Math in Focus: Singapore Math
Downloadable Resources
Publisher Response
2020

Math in Focus: Singapore Math

Publisher
Houghton Mifflin Harcourt
Subject
Math
Grades
K-8
Report Release
10/25/2021
Review Tool Version
v1.5
Format
Core: Comprehensive

EdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.
Alignment (Gateway 1 & 2)
Does Not Meet Expectations

Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.
Usability (Gateway 3)
NE = Not Eligible. Product did not meet the threshold for review.
Not Eligible
Downloadable Resources
Publisher Response
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Additional Publication Details

Title ISBN
International Standard Book Number
Edition Publisher Year
Teacher Assessment Guide Grade 8 9780358105015
Poster Collection Grade 8 9780358105114
Student Edition Set Grade 8 9780358116844
Extra Practice and Homework Set Grade 8 9780358116943
CCSS Teacher Edition Set Grade 8 9780358117049
Transition Guide Courses 1-3 9780358216643
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Showing:
8th Grade

About This Report

Report for 8th Grade

Alignment Summary

The materials reviewed for Math in Focus: Singapore Math Course 3 do not meet expectations for Alignment to the CCSSM. In Gateway 1, the materials partially meet expectations for focus and coherence, and in Gateway 2, the materials do not meet expectations for rigor and practice-content connections.

8th Grade
Gateway 1

Focus & Coherence
11/14
0
7
12
14
Gateway 2

Rigor & Mathematical Practices
10/18
0
10
16
18
Alignment (Gateway 1 & 2)
Does Not Meet Expectations
Gateway 3

Usability
Usability (Gateway 3)
Not Rated
Gateway 1
Gateway 2
Gateway 3
Viewing:
Gateway 1
Overview of Gateway 1

Focus & Coherence

The materials reviewed for Math in Focus: Singapore Math Course 3 partially meet expectations for focus and meet expectations for coherence. For focus, the materials assess grade-level content and partially meet expectionations in providing all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials are coherent and consistent with the CCSSM.

Gateway 1
v1.5
Gateway 1

Focus & Coherence
11/14
0
7
12
14
Criterion 1.1: Focus
04/06
Criterion 1.2: Coherence
07/08
Partially Meets Expectations

Criterion 1.1: Focus
04/06

Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Math in Focus: Singapore Math Course 3 partially meet expectations for focus as they assess grade-level content and partially provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Indicator 1A
02/02

Materials assess the grade-level content and, if applicable, content from earlier grades.

The materials reviewed for Math in Focus: Singapore Math Course 3 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Summative assessments provided by the materials include Chapter Tests, Cumulative Reviews, and Benchmark Assessments and are available in print and digitally. According to the Preface of the Math in Focus: Assessment Guide, "Assessments are flexible, teachers are free to decide how to use them with their students. ... Recommended scoring rubrics are also provided for some short answer and all constructed response items to aid teachers in their marking." The following evidence is based upon the provided assessments and acknowledges the flexibility teachers have in administering them in order to understand their students' learning.

The provided assessments, found in the Assessment Guide Teacher Edition, assess grade-level standards. Examples include:

  • In Chapter Test 1, Section C, Item 11 states, “This question has two parts. Part A: Locate 34\sqrt{34} on a number line. Explain how you found its position on the number line. Show your drawing and explanation in the space below. Part B: Give another irrational number in the form of n\sqrt{n} where n is an integer. The value of this irrational number must be between 34\sqrt{34} and 6.” (8.NS.2)

  • In Cumulative Review 1, Section B, Item 12 states, “Arrange the numbers below in ascending order. 4,2,33,92\sqrt{4}, \sqrt{2}, \sqrt[3]{3}, \frac{\sqrt{9}}{2}” (8.EE.2)

  • In Chapter Test 7, Functions, Section C, Item 10 states, “This question has two parts. Devin has $30 on his bus card. Every time he rides a bus, $1.20 is deducted from the value on his card. The amount of money he has on his card, y dollars, is a function of the number of times he rides a bus, x. Part A: Find an equation in slope-intercept form to represent the function. Find the number of the bus rides Devin has taken if he has $3.60 on his card. Part B: Rebecca has a bus card as well. The amount of money left on her card, y dollars, after taking x rides on the bus can be represented by the function y = -1.4x + 33. After how many rides on the bus would both Devin and Rebecca have the same amount of value on their bus card?” (8.F.4)

  • In Chapter Test 8, Section A, Item 5 states, “What is the vertical height in feet of the cone shown? A. 320\frac{3}{20} B. 15\frac{1}{5} C. 14\frac{1}{4} D. 25\frac{2}{5}” A picture of a cone with the diameter of the base measuring 12 ft and the angled side of the cone measuring 16 feet is shown. (8.G.7)

  • In the End-of-Year Benchmark Assessment, Section A, Item 19 states, “Which is the best description of the association for the scatter plot shown? A. Strong, nonlinear association B. Weak, negative nonlinear association C. Weak, positive linear association D. Strong, negative nonlinear association.” (8.SP.1)

Indicator 1B
02/04

Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Math in Focus: Singapore Math Course 3 partially meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Materials provide opportunities for students to engage in grade-level problems during Engage, Learn, Think, Try, Activity, and Independent Practice portions of the lesson. Engage activities present inquiry tasks that encourage mathematical connections. Learn activities are teacher-facilitated inquiry problems that explore new concepts. Think activities provide problems that stimulate critical thinking and creative solutions. Try activities are guided practice opportunities to reinforce new learning. Activity problems reinforce learning concepts while students work with a partner or small group. Independent Practice problems help students consolidate their learning and provide teachers information to form small group differentiation learning groups. However, students are not provided opportunities to engage in extensive work and full intent with all standards.

The materials provide one or more Focus Cycles of Engage, Learn, Try activities and opportunities for Independent Practice which provide students extensive work with grade-level problems to meet the full intent of grade-level standards. Examples include:

  • In Section 1.2, Introducing the Real Number System, students apply their knowledge of irrational numbers and locate them on a number line. In the Engage activity on page 15, students order numbers in different forms. The activity states, “Using a calculator, locate 179\frac{17}{9}, 2.56 and 7\sqrt{7} on a number line. Now, think of another three numbers of different forms and put them in order.” In the Learn activity on page 15, students order real numbers. The activity states, “1. The real number system is a combination of the set of rational numbers and the set of irrational numbers (a chart is given to show the relationship between real, rational and irrational numbers) 2. To compare different forms of real numbers, it is generally easier to convert any non-decimal to decimal form before comparing.” In Try, Problem 1, page 16, students practice ordering real numbers. The problem states, “Locate the set up real numbers on a number line, and order them from least to greatest using the symbol <. 10,−539,−3π2,6.286ˉ\sqrt{10}, -\frac{53}{9}, -\frac{3\pi}{2}, 6.2\bar{86}.” In Independent Practice, Problem 2, students compare pairs of numbers. The problem states, “Compare each pair of real numbers using the symbol < or <. -4.38 and -12\sqrt{12}.” Students engage with extensive work and full intent of 8.NS.2 (Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions).

  • In Section 2.6 Squares, Square Roots, Cubes, and Cube Roots, students evaluate square roots and cube roots of positive real numbers, solve equations involving squared and cubed variables, and solve real-world problems. In the Engage activity on page 91, students isolate variables on one side of the equation. The activity states, “When you solve an algebraic equation, what rules do you follow to keep the equation ‘balanced’? Use specific examples to explain your thinking. By making a systematic list, show how you can solve the equations x3x^3 = 9 and y3y^3 = 8.” In Learn, Problem 2, page 93, students solve real-world problems involving squares or cubes of unknowns. The problem states, “Theresa wants to put a piece of carpet on the floor of her living room. The floor is a square with an area of 182.25 square feet. How long should the piece of carpet be on each side?” In Try, Problem 2, page 94, students practice solving real-world problems involving squares or cubes of unknowns. The problem states, “Richard bought a crystal cube that has a volume of 1,331 cubic centimeters. Find the length of a side of the crystal cube.” In Independent Practice, Problem 12, students solve an equation involving a variable that is cubed. The problem states, “Solve each equation. Round each answer to the nearest tenth where applicable. x3x^3 = 216729\frac{216}{729}.” Students engage with extensive work and full intent of 8.EE.2 (Use square root and cube root symbols to represent solutions to equations of the form x2x^2 = p and x3x^3 = p, where p is a positive rational number).

  • In Section 7.2, Representing Functions, students use functions to model relationships between quantities and represent functions in different forms. In the Engage activity on page 23, students determine if a relation is a function from a real-world situation. The activity states, “Chris rented a car at $90 per day during his trip to New Orleans. Construct a table to show the rental costs for 1, 2, 3, and 4 days of rental. Is the relation between the number of days of rental and the rental cost a function? Discuss what are the various ways you can represent this relation. Explain your thinking.” In Learn, Problem 2, page 25, students represent functions in different forms. The problem states, “A fire sprinkler sprays water at the rate of 3 gallons per minute. The amount of water sprayed, y gallons, is a function of the number of minutes, x, that the sprinkler sprays water.” In Try, Problem 3, page 28, students practice representing functions in different forms. The problem states, “Amelia goes to the supermarket to buy walnuts for her mother. The walnuts are sold at 15 dollars per pound. The amount of money, y dollars, she pays is a function of the amount of walnuts, x pounds, that she buys.” In Independent Practice, Problem 9b, students practice representing a function verbally, in a table, a graph, and an equation. The problem states, “Sofia has $60 on her bus pass initially. Every time she rides a bus, $1.50 is deducted from the value of her pass. The amount of money, y dollars, she has on her pass is a function of the number of times, x, that she rides a bus. Write an equation to represent the function.” Students engage with extensive work and full intent of 8.F.1 (Understand that a function is a rule that assigns to each input exactly one output).

  • In Section 9.5 Comparing Transformations, students compare translations, reflections, rotations, and dilations. In the Engage activity on page 193, students compare properties of translations, reflections, rotations, and dilation. The activity states, “You have learned four types of transformations: translations, reflections, rotations, and dilations. How do you describe each transformation? What similar and different properties do they have? What properties of lines and figures do they preserve? Complete the table below for reflection, rotation, and dilation.” In Learn, Problem 6, page 194, students compare translations, reflections, rotations, and dilations. The problem states, “Dilate ∠AOB with the origin as the center of dilation and choose a positive or negative scale factor k. Measure ∠AOB after dilation and record your result in the table.” In Try, Problem 1a, page 196, students practice comparing, translations, reflections, rotations, and dilations. The problem states, “ △\triangleABC is mapped onto △\trianglePQR, △\triangleLMN, and △\triangleXYZ by three different transformations. Describe the transformation that maps △\triangleABC onto △\trianglePQR." In Independent Practice, Problem 2a, students practice drawing an image of a figure after a rotation. The problem states, “A curtain is made from a fabric with a pattern of many pentagons. One of the pentagons ABCDE is shown on the coordinate plane. Four transformations described in a to d map ABCDE onto the other pentagons. Draw the image for each transformation. ABCDE is mapped onto FGHIJ by a translation 8 units down.” Students engage with extensive work and full intent of 8.G.1 (Verify experimentally the properties of rotations, reflections, and translations).

  • In Section 12.1, Scatter Plots, students construct, identify, and describe scatter plots. In the Engage activity on page 347, students use data from experiments. The activity states, “The table below shows data from an experiment where x was modified and the corresponding values of y recorded. Plot the points on graph paper. Use 1 centimeter to represent 1 unit on both axes. What do you notice about the position of the plots? Discuss the similarities and differences compared to plotting a linear graph.” A table of x and y values is provided. In the Learn activity, Problem 2, page 351, students observe patterns of explanations of forms and directions of associations. The problem states, “Clustering, The points form clusters, which are distinct regions where the points are close to one another. In this scatter plot, you see two clusters of points.” A pictorial example is provided for students. In Try, Problem 3, page 354, students practice identifying patterns of association between two variables. The problem states, “Describe the association between variables x and y in each scatter plot. ______ association.” In Independent Practice, Problem 10d, students draw a scatter plot, identify and validate the presences of outliers, describe association, and think critically about how the range of bivariate data affects the association. The problem states, “A retailer wanted to know the association between the number of items sold, y, and the number of salespeople, x, in a store. She recorded the data over 16 days in the table below (table provided). If the data for the number of salespeople ranged from 1 to 100, do you think the answer would be different? Explain.” Students engage with extensive work and full intent of 8.SP.1 (Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities). 

Materials do not provide students the opportunity to engage with the full intent of some major work grade-level standards. Examples include:

  • Students do not engage with the full intent of 8.G.5 (Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles). This standard is introduced in Course 2, Section 5.3 and is not addressed in Course 3.

  • Students do not engage with the full intent of 8.EE.6 (Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane, derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b). Part of this standard is not addressed in Course 3. Students derive y = mx and y = mx + b, but students do not use similar triangles to explain why the slope is the same between two distinct points on a non-vertical line in the coordinate plane.

Materials do not provide extensive work with all standards. For example:

  • The materials do not provide extensive work with 8.G.9 (Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems). Students do not spend extensive time working on this standard because they are distracted by finding the volume and surface area of pyramids. For example, in Section 11.3 Finding Volumes and Surface Areas of Pyramids and Cones, Engage activity, page 301, students use nets to find surface area. The activity states, “Draw a net of a rectangular prism and the net of a square pyramid. Cut out the shapes and fold them. How can you find the surface area of the rectangular prism? Explain how you can use this information to find the surface area of a pyramid.” In Learn, Problem 3, page 302, students find the volume and surface area of a pyramid. The problem states, “Find the volume and total surface area of a solid pyramid with a 5-centimenter square base and a height of 6 centimeters.” In Try, Problem 2, page 304, students practice finding the volume and surface area of a pyramid. The problem states, “Find the total surface area of a solid pyramid with a 8-centiment square base and a height of 9 centimeters.” In Independent Practice, Problem 6, students find the height, surface area, and volume of a pyramid given and the dimensions of the pyramid. The problem states, “A candle has the shape of a square pyramid. Its base area is 100 cm2 and its height is 15 cm. a. Find the volume of the candle. b. Find the area of a triangular face. c. Find the surface area of the candle.” Finding the surface area and volume of square pyramids is a Grade 7 standard (7.G.6).

Criterion 1.1: Focus
04/06
Indicator 1A
02/02
Indicator 1B
02/04

Criterion 1.2: Coherence
07/08

Each grade’s materials are coherent and consistent with the Standards.

The materials reviewed for Math in Focus: Singapore Math Grade Course 3 meet expectations for coherence. The majority of the materials, when implemented as designed, address the major clusters of the grade, and the materials have supporting content that enhances focus and coherence simultaneously by engaging students in the major work of the grade. The materials also include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. The materials partially have content from future grades that is identified and related to grade-level work and relate grade-level concepts explicitly to prior knowledge from earlier grades.

Indicator 1C
02/02

When implemented as designed, the majority of the materials address the major clusters of each grade.

The materials reviewed for Math in Focus: Singapore Math Course 3 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

  • There are 12 chapters, of which 9 address major work of the grade, or supporting work connected to major work of the grade, approximately 75%.

  • There are 50 sections (lessons), of which 38 address major work of the grade, or supporting work connected to major work of the grade, approximately 76%.

  • There are 139 days of instruction, of which 100.5 days address major work of the grade, or supporting work connected to the major work of the grade, approximately 72%.

A day-level analysis is most representative of the instructional materials because the days include all instructional learning components. As a result, approximately 72% of the instructional materials focus on major work of the grade.

Indicator 1D
02/02

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The materials reviewed for Math in Focus: Singapore Math Course 3 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Digital materials provide “Math in Focus 2020 Comprehensive Alignment to CCSS: Course 3” located under Discover, Planning. This document identifies the standards taught in each chapter’s section allowing connections between supporting and major work to be seen. Examples include: 

  • In Section 2.6 Squares, Square Roots, Cubes, and Cube Roots, Independent Practice, Problem 10, page 95, students find the square root of an irrational number in an equation which connects the supporting work of 8.NS.2 (Use rational approximations of irrational numbers) with the major work 8.EE.2 (Use square root symbols to represent solutions to equations of the form x2x^2 = p and x3x^3 = p, where p is a positive rational number). Students solve, “Solve each equation. Round each answer to the nearest tenth when applicable. n2n^2 = 350.”

  • In Section 8.1, Understanding the Pythagorean Theorem and Plane Figures, Try, Problem 2, page 84, students make rational approximations of irrational numbers by rounding to the nearest tenth when using the Pythagorean Theorem to find unknown side lengths of right triangles which connects the supporting work of 8.NS.2 (Use rational approximations of irrational numbers to compare the size of irrational numbers) with the major work of 8.G.7 (Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions). Students solve, “Find the length of ACˉ\bar{AC}. Round your answer to the nearest tenth.” A picture shows the length of BAˉ\bar{BA} as 7 in. and the length of BCˉ\bar{BC} as 9.5 in.

  • In Section 11.3 Finding Volume and Surface Areas of Pyramids and Cones, Try, Problem 1, page 308, students find the volume and surface area of a cone which connects the supporting work of 8.G.9 (Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems) with the major work of 8.G.7 (Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions). Students solve, “Find the volume and total surface area of a solid cone of diameter 6 inches and height 7 inches. Use 3.14 as an approximation for π\pi.”

  • In Section 11.5, Real-World Problems: Composite Solids, Independent Practice, Problem 1a, page 327, students find the volume of composite figures which connects the supporting work 8.G.9 (Use the formulas for the volumes of cones, cylinders, and spheres in solving real-world problems) with the major work 8.G.7 (Apply the Pythagorean Theorem to determine an unknown side length in right triangles in real-world and mathematical problems in two and three dimensions). Students solve, “The figure shows a solid glass trophy which is made up of a cone and cylindrical base. The slant height of the cone is 13 centimeters. The height of the cylindrical base is 2 centimeters. The radius of the cone and the base is 5 centimeters. Find the volume of the trophy.”

  • In Section 12.2, Modeling Linear Associations, Try, Problem 1, page 371, students draw, find, and use the equation for a line of best fit which connects the supporting work of 8.SP.2 (Know that straight lines are widely used to model relationships between two quantitative variables) with the major work of 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations). Students solve, “The scatter plot shows the number of eggs hatched per 100 eggs, y, in an incubator with varying temperatures, x°F.” a. Given that the line of best fit passes through (80, 41) and (95, 68), find the equation of the line of best fit. b. Interpret the meaning of the value of the slope.”

Indicator 1E
02/02

Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

The materials reviewed for Math in Focus: Singapore Math Course 3 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Digital materials provide “Math in Focus 2020 Comprehensive Alignment to CCSS: Course 3” found under Discover, Planning. This document identifies the standards taught in each chapter’s section showing connections between supporting to supporting work and major to major work. 

There are connections from supporting work to supporting work throughout the grade-level materials, when appropriate. Examples include:

  • In Section 11.2, Finding Volumes and Surface Areas of Cylinders, Independent Practice, Problem 2 connects the supporting work of 8.G.C (Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres) to the supporting work of 8.NS.A (Know that there are numbers that are not rational, and approximate them by rational numbers). Students solve, “Use 3.14 as an approximation for π\pi, unless stated otherwise. Round your answers to the nearest tenth, where applicable. Find the volume and surface area of each solid cylinder.” A diagram of a cylinder is provided with a radius of 5.2 in. and a height of 10.4 in.

  • In Section 11.3, Finding Volume and Surface Area of Pyramids and Cones, Independent Practice, Problem 4 connects the supporting work of 8.G.C (Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres) to the supporting work of 8.NS.A (Know that there are numbers that are not rational, and approximate them by rational numbers). Students solve, “Use 3.14 as an approximation for π\pi, unless stated otherwise. Round your answers to the nearest tenth, where applicable. Find the volume and surface area of each cone.” A diagram of a cone is provided with a radius of 16 units and a height of 30 units.

  • In Section 11.4, Volume and Surface Areas of Spheres, Independent Practice, Problem 1 connects the supporting work of 8.G.C (Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres) to the supporting work of 8.NS.A (Know that there are numbers that are not rational, and approximate them by rational numbers). Students solve, “Use 3.14 as an approximation for π\pi, unless stated otherwise. Round your answers to the nearest tenth, where applicable. Find the volume and surface area of each sphere.” A diagram of a sphere is provided with a radius of 6 in.

There are connections from major work to major work throughout the grade-level materials, when appropriate. Examples include:

  • In Section 7.3, Understanding Linear and Nonlinear Functions, Try, Problem 1e, page 46, connects the major work of 8.F.A (Define, evaluate, and compare functions) to the major work of 8.F.B (Use functions to model relationships between quantities). Students solve, “A cruise ship traveling at a constant speed consumes 4,000 gallons of gasoline per hour. Before the ship begins its journey, the fuel tank is filled with 330,000 gallons of gasoline. The amount of gasoline, y gallons, that is left in the fuel tank is a function of the traveling time, x hours. Write an equation of the function. Sketch a graph of the function.”

  • In Section 8.3, Understanding the Pythagorean Theorem and Solids, Independent Practice, Problem 2 connects the major work of 8.G.B (Understand and apply the Pythagorean Theorem) to the major work of 8.EE.A (Work with radicals and integer exponents). Students solve, “Solve. Round non-exact answers to the nearest tenth, unless otherwise stated. Find the value of x in the rectangular prism.” A rectangular prism diagram shows a length of 7 cm, width of 8 cm, and rectangular diagonals of the base as 12.2 cm and x cm.

  • In Section 9.1 Translations, Independent Practice, Problem 4 connects the major work of 8.G.A (Understand congruence and similarity using physical models, transparencies, or geometry software) to the major work of 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations). Students solve, “On the coordinate plane, draw and label the image under each translation. ABˉ\bar{AB} is translated 5 units to the right and 1 unit down.”

Indicator 1F
01/02

Content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.

The materials reviewed for Math in Focus: Singapore Math Course 3 partially meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.

Materials relate grade-level concepts to prior knowledge from earlier grades. Recall Prior Knowledge highlights the concepts and skills students need before beginning a new chapter. What have students learned? states the learning objectives and prior knowledge relevant to each chapter. The Scope and Sequence shows the progressions of standards from Course 1 through Course 3. The online materials do not include the standard notation. Examples include: 

  • In Chapter 2, Chapter Overview, Math Background states, “In Chapter 1, students learned about the real number system. In previous courses, students learned how to add, subtract, multiply, and divide integers.”

  • In Chapter 4, Recall Prior Knowledge states, “In Course 1, students learned to express the relationship between two quantities with linear equations. They also learn to represent fractions as repeating decimals. In Course 2, students learned to identify equivalent equations, solve algebraic equations, solve algebraic inequalities, and graph the solution sets on number lines.”

  • In Chapter 6, Learning Continuum, What have students learned? states, “In Chapter 4, students have learned: Solving linear equations with one variable. (8.EE.7b), In Chapter 5 students have learned: Sketching graphs of linear equations. (8.EE.6)”

  • In Chapter 7, Recall Prior Knowledge states, “In Course 2, students learned how to write an algebraic expression to represent an unknown quantity. They also learned how to graph linear equations using a table of values in the previous chapters.”

  • In Chapter 9, Learning Continuum, What have students learned? states, “In Course 1 Chapter 9, students have learned: Naming and locating points on the coordinate plane. (6.NS.8), Finding lengths of horizontal and vertical line segments on the coordinate plane. (6.NS.8), In Course 2 Chapter 6, students have learned: Constructing a triangle with given measures. (7.G.2), Identifying the scale factor. (7.G.1), Calculating lengths and distances from scale drawings. (7.G.1), In Course 3 Chapter 5, students have learned: Writing an equation of a line in the form y = mx or y = mx + b. (8.EE.6)” 

Materials provide grade-level standards of upcoming learning to future grades with no explanation of the relationship to grade-level content. What will students learn next? states the learning objectives from the following chapters of future courses to show the connection between the current chapter and what students will learn next. The Scope and Sequence shows the progressions of standards from Course 1 through Course 3. Examples include:

  • In Chapter 3, Learning Continuum, What will students learn next? states, “In high school, students will make use of the knowledge in this chapter in the following domains: Number and Quantity, Modeling.”

  • In Chapter 5, Learning Continuum, What will students learn next? states, “In Chapter 6, students will learn: Solving systems of linear equations by graphing. (8.EE.8a), In Chapter 7, students will learn: Understanding relations and functions. (8.F.1)”

  • In Chapter 6, Learning Continuum, What will students learn next? states, “In high school, students will make use of the knowledge in this chapter in the following domains: Algebra, Functions, Modeling.”

  • In Chapter 8, Learning Continuum, What will students learn next? states, “In Chapter 11, students will learn: Find the volume and surface area of pyramids.”

  • In Chapter 11, Learning Continuum, What will students learn next? states, “In high school, students will make use of the knowledge in this chapter in the following domains: Modeling, Geometry.”

Indicator 1G
Read

In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification.

The materials reviewed for Math in Focus: Singapore Math Course 3 can be completed within a regular school year with little to no modification to foster coherence between grades.

Recommended pacing information is found in the Teacher’s Edition, Chapter Planning Guide and online under Discover, Planning, Common Core Pathway and Pacing Course 3. Each section consists of one or more Engage-Learn-Try focus cycles followed by Independent Practice. As designed, the instructional materials can be completed in 139 days.

  • There are 12 instructional chapters divided into sections of 103 instructional days.

  • There is one day for each chapter’s instructional beginning consisting of Chapter Opener and Recall Prior Knowledge, for a total of 12 additional days.

  • There is one day for each chapter’s closure consisting of Chapter Wrap-Up, Chapter Review, Performance Task, and Project work, for a total of 12 additional days. 

  • There is one day for each chapter’s Assessment, for a total of 12 additional days.

The online Common Core Pathway and Pacing Course 2 states the instructional materials can be completed in 140 days, one instructional day is added to Section 5.1 in the Teacher’s Edition and two extra days are added in the online pacing guide for Sections 8.3, and 10.3. For the purpose of this review the Chapter Planning Guide provided by the publisher in the Teacher’s Edition was used.

Each section consists of one or more Engage-Learn-Try focus cycles followed by Independent Practice. Instructional minutes are not provided.

Criterion 1.2: Coherence
07/08
Indicator 1C
02/02
Indicator 1D
02/02
Indicator 1E
02/02
Indicator 1F
01/02
Indicator 1G
Read
Overview of Gateway 2

Rigor & the Mathematical Practices

The materials reviewed for Math in Focus: Singapore Math Course 3 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).

Gateway 2
v1.5
Gateway 2

Rigor & Mathematical Practices
10/18
0
10
16
18
Criterion 2.1: Rigor and Balance
07/08
Criterion 2.2: Math Practices
03/10
Does Not Meet Expectations

Criterion 2.1: Rigor and Balance
07/08

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 3 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
01/02

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 3 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 1.1, Introducing Irrational Numbers, Learn, Problem 1, page 7, students locate irrational numbers on a number line using areas of squares. The problem states, “Finding the value of 2\sqrt{2} using a square. Square ABCD is made up of 4 smaller squares. The side length of each small square is 1 inch. Find the area of ABCD.” Students develop conceptual understanding of 8.NS.1 (Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion, for rational numbers show that the decimal expansion repeats eventually).

  • In Section 5.1, Finding and Interpreting Slopes of Lines, Activity, Problems 1-5, page 237, students use conceptual understanding to determine whether the slope of a line is positive, negative, zero, or undefined. The materials state, “1. Using a geometry software, graph the line that passes through each pair of points. Then, fill in the table. (The table provided requires students to identify the slope as positive, negative, zero, or undefined from the graph and calculate y2 - y1 and x2 - x1.) 2. When the signs of y2 - y1 and x2 - x1 are the same, what do you observe about the slope? 3. When the signs of y2 - y1 and x2 - x1 are different, what do you observe about the slope? 4. When the value of y2 - y1 = 0, what do you observe about the slope? 5. When the value of x2 - x1 = 0, what do you observe about the slope?” Students develop conceptual understanding of 8.EE.6 (Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane). 

  • In Section 7.1, Understanding Relations and Functions, Learn, Problem 2, page 4, students construct mapping diagrams to determine relations. The problem states, “You can use a mapping diagram to represent the relation between the inputs and the outputs. The mapping diagram below represents the relation which have the ordered pairs (0,0), (0,1), (0,2), and (1,2). An arrow is used to map each input to one or more outputs. There are four different types of relations.” Students develop conceptual understanding of 8.F.1 (Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output).

  • In Section 7.4, Comparing Two Functions, Engage, page 55, students use tables to make function comparisons. The materials state, “Max is looking for a new phone plan. Plan A charges $35 dollars per month for 100 minutes of local calls and $0.02 per minute after 100 minutes. Plan B provides unlimited text messages, and charges local calls based on the following table (shown). How can Max figure out which is the better plan? Explain.” Students develop conceptual understanding of 8.F.2 (Compare properties of two functions each represented in a different way).

  • In Section 10.2, Understanding and Applying Similar Triangles, Learn, Problems 1-4, page 243, students explore conditions for similarity. ABC and PQR are given. The materials state, “1. Measure the side lengths of ABCand PQR. Are the ratios of the corresponding side lengths equal? Are ABC and PQR similar? Explain. 2. Measure ∠A, ∠B, ∠P,and ∠Q. Which pairs of angles have equal measure? 3. Without measuring ∠Cand ∠R, how can you deduce that they have equal measures? 4. If you only know that two pairs of corresponding angles have equal measures, can you conclude that ABCand PQR are similar?” Students develop conceptual understanding of 8.G.5 (Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles).

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. For example:

  • In Section 7.1, Functions, Independent Practice, Problem 4, page 18, students interpret a mapping diagram and determine the type of relation obtained and determine if the relation is a function. The problem states, “Identify the type of relation represented by each mapping diagram. Determine whether each relation is a function. Explain.” Students are shown a mapping diagram with inputs, outputs, and relations. Students independently practice conceptual understanding of 8.F.1 (Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output).

Indicator 2B
02/02

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 3 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 2.1, Exponential Notation, Learn, Problem 3, page 47, students expand and evaluate expressions in exponential notation. The problem states, “Expand and Evaluate (23)5(\frac{2}{3})^5 = 23\frac{2}{3} ⋅ 23\frac{2}{3} ⋅ 23\frac{2}{3} ⋅ 23\frac{2}{3} ⋅ 23\frac{2}{3} = 32243\frac{32}{243}.” Students develop procedural skill and fluency of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions).

  • In Section 2.3, The Power of a Power, Learn, Problem 3, page 62, students use the power of powers property. The problem states, “Simplify [(27)6]4[(\frac{2}{7})^6]^4 .” Students develop procedural skill and fluency of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions).

  • In Section 2.5, Zero and Negative Exponents, Try, Problem 1, page 84, students practice simplifying expressions with negative exponents. Students “Simplify each expression. Write each answer using a positive exponent.” Problem 1 states, “2.5−72.5^{-7} ÷ 2.5−42.5^{-4}” Students develop procedural skill and fluency of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions). 

  • In Section 4.1, Solving Linear Equations with One Variable, Learn, Problem 2, page 163, students solve linear equations with one variable. The problem states, “Solve the linear equation 4x + 7 = x + 13.” Students develop procedural skill and fluency of 8.EE.7 (Solve linear equations in one variable).

  • In Section 6.2, Solving Systems of Linear Equations Using Algebraic Methods, Learn, Problem 1, page 328, students solve systems of linear equations using substitution. The problem states, “You have learned to use the elimination method to solve systems of linear equations. Look again at the system of linear equations and the bar models representing the equations. x + y = 8 as x = 8 - y. Step 2: Substitute x = 8 - y for x in x + 2y = 10 to find (8 - y) + 2y = 10. Step 3: Combine like terms to find 8 + y = 10. Step 4: Solve for y to find y = 2. Step 5: Substitute y = 2 into either equation to find x = 6.” Students develop procedural skill and fluency of 8.EE.8b (Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations).

Students have opportunities to independently practice procedural skill and fluency during the Try and Independent Practice portion of the lesson. Examples include: 

  • In Section 2.1, Exponential Notation, Independent Practice, Problem 8, page 51, independently expand and evaluate expressions written in exponential notation. The problem states, “Expand and evaluate each expression. (38)4(\frac{3}{8})^4.” Students independently practice procedural skill and fluency of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions).

  • In Section 2.3, The Power of a Power, Independent Practice, Problem 7, page 67, students independently practice using the power of powers property to simplify expressions. The problem states, “Simplify each expression. Write each answer in exponential notation. [(2y)3]8[(2y)^3]^8.” Students develop and independently practice procedural skill and fluency of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions).

  • In Section 2.5, Zero and Negative Exponents, Independent Practice, Problem 16, page 87, students practice using negative exponents to simplify and evaluate expressions. The problem states, “Evaluate each expression. Write each answer using a positive exponent. 5.2−3÷2.6−35.2^{-3} ÷ 2.6^{-3}” Students independently practice procedural skill and fluency of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions). 

  • In Section 4.1, Solving Linear Equations with One Variable, Independent Practice, Problem 3, page 169, students independently solve linear equations with one variable and with no factored terms. The problem states, “2(x - 1) - 6 = 10(1 - x) + 6.” Students independently practice procedural skill and fluency of 8.EE.7 (Solve linear equations in one variable).

Indicator 2C
02/02

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 3 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 6.3 Real-World Problems: Systems of Linear Equations, Independent Practice, Problem 5, page 344, students apply systems of equations to real-world problems. The problem states, “Jasmine saves some dimes and quarters. She has 40 coins in her coin bank, which total up to $6.55. How many of each coin does she have?” Students independently engage in routine application of 8.EE.8c (Solve real-world and mathematical problems leading to two linear equations in two variables).

  • In Section 7.3, Understanding Linear and Nonlinear Functions, Try, Problem 1, page 42, students practice graphing a function, identifying and interpreting the rate of change, identifying and interpreting the initial y value, and writing the linear equation to model the function. The problem states, “The table shows the total distance, y miles, indicated on the odometer of Tyler’s car as a function of the amount of gasoline, x gallons, used on a particular day. a. Graph the function. b. Find the constant rate of change of the function. c. What does the rate of change represent? d. What is the initial value of y? e. What does the initial value of y represent? Write a linear equation to model the function.” Students engage in routine application of 8.F.4 (Construct a function to model a linear relationship between two quantities).

  • In Section 8.1, Understanding the Pythagorean Theorem and Plane Figures, Learn, Problem 1, page 87, students use the Pythagorean Theorem to solve real-world problems. The problem states, “Some students are having a fundraiser. They attach one end of a 15 feet long banner to the top of a pole that is 12 feet tall. They attach the other end to the ground. How far from the base of the pole is the banner attached?” Students engage in routine application of 8.G.7 (Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions).

  • In Section 8.3 Understanding the Pythagorean Theorem and Solids, Independent Practice, Problem 6, page 112, students use the Pythagorean Theorem to solve a real-world problem. The problem states, “The figure shows the dimensions of a tank which is a rectangular prism. A spider sitting in a top corner of the tank starts making a web by spinning a taut length of silk from its corner to the opposite bottom corner of the tank. What is the length of silk the spider has spun?” The tank dimensions shown are 13 in. by 6 in. by 8 in. Students independently engage in routine application of 8.G.7 (Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions).

Students have opportunities throughout the materials to engage in non-routine application of mathematics. Examples include:

  • In Section 4.3, Understanding Linear Equations with Two Variables, Think, page 179, students work in pairs to represent a real-world problem using a linear equation. The materials state, “Owen sells blood pressure monitors. He earns a monthly salary that includes a basic amount of $750 and $4 for each monitor sold. Write a linear equation for his monthly salary. M dollars, in terms of the number, n, of monitors sold. Think about what kind of restrictions the variables should have, then deduce if it is possible for Owen to earn a monthly salary of $832.” Students engage in non-routine application of 8.EE.7b (Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms).

  • In Chapter 6, Math Journal, page 371, students use elimination and substitution in solving systems of linear equations. The materials state, “Explain when it is convenient to use each method of solving a system of linear equations: Elimination or substitution. Give an example of each method.” Students independently engage in non-routine application of 8.EE.8 (Analyze and solve pairs of simultaneous linear equations). 

  • In Chapter 8, Put on Your Thinking Cap! Problem 3, page 115, states, “The longest diagonal in a rectangular prism measuring a units by b units by c units is d units. Find a formula for the longest diagonal d, in terms of a, b, and c.” Students independently engage in non-routine application of 8.G.7 (Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions).

Indicator 2D
02/02

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 3 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 2.4, The Power of a Product and the Power of a Quotient, Independent Practice, Page 77, Problem 6, students use power of a product property to simplify expressions. The problem states, “Simplify each expression. Write each answer in exponential notation. 2.87÷0.772.8^7 ÷ 0.7^7” Students engage in procedural skill and fluency of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions).

  • In Section 3.1, Understanding Scientific Notation, Independent Practice, Problem 3, page 119, students analyze expressions for correct scientific notation. The problem states, “Tell whether each number is written correctly in scientific notation. If it is incorrectly written, state the reason. 0.99 10−310^{-3}.” Students engage in conceptual understanding of 8.EE.3 (Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other).

  • In Section 9.3, Rotations, Try, Problem 2, page 164, students identify the center of rotation and draw the image of a line segment on the coordinate plane. The problem states, “The windshield wiper on a car swept through a counterclockwise rotation from A to A’ about the origin, O. B is the point at (0,3). If m∠AOB = 59°\degree, what is the angle of rotation?” Students engage in the application of 8.G.3 (Describe the effect of dilations, translations, rotations, and reflections on two- dimensional figures using coordinates). 

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. 4.2, Identifying the Number of Solutions to a Linear Equation, Independent Practice, Problem 13, page 177, students write and solve algebraic expressions. The problem states, “Solve. Cabinet A is 5 inches taller than Cabinet B. Cabinet C is 3 inches taller than Cabinet B. The height of Cabinet B is x inches. a) Write algebraic expressions for the heights of cabinets A and C. b) If the total height of the three cabinets is (3x + 8) inches, can you solve for the height of Cabinet B? Explain.” Students develop conceptual understanding and apply the mathematics of 8.EE.7a (Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions).

  • In Section 6.1, Introduction to Systems of Linear Equations, Independent Practice, Problem 8, page 320, students solve a system of equations in a real-world context by making a table of values. The problem states, “Cole and Aaron start driving at the same time from Boston to Paterson. The journey is d kilometers. Cole drives at 100 kilometers per hour and takes t hours to complete the journey. Aaron, who drives at 80 kilometers per hour, is 60 kilometers away from Paterson when Cole reaches Paterson. The related system of equations is the following. 100t = d and 80t = d - 60 Solve the system of linear equations. Then, find the distance between Boston and Paterson.” Students build procedural skill and fluency and apply the mathematics of 8.EE.8 (Analyze and solve pairs of simultaneous linear equations). 

  • In Section 12.3, Two-Way Tables, Try, Problem 1, page 379, students solve problems using two- way tables. The problem states, “1,000 gym members in two age groups were asked whether they did aerobic exercises or weightlifting exercises at the gym, or both. The data are recorded in the two-way table shown below. Some values are missing from the table. a) Find the total number of members aged 20 to 29. Total aged 20 to 29 = Total surveyed - Total aged 30 to 39 = ___ - ___ = ___. The total number of members aged 20 to 29 was ___. b) Find the number of members aged 20 to 29 who did both types of exercises. Number aged 20 to 29 who did both types of exercises = Total number age 20 to 29 - Number aged 20 to 29 who chose aerobics - Number aged 20 to 29 who chose weightlifting = ___ - ___ - ___ = ___. The number of members aged 20 to 29 who did both types of exercises was ___.” A visual model is provided. Students develop conceptual understanding, build procedural skill and fluency, and apply the mathematics of 8.SP.4 (Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table).

Criterion 2.1: Rigor and Balance
07/08
Indicator 2A
01/02
Indicator 2B
02/02
Indicator 2C
02/02
Indicator 2D
02/02

Criterion 2.2: Math Practices
03/10

Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).

The materials reviewed for Math in Focus: Singapore Math Course 3 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
00/02

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 3 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 2, Exponents, Put On Your Thinking Cap! Problem 2, page 98, students simplify an algebraic expression so they can solve for x and y. The problem states, “Find the values of x and y that make the equation 8(x4⋅16y4)[(2y)2]2\frac{8(x^4 ⋅ 16y^4)}{[(2y)^2]^2} = 1296 true.” Teacher guidance states, “Go through the problem using the four-step problem-solving model. Remind students that they must find values for both x and y (in case any students simply equate y0y^0 to 1, and forgot about it in their final answer.” The materials misidentify MP1 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 Chapter 3, Scientific Notation, Put On Your Thinking Cap! Problem 1, page 144, students find the cube root of a number written in scientific notation. The problem states, “Find the cube root of 2.7 × 101010^{10}.” 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.” Materials misidentify MP1 in this problem 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 6, Systems of Linear Equations, Put On Your Thinking Cap! Problem 1, page 372 students write and solve linear equations. The problem states, “In their bank accounts, Lillian has $110 and Jenna has $600. Lillian’s account balance increases by $30 every year. Her account balance will be C dollars in x years. Jenna’s account balance decreases by $40 every year. Her account balance will also be C dollars in x years. a) Write two equations of C in terms of x. b) Solve this system of linear equations to find the amounts in the girls’ account balances when they are equal.” 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 four-step problem-solving method which is repeated throughout the materials.

  • In Chapter 11, Put on Your Thinking Cap!, Problem 1, page 332, students solve real-world problems using formulas for the volume of cones, cylinders, and spheres. The problem states, “A manufacturer of cylindrical tin cans receives an order for a can that can hold 500 millimeters of food content. As an engineer, you are tasked to find the dimensions of the can, in centimeters, such that it has the least surface area. This will minimize the cost of the materials needed to manufacture the cans. a) Let the radius of the cans be r and the height be h. Show that the total surface area, A of the can is A = 2π2\pir2r^2 + 10000r\frac{10000}{r} b) Use the equation A = 2π2\pir2r^2 + 10000r\frac{10000}{r}, complete the table. c) On graph paper, draw a graph of A against r for 1 ≤ r ≤ 7 d) From the graph, determine the value of r that gives the least value of A. e) What is the least surface area? f) With the radius value, find the dimensions of the can that will give the smallest surface area. Are the dimensions you found suitable for canned food?” Teacher guidance states, “These problem-solving exercises involve using strategies such as simplifying the problem and working backwards. Heuristics, Simplify the problem. Solve part of the problem. Draw a diagram.” Teacher guidance gives a generic reference to have students use the four-step problem-solving method 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 6.3, Real-World Problems: Systems of Linear Equations, is noted as addressing MP1 on pages 339-348 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.

  • Chapter 6, Systems of Linear Equations, Chapter Wrap-Up, is noted as addressing MP1 on pages 373-382 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 4.2, Identifying the Number of Solutions to a Linear Equation, Independent Practice, Problem 15, page 178, students interpret a riddle and write an equation to solve it. The problem states, “Grace gave her sister the following riddle. I have a number x. I add 15 to twice of x to obtain A. I subtract 4 from x to obtain B. I multiply B by 3 to obtain C. A is equal to C. Grace’s sister said the riddle cannot be solved but Grace thought otherwise. Who is right? Explain.” Teacher guidance states, “Assesses students’ ability to interpret a complicated riddle and write an equation to solve it. Students must determine that A is 15 + 2x and C is 3(x - 4) and that they are equal. Since the solution is x = 27, the riddle can be solved.” Teacher guidance gives a generic reference to assess students’ abilities which is repeated throughout the materials.  

  • In Section 5.4, Sketching Lines of Linear Equations, Independent Practice, Problem 9, page 278, students reason whether a point lies on the graph. The problem states, “Maria says that the point (4, -2) lies on the graph of the equation y = -14\frac{1}{4}x - 1. Explain how you can find out if she is right without actually graphing the equation.” Teacher guidance states, “Assesses students’ ability to reason whether a point lies on the graph.” Teacher guidance gives a generic reference to assess students’ abilities which is repeated throughout the materials.

  • In Section 7.1, Understanding Relations and Function, Independent Practice, Problem 13, page 21, students apply their understanding of functions to a situation involving area. The problem states, “Is the relation between the side length and the area of a square a function? Explain.” Teacher guidance states, “Assesses students’ ability to extend their thinking beyond the lesson to apply understanding of functions in a situation involving area.” The materials misidentify MP2 in this problem, students do not consider units involved in a problem, attend to the meaning of quantities, nor do students understand the relationships between problem scenarios and mathematical representations

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 1.3, Introducing Significant Digits, is noted as addressing MP2 on pages 19-28 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.

  • Section 3.3, Understanding the Pythagorean Theorem and Solids, is noted as addressing MP2 on pages 107-112 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
02/02

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 3 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. Examples include:

  • In Section 5.3, Writing Linear Equations, Math Talk, page 255, students construct viable arguments when using slope-intercept form to identify slopes and y-intercepts. The materials state, “Eve notices that the ordered pair (0,6) is a solution of y = -2x +6. Is this ordered pair also a solution of y + 2x + 6 = 0? Is any ordered pair (x,y) that is a solution of y = -2x + 6 also a solution of y + 2x - 6 = 0? Why or why not?” Teacher guidance states, “Get students to explain what it means for (0,6) to be a solution of the equation. Have students check and verify that (0,6) satisfies both equations. Get students to discuss the second part of the question. Have some groups share their answers. Conclude that equivalent equations will always have the same set of solutions.”

  • In Section 9.5, Comparing Transformations, Learn, Problem 8, page 194, students construct viable arguments when discussing why perpendicular lines can be preserved when a transformation preserves angle measures. The problem states, “If a transformation preserves angle measures, how does it prove that perpendicular lines will also be preserved?” Teacher guidance states, “Encourage students to discuss why perpendicular lines can be preserved when a transformation preserves angle measures. Students may have observed that two lines form a right angle, so if a transformation preserves right angles, it will also preserve perpendicular lines.”

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, teacher guidance is often repetitive, not specific, and distracts from the intentional development of MP3. Examples include:

  • In Section 5.3, Writing Linear Equations, Independent Practice, Problem 9, page 268, students critique the reasoning of others when comparing slopes. The problem states, “Anna says that the graphs of y = -3x + 7 and y = 3x - 7 are parallel lines. Do you agree? Explain.” Teacher guidance states, “Assesses students’ ability to reason whether two lines are parallel.” While this teacher guidance does not intentionally develop MP3, students do have the opportunity to critique the reasoning of others.

  • In Section 8.1, Understanding the Pythagorean Theorem and Plain Figures, Independent Practice, Problem 18, page 96, students critique the reasoning of others when solving a real-world problem using the Pythagorean Theorem. The problem states, “Julia buys a triangular table. The sides of the table top are 29.4 inches, 39.2 inches, and 49 inches long. She wants to place the table in a corner of a rectangular room but her sister says the table will not fit in the corner. Do you agree with her sister? Explain.” Teacher guidance states, “Assess students’ ability to solve real-world problems using the Pythagorean Theorem.” While this teacher guidance does not intentionally develop MP3, students do have the opportunity to critique the reasoning of others.

Math Journal Activities provide opportunities for students to engage in the intentional development of MP3. Examples include:

  • In Chapter 5, Lines and Linear Equations, Math Journal, page 293, students construct viable arguments when graphing linear equations. The materials state, “Suppose Gabriella shows you her homework. Describe Gabriella’s mistakes. Graph the equation correctly.” The equation y = 2x + 12\frac{1}{2} is shown along with Gabriella’s graph of the equation. Teacher guidance states, “Review with students the various strategies learned in this chapter. You may want to pose these questions to students. How would you draw this graph? How would your graph be different from the graph Gabriella drew?”

  • In Chapter 7, Functions, Math Journal, page 63, students construct viable arguments when determining if straight-line graphs are linear functions. The problem states, d“Give examples of the different types of linear functions you have learned. Use graphs and equations to represent your functions. Diego says that all straight line graphs are linear functions. Do you agree with him? Explain your answer.” Teacher guidance states, “This journal provides opportunities for students to reflect on the concepts learned in this chapter. In a, students are required to explain their answer and support their explanation by going back to the basis to decide whether all straight-line functions are linear functions. In b, students should recall that linear functions have an equation of the slope-intercept form, y = mx + b. Reflect upon what you have learned in this chapter. What are examples of different types of functions? Justify your examples with a graph, table and/or equation. Students might not recognize that a vertical line is a straight line but not a function. Encourage them to draw a sketch of a vertical line and prove that it is not a function. Some students might need to be reminded that the type of relation in addition to whether the function is a line needs to be considered. Remind them of the example x = 2. this is a vertical line, so while it is a straight line, it is not a function.”

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 1.1, Introducing Irrational Numbers, is noted as addressing MP3 on pages 7-14  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 7.4, Comparing Two Functions, is noted as addressing MP3 on pages 51-62  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
00/02

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 3 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 5, Lines and Linear Equations, Put On Your Thinking Cap!, Problem 1, page 294, students draw graphs without being given an equation. The problem states, “Carter and Alexis are both students. Carter has $28 to spend for the whole week, and he decides to spend the same amount every day. Alexis currently does not have any savings. She is given a daily allowance and she decides to save the same amount every day. After four days, both have the same amount of money. The graph shows the amount of money Carter has, y dollars, after x days during one week. a) A copy of the graph is shown below. Draw a line on the graph to represent the amount of money Alexis has after x days. b) Find the slope of Carter’s graph and explain what information it gives about the situation. c) Write an equation to represent the amount of money each person has during that week.” Teacher guidance states, “a) Requires students to be able to draw a graph based on the information given in the question without being given information on the equation itself. b) Requires students to find the slope of Carter’s graph and interpret what it means in the real-world context. Go through the problem using the four-step problem-solving model. c) Requires students to determine the equation of a line from its graph. Go through the problem using the four-step problem-solving model.” Teacher guidance gives a generic reference to have students use the four-step problem-solving method which is repeated throughout the materials. 

  • In Chapter 7, Functions, Put on Your Thinking Cap!, Problem 1, page 64, students solve a real-world problem by comparing functions represented in different forms. The problem states, “Five teachers at a school are taking a group of students to a museum. The museum offers three different admission packages, A, B, and C. The total admission fee, y dollars, for each package is a linear function of the number of students, x.” Students are given necessary information about each package in either a table or statement. Students “a) Find an equation representing each of the three functions. b) Using graph paper, graph the three functions on the same coordinate plane. Use 1 grid square to represent 5 students on the x-axis, for the x interval from 0 to 50. Use 1 grid square to represent $100 on the y-axis, for the y interval from 0 to 1000. For each function, draw a line through the points. c) Use your graph to identify the best deal for 5 teachers and 20 students. Explain. d) Use your graph to identify the best deal for 5 teachers and 50 students. Explain.” Teacher guidance states, “Requires students to solve a real-world problem by comparing functions represented in different forms. What is this problem about? What can we do first? Carry out the plan. a) Looking at the information given for Package A, how can we find the equation of the line? Looking at the information given for Package B, how can we find the equation of the line? Looking at the information given for Package C, how can we find the equation of the line?’ b) Distribute graph papers to students. What do the x-axis and y-axis represent? How should you graph the three equations? What does it mean when the lines intersect? c) Using the graph, guide students to locate 20 on the x-axis and see that Package A is the best deal at $450. Alternatively, students can substitute 20 for x into each of the three equations. d) Using the graph, guide students to locate 50 on the x-axis and see that Package C is the best deal at $800.” Teacher guidance provides scaffolding for students, preventing them from independently modeling the mathematics.

  • In Section 11.2, Finding Volumes and Surface Areas of Cylinders, Activity, page 294, students make a net of a cylinder in pairs. The activity states, “1) A toilet roll tube is a paper cylinder without the two circular end faces. To make the two end faces stand a toiled tube on a piece of thick paper and draw two circles. Cut out the two circles. 2) On the toilet roll tube, draw a straight line from one circular end to the other circular end. Cut up the toilet roll tube along the straight line and lay it flat on the table. What shape do you get? 3) How do you use the two circles and the cut-up toilet roll tube to form a net of the paper cylinder? Draw the net you have formed. 4) compare your net with your classmate’s. How are they different? 5) How is the surface area of the paper cylinder related to its net?” Teacher guidance states, “In this activity students will create a net of a cylinder to discover the relationship between surface area and the area of a net. Ask students to follow the instructions in the activity. Prompt them to think about how the net can be formed for 3. If possible, ask them to show that it is a net of a cylinder by folding it. Ensure that students understood the aim of the activity. How is surface area of a cylinder related to its net?” Teacher guidance provides scaffolding for students, preventing them from independently modeling the mathematics. Additionally, creating cylindrical nets is not connected to grade level standards. 

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 1.1, Introducing Rational Numbers, is noted as addressing MP4 on pages 7-10 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 11.4, Finding Volumes and Surface Areas of Spheres, 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 Lesson 2.5, Zero and Negative Exponents, Activity, Problem 4, page 80, students make a prediction about any non-zero number raised to the zero power and use a calculator for accuracy. The problem states, “Make a prediction about the value of any nonzero number raised to the zero power. Then, use a calculator to check your prediction for several numbers. For example, to raise the number –2 to the zero power, use the following keystrokes (−2)0( - 2 ) ^ 0 enter. Does your prediction hold true?” Teacher guidance states, “Get students to complete the table on page 79. Remind them that they are supposed to leave their answer in exponential notation, and not calculate the value. Get students to do 2 and 3, then work in pairs to discuss 4. Point out to students that the point of this activity is to determine the value of any nonzero raised to the zero power. Get them to see that any nonzero value raised to the zero power always takes on the value of 1. Go through the general form of this rule, that a0a^0 = 1, regardless of the value of a, as long as a ≠ 0. Guide students to conclude the task in ENGAGE.” Students are told what type of tool to use (calculator), what steps to take, and a chart. Therefore, students do not independently choose their tools and strategies.

  • In Section 4.3, Understanding Linear Equations with Two Variables, Activity, page 185, students create tables of values for linear equations with two variables. The activity states, “1) Enter the equation y = xπ\frac{x}{\pi} using the equation screen of a graphing calculator. 2) Set the table function to use values of x starting at 0, with increments of 1. 3) Display the table. It will be in two columns as shown. 4) Repeat 1 to 3 for the question y = -2x + 2\sqrt{2}.” Teacher guidance states, “Let’s use the graphing calculator to create tables of values for linear equations with two variables and an irrational number. Invite students to work in pairs. To specify ‘increments of 1’, guide students to look for ‘△\trianglex’ found in the table setting menu. (This is read ‘the change in x.’) The default value is 1, so it may be correct without a new input value. Tell students to check that they get the table shown on the calculator screen. Alert students that they should use the (-) key instead of the - key for the negative coefficient, -2.” Students are told what type of tool to use (calculator) and what steps to take. Therefore, students do not independently choose their tools and strategies.

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:

  • Section 4.3, Understanding Linear Equations with Two Variables, is noted as addressing MP5 on pages 179-190 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 12.3, Two-Way Tables, is noted as addressing MP5 on pages 377-390 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
01/02

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 3 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 7.3, Understanding Linear and Nonlinear Functions, For Language Development, TE page 34, states, “Make sure that students understand the meaning of rate of change. In particular, make sure they understand that the phrase constant rate of change means ‘constant change in the output valued per unit of input’. Direct students to the line of rate of change values in b. Point out that before being simplified to a single value, each ratio can be simplified to a ratio of the change in the output values to the denominator. 1. Since the denominator shows the change in input values, a constant rate of change is a unit rate per unit of input.”

  • In Chapter 9, Geometric Transformations, Math Journal, page 201, students have the opportunity to use the specialized language of mathematics when describing a real-world example of each of the four transformations. The materials state, “You can see the four transformations you have learned around you in the real world. For example, the clock hands rotate to show the time. A car moving in a straight line translates from one position to another position. Describe at least one real-world example of each of the four transformations.” Teacher guidance states, “This journal requires students to describe one real-world example of each of the four transformations. Review with students the four types of transformations. Encourage students to work independently. You may want to pose these questions to students. The minute hand of a clock moving around its center is an example of a rotation. A car moving on a straight line is an example of translation. Can you think of other examples on rotation and translation? What about reflection and dilation? Give at least one real-world example for each of them. What other real-world examples of transformations can you think of?” 

  • In Section 12.2, Modeling Linear Associations, For Language Development, TE page 370, states, “Make sure that students understand the terms interpolate and extrapolate. Point out the prefixes inter-, meaning ‘between’, and extra-, meaning ‘outside.’ Discuss with students other words that include inter such as intersection, interstate, interpret, intercept or extra- such as extraordinary, extraneous, extraterrestrial.”

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 Chapter 1, The Real Number System, Put On Your Thinking Cap!, Problem 2, page 31, students have the opportunity to attend to the precision of mathematics when finding the area of a circle. The problem states, “In an experiment, Sarah used a ruler to measure the radius of a circular disc, and she measured it at 7.2 centimeters. Using the calculator value of π\pi, she calculated the area of the circular disc as shown. How can Sarah write her answer to better reflect the precision of the measuring instrument used during the experiment?” A picture of Sarah’s work is provided. Teacher guidance states, “Requires students to be able to determine the number of significant digits to round the answer to when the answer is not exact. Go through the problem using the four-step problem-solving model. Encourage students to make use of the formula of the area of a circle to solve the problem.” Teacher guidance prompts students to use the four-step problem-solving model, not use specific mathematical terms to explain their thinking or communicate their ideas.

  • In Section 5.1, Finding and Interpreting Slopes of Lines, Independent Practice, Problem 5, page 240, students have the opportunity to attend to the precision of mathematics when identifying a slope as undefined. The problem states, “Andrew graphs a vertical line through the points (5, 2) and (5, 5). He says the slope of the line is 30\frac{3}{0}. What error is he making?” Teacher guidance states, “Assesses students’ ability to determine whether a statement is correct by knowing the concept of the slope of a vertical line.” Neither teacher guidance nor student directions prompt students to use specific mathematical terms to explain their thinking or communicate their ideas.

  • In Section 12.1, Scatter Plots, Try, Problem 1d, page 358, students have the opportunity to use the specialized language of mathematics when explaining what the values of outliers represent. The problem states, “1. Jack is investigating the effect of the amount of water on the growth of tomato seedlings. He watered each of the 22 plants with a fixed amount of water daily. He recorded their heights, y inches, at the end of two weeks. His data are shown in the two tables. d. Explain what the values of outliers represent. The outlier represents a tomato seedling that grew to a height of __ inches after being given __ fluid ounces of water daily for two weeks.” Teacher guidance states “assess students’ ability in providing rationales for the outliers.” 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 are not identified as aligned to MP6. For example: 

  • In Section 4.3, Understanding Linear Equations with Two Variables, For Language Development, TE page 182, states, “Review the terms solution and ordered pair with students. Explain that the solution to an equation with two variables is not one number, but a set of ordered pair.”

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. For example:

  • In Section 9.4, Dilations, is noted as addressing MP6 on pages 177-192 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
00/02

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 3 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 Chapter 1, The Real Number System, Math Journal, page 29, students create their own tree map diagram to summarize relationships in the real number system. The materials state, “Create your own diagram summarizing the relationship among the types of real numbers that you have learned.” Teacher guidance states, “Use this Math Journal to assess how well students understand the set of real numbers and the structure of rational and irrational numbers. Review with students the various strategies learned in this chapter. You may want to pose these questions to students. What are the different types of numbers that make up the set of real numbers? How can you further split each type? How do you know if one group cannot be divided into smaller groups?” Materials misidentify MP7 in this problem, students do not have the opportunity to look for patterns or structures to make generalizations and solve problems, or look for and explain the structure within mathematical representations. 

  • In Chapter 3, Scientific Notation, Math Journal, page 143, students analyze a table of values written in scientific form and notation. The materials state, “The table shows some numbers written in standard form and in the equivalent scientific notation. Describe the relationship between each pair of variables. a) The value of the positive number in standard form and the sign of the exponent when expressed in scientific notation. b) The sign of the exponent when expressed in scientific notation and the direction the decimal point moves to express the number in standard form.” Teacher guidance states, “Use the Math Journal to assess how well students understand scientific notation. Review with students the various strategies learned in this chapter. You may want to encourage discussion by posing these questions to students. What pattern can you see between each pair of variables? Does this pattern also apply to other numbers written in scientific notation?” Part of the teacher guidance gives a generic reference to have students review strategies they have learned in the chapter which is repeated throughout the materials. 

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. For example:

  • Chapter 5, Wrap-Up, Chapter Review, Performance Task, is noted as addressing MP7 on pages 144-145 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 and teacher guidance is often repetitive, not specific, and activities are scaffolded preventing intentional development of the full intent of MP8. Examples include:

  • Section 2.3, The Power of a Power, Activity Problem 5, page 63, students calculate a power of a power. “Is it correct to assume that using the greatest number drawn as the base will give an expression with the greatest possible value? Explain or give an example.” Teacher guidance states, “Discuss 5 with the students. Ask them to look through the values they have gotten and think about the question. Have them conclude that the assumption is false.” Teacher guidance gives a generic reference to have students think about the question and determine the answer is false. Questions to prompt students to look for and express regularity in repeated reasoning are not provided.

  • In Section 2.5, Zero and Negative Exponents, Learn, Problem 4, page 83, students simplify expressions involving negative exponents. The problem states, “Suppose a represents any nonzero number. How would you write a−3a^{-3} using a positive exponent?” No teacher guidance is provided for this 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. Examples include:

  • Section 5.2, Understanding Slope-Intercept Form, is noted as addressing MP8 on pages 245-254 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.

  • Section 6.1, Introduction to Systems of Linear Equations, is noted as addressing MP8 on page 315-320 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.

Criterion 2.2: Math Practices
03/10
Indicator 2E
00/02
Indicator 2F
02/02
Indicator 2G
00/02
Indicator 2H
01/02
Indicator 2I
00/02
Overview of Gateway 3

Usability

Gateway 3
v1.5
Gateway 3

Usability
NE = Not Eligible. Product did not meet the threshold for review.
NE
0
17
24
27
Criterion 3.1: Teacher Supports
NE = Not Eligible. Product did not meet the threshold for review.
NE
Criterion 3.2: Assessment
NE = Not Eligible. Product did not meet the threshold for review.
NE
Criterion 3.3: Student Supports
NE = Not Eligible. Product did not meet the threshold for review.
NE
Criterion 3.4: Intentional Design
NE = Not Eligible. Product did not meet the threshold for review.
NE
Not Rated

Criterion 3.1: Teacher Supports

NE = Not Eligible. Product did not meet the threshold for review.
NE

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

Indicator 3A
00/02

Materials provide teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Indicator 3B
00/02

Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

Indicator 3C
00/02

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

Indicator 3D
Read

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

Indicator 3E
00/02

Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.

Indicator 3F
00/01

Materials provide a comprehensive list of supplies needed to support instructional activities.

Indicator 3G
Read

This is not an assessed indicator in Mathematics.

Indicator 3H
Read

This is not an assessed indicator in Mathematics.

Criterion 3.1: Teacher Supports
NE = Not Eligible. Product did not meet the threshold for review.
NE
Indicator 3A
00/02
Indicator 3B
00/02
Indicator 3C
00/02
Indicator 3D
Read
Indicator 3E
00/02
Indicator 3F
00/01
Indicator 3G
Read
Indicator 3H
Read

Criterion 3.2: Assessment

NE = Not Eligible. Product did not meet the threshold for review.
NE

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

Indicator 3I
00/02

Assessment information is included in the materials to indicate which standards are assessed.

Indicator 3J
00/04

Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Indicator 3K
00/04

Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

Indicator 3L
Read

Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

Criterion 3.2: Assessment
NE = Not Eligible. Product did not meet the threshold for review.
NE
Indicator 3I
00/02
Indicator 3J
00/04
Indicator 3K
00/04
Indicator 3L
Read

Criterion 3.3: Student Supports

NE = Not Eligible. Product did not meet the threshold for review.
NE

The program includes materials designed for each student’s regular and active participation in grade-level/grade-band/series content.

Indicator 3M
00/02

Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

Indicator 3N
00/02

Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

Indicator 3O
Read

Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

Indicator 3P
Read

Materials provide opportunities for teachers to use a variety of grouping strategies.

Indicator 3Q
00/02

Materials provide strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Indicator 3R
Read

Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

Indicator 3S
Read

Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

Indicator 3T
Read

Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

Indicator 3U
Read

Materials provide supports for different reading levels to ensure accessibility for students.

Indicator 3V
00/02

Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Criterion 3.3: Student Supports
NE = Not Eligible. Product did not meet the threshold for review.
NE
Indicator 3M
00/02
Indicator 3N
00/02
Indicator 3O
Read
Indicator 3P
Read
Indicator 3Q
00/02
Indicator 3R
Read
Indicator 3S
Read
Indicator 3T
Read
Indicator 3U
Read
Indicator 3V
00/02

Criterion 3.4: Intentional Design

NE = Not Eligible. Product did not meet the threshold for review.
NE

The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

Indicator 3W
Read

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

Indicator 3X
Read

Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

Indicator 3Y
Read

The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

Indicator 3Z
Read

Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

Criterion 3.4: Intentional Design
NE = Not Eligible. Product did not meet the threshold for review.
NE
Indicator 3W
Read
Indicator 3X
Read
Indicator 3Y
Read
Indicator 3Z
Read
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