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.

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.

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.