PARCC Grade 8 Mathematics Lesson 10: Eight Mathematical Practices Rationale The Standards for Mathematical Practice describe the habits of mind of an excellent math student. The Standards for Mathematical Practice describe ways in which students should engage in mathematics as they grow in mathematical maturity and expertise throughout the elementary, middle, and high school years. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, and apply the mathematics to practical situations. They may have difficulty explaining the mathematics accurately to other students. The Standards for Mathematical Practice provide students with a flexible base to work from and to help students develop the ability to step back for an overview or to change course from a known procedure to find a shortcut. The goal of all instruction in mathematics should be to find the points of intersection between the Standards for Mathematical Content and the Standards for Mathematical Practice. The content standards which set an expectation of understanding are potential points of intersection. Students will be expected to solve Type I, Type II, and Type III items in the Performance- Based Assessment. Type II items call for written arguments/justifications, critique of reasoning, or precision in mathematical statements (MP.3, MP.6). Type III items call for modeling/application in a real-world context or scenario (MP.4). Other Mathematical Practices are also assessed through these types of problems. Standards from Expressions and Equations, Functions, and Geometry will be assessed on the Performance-Based Assessment in grade 8. Students should have the opportunity to practice Type II and Type III items using these standards. Additional information is provided in the PARCC Evidence documents found on the PARCC website (parcconline.org). Lesson 10: Eight Mathematical Practices Page 1
The Eight Mathematical Practices and Examples for Grade 8 1. Make sense of problems and persevere in solving them. In grade 8, students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, What is the most efficient way to solve the problem?, Does this make sense?, and Can I solve the problem in a different way? 2. Reason abstractly and quantitatively. In grade 8, students represent a wide variety of real world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. They examine patterns in data and assess the degree of linearity of functions. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations. 3. Construct viable arguments and critique the reasoning of others. In grade 8, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like How did you get that?, Why is that true? Does that always work? They explain their thinking to others and respond to others thinking. 4. Model with mathematics. In grade 8, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students solve systems of linear equations and compare properties of functions provided in different forms. Students use scatterplots to represent data and describe associations between variables. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context. 5. Use appropriate tools strategically. Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 8 may translate a set of data given in tabular form to a graphical representation to compare it to another data set. Students might draw pictures, use applets, or write equations to show the relationships between the angles created by a transversal. 6. Attend to precision. In grade 8, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to the number system, functions, geometric figures, and data displays. 7. Look for and make use of structure. Students routinely seek patterns or structures to model and solve problems. In grade 8, students apply properties to generate equivalent expressions and solve equations. Students examine patterns in tables and graphs to generate equations and describe relationships. Additionally, students experimentally verify the effects of transformations and describe them in terms of congruence and similarity. 8. Look for and express regularity in repeated reasoning. In grade 8, students use repeated reasoning to understand algorithms and make generalizations about patterns. Students use iterative processes to determine more precise rational approximations for irrational numbers. They analyze patterns of repeating decimals to identify the corresponding fraction. During multiple opportunities to solve and model problems, they notice that the slope of a line and rate of change are the same value. Students flexibly make connections between covariance, rates, and representations showing the relationships between quantities. http://www.ncpublicschools.org/acre/standards/common-core-tools/#unmath Lesson 10: Eight Mathematical Practices Page 2
Goals Objectives Materials To provide the students with an opportunity to experience problems that model each Mathematical Practice To provide students with the opportunity to identify each Mathematical Practice by matching the appropriate description To understand the expectations of Type II and Type III items To solve Type II and Type III problems To identify items that present the greatest challenge for each student and the class as a whole Students will understand the eight Mathematical Practices. Students will recognize the relationship between a Mathematical Practices and a description which represents the practice. Students will be able to explain each Mathematical Practice in their own words. Students will discuss and solve Type II and Type III problems. Eight Mathematical Practices title cards Description cards Type II and Type III problem-solving cards Rubric Class folder labeled Lesson 10: Mathematical Practices (At the end of the lesson, place the class papers in the folder). Procedures for the Performance-Based Practice Pass out Eight Mathematical Practice title cards and the Description cards. Students work in pairs to match the description with the Mathematical Practice. Whole group discussion of students responses. Ask students to share their findings. Pass out Type II and Type III problem-solving cards. Students work in pairs to solve the two problems. They must choose one Type II and one Type III. Choose students to share their solutions to the problems. Ask students to identify the Mathematical Practices used when solving the problems. Students should be able to justify their choices. Pass out Rubric. Discuss with students. Discuss description of Type II and Type III problems and rubrics. Lesson 10: Eight Mathematical Practices Page 3
Teacher & Teachers Aide Observations During the Performance-Based Practice Be sure to circulate the classroom and monitor students while they are completing the activity. Which students are using their time wisely? Which students are using mathematical language and vocabulary in their discussion and writings? Which students are having difficulty with the task? Assessment or Check for Understanding Whole group discussion to check for understanding. Pay particular attention to the students correct use of vocabulary and clarity in their explanations. Follow-Up Ask students to create a graffiti wall or word wall with their interpretive descriptions of the Mathematical Practices. Create additional Type II and Type III items to coincide with mathematics taught in current unit. PARCC Technology Tips PARCC equation editors are provided as the answer boxes for responses that include math, utilizing special mathematical functions. In this lesson, a written response is needed and the Open Response Equation Editor is provided. It is possible to respond with a combination of words and math. It is suggested that letters, numbers, and punctuation symbols from the standard keyboard be used. Any part of the response that indicates mathematical processes can be described using the function keys provided in the Open Response Equation Editor. When the pointer (curser) hovers over a function key, the name of the function appears as a pop-up; the question mark function represents the unknown in an equation. It is NOT possible to create diagrams, models and/or step-by-step solution processes in a vertical format (such as solving with an algorithm). To provide students with paper/pencil practice with the Open Response Equation Editor, a worksheet follows. Using the worksheet will allow the editor to become familiar to the students. It is highly recommended that the students practice with the editor provided on the PARCC website. Lesson 10: Eight Mathematical Practices Page 4
Problem: Lesson 10: Eight Mathematical Practices Page 5
Eight Mathematical Practices Lesson 10: Eight Mathematical Practices Page 6
Eight Mathematical Practices Title Cards 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Lesson 10: Eight Mathematical Practices Page 7
E I need to make a plan to solve my problem. If my plan doesn t work, I need to make changes. I can t give up!! C I can use reasoning to solve problems using numbers, symbols and words. B I can justify my answers and analyze the reasoning of others. D I can use what I know about math to solve real-life problems. I can use tables and diagrams to explain my thinking. F Certain tools make it easier to solve and explain my math. H I need to be very clear in showing and explaining my math. There is an organization of numbers and shapes in math. It makes it so much easier to remember how to solve problems. G I can learn steps in solving problems that I can use again and again in math. Lesson 10: Eight Mathematical Practices Page 8 A.
Type II Card 1. The minimum salary for a major league baseball player is $500,000. Ryan Howard s salary is 2.5 x 10⁷. Ryan Howard s salary is how many times as much as the minimum salary? Write your answer in scientific notation and standard form. Explain how you determined your answer. 8.EE.4 Type III Card 2. Describe the sequence of transformations that could be used to determine the congruency of triangle A and triangle A¹. A A¹ 8.G.4 Lesson 10: Eight Mathematical Practices Page 9
Type II Card 3. Copy the table below. Fill in each x-value and y-value in the table below to create a relation that is a function. Justify your answer. x y 8.F.1 Type III Card 4. Compare the scenarios to determine which represents the greater speed. Write a description of each scenario including the unit rate in your explanation. Scenario 1 Scenario 2 Distance (Miles) 500 400 300 200 100 0 0 1 2 3 4 5 Time (Hours) y = 150x x = time in hours y = distance in miles 8.EE.5 Lesson 10: Eight Mathematical Practices Page 10
PARCC Grade 8 Mathematics Lesson 10: Eight Mathematical Practices Rubric Score 3 # 1 Description Student response includes each of the following 3 elements: The student correctly determines that Ryan Howard s salary is 50 times as much as the minimum salary. Valid representations of standard form and Scientific Notation. Valid explanation of solution process used. Sample Student Response: To compare the values, the values need to be expressed in the same form, either in standard form or Scientific Notation. I used Scientific Notation; I rewrote 500,000 as 5 x 10 5. Next I divided the values because the problem asks how many times as much and division is the inverse of multiplication. 2.5 x 10 7 = 0.5 x 10²(Scientific Notation) or 5 x 10¹ = 50 (standard form) 5 x 10 5 2 Student Response includes 2 of the 3 elements. 1 Student Response includes 1 of the 3 elements. 0 Student Response is incorrect or irrelevant. Score Description # 2 1 The student describes the sequence: reflect over the x-axis, rotate 90, reflect over y-axis. 0 Student Response is incorrect or irrelevant. Lesson 10: Eight Mathematical Practices Page 11
Score 3 # 3_ Description Student response includes each of the following 3 elements: Complete table correctly representing a relation that is a function. Correct description of a relation and a function. Valid justification. Sample Student Response: A relation is a set of ordered pairs. The first value in the ordered pair is calle the input and the corresponding value is the output. For a relation to be a function each input must be paired with exactly one output. In the table I completed each input (x value) is paired with one output (y value). If an input is corresponding to two different outputs, the relation would not be a function. 2 Student Response includes 2 of the 3 elements. 1 Student Response includes 1 of the 3 elements. 0 Student Response is incorrect or irrelevant. Score # 4_ Description Student response includes each of the following 3 elements: Scenario 1 shows a unit rate of 200 miles/ hour. Scenario 2 shows a unit rate of 150 miles/hour. Scenario 1 shows the greater speed. 3 2 Student Response includes 2 of the 3 elements. 1 Student Response includes 1 of the 3 elements. 0 Student Response is incorrect or irrelevant. Lesson 10: Eight Mathematical Practices Page 12