1. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Bridging the Gap Grades 6-9

2. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Session Objectives Learn and experience efficient ways to assess and remediate prerequisite knowledge: Assessing Conceptual Understanding Remediating Conceptual Understanding Gaps Assessing Fluency Remediating Fluency Gaps

3. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Assessing Prerequisite Knowledge – Conceptual Understanding The 4 basic operations and their models Properties of operations The equal sign The inequality signs Fractions; operations with fractions; fractions as division Operations with negative numbers Exponentiation Systems of Linear Equations

4. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations - Addition

5. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Remediation Strategy Assess Discuss Repeat

6. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations - Addition Addition means putting together Model 2: Comparison Model, e.g. “3 more than” Give Joe 5 Starburst. Now give Max enough Starbursts so that he has 3 more than Joe. How many does Max have? Give students 17 Starbursts and ask: show me how to split up these Starbursts so that Max gets 3 more than Joe. If Max has 1.7 more feet of string than Joe and all together they have 9.3 feet of string, how much string does each boy have?

7. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations - Addition Addition means putting together Model 2: Comparison Model, e.g. “3 more than” Max’s string 9.3 feet Joe’s string 2 units = 7.6 feet; 1 unit = 3.8 feet If Max has 1.7 more feet of string than Joe and all together they have 9.3 feet of string, how much string does each boy have? 1.7 feet

8. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Remediation Strategy Assess Discuss and/or Model (Concrete  Pictorial  Visualization) Repeat

9. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations - Subtraction

10. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations - Subtraction Subtraction means taking apart or taking away Model 2: Comparison, e.g. “3 fewer than” Give Joe 12 Starburst. Now give Max enough Starbursts so that he has 3 fewer than Joe. How many does Max have? Give students 17 Starbursts and ask: show me how to split up these Starbursts so that Max gets 3 fewer than Joe. If Max has 1.7 less feet of string than Joe and all together they have 9.3 feet of string, how much string does each boy have?

11. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations - Subtraction Subtraction means taking apart or taking away Model 2: Comparison, e.g. “3 fewer than” Max’s string 9.3 feet Joe’s string If Max has 1.7 less feet of string than Joe and all together they have 9.3 feet of string, how much string does each boy have? 1.7 feet

12. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations – Multiplication

13. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations – Multiplication Multiplication means putting together equal groups Model 2: Array model Is it true that 5 3 will have the same value as 3 5? How can I prove it will work for any two numbers I pick? Why should it be obvious that the number of dots here: Should be the same as the number here?

14. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations – Multiplication Multiplication means putting together equal groups Model 3: Area model What does area mean? How do I find it? Write me a word problem where I am trying to find the area of something.

15. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations – Multiplication Multiplication means putting together equal groups Model 4: Comparison model Amy has 5 times as many Starbursts as Meg. They have 24 Starbursts all together. How many Starbursts does each girl have? Meg 24 Amy 6 units = 24; 1 unit = 4

16. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations – Division

17. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations – Division Another approach to differentiating between first two models: Act out the process of the problem you wrote (for model 1). Let’s compare that to my problem. Act out the process of the problem I wrote. What do you notice? There are two ways to perform the division problem, 12 ÷3, grabbing groups of 3 (repeated subtraction), vs. giving one to each of 3 groups until there are none left. Write me a problem where you are asking to find the number of groups (not the number in each group).

18. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations – Division To reinforce the understanding of the “how many groups” model, change our language: 12÷3 Instead of “Twelve divided by three”… …“How many three’s are in twelve?” How many one-halves are in 12?

19. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions The 4 Basic Operations – Division Division means separating into equal groups Model 3: Array model – Finding the number of rows given the number of columns (or vice versa). Model 4: Area model – Finding a missing side length, given the area and a side length. Write me a word problem about area of a rectangle in which you need to find 12 ÷ 3 to solve the problem.

20. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions

21. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions

22. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Equal and Inequality Signs The equal sign What value would make this statement true? 11 – 5 = + 2 The inequality signs Give me a value that would make this statement true: 14 – 6 < + 3

23. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Fractions

24. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Fractions

25. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Fractions

26. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Dividing a Fraction by a Fraction Write me a word problem where you have to compute 2/3 ÷1/6. Precede this challenge with the development on the next slide.

27. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Dividing a Fraction by a Fraction A Progression for students: Use tape diagram to demonstrate the answer to the following: How many ½’s are in 6? How many 1/3’s are in 6? How many 1/3’s are in 1? How many 1/3’s are in 2/3? How many 1/3’s are in ½? How many 5/2’s are in 2/3?

28. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Operations with Negatives Why should 4 – (-3) = 7 be true? Why is (5)(-3) negative? Why is (-5)(3) negative? Why should a negative x a negative = a positive?

29. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Exponentiation Make up a word problem… … in which the expression 1.1 3 will be used in solving it. … in which the expression 1.1 5 will be used in solving it. … in which the expression 3 5 will be used in solving it.

30. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Solving Systems of Equations Consider the following question:

31. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Solving Systems of Equations Here is the solution according to the answer key for the test:

32. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Solving Systems of Equations A-REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

33. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Solving Systems of Equations Here is a graph of the two equations:

34. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Correcting the Misconception

35. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Correcting the Misconception

36. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Remediating Prerequisite Knowledge First address conceptual understanding: Conceptual Questioning / Discussion / Models 15 minute sessions or whole class sessions? All at the beginning of the year or throughout the year? Then address fluency: Rapid White-Board Exchanges (first) Sprints (second, if feasible)

37. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Fluency – Rapid White Board Exchanges Do 10-20 problems depending on how long each problem will take. Fluency work should take from 5-12 minutes of class All students will need a personal white board, white board marker, and a means of erasing their work. Prepare/post the questions in a way that allows you to reveal them to the class one at a time.

38. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Fluency – Rapid White Board Exchanges

39. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Fluency – Rapid White Board Exchanges Reveal or say the first problem followed by “Go”. Students work the problem on their boards and hold their work up for their teacher to see their answers as soon as they have the answer ready. Give immediate feedback to each student, pointing and/or making eye contact and affirm correct with, “Good job!”, “Yes!”, or “Correct!”, or gentle guidance for incorrect work such as “Look again,” “Try again,” “Check your work,” etc. If many students struggled, go through the solution of that problem as a class before moving on to the next problem in the sequence.

40. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Functions Fluency – Sprints Your class is ready for a sprint when students are able to make it through a set of rapid white board exchanges in which every student got some correct, and only one or two needed to be done as a class. Sprints are done in pairs – both sprints have very similar problems that progress from easy enough that all students will get some correct in the first ¼ to hard enough that even the best students are challenged in the last ¼. Typically 44 problems on a sprint. Always 60 seconds to complete one sprint. Follow the guidance in How to Implement A Story of Units and/or the 6-8 Fluencies