7th Grade - Gateway 2
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Rigor & Mathematical Practices
Rigor & the Mathematical PracticesGateway 2 - Does Not Meet Expectations | 55% |
|---|---|
Criterion 2.1: Rigor and Balance | 7 / 8 |
Criterion 2.2: Math Practices | 3 / 10 |
The materials reviewed for Math in Focus: Singapore Math Course 2 do not meet expectations for rigor and balance and practice-content connections. The materials reflect the balances in the Standards and help students develop conceptual understanding, procedural skill and fluency, and application. However, the materials do not make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).
Criterion 2.1: Rigor and Balance
Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.
The materials reviewed for Math in Focus: Singapore Math Course 2 meet expectations for rigor. The materials give attention throughout the year to procedural skill and fluency, spend sufficient time working with engaging applications of mathematics, and do not always treat the three aspects of rigor together or separately. The materials partially develop conceptual understanding of key mathematical concepts.
Indicator 2a
Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
The materials reviewed for Math in Focus: Singapore Math Course 2 partially meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.
Students have opportunities to develop conceptual understanding of mathematical concepts during the Engage and Learn portions of the lessons. Examples include:
In Section 1.3, Adding Integers, Learn, Problem 1, page 38, students use counters to model addition. The problem states, “Suppose the temperature was -8°F at 7 a.m. Five hours later, the temperature has risen 10°F. Find the new temperature.” Students develop conceptual understanding of 7.NS.1 (Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram).
In Section 2.4, Expanding Algebraic Expressions, Engage, page 153, students use algebra tiles to expand expressions. The materials state, “Use (green tile shown) to represent +x and (orange tile shown ) to represent +1. Show how you expand 2(2x + 6) and $$\frac{1}{2}$$(2x + 6). Share your method.” Students develop conceptual understanding of 7.EE.1 (Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients).
In Section 5.4, Interior and Exterior Angles, Activity, page 47, students work in pairs to discover the property of interior angles of a triangle. The materials state, “1) Draw and cut out a triangle. Label the three interior angles of the triangle as 1, 2, and 3. 2) Cut out the three angles. Then, arrange them on a straight line. What do you notice about the sum of the measures of the three interior angles?” Students develop conceptual understanding of 7.G.5 (Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure).
In Section 8.1, Random Sampling Methods, Learn, Problem 3c, page 215, students use a random number table to determine a random sample. The problem states, “Suppose you want to pick a random sample of 30 people from a town that has 500 residents. You can first assign a unique 3-digit number from 001 to 500 to each resident. Then, use a random number table like the one shown below to select the members of the sample.” Students develop conceptual understanding of 7.SP.1 (Understand that statistics can be used to gain information about a population by examining a sample of the population).
In Section 8.4, Finding the Probability of Events, Engage, page 245, students use pictures of cards to determine the probability of an event occurring. The materials state, “Suppose you randomly pick two of the cards below. What is the probability of picking two cards with the same letter? Explain your answer.” Four pink cards labeled A-D and four purple cards labeled A-D are shown. Students develop conceptual understanding of 7.SP.5 (Summarize numerical data sets in relation to their context).
Students have opportunities to demonstrate conceptual understanding through Try activities, which are guided practice opportunities to reinforce new learning. The Independent Practice provides limited opportunities for students to continue the development of conceptual understanding. Examples include:
In Section 5.4, Interior and Exterior Angles, Independent Practice, Problem 2, page 55, students use the angle sum of a triangle to find unknown angle measures. The problem states, “The diagrams may not be drawn to scale. Find the value of y. In the diagrams for 4 to 6, ACis a straight line.” Students are shown a triangle with interior angles labeled 18° and 26°. Students independently practice conceptual understanding of 7.G.5 (Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure).
In Section 8.1, Random Sampling Methods, Independent Practice, Problem 7, page 220, students describe how to implement each of the three sampling methods for a given situation. The problem states, “2,000 runners participated in a marathon. You want to randomly choose 60 of the runners to find out how long it took each one to run the race. Describe how you would select the 60 runners if you use a a) random sampling method. b) systematic sampling method. c) stratified sampling method.” Students independently practice conceptual understanding of 7.SP.1 (Understand that statistics can be used to gain information about a population by examining a sample of the population, generalizations about a population from a sample are valid only if the sample is representative of that population).
Indicator 2b
Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency.
The materials reviewed for Math in Focus: Singapore Math Course 2 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.
Students have opportunities to develop procedural skill and fluency during the Engage and Learn portions of the lessons. Examples include:
In Section 1.2, Writing Rational Numbers as Decimals, Learn, Problem 1, page 23, students write rational numbers as repeating decimals using long division. The problem states, “Since $$\frac{1}{3}$$ means 1 divided by 3, you can write $$\frac{1}{3}$$ as a decimal using long division. When you divide 1 by 3, the division process will not terminate with a remainder of 0. The digit 3 keeps repeating infinitely. A decimal, such as 0.333…, is called a repeating decimal. For the repeating decimal 0.333…, the digit 3 repeats itself. you can write 0.333… as $$0.\bar{3}$$, with a bar above the repeating digit 3. So, 0.333… = $$0.\bar{3}$$.” Students develop procedural skill and fluency of 7.NS.2d (Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats).
In Section 3.3, Real-World Problems: Algebraic Equations, Learn, Problem 4, page 235, students translate descriptions into algebraic equations. The problem states, “Ivan has 12 more comic books than Hana. If they have 28 comic books altogether, how many comic books does Ivan have? Let the number of comic books that Hana has be x. Then, the number of comic books that Ivan has is x + 12. Because they have 28 books altogether, x + (x + 12) = 28. Number of books that Ivan has: x + 12 = 8 + 12 = 20. Ivan has 20 comic books.” Students develop procedural skill and fluency of 7.EE.B.4a (Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently).
In Section 4.1, Identifying Direct Proportion, Learn, Problem 3, page 291, students identify a constant of proportionality from a verbal description. The problem states, “Ana is buying some baseball caps. Each cap costs $8. The amount Ana pays for the caps is directly proportionally to the number of caps she buys. Write an equation that represents the direct proportion.” Students develop procedural skill and fluency of 7.RP.2b (Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships).
In Section 5.1, Complementary, Supplementary, and Adjacent Angles, Learn, Problem 2, page 10, students calculate an unknown angle when one of two angles is known in a right angle. The problem states, “In the diagram, ∠PQS and ∠SQR are adjacent angles. They share a common vertex, Q, and a common side, QS, QP is perpendicular to QR. Find the measure of ∠SQR.” Students are shown a right angle and given the measurement of 41° for ∠PQS. Students develop procedural skill and fluency of 7.G.5 (Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure).
In Section 9.2, Probability of Compound Events, Learn, Problem 2, page 328, students find probability using tree diagrams. The problem states, “Suppose that it is equally likely to rain or not rain on any given day. Draw a tree diagram and use it to find the probability that it rains exactly once on two consecutive days. P(rain exactly once on two consecutive days) = $$\frac{2}{4}$$ = $$\frac{1}{2}$$.” Students develop procedural skill and fluency of 7.SP.8 (Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation).
Students have opportunities to independently practice procedural skill and fluency during the Try and Independent Practice portion of the lesson. Examples include:
In Section 3.3, Real-World Problems: Algebraic Equations, Independent Practice, Problem 1, page 239, students independently solve a perimeter problem algebraically. The problem states, “Two sections of a garden are shaped like identical isosceles triangles. The base of each triangle is 50 feet, and the other two sides are each x feet long. If the combined perimeter of both gardens is 242 feet, find the value of x.” Students independently practice procedural skill and fluency of 7.EE.B.4a (Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently).
In Section 4.1, Identifying Direct Proportion, Independent Practice, Problem 18, page 296, students independently interpret verbal descriptions to write the direct proportion equation. The problem states, “Owen hikes 3 miles in 45 minutes. Given that the distance is directly proportional to the duration he walks, find the constant of proportionality and write an equation to represent the direct proportion.” Students independently practice procedural skill and fluency of 7.RP.2b (Identify the constant of proportionality [unit rate] in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships).
In Section 5.1, Complementary, Supplementary, and Adjacent Angles, Independent Practice, Problem 34, students independently apply concepts of complementary and supplementary angles to solve. The problem states, “The diagrams may not be drawn to scale. The ratio a:b = 2:3. Find the values of a and b. The measure of ∠PQR = 90$$\degree$$.” Students independently practice procedural skill and fluency of 7.G.5 (Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure).
In Section 9.2, Probability of Compound Events, Independent Practice, Problem 2, page 333, students independently draw a tree diagram and calculate the probability of a compound event. The problem states, “A letter is randomly chosen from the word FOOD, followed by randomly choosing a letter from the word DOT. Use a tree diagram to find the probability that both letters chosen are the same.” Students independently practice procedural skill and fluency of 7.SP.8 (Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation).
Indicator 2c
Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.
The materials reviewed for Math in Focus: Singapore Math Course 2 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.
Students have opportunities throughout the materials to engage in routine application of mathematics. Examples include:
In Section 1.7, Operations with Fractions and Mixed Numbers, Learn, Problem 1, page 91, students add, subtract, multiply, and divide rational numbers in a real-world situation. The problem states, “Mr. Turner has a partial roll of wire $$18\frac{1}{4}$$ feet long. He needs $$25\frac{1}{2}$$ feet of wire for a remodeling project. How much wire is he short?” Students engage in routine application of 7.NS.3 (Solve real-world and mathematical problems involving the four operations with rational numbers).
In Section 3.5, Real-World Problems: Algebraic Inequalities, Try, Problem 3, Page 262, students solve real-world problems involving algebraic inequalities. The problem states, “Ms. Cooper pays $200 in advance on her account at a health club. Each time she visits the club, $8 is deducted from the account. If she needs to maintain a minimum amount of $50 in the account, how many visits can Ms. Cooper make before she needs to top up the account again?” Students engage in routine application of 7.EE.4b (Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers).
In Section 4.3, Real-World Problems: Direct Proportion, Independent Practice, Problem 10, page 315, students write a proportion to find an unknown quantity given one of the quantities. The problem states, “It costs $180 to rent a car for 3 days. Find the cost of renting a car for 1 week.” Students independently engage in routine application of 7.RP.2 (Recognize and represent proportional relationships between quantities).
In Section 7.6, Real-World Problems: Surface Area and Volume, Independent Practice, Problem 3, page 188, students find the volume and surface area of a composite solid made up of a triangular prism and a rectangular prism. The problem states, “Mr. Turner builds a shed to store his tools. The shed has a roof that is in the shape of a triangular prism. a. Find the amount of space the shed occupies. b. Find the surface area of the shed, including its floor.” A diagram with dimensions is provided. Students independently engage in routine application of 7.G.6 (Solve real-world and mathematical problems involving area, volume and surface area of two-and three- dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms).
Students have opportunities throughout the materials to engage in non-routine application of mathematics. Examples include:
In Section 1.5, Multiplying and Dividing Integers, Engage, page 63, students represent a multiplication situation using counters. The materials state, “Show 2 × 4 using (picture of yellow and orange counter shown). How do you show 2 × (-4)? Create a real-world problem to model each situation. Share your real-world problems.” Students engage in non-routine application of 7.NS.2 (Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers).
In Section 1.7, Operations with Fractions and Mixed Numbers, Engage, page 91, students apply operations with rational numbers in a real-world context. The materials state, “A clock’s battery is running low. Every 6 hours, the clock slows down by $$\frac{1}{2}$$ hour. How do you find out how much time the clock slows down by in 1 hour? Share your method.” Students engage in non-routine application of 7.NS.3 (Solve real-world and mathematical problems involving the four operations with rational numbers).
In Section 2.7, Real-World Problems: Algebraic Reasoning, Enage, page 192. students work in pairs to solve a percent problem involving algebraic expressions. The materials state, “Ms. Evans bought a total of 60 pens and pencils. There was an equal number of pens and pencils. She gave x percent of the pens and y percent of the pencils to her students. Use algebraic reasoning to write an algebraic expression for the number of pens and pencils that she gave her students. Share your reasoning.” Students engage in non-routine application of 7.EE.3 (Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form).
In Chapter 8, Performance Task, Problem 2, page 293, students find the probability of winning if they draw two number cards and the difference is 3 or more. The problem states, “In a game, you and your friend are asked to each select a card from a deck of ten cards with the numbers 1 to 10. Your friend selects a card from the deck. Then, you select a card from the ones remaining in the deck. You do not know your friend’s number. You win if the difference between your number and your friend’s number is at least 3. a) For which of your friend’s numbers do you have the greatest chance of winning? b) For which of your friend’s numbers do you have the least chance of winning? c) What is the probability that you will win?” Students independently engage in non-routine application of 7.SP.8 (Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation).
Indicator 2d
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.
The materials reviewed for Math in Focus: Singapore Math Course 2 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.
All three aspects of rigor are present independently throughout the materials. For example:
In Section 1.7, Operations with Fractions and Mixed Numbers, Learn, Problem 2, page 86, students multiply rational numbers. The problem states, “Evaluate $$-\frac{3}{7} ⋅ \frac{8}{15}$$.”Students engage in procedural skill and fluency of 7.NS.2c (Apply properties of operations as strategies to multiply and divide rational numbers).
In Section 3.5, Real-World Problems: Algebraic Inequalities, Independent Practice, Problem 10, page 266, students write and solve inequalities. The problem states, “A cab company charges $0.80 per mile plus $2 for tolls. Rachel has at most $16 to spend on her cab fare. Write and solve an inequality for the maximum distance she can travel. Can she afford to take a cab from her home to an airport that is 25 miles away?” Students engage in application of 7.EE.4b (Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem).
In Section 8.3, Defining Outcomes, Events and Sample Space, Try, Problem 1, page 237, students practice identifying outcomes, samples spaces, and events. The problem states, “Jake spun the spinner on the right and recorded the numbers where the spinner lands. a) List all the possible outcomes. b) State the number of outcomes in the sample space.” Students engage in conceptual understanding of 7.SP.7a (Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events).
Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the materials. For example:
In Section 1.4, Subtracting Integers, Try, Problem 3, page 53, students use the additive inverse property to subtract integers. The problem states, “A fishing boat drags its net 35 feet below the ocean’s surface. Then, it lowers the net by an additional 12 feet. Find the fishing net’s new position relative to the ocean’s surface.” Students develop procedural skill and fluency and apply the mathematics of 7.NS.1c (Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q)).
In Section 2.1, Adding Algebraic Terms, Learn, Problem 1, page 132, students simplify algebraic expressions with decimal or fractional coefficients by adding like terms. The problem states, “Simplify the expressions 0.9p + 0.7p. Represent the term 0.9p with nine 0.1p sections and the term 0.7p with seven 0.1p sections. From the bar model, 0.9p + 0.7p = 1.6p. The sum is the total number of colored sections in the bar model.” Students develop conceptual understanding and build procedural skill and fluency of 7.EE.1 (Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients).
In Section 7.3, Real-World Problems: Circles, Try, Problem 1, page 156, students use the formula for area of a circle to solve real-world problems. The problem states, “Alex recycled some old fabric to make a rug. He cut out a quadrant and two semicircles to make the rug. Find the area of the rug. Use $$\frac{22}{7}$$ as an approximation for $$\pi$$.” A diagram of the rug with dimensions is provided. Students develop procedural skill and fluency and apply the mathematics of 7.G.4 (Know the formulas for the area and circumference of a circle and use them to solve problems).
Criterion 2.2: Math Practices
Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).
The materials reviewed for Math in Focus: Singapore Math Course 2 do not meet expectations for practice-content connections. The materials support the intentional development of MP3 and partially support the intentional development of MP6. The materials do not support the intentional development of MPs 1, 2, 4, 5, 7, and 8.
Indicator 2e
Materials support 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.
The materials reviewed for Math in Focus: Singapore Math Course 2 do not 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.
Students have limited opportunities to make sense of problems and persevere in solving them in connection to grade-level content, identified as mathematical habits in the materials. Student materials do not provide guidance, teacher guidance is repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP1. Examples include:
In Chapter 3, Algebraic Equations and Inequalities, Put On Your Thinking Cap! Problem 1, page 268, students solve real-world problems using algebraic equations. The problem states, “Jamar is five times as old as Kylie. Larissa is five times as old as Jamar. Mitchell is twice as old as Larissa. The sum of their ages is the age of Nora. Nora just turned 81. How old is Jamar?” Teacher guidance states, “You may want to guide students on applying the various heuristics using the problem-solving heuristics poster. Refer students to the corresponding teacher resources for prompts and worked out solutions.” Teacher guidance gives a generic reference to have students use the heuristics poster which is repeated throughout the materials.
In Chapter 4, Proportion and Percent of Change, Put On Your Thinking Cap! Problem 1, page 358, students solve real-world problems using proportional relationships. The problem states, “Ms. Davis plans to drive from Town P to Town Q, a distance of 350 miles. She hopes to use only 12 gallons of gasoline. After traveling 150 miles, she checks her gauge and estimates that she has used 5 gallons of gasoline. At this rate, will Ms. Davis arrive at Town Q before stopping for gasoline? Justify your answer.” Teacher guidance states, “You may want to guide students on applying the various heuristics using the problem-solving heuristics poster. Refer students to the corresponding teacher resources for prompts and worked out solutions.” Teacher guidance gives a generic reference to have students use the heuristics poster which is repeated throughout the materials.
In Chapter 7, Put on Your Thinking Cap!, Problem 2, page 190, students solve a real-world problem using their understanding of circumference of circles and quadrants to find the distance of a shaded part. The problem states, “A cushion cover design is created from a circle of radius 7 inches and 4 quadrants. Find the total area of the shared parts of the design. Use $$\frac{22}{7}$$ as an approximation for $$\pi$$.” Teacher guidance states, “Go through the problem using the four-step problem-solving model. Students may need some help getting started after they have understood the problem. Suggest to students that they start by studying the figure and determining what shapes are involved. They can then make a plan as to which formula to use and in what order.” Teacher guidance gives a generic reference to have students use the four-step problem-solving method which is repeated throughout the materials.
In Chapter 9, Probability of Compound Events, Put on Your Thinking Cap! Problem 1, page 364, students solve real-world problems using their understanding of dependent events to find the probability of an event, without replacement. The problem states, “If there are 12 green and 6 red apples, find the probability of randomly choosing three apples of the same color in a row, without replacement. Show your work.” Teacher guidance states, “You may want to guide students on applying the various heuristics using the problem-solving heuristics poster. Refer students to the corresponding teacher resources for prompts and worked out solutions.” Teacher guidance gives a generic reference to have students use the heuristics poster which is repeated throughout the materials.
Materials identify focus Mathematical Habits for MP1 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are not intentionally addressed in the activities and problems. Examples include:
Section 2.7, Real-World Problems: Algebraic Reasoning, is noted as addressing MP1 on pages 187-196 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to make sense of problems and persevere in solving them are provided.
Section 6.4, Real-World Problems: Percent Increase and Decrease, is noted as addressing MP1 on pages 345-356 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to make sense of problems and persevere in solving them are provided.
Students have limited opportunities to reason abstractly and quantitatively in connection to grade-level content, identified as mathematical habits in the materials. Student guidance is not provided in the materials, teacher guidance is repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP2. Examples include:
In Section 2.2, Subtracting Algebraic Terms, Independent Practice, Problem 21, page 144, students write algebraic expressions for finding the area of rectangles, and then simplify the expression with rational coefficients by subtracting like terms. The problem states, “Luke simplified the algebraic expression $$\frac{3}{2}x - \frac{1}{3}x$$ as shown below: $$\frac{3}{2}x$$ - $$\frac{1}{3}x$$ = $$\frac{18}{12}x$$ - $$\frac{4}{12}x$$ = $$\frac{14}{12}x$$ Is Luke’s simplification correct? Why or why not?” Teacher guidance states, “Assesses students’ ability to simplify an algebraic expression with unlike fractional coefficients. They are required to recognize that the given solution is correct but is not in simplest form.” The materials misidentify MP2, as students do not consider units involved in a problem, attend to the meaning of quantities, nor understand the relationships between problem scenarios and mathematical representations.
In Section 6.1, Constructing Triangles, Independent Practice, Problem 10, page 90, students recognize that many triangles can be created given the sum of measures of three angles. The problem states, “Suppose you are given three angle measures with a sum of 180. Can you construct a triangle given this information? Can you construct other different triangles? Explain.” Teacher guidance states, “Assesses students’ ability to recognize that many triangles can be created given the sum of measures of three angles. Changing the side lengths of this kind of triangle will make many similar triangles.” Teacher guidance gives a generic reference to assess students’ abilities which is repeated throughout the materials.
In Section 8.6, Developing Probability Models, Activity, Problem 6, page 283, students compare the theoretical and experimental probability of randomly selecting a number from 0 to 9. The problem states, “Compare each of the experimental probability models you made with the theoretical probability model at the beginning of this activity. What effect does increasing the number of selected digits have on the experimental probabilities? Which experimental probability model resembles the theoretical probability model more closely? Explain.” Teacher guidance states, “In 6, students compare the two experiments with the theoretical model and graph. Pose the following question to students and prompt them for their reasoning. Does the second experiment more closely resemble the theoretical probability? Why? You may want to conclude the activity by having students share their responses with the class.” The materials misidentify MP2 in this problem, students do not consider units involved in a problem, attend to the meaning of quantities, nor understand the relationships between problem scenarios and mathematical representations.
In Section 9.1, Compound Events, Independent Practice, Problem 13c, page 326, students recognize outcomes and realize that two simple events can be switched and the number of outcomes still remains the same. The problem states, “For a game, Jesse first rolls a fair four-sided number die labeled 1 to 4. The result recorded is the number facing down. Then, he randomly draws a ball from a box containing two different colored balls. If Jesse first draws a colored ball and then rolls the four-sided number die, will the number of possible outcomes be the same? Explain your reasoning.” Teacher guidance states, “Assess students’ ability to draw a tree diagram, recognize the outcomes, and realize that the two simple events can be switched and the number of outcomes still remains the same.” Teacher guidance gives a generic reference to assess students’ abilities which is repeated throughout the materials.
Materials identify focus Mathematical Habits for MP2 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are not intentionally addressed in the activities and problems. Examples include:
Section 1.8, Operations with Decimals, is noted as addressing MP2 on pages 97-110 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to reason abstractly and quantitatively are provided.
Chapter 6, Geometric Construction, Performance Task, is noted as addressing MP2 on pages 119-120 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to reason abstractly and quantitatively are provided.
Indicator 2f
Materials support 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 reviewed for Math in Focus: Singapore Math Course 2 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.
Students have the opportunity to construct viable arguments in connection to grade-level content leading to the intentional development of MP3, identified as mathematical habits in the materials. However, the teacher guidance is often repetitive, not specific, and often distracts from the intentional development of MP3. Examples include:
In Section 1.1, Representing Rational Numbers on the Number Line, Activity, Problem 3, page 15, students construct viable arguments when locating rational numbers on a number line. “What is another way to locate the rational numbers on the number line? Explain your answer.” Teacher guidance states, “Display the task and give students time to work on it in pairs. Invite students to share their completed number line. Prompt them to share explanations and emphasize: how to convert fractions to decimals and vice versa. How to determine additional segments needed to accurately place the rational numbers. The correct placement of positive and negative numbers. Reasonable approximations, such as 3.6 is a little more than 3.5, so 3.6 is a little more than halfway between 3 and 4. Identify positions of opposites, for instance, -3.6 needs to be the same distance from 0 as 3.6 is from 0.”
In Section 4.3, Real-World Problems: Direct Proportion, Independent Practice, Problem 20, page 316, students construct viable arguments. The problem states, “Laila wants to buy some blackberries. Three stores sell blackberries at different prices. Which store has the best deal? Explain.” Three bowls of blackberries with different prices per pound are shown. Teacher guidance states, “assesses students’ ability to compare unit rates to determine the best deal, They may need to be reminded that there are 16 ounces in a pound.” While this teacher guidance does not intentionally develop MP3, students do have the opportunity to construct a viable argument.
Students have the opportunity to critique the reasoning of others in connection to grade-level content leading to the intentional development of MP3, identified as mathematical habits in the materials. However, the teacher guidance is often repetitive, not specific, and distracts from the intentional development of MP3. Examples include:
In Section 3.1, Identifying Equivalent Equations, Independent Practice, Problem 12, page 220, students critique the reasoning of others when finding equivalent equations. The problem states, “Chris was asked to write an equation equivalent to $$\frac{2}{3}$$x = 3 - x. He wrote the following: $$\frac{2}{3}$$x = 3 - x, $$\frac{2}{3}$$x ⋅ 3= 3 ⋅ 3 - x, 2x = 9 - 3x. Chris concluded that $$\frac{2}{3}$$x = 3 - x and 2x = 9 - x are equivalent equations. Do you agree with his conclusion? Give a reason for your answer.” Teacher guidance states, “assesses students’ ability to identify a mistake in the expansion on the right side of the equation. This is a good opportunity to point out the importance of using parentheses when distributing.” While this teacher guidance does not intentionally develop MP3, students do have the opportunity to critique the reasoning of others.
In Section 4.5, Percent Increase and Decrease, Math Sharing, page 342, students critique the arguments of others when solving direct proportion problems involving percent. The materials state, “Caleb has 40 magnets. Zara has 50 magnets. Zara says that she has 25% more magnets than Caleb, hence, Caleb has 25% fewer magnets than her. Do you agree with Zara? Discuss.” Teacher guidance states, “Pose the problem to students. Have students express the number of Zara’s magnets as a percent of the number of Caleb’s magnets, $$\frac{50}{40}$$ ⋅ 100% = 125%. Lead them to see that Zara has 25% more magnets than Caleb. Prompt them to explain why they have to use 40 as a base when calculating the percent. Then, have students express the number of Caleb’s magnets as a percent of the number of Zara’s magnets, $$\frac{40}{50}$$ ⋅ 100% = 80%. Lead them to see that Caleb has 20% fewer magnets than Zara. Prompt them to explain why they have to use 50 as a base when calculating the percent in this instance. Reiterate the importance of using the correct base when calculating a percent of one quantity over another.”
Math Journal Activities provide opportunities for students to engage in the intentional development of MP3. However, the teacher guidance is often repetitive, not specific, and distracts from the intentional development of MP3. Examples include:
In Chapter 2, Algebraic Expressions, Math Journal, page 197, students construct viable arguments and critique the reasoning of others when simplifying expressions. The materials state, “Brielle expanded and simplified the expression 6(x+3) - 2(x+1) + 5 as follows: 6(x + 3) − 2(x + 1) + 5 = 6x + 3 − 2x + 1 + 5 = 6x − 2x + 3 + 1 + 5 = 4x + 9. Explain to Briella her mistakes and show the correct solution.” Teacher guidance states, “Review with students the various strategies learned in this chapter. Encourage students to work independently. Error analysis is a useful activity because it leads to fewer mistakes in the future.” While this teacher guidance does not intentionally develop MP3, students do have the opportunity to construct a viable argument and critique the reasoning of others.
In Chapter 4, Proportion and Percent of Change, Math Journal, page 357, students construct viable arguments and critique the reasoning of others. The materials state, “Ameila has a box of 150 beads that are either red, purple, or yellow. 20% of the beads are read and $$\frac{2}{5}$$ of the remaining beads are yellow. Ameila worked out the number of yellow beads as follows: 100% - 20% = 80%, $$\frac{2}{5}$$ ⋅ 150 = 60, There were 60 yellow beads. Explain to Ameila her mistake and show her the correct solution.” Teacher guidance states, “Review with students the various strategies learned in this chapter. Encourage students to write and explain their steps clearly to avoid such mistakes.” While this teacher guidance does not intentionally develop MP3, students do have the opportunity to construct a viable argument and critique the reasoning of others.
Materials identify focus Mathematical Habits in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and in the Section Objectives. However, these mathematical habits are not intentionally addressed in the activities and problems. Examples include:
Section 5.4, Interior and Exterior Angles, is noted as addressing MP3 on pages 47-58 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no opportunities for students to construct arguments or critique the reasoning of others is provided in the lesson.
Section 7.1, Radius, Diameter, and Circumference of a Circle, is noted as addressing MP3 on pages 127-142 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no opportunities for students to construct arguments or critique the reasoning of others is provided in the lesson.
Indicator 2g
Materials support 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.
The materials reviewed for Math in Focus: Singapore Math Course 2 do not meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.
Students have limited opportunities to model with mathematics in connection to grade-level content, identified as mathematical habits in the materials. Additionally, MP4 is referred to as “use mathematical models” in the student and teacher materials. Students are told which models to use and teacher guidance is often repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP4. Examples include:
In Chapter 2, Algebraic Expressions, Put On Your Thinking Cap, Problem 2, page 199, students convert degrees Fahrenheit to degrees Celsius. The problem states, “Steven and his father are from Singapore, where the temperature is measured in degrees Celsius. While visiting downtown Los Angeles, Steven saw a temperature sign that read 72°F. He asked his father what the equivalent temperature was in °C. His father could not recall the Fahrenheit-to-Celsius conversion formula. C = $$\frac{5}{9}$$(F - 32) However, he remembered that water freezes at 0$$\degree$$C or 32$$\degree$$F and boils at 100$$\degree$$C or 212$$\degree$$F. Using these two pieces of information, would you be able to help Steven figure out the above conversion formula? Explain.” Teacher guidance states, “Requires students to solve a problem using algebraic reasoning. Go through the problem using the four-step problem-solving model. Students may need some help getting started after they have understood the problem. If students seem stuck, tell them that a diagram with the given information might help. Encourage them to think of the two scales a double number line, in which 0$$\degree$$C equals 32$$\degree$$F and 100$$\degree$$C to 212$$\degree$$F. So a change in 180 degrees in Fahrenheit correspond to 100 degrees in Celsius. 1 unit in Celsius = $$\frac{180}{100}$$ or $$\frac{9}{5}$$ units in Fahrenheit, and 1 unit in Fahrenheit equals $$\frac{100}{180}$$ or $$\frac{5}{9}$$ units in Celsius. Remind students that scales start at different places. 0$$\degree$$C equals 32$$\degree$$F. If they continue to have trouble, help them think through the expression F = $$\frac{9}{5}$$C + 32 or C = $$\frac{5}{9}$$(F - 32).” Students are told what type of model to use (double number line), and teacher guidance gives a generic reference to have students use the four-step problem-solving method which is repeated throughout the materials.
In Chapter 3, Algebraic Equations and Inequalities, Put On Your Thinking Cap!, Problem 2, page 269, states, “Sara can buy 40 pens with a sum of money. She can buy 5 more pens if each pen costs $0.05 less. a. How much does each pen cost? b. If Sara wants to buy at least 10 more pens with the same amount of money how much can each pen cost at most?” Teacher guidance states “requires students to solve a real-world problem using algebraic reasoning. The challenge is in writing an equation and inequality to represent the situation. Go through the problem using the four-step problem-solving model. Have students work in pairs or small groups. Encourage them to discuss and share strategies.” Teacher guidance gives a generic reference to have students use the four-step problem-solving method which is repeated throughout the materials.
In Chapter 9, Probability of Compound Events, Put On Your Thinking Cap!, Problem 3, page 365, students find the probability of dependent events. The problem states, “Diego plans to visit Australia for a vacation, either alone or with a friend. Whether he goes alone or with a friend is equally like. If he travels with a friend, there is a 40% chance of him joining a guided tour. If he travels alone, there is an 80% chance of him joining a guided tour. a) What is the probability of Diego traveling with a companion and not joining a guided tour? b) What is the probability of Diego joining a guided tour?” Teacher guidance states, “requires students to recognize that the first event is whether Diego goes alone or with someone, and the second event is dependent on the occurrence of the first event. Students need to use the concept of complementary events to find the probability. Go through the problem using the four-step problem-solving model. Students may need some help getting started after they have understood the problem. Suggest to students that they start by drawing a tree diagram.” Teacher guidance gives a generic reference to have students use the four-step problem-solving method which is repeated throughout the materials. Additionally, students are encouraged to use a tree diagram.
Materials identify focus Mathematical Habits for MP4 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are never intentionally addressed in the activities and problems. Examples include:
Section 1.3, Adding Integers, is noted as addressing MP4 on pages 35-40 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to model mathematics are provided.
Section 7.3, Real-World Problems: Circles, is noted as addressing MP4 on pages 151-158 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to model mathematics are provided.
Students have limited opportunities to use appropriate tools strategically in connection to grade-level content, identified as mathematical habits in the materials. Teacher guidance is repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP5. Examples include:
In Chapter 6, Geometric Construction, Put On Your Thinking Cap!, Problem 1, page 112, students find the area of an enlarged triangle. The problem states, “Construct triangle ABC, where AB = AC, BC = 8cm and m∠ABC= 37$$\degree$$. Triangle ABC is enlarged to produce triangle DEF by a scale factor of 2.5. Find the area of triangle DEF.” Teacher guidance states, “You may want to guide students on applying the various heuristics using the problem-solving heuristics poster. Refer students to the corresponding teacher resources for prompts and worked out solutions. Requires students to find the area of an enlarged triangle. What are we required to do? What strategies can we use to find the area of triangle DEF? Alert students that the height is measured 4 centimeters. Encourage students to work with a partner to determine the area of the constructed triangle and then multiply it by the scale factor squared.” Teacher guidance gives a generic reference to have students use the heuristics poster which is repeated throughout the materials.
In Section 9.3, Probability of Compound Events, Independent Practice Problem 11, page 351, students solve probability problems. The problem states, “Hunter tosses a fair six-sided die twice. What is the probability of tossing an even number on the first toss and a prime number on the second toss?” Teacher guidance states, “Assesses students’ ability to find the probability of a compound event that consists of tossing an even number followed by a prime number when a six-sided number die is tossed twice.” Teacher guidance gives a generic reference to assess students’ abilities which is repeated throughout the materials.
Students do not have the opportunity to use appropriate tools strategically. Materials identify focus Mathematical Habits for MP5 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are never intentionally addressed in the activities and problems. Examples include:
Chapter 6, Recall Prior Knowledge, is noted as addressing MP5 on pages 76-77 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to use tools strategically are provided.
Section 7.1, Radius, Diameter, and Circumference of a Circle, is noted as addressing MP5 on pages 131-133 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to use tools strategically are provided.
Indicator 2h
Materials attend to 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.
The materials reviewed for Math in Focus: Singapore Math Course 2 partially 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.
Students have some opportunities to attend to precision and the specialized language of mathematics in connection to grade-level content, identified as mathematical habits in the materials. However, little to no student guidance is provided. Examples include:
In Section 6.2, Scale Drawings and Lengths, For Language Development, TE page 92, teacher guidance includes, “Make sure that students understand the meaning of ‘scale’. You may want to share some examples of scale drawings. Show maps and architectural drawings, pointing out the scales, and explaining that the region or building is ‘scaled down’ in the scale drawing.”
In Section 9.2, Probability of Compound Events, For Language Development, TE page 328, teacher guidance includes,“Be sure students understand the words and the concept of ‘favorable outcomes’. Favorable does not mean ‘good’ or ‘bad’. It refers to the elements of which we are trying to find the probability.”
In Chapter 9, Probability of Compound Events, Math Journal, Problem 1, page 363, students have the opportunity to use the specialized language of mathematics to explain outcomes. The problem states, “Use an example to explain the difference between possible outcomes and different outcomes.” Teacher guidance states, “Requires students to explain the difference between possible outcomes and different outcomes. Encourage them to give examples such as tossing a coin twice, which has four outcomes. Are all the outcomes different? If order does not matter, then the outcome (H, T) is identical to the outcome (T, H). If we draw one piece of fruit from a bag of apples and oranges, we have two mutually exclusive outcomes (apple and orange) and the two outcomes are different. Review with students the various strategies to explain the difference between the probability terms. Encourage students to work independently. You may want to pose the following question to students who are struggling with using precise mathematical language. What strategy would you use to explain the difference between probability terms?”
Students have some opportunities to attend to precision and the specialized language of mathematics in connection to grade-level content, identified as mathematical habits in the materials. However, there is no student guidance and teacher guidance is repetitive and not specific, preventing intentional development of the full intent of MP6. Examples include:
In Section 1.4, Subtracting Integers, Independent Practice, Problem 21, page 60, Problem 21, students have the opportunity to attend to the specialized language of mathematics as they subtract integers. The problem states, “Ms. Davis has only $420 in her bank account. Describe how to find the amount in her account after she writes a check for $590.” Teacher guidance states, “Assess students’ ability to explain the steps involved in subtracting a negative integer from a positive one.” Neither teacher guidance nor student directions prompt students to use specific mathematical terms to explain their thinking or communicate their ideas.
In Section 3.2, Solving Algebraic Equations, Independent Practice, Problem 13, page 229, students have the opportunity to attend to precision looking for an error another student made, “Tara was asked to solve the equation -4p + 5 = 7 Her solution is shown. (An index card is shown with Tara’s steps.) Tara concluded that p = $$\frac{1}{2}$$ is the solution of the equation -4p + 5 = 7. Describe and correct the error that Tara made.” Teacher guidance states, “assesses students’ ability to identify a mistake that involves dividing by a negative coefficient.” Neither teacher guidance nor student directions prompt students to attend to precision. Furthermore, this problem lends itself to MP3.
In Section 4.2, Representing Direct Proportion Graphically, Independent Practice, Problem 6, students have the opportunity to use the specialized language of mathematics to explain a direct proportion. The problem states, “Explain how you can determine whether a line represents a direct proportion.” Teacher guidance states, “Assesses students ability to explain what determines a direct proportion graph.” Neither teacher guidance nor student directions prompt students to use specific mathematical terms to explain their thinking or communicate their ideas.
There are some instances when the materials attend to the specialized language of mathematics; however, these lessons were not identified as aligned to MP6. For example:
In Section 2.1, Adding Algebraic Terms, For Language Development, TE page 130, states, “Be sure students understand the meaning of like terms. Explain that like terms are terms that have the same variable part. Constant terms are also like terms. Give examples of like terms such as 2, $$\frac{1}{4}$$, 0.3, and -5; a, 3a, $$\frac{1}{2}$$ a, 2.4a, and -7a. List a variety of ten terms (both like terms and unlike terms) on the board and invite volunteers to identify like terms.”
Materials identify focus Mathematical Habits in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and in the Section Objectives. However, this mathematical habit is not intentionally addressed in the activities and problems. Examples include:
Section 2.6, Writing Algebraic Expressions, is noted as addressing MP6 on pages 171-186 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no opportunities for students to attend to precision and attend to the specialized language of mathematics are identified in the lesson.
Section 5.1, Complementary, Supplementary, and Adjacent Angles, is noted as addressing MP6 on pages 5-18 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no opportunities for students to attend to precision and attend to the specialized language of mathematics are identified in the lesson.
Indicator 2i
Materials support 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 the grade-level content standards, as expected by the mathematical practice standards.
The materials reviewed for Math in Focus: Singapore Math Course 2 do not 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.
Students have minimal opportunities to look for and make use of structure in connection to grade-level content, identified as mathematical habits in the materials. There is no student guidance, teacher guidance is often repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP7. Examples include:
In Section 1.4, Subtracting Integers, Learn, Problem 5, page 49, students use counters to subtract integers. The problem states, “Based on your results in 1 to 4, explain how you can subtract integers.” In Question 1, students used counters to evaluate 5 - (+2) compared with 5 + (-2). In Question 4, students used counters to evaluate 7 - (-3) and 7 + 3. Teacher guidance states, “Conclude the activity by asking students to generalize what they have observed about subtraction. You may want to consider each case: a positive minus a positive, a negative minus a positive, a positive minus a negative, a negative minus a negative. Guide students to conclude the task in ENGAGE.” Materials scaffold MP7 in this problem which prevents the opportunity to identify structure. Students are provided the model of using counters for each problem and guided through its development. Therefore students do not independently look for and make use of structure to generalize their understanding of subtraction.
In Chapter 1, Rational Numbers, Put On Your Thinking Cap!, Problem 1, page 112, students write an equivalent expression using the distributive property. The problem states, “The 4 key on your calculator is not working. Show how you can use the calculator to find 321 × 64.” Teacher guidance states, “Requires students to solve a problem that involves writing an equivalent expression, and evaluating it using the distributive property. Step 1. Understand the problem: What information can we gather from the problem? What are we asked to find? Step 2: Think of a plan: What can we do to help us solve the problem? What number do we have to change in the expression, and how? What is equivalent to 64? Step 3. Carry out the plan: So, what is an equivalent expression of 321 ⋅ 64? Are all these expressions equivalent? Why? Which expression will be easier to evaluate? Why? Invite volunteers to share their equivalent expressions, and ask students to evaluate each of them. Ensure that students are able to apply the distributive property correctly. For example, 321 ⋅ (65 − 1) = 321 ∙ 65 − 321 ∙ 1. If necessary, suggest that students write the subtraction within the parentheses as adding the opposite to help them keep track of the signs. Prompt students to see that the equivalent expressions all have the same result of 20,544. Step 4. Check the answer.” Teacher guidance gives a generic reference to have students use the four-step problem-solving method which is repeated throughout the materials and does not require students to look for or use structure in solving.
Materials identify focus Mathematical Habits for MP7 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are not intentionally addressed in the activities and problems. Examples include:
Chapter 5, Wrap-Up, Chapter Review, Performance Task, is noted as addressing MP7 on pages 270-276 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to make use of structure are provided in the lesson.
Section 8.5, Approximating Probability and Relative Frequency, is noted as addressing MP7 on pages 261-276 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to make use of structure are provided in the lesson.
Students have minimal opportunities to look for and express regularity in repeated reasoning in connection to grade-level content, identified as mathematical habits in the materials. There is no student guidance, teacher guidance is often repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP8. Examples include:
In Section 3.4, Solving Algebraic Inequalities, Learn, Problem 1, page 248, Problem 1, students analyze a table of values involving dividing inequalities with positive and negative integers. The problem states, “Fill in the table. Use the symbols > or <. a) What happens to the direction of the inequality symbol when you divide by a positive number? Based on your observation, write a rule for dividing both sides of an inequality by a positive number. b) What happens to the direction of the inequality symbol when you divide by a negative number? Based on your observation, write a rule for dividing both sides of an inequality by a negative number. Teacher guidance states, “Summarize that the inequality symbol remains the same when we multiply or divide by a positive number, and the direction of the symbol is reversed when we multiply or divide by a negative number. Help students to visualize why this occurs. Ask them to consider x > y. In other words, when we multiply by a negative number, we flip the number line, moving to the left.” Students are provided problems with answers (in the table) and are only responsible for comparing the answers. They do not independently look for and make use of structure in generalizing an understanding of inequalities.
In Section 6.3, Scale Drawings and Areas, Learn, Problem 5, page 104, students explore the relationship between scale factor and corresponding area. The problem states, “Compare the side lengths and the areas for the various scale factors. What pattern do you observe? What relationship between scale factor and area can you deduce?” Teacher guidance states, “In 5, encourage students to look for patterns, in particular the relationship between the scale factor and the area. What is the relationship? Emphasize that this property applies to the areas of other two dimensional figures as well. You may want to conclude the activity by discussing the activity in terms of inductive reasoning, as described in the Best Practice below.” Materials scaffold MP8 in this problem which prevents the opportunity to look for and express regularity in repeated reasoning. Students are provided models to find areas of scaled images and a partially completed table designed to organize results. Students look for and express regularity in repeated reasoning to understand the relationship between scale factors and corresponding areas with teacher assistance, including a heavily scaffolded problem.
Materials identify focus Mathematical Habits for MP8 in the Chapter Planning Guide and in the Section objectives. However, these Mathematical Habits are not intentionally addressed in the activities and problems. For example:
Section 7.4, Area of Composite Figures, is noted as addressing MP8 on page 164 in the Standards for Mathematical Practice Chart, Chapter Planning Guide, and Section Objectives. However, no identified opportunities for students to look for and express regularity in repeated reasoning are provided in the lesson.