, a Standards for Mathematical Practice correlation chart is provided at the end of each Course Overview.

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, use strategies that make sense to them, monitor and evaluate their progress, determine whether their answers are reasonable, reflect on and revise their approaches, and increasingly devise strategies independently. 

Examples include:

• Grade 6, Unit 4: Dividing Fractions, Lesson 4, Screen 4, Two-Thirds of a Planter, students make sense of division problems by using a variety of strategies to make sense of the problem. The materials state, “Brianna also put flowers in a big planter. 12 flowers fill \\frac{2}{3} of a big planter. How many flowers fill 1 big planter?” The Teacher Moves, Monitor states: “Look for a variety of student strategies as they make sense of the problem. (MP1)” Students make sense of the problems and persevere in solving them as they analyze the relationship between the part (12 flowers), the fractional portion of the planter \\frac{2}{3}, and the whole (1 full planter), using strategies such as unit rates, visual models, equations, or logical reasoning to determine how many flowers would fill the entire planter.

• Grade 7, Unit 8: Probability and Sampling, Lesson 10, Screen 7, See Some Samples, students use a slider to examine the relationship between the sample mean and population mean. The materials state, “Drag the point to collect crabs and see the mean width for the sample. Discuss: What are advantages and disadvantages of using a large sample.” The Teacher Moves, Purpose states: “Students analyze how sample size can affect the quality of the data we collect. (MP1)” Connect, Consider asking: “What happened to the sample mean as the sample size increased?  Why might it be challenging in real life to use a large sample? What other questions might you be interested in asking where surveying the whole population would be challenging?” Students make sense of the relationship between the sample mean and the population mean as they think about the advantages and disadvantages of using a large sample, as well as the role of randomness in sampling.

• Grade 8, Unit 5: Functions and Volume, Lesson 6, Screen 6, Water in the Bowl, students apply previous strategies to draw a graph of a function. Teacher’s Edition, Today’s Goals states, “Students draw graphs of functions based on short videos they watch of real-world situations, identifying important features to consider when modeling a situation with a graph. They consider the qualitative features of a function, such as whether it is increasing, decreasing, linear, or non-linear, and interpret specific points in context. Students make sense of different strategies to support graphing functions to represent different events in a story. (MP1, MP4)” Screen 6 states, “Sketch a graph representing the relationship between the volume of water in the 5-liter bowl and time.” The Teacher Moves, Launch states: “Invite students to think about the new context involving a volume of water poured into a bowl. Pause briefly to play the video for the whole class. Use the Notice and Wonder routine to promote curiosity and help students make sense of water poured into a bowl. See the Routine Facilitation Guide for more information. (MP1)” Students make sense of how real-world events unfold over time by analyzing key features of a scenario and use that understanding to represent the situation with a graph, considering how volume relates to time and how the shape of the graph reflects whether the function is increasing steadily, slowing down, or leveling off.

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 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. It is explicitly identified for teachers in the Unit Teacher Edition. The Math of the Unit, Focus identifies the MP standards used and describes how students will engage with the standards. The Math Practices are also identified with a description of how the practice standard is developed in Focus and Coherence, Today’s Goal’s for each lesson with development identified within Warm-Ups and Activities. Course Overview, Grade Overview lists math practices correlated to each lesson.  Additionally a Standards for Mathematical Practice correlation chart is provided at the end of each Course Overview.

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, Unit 2: Introducing Ratios, Lesson 9, Screen 6, A Darker Red, students use strategies to compare two ratios in context. The materials state, “Here is your ratio from the previous screen: 1 ounce red : 2 gallons white. Drag the points to create a new ratio. a. Can you find two different ways to make a darker red? b. Can you find two different ways to make a lighter red?” The Teacher Moves, Monitor states: “Invite students to adjust the amount of white paint and colored tint and share what they notice with a partner. Note: This screen will use the same color the student selected in Screen 5. Pause briefly to discuss multiple strategies for creating a darker color. If it doesn’t come up naturally, consider asking: ‘What happens if you keep the tint the same and change the amount of white paint?’ (MP2) ‘How can we change our ratio to create a darker color? A lighter color?’ (MP2) ‘How could I make a larger quantity of the same color?’ Use ratios equivalent to 1:2 such as 2:4, 3:6, 4:8 etc.” Students reason quantitatively as they compare the ratios using equivalent ratios and other strategies.

• Grade 7, Unit 1: Scale Drawings, Lesson 7, Screens 2 and 3, Activity 1 Launch & Connect, students use the measurements of a scale drawing to calculate the actual dimensions of a figure. Screen 2 states, “Karima heard from students that they would like a basketball court in their community park. When Karima presented the idea to the park’s board of directors, they approved building the court in a 20-by-20-meter area of the park. Make a prediction: Will the basketball court fit in the designated area?” The Teacher Moves, Purpose states: “ Students analyze a scale drawing to determine whether a basketball court will fit in a location. (MP2)” Screen 3, states, “Will Karima’s court fit in the 20-by-20-meter square area the park directors designated for the court?” Students find the scale and actual measurements of the Length of the Court, L, the Width of the Court, W, the Hoop to 3-Point Line, H, and the 3-Point Line to the Side Line, S. Students reason abstractly and quantitatively as they find the measurement of the scale drawing and convert to the dimensions of the actual court. 

• Grade 8, Unit 4: Linear Equations and Linear Systems, Lesson 8, Screen 2, Distance and Time, students write an equation with one variable to show when two quantities are the same in a given situation. Teacher Edition, Today’s Goals states, “In this lesson, students apply their knowledge of solving equations by considering two vehicles moving on a road. Using the rate of speed and the initial position of the vehicles, they are asked to determine when the vehicles will meet. Students recognize that they can create two expressions and set them equal, solve the equation for the unknown, and interpret their solution in the context of the situation. (MP2)” Screen 2 states, “The table shows each vehicle's position at certain times. The vehicles are moving at a constant rate. Fill in the missing information in the table.” A table is provided that shows time (in seconds) at intervals of 0, 1, 2, 3, 4, …, and t. It includes the truck’s position (in meters) as 0, 15, 30, and the car’s position (in meters) as 18, 29, 40. The Teacher Moves, Purpose: “Students reason abstractly to generate expressions representing the position of two vehicles for a given time and determine when they will meet. (MP2) Launch: Display the table and invite students to describe how it connects with the animation in the Warm-Up. Consider asking, ‘How can we see the constant rate in the table?’ Note: Ensure students understand that when two vehicles meet, the front of each vehicle will be in the same position. Encourage students to use paper or their Student Edition to help them show their thinking. Consider asking, ‘What will it look like in the table when the vehicles meet?’” Students reason abstractly and quantitatively as they represent each vehicle’s position with an expression and solve for when the positions are equal.

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 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. It is explicitly identified for teachers in the Unit Teacher Edition. The Math of the Unit, Focus identifies the MP standards used and describes how students will engage with the standards. The Math Practices are also identified with a description of how the practice standard is developed in Focus and Coherence, Today’s Goal’s for each lesson with development identified within Warm-Ups and Activities. Course Overview, Grade Overview lists math practices correlated to each lesson.  Additionally a Standards for Mathematical Practice correlation chart is provided at the end of each Course Overview.

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 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, Unit 7: Positive and Negative Numbers, Lesson 11, Screen 6, Decide and Defend, students place points with both positive and negative numbers on coordinate grids that use different scales. In the Teacher Edition, Today’s Goal states, “Students practice plotting points in all four quadrants of the coordinate plane within the context of solving mazes. They are introduced to grids that use different scales for the axes and use structure to identify patterns in coordinates that differ only by the sign. Students should gain confidence in identifying coordinates with positive and negative numbers and be able to critique the reasoning of others’ claims about the location of points when reflected across one or both axes. (MP3, MP7)” Screen 6 states, “Trinidad and Annika were working on the previous maze. Trinidad says: The star's coordinates are (1, -2). Annika says: The coordinates are (1,-10). Whose claim is correct?” The Teacher Moves, Connect states: “Key Takeaway: Sometimes axes have a scale that is not equal to 1. In these cases, the interval is still consistent (e.g., goes by 2s). It can be helpful to look at the axis scale before plotting any points. Display the distribution of responses using the dashboard’s Teacher View. Use the Decide and Defend routine to support students in making arguments and critiquing the reasoning of others. (MP3) Invite students who made different choices to use reasoning to convince their classmates. Consider asking: ‘What reasoning or evidence supports your claim?’ ‘What evidence or language was most convincing to you?’” Students construct viable arguments and critique the reasoning of others as they analyze the placement of points on coordinate grids with varying scales and use mathematical evidence to justify which coordinates are correct. 

• Grade 7, Unit 4: Proportional Relationships and Percentages, Lesson 7, Screen 6, Decide and Defend, students solve multistep percent problems in a common context such as sales tax and tip. The materials state, “Kiran and Ava had a bill for $100. The tax rate is 5% and they want to tip 20%. Kiran calculates the tip after tax is added to the bill. Ava calculates the tip before tax is added to the bill. Whose strategy would result in a greater tip? Kiran’s, Ava’s, or They’re the same.” The Teacher Moves, Monitor states: “Invite students to spend 1–2 minutes thinking independently about the problem before discussing it with a partner. Display the distribution of responses using the dashboard’s Teacher View. Use the Decide and Defend routine to support students in making arguments and critiquing the reasoning of others. (MP3) Consider asking, ‘Why does the order we calculate tax and tip matter for the total bill?’” Students construct viable arguments and critique the reasoning of others, as they analyze different strategies for calculating percent increases and explain how the order of operations affects the total cost.

• Grade 8, Unit 8: The Pythagorean Theorem and Irrational Numbers, Lesson 4, Screen 4, Reflection students identify the two whole number values that a square root is between and explain the reasoning. The materials state, “Esi thinks that the description ‘The value of z when z^2=80does not belong in either category. a. What two whole numbers is the value of z between? A. 4 and 5. B. 6 and 7. C. 7 and 8. D. 8 and 9.” Students are given a visual of three cards: one between 7 and 8, one between 4 and 5, and one the value of z^2=80. “b. Of those two numbers, which would z be closer to? Explain your thinking.” The activity continues on screen 5 and 6. Screen 6 states, “The numbers x and y are positive.  x^2=3 and y^2=35. a. Plot x and y on the number line. b. Discuss: How did you decide to plot each point?” The Teacher Moves, Connect states: “Key Takeaway: A square root represents the exact solution to an equation in the form x^2=p. Using square roots of perfect squares helps identify the two whole numbers a square root is between.  Display the captured student responses. Invite students to share their thinking and critique the reasoning of others. (MP3) To surface the Key Takeaway, consider asking: ‘How can we determine the solution to an equation in the form x^2=p?’ ‘What is helpful to remember when comparing the values of square roots and plotting them on a number line?’” Students construct viable arguments and critique the reasoning of others, as they estimate the value of square roots, justify which two whole numbers the root falls between, and explain their reasoning using perfect squares and number line placement.

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 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. It is explicitly identified for teachers in the Unit Teacher Edition. The Math of the Unit, Focus identifies the MP standards used and describes how students will engage with the standards. The Math Practices are also identified with a description of how the practice standard is developed in Focus and Coherence, Today’s Goal’s for each lesson with development identified within Warm-Ups and Activities. Course Overview, Grade Overview lists math practices correlated to each lesson.  Additionally a Standards for Mathematical Practice correlation chart is provided at the end of each Course Overview.

Across the grades, students participate in tasks that support key components 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, Unit 1: Area and Surface Area, Lesson 14, Activity 2, students apply their understanding of surface area to design three-dimensional to-go containers. Teacher Edition, Today’s Goals states, “Students design a three-dimensional container to hold one item at a fictional restaurant, draw a two-dimensional pattern for their container, and then calculate how much material they will need to create their container. They conclude by making their own model of the to-go container. (MP1, MP4)” Student Edition states, “Problem 7. Share your design with a partner. Discuss how you might improve your design and write down what adjustments you want to make. Problem 8. Draw your revised pattern on blank paper using the measurements you designed. Problem 9. Cut out and fold your pattern to create your container.” Screen 5, The Teachers Moves, Monitor states: “To support students getting started, consider asking, ‘What revisions are you making to your design? How do these revisions make your container better?’  Math Identity and Community: Listen for students who identify a variety of considerations for their design and celebrate them as thinking like a ‘math modeler.’ Consider sharing the importance of this way of thinking for solving complex problems. (MP4)” Students model with mathematics as they apply their understanding of surface area to design, revise, and construct a functional three-dimensional container, using measurements and calculations to determine how much material is needed.

• Grade 7, Unit 8: Probability and Sampling, Lesson 7, Screen 9, Challenge #1, students estimate the probability of a compound event using a simulation. Teacher Edition, Today’s Goals states,“Students apply what they have learned about probability tools and repeated experiments to simulate compound real-world events. They use spinners and blocks in bags to model and simulate the chances that (MP4)“ Screen 9 states, “Try to make a simulation for a forecast that meets this criteria: The chance that it rains all three days is between 50% and 75%. Press ‘Simulate’ to check your work.” The Teachers Moves, Purpose states: “Students make connections between real-world situations and probability tools that could be used to model and simulate those situations. (MP4) Launch:  Share that this activity provides the opportunity to practice designing simulations and interpreting the results. Encourage students to experiment by pressing the ‘Simulate’ button to see their results. Students using print will need access to the digital simulation. Consider pairing these students with a student using digital or conducting the simulations as a class. Connect: Display unique or creative simulations using the dashboard’s Teacher View. Invite students to share their strategy for choosing the probabilities for each day to model the given criteria. (MP4)” Students model with mathematics as they design and run simulations to estimate the probability of a compound event, using tools like spinners or bags of blocks to represent and analyze real-world scenarios.

• Grade 8, Unit 2: Dilations, Similarity, and Slope, Lesson 8, Activity 2, students use real-world measurements in triangles to model with mathematics. Teacher Edition, Today’s Goals, “Students discover that similar triangles share common proportions between and among their corresponding side lengths and use these ratios to solve for missing side lengths. They apply their knowledge of similarity to reason abstractly about similar triangles in the context of shadows for different real life objects. Students use their knowledge of similar triangles to create models that help determine missing side lengths. (MP2, MP4)” Student Edition, Problem 7 states, “a. Determine all the side lengths using as few hints as you can. You can ask for the measure of up to two side lengths, if needed. b. What was your strategy?” The Teachers Moves, Purpose states: “Students build fluency with using relevant expressions to model and solve for missing side lengths of similar triangles. (MP4)” Students model with mathematics as they apply proportional reasoning and properties of similar triangles to determine missing side lengths using real-world measurements, such as the lengths of shadows.

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 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. It is explicitly identified for teachers in the Unit Teacher Edition. The Math of the Unit, Focus identifies the MP standards used and describes how students will engage with the standards. The Math Practices are also identified with a description of how the practice standard is developed in Focus and Coherence, Today’s Goal’s for each lesson with development identified within Warm-Ups and Activities. Course Overview, Grade Overview lists math practices correlated to each lesson.  Additionally a Standards for Mathematical Practice correlation chart is provided at the end of each Course Overview.

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, Unit 6: Expressions and Equations, Lesson 4, Activity 3, students build fluency in solving equations with decimals and fractions by using strategies such as inverse operations, reasoning about solutions, and connecting equations to visual models like balanced hangers and tape diagrams. Teacher Edition, Today’s Goals states, “Students develop fluency with solving equations, particularly equations that include decimals and fractions. They revisit connections between equations, balanced hangers, and tape diagrams, they consider strategies like reasoning about what number makes an equation true and using inverse operations to solve equations. Students use a variety of tools, including strategies with and without diagrams, to solve equations. (MP5)” Student Edition Problem 12 states, “Decide with your partner who will complete Column A and who will complete Column B. • The solutions in each row should be the same. Compare your solutions, then discuss and resolve any differences. • Solve as many equations as you have time for. Sense-making is more important than speed. Column A: a. 36=4x, b. 13=x+5, c. \\frac{1}{3}=2x, d. x+6.17=9 e. x+1.8=1.7, f. \\frac{1}{2}x=16, g. \\frac{7}{8}=x+\\frac{1}{4}. Column B: a. 7x=63, b. 21=x+13, c. 3x=\\frac{1}{2}, d. 12.22=x+9.39, e. x+5.3=18.2, f. 4=\\frac{1}{8}x, g. x+\\frac{1}{16}=\\frac{1}{16}.” Teachers Edition, Monitor, “Invite students using print to solve problems in any order or to choose any four problems to solve. Differentiation, Use a variety of strategies like: Creating a tape diagram. Using number sense. Using inverse operations. (MP5)” Students use appropriate tools strategically as they select and apply methods such as tape diagrams, number sense, and inverse operations to solve equations involving decimals and fractions, and verify their solutions through comparison and discussion.

• Grade 7, Unit 3: Measuring Circles, Lesson 3, Activity 1, students use a variety of tools to measure the circumference of circular objects and discover a proportional relationship between a circle’s diameter and its circumference. Student Edition, Problem 2 states, “Measure the diameter and circumference of at least three circular objects. Record your results in the table.” An image shows “Three Ways to Measure the Circumference, String, Tape Measure, Roll.” A table is provided for students to record their three objects, diameter (cm) and circumference (cm). Teacher Edition, Launch states, “Demonstrate measuring a circular object using each of the methods shown in the image. Consider asking: ‘Why might you get two different answers for the diameter of the same object? What should you do if that happens?’ Encourage students to consider the limitations and advantages of each method. (MP5) Distribute the circular objects, measuring tools, and string to each pair. Monitor: Encourage students to measure each object using two different methods and come to consensus about which measurement to record. (MP5)” Students use appropriate tools strategically as they select and compare different methods for measuring the circumference and diameter of circular objects, evaluate the accuracy of their measurements, and determine which tools yield the most reliable results.

• Grade 8, Unit 1: Rigid Transformations and Congruence, Lesson 4, Activity 1, students use tools to translate figures on the coordinate plane. Student Edition states, “Perform each transformation. Then label the points in the image to correspond with the points in the pre-image. Problem 2. Translate Figure ABC 3 units right and 1 unit down.” Teacher Edition, Launch states, “Distribute geometry tools for students to choose from, including tracing paper and straightedges. (MP5) Monitor: Look for a variety of strategies to share during the Connect. Use a variety of strategies such as: Using tracing paper. Using a ruler to measure distance between corresponding points. Using the grid to help decide where to draw the transformed figures.” Students use appropriate tools strategically as they choose methods such as tracing paper, rulers, or the coordinate grid to accurately perform and represent translations on the coordinate plane.

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 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. It is explicitly identified for teachers in the Unit Teacher Edition. The Math of the Unit, Focus identifies the MP standards used and describes how students will engage with the standards. The Math Practices are also identified with a description of how the practice standard is developed in Focus and Coherence, Today’s Goal’s for each lesson with development identified within Warm-Ups and Activities. Course Overview, Grade Overview lists math practices correlated to each lesson.  Additionally a Standards for Mathematical Practice correlation chart is provided at the end of each Course Overview.

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, Unit 3: Unit Rates and Percentages, Lesson 14, Screen 2, Indonesia’s Population states, “The population of Indonesia was about 272 million people in 2020. How many people in Indonesia had each of these characteristics?” Students are shown cards with the following information: Population of Indonesia: 272 million, 2 out of 25 people own a car., 54% have access to the internet., 71 million people are under 15 years old., 39 out of 100 people speak Javanese.” Students use this information to fill out a table and convert the information to the number of people in the millions. Teacher Edition, Activity 1 Monitor states, “Look and listen for a variety of strategies. Consider sharing that the number of people is measured in millions and practice interpreting some values like 0.1, 1, and 100. (MP6) Invite students to use a ratio table or draw a double number line to visualize their thinking. (MP6)” Students attend to precision as they solve problems involving finding the whole, given a part and the percent.

• Grade 7, Unit 1: Scale Drawings, Lesson 11, Activity 2, students create logos and apply a scale factor to produce an accurate resized version that fits a specific object. Teacher Edition, Today’s Goals states, “Students work in teams to create a logo and a precise scaled copy of their logo, then consider how to scale it to fit in a much larger space. This lesson combines personal expression with the skills and understandings gained throughout the unit. (MP6)” Student Edition states, "Imagine that you decide to print your team logo on a sticker that measures 9 centimeters by 12 centimeters. You need to scale the logo to make sure it is a good size for the sticker. Problem 6. Discuss: What is a scale factor that would make your drawing too large to fit on the sticker? What is a scale factor that would make your drawing too small for the sticker? Problem 7. What is a scale factor that would make your drawing just right for the sticker? Explain your thinking. Problem 8. Use the scale factor you chose to draw a scaled copy of your logo within the 9-by-12 centimeter space below. (Each grid line is \\frac{1}{2} centimeter.) Label your logo’s height and width.” Teacher Edition, Purpose states , “Students discuss how to use scale factor to resize their logo and draw a precise copy to fit on a specific object. (MP6) Monitor: Listen for student ideas to highlight during the Connect, such as: Counting grid units, then using multiplication or division to identify a scale factor that is too large. Naming that scale factors less than 1 will result in a lot of blank space on the sticker. Rotating their logo or sticker for a better fit.” Students attend to precision as they create a scaled drawing accurately.

• Grade 8, Unit 3: Proportional and Linear Relationships, Lesson 10, Activity 2, students write equations of horizontal and vertical lines. Teacher Edition, Today’s Goals states, “Students look for and make use of the structure of points to write the equations of vertical and horizontal lines. They attend to precision as they write equations of lines with different slopes to capture coins on a coordinate plane, and they justify whether horizontal or vertical lines have a slope of zero. (MP3, MP6, MP7)”  Student Edition states, "Problem 6, a. Draw lines through the coins to ‘capture’ them. Try to draw as few lines as possible. b. Write an equation for each line you drew. Problem 7. Create your own Coin Capture challenge! a. Make It! Use the Activity 2 Sheet to create your challenge. b. Solve It! On this page, write equations for the lines you would use to capture all the coins in your challenge. Try to use as few lines as you can. c. Swap It! Trade graphs with a partner and solve each other’s challenges.” Teacher Edition Purpose states, “Students attend to precision when writing equations of lines in challenges designed by their classmates. (MP6) Connect: Display different challenges and invite students to make sense of strategies that helped them complete these challenges. Consider asking: ‘How did you approach solving different challenges?’ ‘When did it make sense to use horizontal lines? Vertical lines? Diagonal lines?’” Students attend to precision as they write accurate equations for horizontal and vertical lines to capture points on a coordinate grid, justify their choices using slope and structure, and clearly communicate their reasoning when solving and creating challenges.

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 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. It is explicitly identified for teachers in the Unit Teacher Edition. The Math of the Unit, Focus identifies the MP standards used and describes how students will engage with the standards. The Math Practices are also identified with a description of how the practice standard is developed in Focus and Coherence, Today’s Goal’s for each lesson with development identified within Warm-Ups and Activities. Course Overview, Grade Overview lists math practices correlated to each lesson.  Additionally a Standards for Mathematical Practice correlation chart is provided at the end of each Course Overview.

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.

• Grade 6, Unit 7: Positive and Negative Numbers, Lesson 10, Activity 3, students use their knowledge of coordinate signs and quadrant locations to identify points on various planes, including those without visible gridlines. Teacher Edition, Today’s Goals states, “Students develop an understanding of negative numbers on the coordinate plane and use the context of searching for sand dollars to move between locations and identify the coordinates. Students also use structure to construct an argument for the location of a point on the coordinate plane using the signs of its coordinates. (MP3, MP7)” Student Edition states, “You will use support cards for this activity. Problem 7. Round 1: There is a sand dollar on each of these islands. When it is your turn to find the sand dollar, label each guess and the feedback you get until you find all three sand dollars. Problem 8. Round 2: Here are three new islands. Label each guess and the feedback you get until you find all three sand dollars. Problem 9.Which of these could be the location of the sand dollar? A. (4, 6) B. (4, ‐6) C. (‐6, ‐4) D. (‐6, 4) Explain your thinking.” Screen 9, The Teachers Moves, Connect: “Invite students to share what is alike and different about each of the ordered pairs. Display the distribution of responses. Consider asking, ‘Based on where we see the sand dollar, what do you know about the sign of the x-coordinate? The y-coordinate?’ To surface the Key Takeaway, consider asking, ‘How does the structure of the ordered pair help us determine which quadrant the point is in?’” Students look for and make use of structure as they analyze the signs of coordinates to determine the quadrant in which a point lies, justify their reasoning using quadrant structure, and apply that understanding to locate points on coordinate planes with and without gridlines.

• Grade 7, Unit 2: Introducing Proportional Relationships, Lesson 7, Screen 3, Are They Proportional?, students decide if equations represent proportional relationships. Teacher Edition, Today’s Goal states, “Students return to comparing proportional and non-proportional relationships, focusing on the connection between the structure of the equation and the type of relationship it represents (MP7)” Screen 3 states, “Here are the equations that represent the four stories. a. Select all the equations that represent a proportional relationship. b. Explain one way to decide if an equation represents a proportional relationship.” The screen contains 4 equations: y=12x, y=500+35x, y=\\frac{1}{2}x, y=x^2. Screen 3, The Teachers Moves, Monitor states: “Invite students to return to the digital student screens to share their thinking after comparing with a partner. Capture student explanations to highlight during the Connect, such as: Identifying the constant of proportionality. Looking at the equation and making generalizations about the structure (e.g., ‘It has the form y=(something)x.’). (MP7)Connect: Key Takeaway: If an equation represents a proportional relationship, then one variable is multiplied or divided by the constant of proportionality. Share the captured student explanations to highlight different strategies for deciding if an equation represents a proportional relationship. To surface the Key Takeaway, consider asking: ‘How can you tell from the equation if a relationship is proportional?’ ‘Where can you see the constant of proportionality in the equation?’ ‘What does the constant of proportionality tell us about the story?’ (MP7)” Students look for and make use of structure to generalize which types of equations do or do not represent proportional relationships.

• Grade 8, Unit 8: The Pythagorean Theorem and Irrational Numbers, Lesson 12, Activity 2 Connect, students use long division to convert unit fractions to decimals, determine whether the decimal representation terminates or repeats, and develop fluency in writing unit fractions as decimals. Teacher Edition, Today’s Goals states, “Students look for and make use of structure to explore connections between unit fractions and their decimal representations. They use long division to write unit fractions as decimals and determine whether the decimal representation terminates, repeats, or neither. (MP7)” Student Edition, Problem 8 states, “How can you predict whether a unit fraction will terminate, repeat, or neither when written as a decimal?” Teacher Edition Connect states, “Invite students to share their strategies for Problem 8, highlighting those that are similar to the given sample responses. Note: If it does not come up naturally, consider sharing that long division can be used to predict whether a unit fraction will terminate, repeat, or neither. If long division results in a 0, the decimal terminates, but if it does not result in 0, the decimal repeats. Key Takeaway: Long division is a strategy that is used to determine the decimal representation of any fraction. When converting a fraction to its decimal representation, make use of the structure of the denominator to predict if the decimal will terminate, repeat, or neither.” Students look for and make use of structure as they analyze the relationship between the denominator of a unit fraction and its decimal representation to determine whether the decimal will terminate or repeat.

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 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. It is explicitly identified for teachers in the Unit Teacher Edition. The Math of the Unit, Focus identifies the MP standards used and describes how students will engage with the standards. The Math Practices are also identified with a description of how the practice standard is developed in Focus and Coherence, Today’s Goal’s for each lesson with development identified within Warm-Ups and Activities. Course Overview, Grade Overview lists math practices correlated to each lesson.  Additionally a Standards for Mathematical Practice correlation chart is provided at the end of each Course Overview.

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, recognize patterns, and develop 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.

• Grade 6, Unit 8: Describing Data, Lesson 7, Activity 1, students are introduced to the concept of mean as the average, understood as distributing a total equally among a group. Teacher Edition, Today’s Goals states, “Students are introduced to the mean of a data set as the amount each member of the set gets if everything is distributed equally. Students determine the mean by calculating the (arithmetic) average of the data set and explore the mean as a statistic that summarizes the data set using a single number. Students also make a connection between the mean and the idea of ‘typical,’ or the center of a distribution, and make sense of why it is called a measure of center for a distribution. (MP8)” Student Edition states, “Problem 2. 4 friends played this game at the arcade. Here are the tickets each friend won. They decided to share the tickets equally. How many tickets should each friend get? Explain your thinking. Problem 3. Here is Ava’s work for determining how many tickets each friend gets. 4+2+1+5=12. 12\\div4. Discuss: What was Ava’s strategy? Problem 4. The mean, or average, is the number of tickets each friend gets if the tickets are distributed equally. Here are the tickets that 6 other friends won. Calculate the mean number of tickets.” Teachers Edition, Connect states: “Discuss Ava’s strategy. Consider asking, ‘Why did Ava add all the tickets? Why did Ava divide by 4?’ Share captured strategies for calculating the mean. Invite students to describe a general strategy for determining the mean no matter how many tickets or players there are. (MP8)” Students look for and express regularity in repeated reasoning as they recognize a general process for finding the mean and apply this reasoning across different data sets.

• Grade 7, Unit 7: Angles, Triangles, and Prisms, Lesson 5, Screen 2, Make a Prediction, students reason about the conditions under which three segment lengths can form a triangle, investigating how changes to a third segment affect the possibility of triangle construction when two segment lengths are fixed. Screen 2 states, “Will these three line segments form a triangle? Yes, No, I’m not sure.” Students are provided three lines with the numbers 3, 4, and 8. The Teacher Moves, Purpose states: “Students explore different combinations of segment lengths, then use repeated reasoning to describe the conditions under which a triangle is formed. (MP8) Launch: Share that in this activity students will develop a strategy for how to determine whether three line segments create a triangle. Distribute the Line Segments Sheet and rulers to students using print. Students can use linkage strips if they are available.” Students look for and express regularity in repeated reasoning as they test combinations of segment lengths and develop a general rule for determining when three lengths can or cannot form a triangle.

• Grade 8, Unit 7: Exponents and Scientific Notation, Lesson 6, Activity 2, students generalize the properties of exponents as they write rules for how exponents work. Teacher Edition, Today’s Goal states, “Students generalize properties of exponents they explored in prior lessons. They develop rules for rewriting exponential expressions. Students examine patterns, draft rules, revise their rules, and use mathematical reasoning to explain how they know these rules will always work. (MP3, MP8)” Student Edition states, “Problem 4. You will use the cards from the Warm-Up and the instructions on the screen to create your own rules for three more exponent situations.” Examples include: 3^{-6}=\\frac{1}{3^{6}}for Negative Exponents. 7^{3}\\cdot3^{3}=21^{3}for Powers With Different Bases. 10^{0}=1for Zero Exponents. “Problem 5. Choose any one of the three rules. Show or explain how you know the rule always works. Problem 6. Describe how exponent rules can help you rewrite exponential expressions.” Teacher Edition, Purpose states: “Students work together and use patterns to write descriptions of the properties of negative and zero exponents and the ‘Powers With Different Bases’ rule.” Students look for and express regularity in repeated reasoning as they identify patterns in exponent operations, develop general rules, and explain why those rules hold true in all cases.