Problem CCSSM Mathematical Practices 3.4 7.G.A.2, 7.G.B.5 MP1 MP3 Mathematical Reflection 7.EE.A.2, 7.EE.B.3, 7.EE.B.4, 7.EE.B.4.A, 7.G.B.5 MP1 MP3 MATHEMATICAL PRACTICES AND HABITS OF MIND DESCRIPTION In the Connected Mathematics® curriculum, we develop understanding of important mathematical ideas by solving problems and reflecting on the mathematics that is embedded in the problem. Every day, we will use “habits of mind” to make sense of problems and apply what we learn to new situations. Some of these habits are described by the Common Core State Standards for Mathematical Practice (MP) (NGAC & CCSSO 2010) and by the National Council of Teachers of Mathematics (NCTM) in Principles to Actions (2014). In 2014, NCTM wrote, An excellent mathematics program includes curriculum that develops important mathematics along coherent learning progressions and develops connections among areas of mathematical study and between mathematics and the real world. In Connected Mathematics® 4 we have organized the CCSS Mathematical Practices and Habits of Mind and NCTM’s Principles to Actions into two main goals: Solving Problems and Connecting Knowledge. Solving Problems MP1. Make sense of problems and persevere in solving them. When using mathematics to solve a problem, it helps to think carefully about • data and other facts we are given and what additional information we need to solve the problem; • strategies we have used to solve similar problems and whether we could solve a related simpler problem first; and • how we could express the problem with equations, tables, diagrams, or graphs. MP2. Reason abstractly and quantitatively. When solving a problem, it often helps to • focus first on the key mathematical ideas; • check that our answer makes sense in the problem setting; and • use what we know about the problem setting to guide our mathematical reasoning. Correlations 239 SAMPLE
MP3. Construct viable arguments and critique the reasoning of others. When we are asked to explain why a conjecture is correct, we can • show some examples that fit the claim and explain why they fit; and • show how a new result follows logically from known facts and principles. When we believe a mathematical claim is incorrect, we can • show one or more counterexamples of cases that don’t fit the claim; and • find steps in the argument that do not follow logically from prior claims. MP4. Model with mathematics. When we are asked to solve problems, it often helps to • think carefully about the numbers or geometric shapes that are the most important factors in the problem, then ask ourselves how those factors are related to each other; and • express data and relationships in the problem with tables, graphs, diagrams, or equations and check our result to see if it makes sense. MP5. Use appropriate tools strategically. When working on mathematical questions, we should always • decide which tools are most helpful for solving the problem and why; and • try a different tool when we get stuck. MP6. Attend to precision. In every mathematical exploration or problem-solving task, it is important to • think carefully about the required accuracy of results; is a number estimate or geometric sketch good enough, or is a precise value or drawing needed?; and • report our discoveries with clear and correct mathematical language that can be understood by those to whom we are speaking or writing. MP7. Look for and make use of structure. In mathematical explorations and problem-solving, it is often helpful to • look for patterns that show how data points, numbers, or geometric shapes are related to each other; and • use patterns to make predictions. MP8. Look for and express regularity in repeated reasoning. When results of a repeated calculation show a pattern, it helps to • express that pattern as a general rule that can be used in similar cases; and • look for shortcuts that will make the calculation simpler in other cases. 240 Shapes and Designs SAMPLE
Connecting Knowledge Build on and connect to prior knowledge to build deeper understandings and new insights. When we finish solving a problem, think about • what mathematical understanding is embedded in the problem; and • how this understanding builds on and connects to the mathematics we learned in prior problems. We will use all the Mathematical Practices in this unit. Sometimes, when we look at a problem, it is obvious which practice is most helpful. At other times, we will decide on a practice to use during class explorations and discussions. After completing each problem, ask the following: • What mathematics have I learned by solving this problem? • What Mathematical Practices were helpful in learning this mathematics? MATHEMATICAL PRACTICES AND HABITS OF MIND EXAMPLES As students worked on the problems in this investigation, they used prior mathematical knowledge to make sense of them. They also applied Mathematical Practices to solve the problems. In Problem 2.3, Marko claimed that his group noted that the angle with measure c and the 150º angle were adjacent angles and supplementary angles. “Their sum is 180º. I wrote 150º + c =180º. So c is 30º. Another person in our group started with vertical angles. She noted that the angle with measure b and the 150º angle were vertical. She wrote 150º = b. So b is also 1500. We were both correct.” L1 L2 L3 a b c g d f e 150° • What Mathematical Practice(s) did Marko’s group use to solve this problem? Answers will vary. Students may relate their work to MP1 (Make sense of problems and persevere in solving them) or MP6 (Attend to precision). For Problem 3.1, Jake’s group noticed there was a relationship between the number of sides of a regular polygon and the measure of each angle. “We recognized a pattern in our table and were able to use it to fill in the table for polygons with 7, 8, 9, and 10 sides. Then, we wrote a formula to find the measure of each angle for any regular polygon with n sides.” Correlations 241 SAMPLE
• What Mathematical Practice(s) did Jake’s group use to solve this problem? Answers will vary. Students may relate their work to MP1 (Make sense of problems and persevere in solving them) or MP8 (Look for and express regularity in repeated reasoning). References National Governors Association Center for Best Practices and Council of Chief State School Officers. Common Core State Standards Initiative: Common Core State Standards for Mathematics. Washington, DC: Authors, 2010. National Council of Teachers of Mathematics. Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: Authors, 2014. ASSESSMENT CORRELATION WITH COMMON CORE STATE STANDARDS OF MATHEMATICS (CCSSM) Checkup 1 Questions CCSSM Student Edition Problem Correlation 1–6 7.G.A.2 1.1, 1.2 Partner Quiz Questions CCSSM Student Edition Problem Correlation 1 7.G.A.2, 7.G.B.5 1.1, 2.1, 2.2, 2.3 2 7.G.A.2 2.1 3 7.G.A.2 2.1 4 7.G.B.5 2.3 5 7.G.A.2, 7.EE.B.3, 7.EE.B.4 2.1, 2.2 Unit Test Questions CCSSM Student Edition Problem Correlation 1 7.G.B.5 2.2, 2.3 2 7.G.A.2 1.1 3 7.G.B.5 2.3 4 7.G.B.5, 7.EE.B.4 2.3, 3.1 5 7.G.B.5 2.3, 3.3 6 7.G.A.2 1.1, 2.1, 2.2, 3.4 242 Shapes and Designs SAMPLE