8th Grade - Gateway 2
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Rigor & Mathematical Practices
Rigor & the Mathematical PracticesGateway 2 - Does Not Meet Expectations | 55% |
|---|---|
Criterion 2.1: Rigor and Balance | 7 / 8 |
Criterion 2.2: Math Practices | 3 / 10 |
The materials reviewed for Math in Focus: Singapore Math Course 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).
Criterion 2.1: Rigor and Balance
Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.
The materials reviewed for Math in Focus: Singapore Math Course 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
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 $$\sqrt{2}$$ 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
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 $$(\frac{2}{3})^5$$ = $$\frac{2}{3}$$ ⋅ $$\frac{2}{3}$$ ⋅ $$\frac{2}{3}$$ ⋅ $$\frac{2}{3}$$ ⋅ $$\frac{2}{3}$$ = $$\frac{32}{243}$$.” 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 $$[(\frac{2}{7})^6]^4$$ .” 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, “$$2.5^{-7}$$ ÷ $$2.5^{-4}$$” 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. $$(\frac{3}{8})^4$$.” 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. $$[(2y)^3]^8$$.” 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. $$5.2^{-3} ÷ 2.6^{-3}$$” 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
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
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. $$2.8^7 ÷ 0.7^7$$” 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 $$10^{-3}$$.” 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 2.2: Math Practices
Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).
The materials reviewed for Math in Focus: Singapore Math Course 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
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 $$\frac{8(x^4 ⋅ 16y^4)}{[(2y)^2]^2}$$ = 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 $$y^0$$ 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 × $$10^{10}$$.” 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 = $$2\pi$$$$r^2$$ + $$\frac{10000}{r}$$ b) Use the equation A = $$2\pi$$$$r^2$$ + $$\frac{10000}{r}$$, 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 = -$$\frac{1}{4}$$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
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 + $$\frac{1}{2}$$ 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
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 $$( - 2 ) ^ 0$$ 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 $$a^0$$ = 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 = $$\frac{x}{\pi}$$ 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 + $$\sqrt{2}$$.” 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 ‘$$\triangle$$x’ 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
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 $$\pi$$, 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 $$\frac{3}{0}$$. 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
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^{-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.