Math practices Secondary

1. Math practicesSecondary Dr. Frank Rodriguez RISE Educational Services

2. Mathematical Practices Overview • There are 8 Mathematical Practices that are consistent from kindergarten through 12th grade. • The mathematical practices are presented in the beginning of the standards handbook. They are not explicitly stated within the standards. Teachers will have to decide when and how to teach and practice these skills.

3. When To Teach the Math Practice Standards • “The MP standards must be taught as carefully and practiced as intentionally as the Mathematical Content Standards. Neither should be isolated from the other; effective mathematics instruction occurs when these two halves of the CCSSM come together in a powerful whole.” California’s Common Core Standards for Mathematics http://www.cde.ca.gov/re/cc/

4. Overview Continued • The Standards for Mathematical Practice describe varieties of expertise students should be taught. These practices are based on important “processes and proficiencies” with longstanding importance in mathematics education. The practices were created from two sources: -http://www.corestandards.org/Math/Practice

5. Overview Continued Standards for Mathematical Practice • The second source is the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up • The first source is the National Council of Teachers of Mathematics (NCTM) process standards of problem solving, reasoning and proof, communication, representation, and connections. -http://www.corestandards.org/Math/Practice

6. Standards for Mathematical Practice The Common Core State Standards for Mathematics Kindergarten – Grade 5

7. Math Practices and Standards Connection Standards for Mathematical Content: Skills and understandings students will learn Identified by grade level or course Standards for Mathematical Practice: Processes and proficiencies that students show when engaged in mathematics Identified for students across all grade levels (K–12) Brokers of Expertise State of California Department of Education: CCSS Mathematics: K-12 Standards for Mathematical Practice. http://myboe.org/portal/default/Content/Viewer/Content?action=2&scId=306591&sciId=11787

8. Making Math Practices Practical

9. 1) Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

10. MP 1: Make Sense of Problems and Persevere in Solving Them • Explain the meaning of a problem and look for entry points • Analyze givens, constraints, relationships, and goals • Plan a solution pathway • Monitor and evaluate progress • Check answer using a different method (asking does this make sense) • Understand the approaches of others and identify correspondences between different approaches

12. Learning Objectives Math Practice 1 Repeating Topics • Create a coherent representation of the problem (consider units involved, attend to meanings of quantities, know and use different properties of operations and objects). (MP 2) • Evaluate the reasonableness of intermediate results while solving a problem (MP 8) • Interpret results and decide whether the results make sense, improving the model if necessary. (MP 4) • Listen to or read the arguments of others, decide whether they make sense, and ask questions to clarify or improve the argument (MP 3) • Explain the meaning of a problem and look for entry points • Analyze givens, constraints, relationships, and goals • Plan a solution pathway • Monitor and evaluate progress • Check answer using a different method (asking does this make sense) • Understand the approaches of others and identify correspondences between different approaches

13. Math Practice Learning Objectives 1. Analyze the information in a problem 2. Plan a pathway for solving a problem 3. Represent relationships graphically 4. Analyze relationships mathematically to draw conclusions 5. Identify and explain mathematical patterns 6. Solve problems using skills you know 7. Evaluate the reasonableness of intermediate results 8. Analyze when to use grade level appropriate tools, recognizing insights to be gained and limitations 9. Attend to details while solving a problem 10. Make conjectures about a problem 11. Explain your reasoning 12. Analyze the work/arguments/reasons of others

14. Lessons + Layered Activities • Content Standards • Math Practices • BBDI lessons with content standards from previously taught lessons • Pacing • BBDI lessons with MPs from previously taught lessons

15. How Do I Make This Work in My Class? • “In the higher mathematics courses, the levels of sophistication of each MP standard increases as students integrate grade appropriate mathematical practices with the content standards.” www.cde.ca.gov Mathematics Framework: Overview of the Standards Chapter, Pg 24 Teach and practice the MPs at an appropriate level for your grade/students. The application and expectation may differ from grade to grade, but the students should still be held accountable for practicing the MPs in a way that makes sense for their age group.

16. What does this look like in my classroom?

17. Sample Year at a Glance

18. Sample Year at a Glance Cont.

19. Sample Year at a Glance Cont.

20. Share with a partner what math content you will be teaching in November. Discuss what Math Practice you would want to teach before teaching the content.

21. Use of Sentence Stems and Frames www.cde.ca.gov: Mathematics Framework: Overview of the Standards Chapter, Page 14 of 27

22. You wouldn’t want to start a trip without planning what clothes to pack, how much money to bring, what music to listen to….. Similarly, you wouldn’t want to start a math problem without planning it out, especially a complicated word problem. You want to plan your path before you start.

23. Objective Plan a pathway for solving a problem

24. Big Idea What: A solution pathway is a series of steps you plan in order to meet the goal of a word problem. Why: You should plan your pathway before you start out. How: Use a flow map or a K-W-L chart to create your pathway.

25. Model Amanda and her best friend found some money buried in a field. They split the money evenly, each getting $24.28. How much money did they find?

26. Model Steps:

27. Model Steps:

28. Model Steps:

29. Model Steps:

30. Model Maria bought seven boxes. A week later, half of her boxes were lost in a fire. She now has only 22 boxes left. How many boxes did she start with? Steps 1. Read the problem 2. Determine goal 3. Determine givens 4. Determine operations/relationships 5. Determine

31. Your turn… Steps 1. Read the problem 2. Determine goal 3. Determine givens 4. Determine operations/relationships 5. Determine A train traveled 250 miles in 2 hours. If it traveled at the same rate of speed, how long would it take the train to travel 600 miles?

32. Your turn… Vince uses 2/3 cup of butter for each batch of cookies he makes. He has 5 ½ cups of butter. He estimated 6 x 3/2 =9, so he can make about 9 batches of cookies with 5 ½ cups of butter. Is his estimate as overestimate or an underestimate? How do you know? Steps 1. Read the problem 2. Determine goal 3. Determine givens 4. Determine operations/relationships 5. Determine

33. Closure Questions 1. What did we learn to do today? 2. What is a solution pathway? Why does it matter? 3. What is the process for determining a solution pathway? 4. Why does it matter that you determine the goal of the problem first?

34. Independent Practice • 6-8 word problems

35. “Just because” or “Because I said so” is not a reason you would want to hear from your teacher, or your parents. In the same way, your math teacher does not want to hear “just because” when they ask you to explain how you found your answer. So today, we are going to learn how to ……

36. Objective Explain your reasoning after solving a problem

37. Big Idea What: When you are asked to explain your reasoning, it means one of two things: • Explain your process (how you did the problem- what operations you chose, what order) • Justify your response (why do you think the process is correct? Why does your answer make sense?)

38. Big Idea Why: You will frequently be asked to either explain your process or justify your response. This is because your teacher wants to make sure that you have thought through what you’re doing, and that you’ve stopped and asked yourself whether it makes sense.” Good mathematicians are able to explain their process and reasoning.

39. Model • Steve’s age is twice Bart’s age plus 3. Steve is 13 years old. How old is Bart?

40. Model • Analyze problem and determine solution pathway • Solve and check for reasonableness as you are going • Determine if you are being asked to explain process or justify reasoning. • Use appropriate sentence frame: First, I ____________, then I ……… The reason that I believe I am correct is…….

41. Model • A fundraiser is being held to raise money for a new school playground. Of every $20 raised, 416 will be spend on playground equipment. If the goal of the fundraiser is $500.00 for playground equipment, how much total money will it need to raise? Steps Analyze problem and determine solution pathway Solve and check for reasonableness as you are going Determine if you are being asked to explain process or justify reasoning. Use appropriate sentence frame: First, I ____________, then I ……… The reason that I believe I am correct is…….

42. Your turn… Caleb and Winona both travel by car to their friend’s home. The distance Winona traveled was 124 miles less than twice the distance Caleb traveled. If Winona traveled 628 miles, how far did Caleb travel? Steps Analyze problem and determine solution pathway Solve and check for reasonableness as you are going Determine if you are being asked to explain process or justify reasoning. Use appropriate sentence frame: First, I ____________, then I ……… The reason that I believe I am correct is…….

43. Your turn… Stephan is planning a hiking trip at Kings Canyon National Park. He plan to hike 14 miles every 2 days. If he hikes 42 miles, how many days will he hike? Steps Analyze problem and determine solution pathway Solve and check for reasonableness as you are going Determine if you are being asked to explain process or justify reasoning. Use appropriate sentence frame: First, I ____________, then I ……… The reason that I believe I am correct is…….

44. Closure Questions • What did we learn to do today? 2. What does it mean to explain your process? How is that different than justifying your response? 3. What are the steps in explaining or justifying?

45. Independent Practice • 6-8 word problems