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1. Bridging the Gap Grades 6-9

2. 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. 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. The 4 Basic Operations - Addition • Addition means putting together (like objects or like quantities) • Model 1: Part-part whole: finding the whole Write me a word problem… … in which you need to find to solve the problem. …in which you need to use the expression to solve the problem. …in which you need to use the expression to solve the problem.

5. Remediation Strategy • Assess • Discuss • Repeat

6. 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. The 4 Basic Operations - Addition • 1.7 feet • 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?

8. Remediation Strategy • Assess • Discuss and/or Model • (Concrete  Pictorial Visualization) • Repeat

9. The 4 Basic Operations - Subtraction • Subtraction means taking apart or taking away • Model 1: Part-part whole: finding one part Write me a word problem… …in which you need to find 12.12 – 3.5 to solve the problem. … in which you need to use the expression to solve the problem. … in which you need to use the expression to solve the problem.

10. 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. The 4 Basic Operations - Subtraction • 1.7 feet • 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?

12. The 4 Basic Operations– Multiplication • Multiplication means putting together equal groups • Model 1: Equal groups model Write me a word problem… …in which you need to find 12 • 3 to solve the problem. …in which you need to use the expression to solve the problem. … in which you need to use the expression to solve the problem.

13. 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. 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. 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. The 4 Basic Operations– Division • Division means separating into equal groups • Write me a word problem in which you need to find 12 ÷ 3 to solve the problem • Write me a word problem in which you need to compute to solve the problem. Model 1: Finding the number in each group (knowing the number of groups) • Model 2: Finding the number of groups (knowing the number in each group)

17. 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. 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. 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. 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

21. Fractions • What is ? • Write me a word problem that requires computing in order to solve the problem.

22. Fractions • What is of 60? • Write me a word problem that requires finding in order to solve the problem.

23. Fractions • What is 3 5? (Write your answer as a fraction.) • How many ’s are in 5? • Write me a word problem that requires finding .

24. 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.

25. 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?

26. 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?

27. Exponentiation Make up a word problem… … in which the expression 1.13 will be used in solving it. … in which the expression 1.15 will be used in solving it. … in which the expression 35 will be used in solving it.

28. Solving Systems of Equations Consider the following question:

29. Solving Systems of Equations Here is the solution according to the answer key for the test:

30. 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.

31. Solving Systems of Equations Here is a graph of the two equations:

32. Correcting the Misconception Sketch the graph of each equation in the following system: Replace one equation with the sum of the first equation and the second equation. Sketch a graph of the new equation.

33. Correcting the Misconception “The graph of the new equation will also pass through (or contain) the intersection point (the solution point). Suppose the new equation is . The graph of that equation passes through the solution point, therefore the solution point must have an -coordinate of . This is helpful, let’s be strategic about how we replace one equation with the sum of itself and a multiple of the other”

34. 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)

35. 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.

36. Fluency – Rapid White Board Exchanges

37. 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.

38. 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