1. Bridging the Gap Grades 6-9
August Network Team Institute
TIME ALLOTTED FOR THIS SLIDE:
3 minutes
MATERIALS NEEDED:
Powerpoint
Projector
Participant Binder
Bridging the Gap
Grades 6-9

2. Bridging the Gap Grades 6-9
August Network Team Institute
TIME ALLOTTED FOR THIS SLIDE:
2 minutes (5)
MATERIALS NEEDED:
none
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
August Network Team Institute
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 Equations
I have outlined on this slide some conceptual understanding that is foundational to 6-9 on the whole.
I’ve chosen to address conceptual understanding gaps before fluency gaps, because it is important to establish conceptual understanding before addressing fluency with the related skills.

4. The 4 Basic Operations - Addition
August Network Team Institute
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 𝑥 + 3 to solve the problem.
…in which you need to use the expression 𝑥 + 𝑦 to solve the problem.
What does addition mean? Allow for participant response
Addition means putting together, putting together like objects or like quantities.
Each of the next few slides shows a progression of possible problems to give to students based on what grade level they are entering and/or where you feel you can most appropriately access your specific student population. The most basic version of this question is simply, “Write me a word problem in which you need to add to solve the problem.” You’ll want to use variations of each question that will assess the student’s understanding of this model and this operation without seeming overly simplex.

5. Remediation Strategy
Assess
Discuss
Repeat

6. The 4 Basic Operations - Addition
August Network Team Institute
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?
The comparison model of addition (and subtraction) represents students earliest exposure to algebraic thinking.

7. The 4 Basic Operations - Addition
August Network Team Institute
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
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?
The comparison model of addition (and subtraction) represents students earliest exposure to algebraic thinking.
1.7 feet

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

9. The 4 Basic Operations - Subtraction
August Network Team Institute
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 – 3.5 to solve the problem.
… in which you need to use the expression 𝑥 – 4 to solve the problem.
… in which you need to use the expression 𝑥 – 𝑦 to solve the problem.

10. The 4 Basic Operations - Subtraction
August Network Team Institute
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
August Network Team Institute
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?
The comparison model of addition (and subtraction) represents students earliest exposure to algebraic thinking.
1.7 feet

12. The 4 Basic Operations – Multiplication
August Network Team Institute
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 12𝑥 to solve the problem.
… in which you need to use the expression 𝑥𝑦 to solve the problem.
They may use one of the more advanced models of multiplication and that’s ok.

13. The 4 Basic Operations – Multiplication
August Network Team Institute
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 of 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
August Network Team Institute
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
August Network Team Institute
The 4 Basic Operations – Multiplication
Multiplication means putting together equal groups
Model 4: Comparison model
Amy and Meg have 24 Starbursts all together. Amy has 5 times as much as Meg. How many Starbursts does each girl have?

16. The 4 Basic Operations – Division
August Network Team Institute
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 12÷ 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
August Network Team Institute
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
August Network Team Institute
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?”

19. The 4 Basic Operations – Division
August Network Team Institute
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. Bridging the Gap Grades 6-9
August Network Team Institute
1987 McGraw-Hill

21. Bridging the Gap Grades 6-9
August Network Team Institute

22. Equal and Inequality Signs
August Network Team Institute
Equal and Inequality Signs
The equal sign
What value would make this statement true?
11 – 5 =
The inequality signs
Give me a value that would make this statement true:
14 – 6 <
What value would make this a true statement? 11 – 5 =
Show me a value that would make this a true statement: 14 – 6 < + 3

23. August 2014 Network Team Institute
Fractions
What is ?
Write me a word problem that requires computing 1 2 − in order to solve the problem.

24. August 2014 Network Team Institute
Fractions
What is of 60?
Write me a word problem that requires finding 2 3 ∙60 in order to solve the problem.
What is 3 ÷ 5 (write your answer as a fraction)?
How many 1 3 ’s are in 5?
Write me a word problem that requires finding 5 ÷ 1 3 .

25. Fractions What is 3 ÷ 5 (write your answer as a fraction)?
August Network Team Institute
Fractions
What is 3 ÷ 5 (write your answer as a fraction)?
How many 1 3 ’s are in 5?
Write me a word problem that requires finding 5 ÷ 1 3 .

26. Dividing a Fraction by a Fraction
August Network Team Institute
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.
Do we need a common denominator to divide a fraction by a fraction?

27. Dividing a Fraction by a Fraction – Measurement Model
August Network Team Institute
Dividing a Fraction by a Fraction – Measurement Model
A Progression for students: Use tape diagram to demonstrate the answer to the following:
How many 1/2’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 1/2?
How many 2/5’s are in 3/4?
How many 5/2’s are in 2/3?
Do we need a common denominator to divide a fraction by a fraction?

28. Dividing a Fraction by a Fraction – Partitive Model
August Network Team Institute
Dividing a Fraction by a Fraction – Partitive Model
A Progression for students: Use tape diagram to demonstrate the answer to the following:
How many 1/2’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 1/2?
How many 2/5’s are in 3/4?
How many 5/2’s are in 2/3?
Do we need a common denominator to divide a fraction by a fraction?

29. Operations with Negatives
August Network Team Institute
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?

30. Bridging the Gap Grades 6-9
August Network Team Institute
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.

31. Solving Systems of Equations
August Network Team Institute
Solving Systems of Equations
Consider the following question:
Before we move in to topic D, I’d like to spend some time studying what we are doing when we solve a system of equations by elimination. Have a look at this example (from MAO test).

32. Solving Systems of Equations
August Network Team Institute
Solving Systems of Equations
Here is the solution according to the answer key for the test:

33. Solving Systems of Equations
August Network Team Institute
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.

34. Solving Systems of Equations
August Network Team Institute
Solving Systems of Equations
Here is a graph of the two equations:

35. Correcting the Misconception
August Network Team Institute
Correcting the Misconception
Sketch the graph of each equation in the following system: 3𝑥 −𝑦=−6 𝑥+2𝑦=5 Replace one equation with the sum of the first equation and the second equation. Sketch a graph of the new equation.

36. Correcting the Misconception
August Network Team Institute
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 𝑥 = 3. The graph of that equation passes through the solution point, therefore the solution point must have an 𝑥-coordinate of 3. This is helpful, let’s be strategic about how we replace one equation with the sum of itself and a multiple of the other”

37. Remediating Prerequisite Knowledge
August Network Team Institute
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)
Conc Und: Questioning / debating / discussing – tie this back in with out asking questions accesses the brain differently than a passive review of information… how in assessing the conceptual understanding we are simultaneously remediating it.
There are two techniques for working on fluency of skills. One of them is a Sprint… how many of you have used a Sprint in the classroom before? Heard of a Sprint?
You don’t want to hand out a sprint on a topic that you haven’t first assessed with a white-board exchange. A sprint builds speed and accuracy of a skill for which there already exists a basic level of competence.
I brought some Sprints with me…
Fluency: Rapid White Board Exchanges progressing to Sprints.

38. Fluency – Rapid White Board Exchanges
Do 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.

39. 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. 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