7th Grade - Gateway 2

The materials reviewed for Math Nation Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Materials develop conceptual understanding throughout the grade level and materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. There are opportunities for students to develop their conceptual understanding in the various parts of each lesson: Warm-up, Exploration Activities, Lesson Synthesis, Cool Down, Check Your Understanding, and Practice Problems. Additionally, students’ conceptual understanding was assessed on Mid-Unit Assessments and End-of-Unit Assessments.

Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. Course Guide, “Concepts Develop from Concrete to Abstract, Mathematical concepts are introduced simply, concretely, and repeatedly, with complexity and abstraction developing over time. Students begin with concrete examples, and transition to diagrams and tables before relying exclusively on symbols to represent the mathematics they encounter.” Examples include:

• Unit 2, Lesson 3, 2.3.3 Exploration Extension, students develop conceptual understanding by representing proportional relationships between quantities and explaining how to convert one quantity to another (7.RP.2). “1. How many square millimeters are there in a square centimeter? 2. How do you convert square centimeters to square millimeters? How do you convert the other way?”

• Unit 4, Lesson 12, 4.12.1 Warm-Up, students develop conceptual understanding by using tape diagrams to solve problems involving parts of a whole within real-world contexts (7.RP.3). “What percentage of the car price is the tax?  What percentage of the food cost is the tip? What percentage of the shirt cost is the discount?” A fraction tape diagram is provided for each question. 

• Unit 5, Lesson 1, Cool Down, students develop conceptual understanding as students place positive and negative numbers on a number line and then interpret these values within the context of different real-world scenarios (7.NS.1). “Here is a set of signed numbers: 7, -3, \frac{1}{2}, -0.8, 0.8, −\frac{1}{10}, -2 1. Order the numbers from least to greatest. 2. If these numbers represent temperatures in degrees Celsius, which is the coldest? 3. If these numbers represent elevations in meters, which is the farthest away from sea level?” 

The materials provide students with opportunities to engage independently with concrete and semi-concrete representations while developing conceptual understanding. Examples include:

• Unit 2, Lesson 8, 2.8.5 Exploration Activity, students develop conceptual understanding as they identify proportional and non-proportional relationships represented in equations and tables (7.RP.1 and 7.RP.2). “Here are six different equations. y=4+x, y=4x, y=\frac{4}{x}, y=\frac{x}{4}, y=4^x, y=x^4 1. Predict which of these equations represent a proportional relationship. 2. Complete each table using the equation that represents the relationship. 3. Do these results change your answer to the first question? Explain your reasoning. 4. What do the equations of the proportional relationships have in common?” 

• Unit 6, End-of-Unit Assessment (A), Question 6, students develop conceptual understanding by performing an error analysis on another student’s work and then correctly simplify the expression (7.EE.1).“Tyler is simplifying the expression 6-2x+5+4x. Here is his work: 6-2x+5+4x; (6-2)x+(5+4)x; 4x+9x; 13x; a. Tyler’s work is incorrect. Explain the error he made. b. Write an equivalent expression to 6-2x+5+4x that only has two terms.”

• Unit 7, Lesson 8, 7.8.1 Warm-Up, students develop conceptual understanding by comparing and contrasting a set of triangles with given side lengths and a separate set of triangles with given angle measures (7.G.2).“Examine each set of triangles. What do you notice? What is the same about the triangles in the set? What is different?” Students are provided with two sets of triangles, the first set has six triangles, and the second set has four triangles.

The materials reviewed for Math Nation Grade 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

There are opportunities for students to develop their procedural skills and fluency throughout the grade levels in each lesson, these opportunities can be found in the Warm-up, Exploration Activities, and Practice Problems. Examples include:

• Unit 1, End-of-Unit Assessment (B), Question 2, students develop procedural skills and fluency as they solve problems involving scale drawings of geometric figures (7.RP.2a). “Rectangle A measures 8 inches by 2 inches. Rectangle B is a scaled copy of Rectangle A. Select all of the measurement pairs that could be the dimensions of Rectangle B. A. 40 inches by 10 inches B. 10 inches by 2.5 inches C. 9 inches by 3 inches D. 7 inches by 1 inch E. 6.4 inches by 1.6 inches” 

• Unit 5, Lesson 15, 5.15.2 Exploration Activity, students develop procedural skills and fluency as they solve equations involving negative numbers (7.EE.4a). “Match each equation to a value that makes it true by dragging the answer to the corresponding equation. Be prepared to explain your reasoning.” Equation options given: \frac{1}{2}x=-5; -2x=-9; -\frac{1}{2}x=\frac{1}{4}; -2x=7; x+(-2)=-6.5; -x+x=\frac{1}{2}. Solution options give x=-4.5; x=-\frac{1}{2}; x=-10; x=4.5; x=2\frac{1}{2}; x=-3.5"  

• Unit 7, Lesson 2, 7.2.6 Practice Problems, Question 1, students develop procedural skills and fluency as they use facts about supplementary angles to solve simple equations for an unknown angle in a figure (7.G.5). “Angles A and C are supplementary. Find the measure of angle C.” An image of two angles is provided: Angle A is equal to 74°, and Angle C is unknown. 

The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:

• Unit 2, Lesson 9, 2.9.6 Practice Problems, Question 3, students calculate the resulting sides of a quadrilateral after it has a scale factor applied (7.G.1). “Quadrilateral A has side lengths 3, 4, 5, and 6. Quadrilateral B is a scaled copy of Quadrilateral A with a scale factor of 2. Select all of the following that are side lengths of Quadrilateral B.” Answer options are the following: 5, 6, 7, 8, and 9.

• Unit 4, Lesson 9, Cool-down, students use proportional relationships to solve percent problems (7.RP.3). “Find each percentage of 75. Explain your reasoning. 1. What is 10% of 75? 2. What is 1% of 75? 3. What is 0.1% of 75? 4. What is 0.5% of 75?“ 

• Unit 6, Mid-Unit Assessment (B), Problem 4, students independently demonstrate procedural skills and fluency by solving equations (7.EE.4a). “Solve each equation. 1. 10-\frac{1}{4}x=7. 2. 3(x+8)=-21.”

The materials reviewed for Math Nation Grade 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Materials provide opportunities for students to work with multiple routine and non-routine applications of mathematics throughout the grade level and independently. Applications of mathematics occur throughout a lesson in the exploration activities, practice problems, and assessments.

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 4, 2.4.7 Practice Problems, Question 4, students solve a routine word problem involving ratios of scale drawings (7.G.1). “A map of Colorado says that the scale is 1 inch to 20 miles or 1 to 1,267,200. Are these two ways of reporting the scale the same? Explain your reasoning.” 

• Unit 3, Lesson 6, 3.6.3 Exploration Activity, students estimate the area of a state through a strategy of their choosing (7.G.B). “Estimate the area of Nevada in square miles. Explain or show your reasoning.” An image of the state of Nevada is given with three measurements provided.

• Unit 6, Mid-Unit Assessment (A), Question 2, students select situations that could represent a tape diagram (7.EE.4). “Select all the situations that can be represented by the tape diagram. A. Clare buys 4 bouquets, each with the same number of flowers. The florist puts an extra flower in each bouquet before she leaves. She leaves with a total of 99 flowers. B. Andre babysat 5 times this past month and earned the same amount each time. To thank him, the family gave him an extra $4 at the end of the month. Andre earned $99 from babysitting. C. A family of 5 drove to a concert. They paid $4 for parking, and all of their tickets were the same price. They paid $99 in total. D. 5 bags of marbles each contain 4 large marbles and the same number of small marbles. Altogether, the bags contain 99 marbles. E. Han is baking five batches of muffins. Each batch needs the same amount of sugar in the muffins, and each batch needs four extra teaspoons of sugar for the topping. Han uses 99 total teaspoons of sugar.”

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of mathematics throughout the grade level. Examples include:

• Unit 2, Lesson 2, 2.2.5 Exploration Extension, students do research and use proportional relationships to determine if the value of a penny was worth more in 1982 (7.RP.2). “Pennies made before 1982 are 95% copper and weigh about 3.11 grams each. (Pennies made after that date are primarily made of zinc). Some people claim that the value of the copper in one of these pennies is greater than the face value of the penny. Find out how much copper is worth right now, and decide if this claim is true.”

• Unit 5, Lesson 17, 5.17.4 Exploration Activity, students look at the difference in stock prices over a three-month time frame. (7.NS.3 and 7.EE.3). “Your teacher will give you a list of stocks. 1. Select a combination of stocks with a total value close to, but no more than, $100. 2. Using the new list, how did the total value of your selected stocks change?” Students are given a list of stock prices before answering Question 1. Then students receive a new list showing the changes in stock prices after three months before answering Question 2. 

• Unit 7, Lesson 16, Cool-Down, students solve a real-world problem by applying their understanding of area  (7.G.6). “Andre is preparing for the school play and needs to paint the cardboard castle backdrop that measures 14\frac{1}{4} feet by 6 feet. 1. How much cardboard does he need to paint? 2. If one bottle of paint covers an area of 40 square feet, how many bottles of paint does Andre need for his backdrop?”

The materials reviewed for Math Nation Grade 7 meet expectations 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 level.

All three aspects of rigor are present independently throughout each grade level. Examples include:

• Unit 2, Lesson 2, 2.2.9 Check Your Understanding, Question 2, students solve a real-world  application problem that requires them to work with proportions (7.RP.2a). “The table below shows the relationship between the amount of candy bought (in pounds) and the total cost of the candy (in dollars) at a movie theater. Which statement about the table is true? A. It does not have a constant of proportionality. B. The constant of proportionality is 0.36. C. The constant of proportionality is 2.75. D. The constant of proportionality is 4.”

• Unit 4, Lesson 8, 4.8.7 Practice Problems, Question 3, students develop procedural skills and fluency by identifying the equation that matches a given situation (7.EE.2). “In a video game, Clare scored 50% more points than Tyler. If c is the number of points that Clare scored and is the number of points that Tyler scored, which equations are correct?  Select all that apply.” Choices include: c=1.5t; c=t+0.5; c=t+0.5t; c=t+50; c=(1+0.5)t

• Unit 6, Lesson 4, 6.4.2 Exploration Activity, Questions 1 and 2, students build conceptual understanding as they use tape diagrams to represent given situations (7.EE.A). “Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable. 1. Diego has 7 packs of markers. Each pack has x markers in it. After Lin gives him 9 more markers, he has a total of 30 markers. 2. Elena is cutting a 30-foot piece of ribbon for a craft project. She cuts off 7 feet, and then cuts the remaining piece into 9 equal lengths of x feet each.” 

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study. Examples include:

• Unit 2, Lesson 6, 2.6.2 Exploration Activity, students develop conceptual understanding, procedural skills and fluency, and application as they solve problems involving ticket sales for a concert (7.RP.2c). “A performer expects to sell 5,000 tickets for an upcoming concert. They want to make a total of $311,000 in sales from these tickets. 1. Assuming that all tickets have the same price, what is the price for one ticket? 2. How much will they make if they sell 7,000 tickets? 3. How much will they make if they sell 10,000 tickets? 50,000? 120,000? a million? x tickets? 4. If they make $379,420, how many tickets have they sold? 5. How many tickets will they have to sell to make $5,000,000?” 

• Unit 3, Lesson 4, 3.4.7 Practice Problems, Question 1, students develop procedural skills and fluency as they apply formulas to solve problems dealing with circumference (7.G.4). “Here is a picture of a Ferris wheel. It has a diameter of 80 meters. A. On the picture, draw and label a diameter. B. How far does a rider travel in one complete rotation around the Ferris wheel?” 

• Unit 7, Lesson 5, 7.5.4 Exploration Extension, students develop procedural and fluency and conceptual understanding as they find the exact measurement of three angles (7.G.5). “The diagram contains three squares. Three additional segments have been drawn that connect corners of the squares. We want to find the exact value of a+b+c. 1. Use a protractor to measure the three angles. Use your measurements to conjecture about the value of a+b+c. 2. Find the exact value of a+b+c by reasoning about the diagram.”

The materials reviewed for Math Nation Grade 7 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Mathematical practices are identified in the Teacher Edition, Lesson Narrative, and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 7, Course Guide). 

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to analyze and make sense of problems, work to understand the information in problems, and use a variety of strategies to make sense of problems. Examples include:

• Unit 3, Lesson 2, 3.2.1 Warm-Up, students make sense of a problem as they decide what a sketch for Figure C could look like. “Here are two figures. Figure C looks more like Figure A than like Figure B. Sketch what Figure C might look like. Explain your reasoning.” Given is a picture of Figure A and Figure B on a grind. Figure A is a square and Figure B is an oval. This problem attends to the full intent of MP1 as students need to make sense of a situation as they decide which characteristics of Figures A and B to use to satisfy the criteria for Figure C.

• Unit 5, Lesson 14, 5.14.1 Warm-up, students look at a group of equations and remove the one(s) that does not belong. “Which equation does not belong? \frac{1}{2}x=-50; -60t = 30; x + 90 = -100; -0.01=-0.001x.” This Warm-up attends to the full intent of MP1 as students need to make sense of the equations to pick the one that does not belong.

• Unit 7, Lesson 13, 7.13.1 Warm-up, students explain which solids are prisms, and draw a cross-section of the prism parallel to the base. “1. Which of these solids are prisms? Explain how you know. 2. For each of the prisms, what does the base look like? a. Shade one base in the picture. b. Draw a cross-section of the prism parallel to the base.” Students are given six shapes. This activity intentionally develops the full intent of MP1 as students need to make sense of the problem to identify which shapes would be prisms and draw the cross-sections

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to represent situations symbolically, attend to the meaning of quantities, and understand relationships between problem scenarios and mathematical representations. Examples include:

• Unit 1, Lesson 10, 1.10.2 Exploration Activity, students find the area of a triangle using a scale that they are assigned. “Here is a map showing a plot of land in the shape of a right triangle. 1. Your teacher will assign you a scale to use. On centimeter graph paper, make a scale drawing of the plot of land. Make sure to write your scale on your drawing. 2. What is the area of the triangle you drew? Explain or show your reasoning. 3. How many square meters are represented by 1 square centimeter in your drawing? 4. After everyone in your group is finished, order the scale drawings from largest to smallest. What do you notice about the scales when your drawings are placed in this order?” This problem attends to the full intent of MP2 as students need to reason abstractly and quantitatively as they order the scale drawings from largest to smallest and convert their scaling.

• Unit 4, Lesson 2, 4.2.2 Exploration Activity, students use given information about a train to calculate the unit rate and answer questions about distance. “A train is traveling at a constant speed and goes 7.5 kilometers in 6 minutes. At that rate: 1. How far does the train go in 1 minute? 2. How far does the train go in 100 minutes?” This problem intentionally develops the full intent of MP2 as students need to reason about the meaning of quantities given the context of the problem.

• Unit 6, Lesson 12, 6.12.1 Warm-Up, students select which expression(s) could represent a scenario. “An item costs x dollars and then a 20% discount is applied. Select all the expressions that could represent the price of the item after the discount. 1. \frac{20}{100}x 2. x-\frac{20}{100}x 3. (1-0.20)x 4. \frac{100-20}{100}x 5. 0.80x 6. 100-20x” This Warm-up attends to the full intent of MP2 as students reason abstractly and quantitatively about the expression(s) that could be the price of the item after the discount.

The materials reviewed for Math Nation Grade 7 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. The materials provide opportunities for students to construct viable arguments and critique the reasoning of others in whole class and small group settings (i.e. exploration activities) and independent work settings (i.e. practice problems and assessments). 

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Throughout the grade level, students are required to construct mathematical arguments by explaining/justifying their strategies and thinking, performing error analysis of provided student work/solutions, listening to the arguments of others and deciding if it makes sense, asking useful questions to better understand, and critiquing the reasoning of others. Examples include:

• Unit 2, Lesson 7, 2.7.6 Practice Problems, Question 5, students use proportional relationships to determine whether either student has a correct argument.“Kiran and Mai are standing at one corner of a rectangular field of grass looking at the diagonally opposite corner. Kiran says that if the field were twice as long and twice as wide, then it would be twice the distance to the far corner. Mai says that it would be more than twice as far, since the diagonal is even longer than the side lengths. Do you agree with either of them?” This activity intentionally develops MP3 as students critique the reasoning of two students and decide who is correct. 

• Unit 3, Lesson 2, 3.2.4 Exploration Activity, students use their knowledge of circles to determine which parts of the circle three students measured. “Priya, Han, and Mai each measure one of the circular objects from earlier. Priya says that the bike wheel is 24 inches. Han says that the yo-yo trick is 24 inches. Mai says that the glow necklace is 24 inches. 1. Do you think that all these circles are the same size? 2. What part of the circle did each person measure? Explain your reasoning.” This activity intentionally develops MP3 as students critique the reasoning of three students while explaining their reasoning.

• Unit 6, Lesson 2, 6.2.2 Exploration Activity, Question 2, students explain how a tape diagram represents a situation. “...With your group, decide who will go first. That person explains why the diagram represents the story. Work together to find any unknown amounts in the story. Then, switch roles for the second diagram and switch again for the third… 2. To thank her five volunteers, Mai gave each of them the same number of stickers. Then she gave them each two more stickers. Altogether, she gave them a total of 30 stickers.” Full Lesson Plan, Teacher Guidance: “Present an incorrect statement for the second situation that reflects a possible misunderstanding from the class. For example, ‘Mai gave 6 stickers to each of the volunteers because 30 divided by 5 is 6. So y is 6.’ Prompt students to identify the error, and then write a correct version. This helps students evaluate, and improve on, the written mathematical arguments of others…” This activity intentionally develops MP3 as students explain why the diagram represents the story they constructed and helps students build on their mathematical arguments.

The materials reviewed for Math Nation Grade 7 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 7, Course Guide). 

Examples where and how the materials use MPs 4 and/or 5 to enrich the mathematical content and demonstrate the full intent of the mathematical practices include:

• Unit 2, Lesson 1, 2.1.1 Warm Up, students fill in a double number line and think of a situation that could be represented by the model.  “Complete the double number line diagram with the missing numbers. What could each of the number lines represent? Invent a situation and label the diagram. Make sure your labels include appropriate units of measure.” This activity intentionally develops (MP4) as students use a representation to model a situation, and (MP5) as students have to use the double number line strategically in order to create an appropriate situation. 

• Unit 3, Lesson 6, 3.6.6 Practice Problems, Question 2, students use polygons to “cover” a map of the state of Virginia to approximate the area of the state. “A. Draw polygons on the map that could be used to approximate the area of Virginia. B. Which measurements would you need to know in order to calculate an approximation of the area of Virginia? Label the sides of the polygons whose measurements you would need. (Note: You aren't being asked to calculate anything.)” This activity intentionally develops (MP4) as students use the polygons to approximate area, and (MP5) as students have to choose the best polygons in order to have the closest estimated approximation.

• Unit 6, Lesson 19, 6.19.2 Exploration Activity, students write equivalent expressions either in factored or expanded form. “In each row, write the equivalent expression. If you get stuck, use a diagram to organize your work. The first row is provided as an example. Diagrams are provided for the first three rows.” (Students are given a table with factored expressions in the left column and expanded expressions in the right column. First three rows of the factored (left) column include: -3(5-2y), 5(a-6), blank row. First three rows of the expanded (right) column include: -15+6y, blank row, 6a-2b.) This activity intentionally develops (MP4) as students write equivalent expressions in either expanded or factored form, and (MP5) as students create diagrams of their choice to assist them in correctly writing the expression.

• Unit 7, Lesson 10, 7.10.6 Practice Problems, Question 1, students draw triangles to fit certain characteristics. “A triangle has sides of length 7 cm, 4 cm, and 5 cm. How many unique triangles can be drawn that fit that description?  Explain or show your reasoning.” This activity intentionally develops (MP4) as students create models of the triangles that fit the criteria, and (MP5) as students are to use any tool or strategy to help them create the triangles.

The materials reviewed for Math Nation Grade 7 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 7, Course Guide). 

There is intentional development of MP6 to meet its full intent in connection to grade-level content and the instructional materials attend to the specialized language of mathematics. Students communicate using grade-level appropriate vocabulary and conventions and formulate clear explanations when engaging with course materials. Students must calculate accurately and efficiently, specify units of measure, and use and label tables and graphs appropriately when engaging with course materials. Teacher guidance very clearly develops the specialized language of mathematics as teachers are explicitly prompted when to introduce content-related vocabulary and use accurate definitions when communicating mathematically. Examples include:

• Unit 1, Lesson 4, 1.4.1 Warm-Up, students name pairs of corresponding angles and confirm their prediction by measuring the angles. “Each of these polygons is a scaled copy of the others. 1. Name two pairs of corresponding angles. What can you say about the sizes of these angles? 2. Check your prediction by measuring at least one pair of corresponding angles using a protractor. Record your measurements to the nearest 5°.” This activity intentionally develops MP6 as students need to be precise in recording their measurements and attend to the specialized language of mathematics as students must understand the definition of corresponding angles.

• Unit 2, Lesson 9, 2.9.6 Practice Problems, Question 1, students determine whether a scenario represents a proportional relationship and explain their reasoning. “For each situation, explain whether you think the relationship is proportional or not. Explain your reasoning. a. The weight of a stack of standard 8.5×11 copier paper vs. number of sheets of paper. b. The weight of a stack of different-sized books vs. the number of books in the stack.” This activity attends to the full intent of MP6 and the specialized language of mathematics as students have to explain their choice in the context of proportionality. 

• Unit 2, End-of-Unit Assessment (A), Problem 7, students write an equation for a situation representing a proportional relationship and explain the context of a point based on the situation. “A recipe for salad dressing calls for 3 tablespoons of oil for every 2 tablespoons of vinegar. The line represents the relationship between the amount of oil and the amount of vinegar needed to make salad dressing according to this recipe. The point  (1, 1.5) is on the line. 1. Label the axes appropriately. 2. Write an equation that represents the proportional relationship between oil and vinegar. Indicate the meaning of each variable. 3. Explain the meaning of the point (1, 1.5) in terms of the situation.” This activity attends to the full intent of MP6 and the specialized language of mathematics as students formulate clear explanations, label graphs appropriately, and specify units of measure.

• Unit 7, Lesson 3, 7.3.6 Practice Problems, Question 3, students use what they know about supplementary and complementary angles to see if a given angle pair relationship is possible. “If two angles are both vertical and supplementary, can we determine the angles? Is it possible to be both vertical and complementary? If so, can you determine the angles? Explain how you know.” This activity attends to the full intent of MP6 and the specialized language of mathematics as students are required to use definitions accurately and mathematical language to explain their answers.

The materials reviewed for Math Nation Grade 7 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Mathematical practices are identified in the Teacher Edition, Lesson Narrative and Teacher Edition, Activities Narrative. Explanations are provided in terms of how the mathematical practices “come into play” within the lesson and activity narratives (Grade 7, Course Guide). 

There is intentional development of MP7 to meet its full intent in connection to grade-level content.  Materials provide opportunities for students to look for patterns or structures to make generalizations and solve problems, look for and explain the structure of mathematical representations, and look at and decompose “complicated” into “simpler.”  Examples include:

• Unit 3, Lesson 3, 3.3.2 Exploration Activity, students measure the diameter and circumference of circular objects, plot these values, and identify a relationship between the two. “ Your teacher will give you several circular objects. 1. Explore the applet to find the diameter and the circumference of three circular objects to the nearest tenth of a unit. Record your measurements in the table. 2. Plot the diameter and circumference values from the table on the coordinate plane. What do you notice? 3. Plot the points from two other groups on the same coordinate plane. Do you see the same pattern that you noticed earlier?” This activity attends to the full intent of MP7 as students make use of the structure of the graph to look for patterns in order to answer the question.

• Unit 5, Lesson 5, 5.5.3 Exploration Activity, students match expressions and number line diagrams and then add and subtract numbers to see that subtracting a number is the same as adding the opposite. “1. Match each diagram to one of these expressions: 3+7, 3-7, 3+(-7), 3-(-7). (Students are given four number line diagrams to match.) 2. Which expressions in the first question have the same value? What do you notice? 3. Complete each of these tables. (Students find the value corresponding to each of the following expressions for one of the tables: 8+(-8), 8-8, 8+(-5), 8-5, 8+(-12), 8-12.) What do you notice?” This activity attends to the full intent of MP7 as students look for structures to make some general claims of add and subtracting numbers with opposites. 

• Unit 6, Lesson 19, 6.19.1 Warm-Up, students use the distributive property to decompose and compose different expressions in order to evaluate. “Find the value of each expression mentally. 2+3\cdot4, (2+3)(4), 2-3\cdot4, 2-(3\cdot4).” This warm-up attends to the full intent of MP7 as students make use of the structure of the layout of the expressions in order to solve them mentally.

There is intentional development of MP8 to meet its full intent in connection to grade-level content. 

Materials provide opportunities for students to notice repeated calculations to understand algorithms and make generalizations or create shortcuts, evaluate the reasonableness of their answers and their thinking, and create, describe, or explain a general method/formula/process/algorithm. Examples include:

• Unit 3, Lesson 8, 3.8.7 Practice Problems, Question 1, students examine a circle and its rearranged parts to explain the area formula for a circle. “The picture shows a circle divided into 8 equal wedges which are rearranged. The radius of the circle is r and its circumference is 2πr. How does the picture help to explain why the area of the circle is 𝜋𝑟²?”This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning as students repeatedly use the information given about the radius and circumference to explain how the formula for area of a circle works. 

• Unit 4, Lesson 8, 4.8.1 Warm-Up, students explain a process for applying a percent increase or decrease. “How do you get from one number to the next using multiplication or division? From 100 to 106? From 100 to 90? From 90 to 100? From 106 to 100?” This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning as students use repeated calculations of multiplication and division to generalize how percent increase and decrease works.

• Unit 8, Lesson 4, 8.4.3 Exploration Activity, students consider the reasonableness of certain outcomes in simulation scenarios. “1. For each situation, do you think the result is surprising or not? Is it possible? Be prepared to explain your reasoning. a. You flip the coin once, and it lands heads up. b. You flip the coin twice, and it lands heads up both times. c. You flip the coin 100 times, and it lands heads up all 100 times. 2. If you flip the coin 100 times, how many times would you expect the coin to land heads up? Explain your reasoning. 3. If you flip the coin 100 times, what are some other results that would not be surprising? 4. You’ve flipped the coin 3 times, and it has come up heads once. The cumulative fraction of heads is currently \frac{1}{3}. If you flip the coin one more time, will it land heads up to make the cumulative fraction \frac{2}{4}?” This activity attends to the full intent of MP8, look for and express regularity in repeated reasoning, as students think about the repeated coin flips and what would be surprising and not surprising if they were to flip it 100 times.