8th Grade - Gateway 1

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 $$\frac{17}{9}$$, 2.56 and $$\sqrt{7}$$ 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 <. $$\sqrt{10}, -\frac{53}{9}, -\frac{3\pi}{2}, 6.2\bar{86}$$.” 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 -$$\sqrt{12}$$.” 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 $$x^3$$ = 9 and $$y^3$$ = 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. $$x^3$$ = $$\frac{216}{729}$$.” 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 $$x^2$$ = p and $$x^3$$ = 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, “ $$\triangle$$ABC is mapped onto $$\triangle$$PQR, $$\triangle$$LMN, and $$\triangle$$XYZ by three different transformations. Describe the transformation that maps $$\triangle$$ABC onto $$\triangle$$PQR." 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).

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.”