1. 2010 VDOE Mathematics Institute
Grades 6-8
Focus: Patterns, Functions, and Algebra

2. Content Focus Key changes at the middle school level:
Properties of Operations with Real Numbers
Equations and Expressions
Inequalities
Modeling Multiplication and Division of Fractions
Understanding Mean: Fair Share and Balance Point
Modeling Operations with Integers

3. Supporting Implementation of 2009 Standards
Highlight key curriculum changes.
Connect the mathematics across grade levels.
Model instructional strategies.

4. Properties of Operations

5. Properties of Operations: 2001 Standards
7.3 The student will identify and apply the following properties of
operations with real numbers:
a) the commutative and associative properties for addition and
multiplication;
b) the distributive property;
c) the additive and multiplicative identity properties;
d) the additive and multiplicative inverse properties; and
e) the multiplicative property of zero.
8.1 The student will
a) simplify numerical expressions involving positive exponents,
using rational numbers, order of operations, and properties of
operations with real numbers;
3.20a&b; 4.16b
5.19
6.19a
6.19c
6.19b

6. Properties of Operations: 2009 Standards
3.20
b) Identify examples of the identity and commutative properties for addition and multiplication.
4.16b
b) Investigate and describe the associative property for addition and multiplication.
5.19
Investigate and recognize the distributive property of multiplication over addition.
6.19
Investigate and recognize
a) the identity properties for addition and multiplication;
b) the multiplicative property of zero; and
c) the inverse property for multiplication.
7.16
Apply the following properties of operations with real numbers:
a) the commutative and associative properties for addition and multiplication;
b) the distributive property;
c) the additive and multiplicative identity properties;
d) the additive and multiplicative inverse properties; and
e) the multiplicative property of zero.
8.1a
a) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;
8.15c
c) identify properties of operations used to solve an equation.
focused on identifying; that has now moved down to grades 3-6, and the 2009 SOL focus in middle school is on applying the properties.
8.15c represents a new (and appropriate) change that is representative of the overall changes in the 2009 Standards.

7. 3.20a&b: Identity Property for Multiplication
x,÷ 1 2 3 4 5 6 7 8 9 10 11 12 14 16 18 20 22 24 15 21 27 30 33 36 28 32 40 44 48 25 35 45 50 55 60 42 54 66 72 49 56 63 70 77 84 64 80 88 96 81 90 99
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The first row and column of products in a multiplication chart illustrate the identity property.
A multiplication chart can be used for more than just finding products:
It’s a natural extension of the 100’s chart that students use in grades K-2.
It can help students understand the meaning of multiplication and its properties if teachers highlight its structure.

8. 3.20a&b: Commutative Property for Multiplication
x,÷ 1 2 3 4 5 6 7 8 9 10 11 12 14 16 18 20 22 24 15 21 27 30 33 36 28 32 40 44 48 25 35 45 50 55 60 42 54 66 72 49 56 63 70 77 84 64 80 88 96 81 90 99
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Why does the diagonal of perfect squares form a line of symmetry in the chart?
This slide is connected to the following slide.
Might want to make sure that participants are in fact familiar/comfortable with this idea of the perfect squares as a symmetry line.

9. 3.20a&b: Commutative Property for Multiplication
x,÷ 1 2 3 4 5 6 7 8 9 10 11 12 14 16 18 20 22 24 15 21 27 30 33 36 28 32 40 44 48 25 35 45 50 55 60 42 54 66 72 49 56 63 70 77 84 64 80 88 96 81 90 99
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The red rectangle (4x6) and the blue rectangle (6x4) both cover an area of 24 squares on the multiplication chart.
If you imagine the multiplication as Geoboard, you could stretch a rubber band from the upper-left corner (0 x 0 = 0) to form rectangles of different areas.
The number in the lower right-hand corner of each rectangle corresponds to the area of the rectangle.
For any pair of factors, it doesn’t matter if you stretch horizontally first or vertically first – the area (product) will be the same, just as if you had rotated the rectangle the way we did with the base ten blocks.

10. 6.19: Multiplicative Property of Zero
x,÷ 1 2 3 4 5 6 7 8 9 10 11 12 14 16 18 20 22 24 15 21 27 30 33 36 28 32 40 44 48 25 35 45 50 55 60 42 54 66 72 49 56 63 70 77 84 64 80 88 96 81 90 99
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6 x 0 = 0
0 x 6 = 0
Area multiplication is based on rectangles. If one factor is zero, then the number sentence doesn’t describe a rectangle, it describes a line segment, and the product (the “area”) is zero.
Highlights the relationship between the C&E and Geometry strands:
One-dimensional line segments have no area; two-dimensional rectangles define an area (a product) based on the lengths of their sides (their factors).

11. Meanings of Multiplication
For 5 x 4 = 20…
Repeated Addition: “4, 8, 12, 16, 20.”
Groups-Of: “Five bags of candy with four pieces of candy in each bag.”
Rectangular Array: “Five rows of desks with four desks in each row.”
Rate: “Dave bought five raffle tickets at $4.00 apiece.” or “Dave walked four miles per hour for five hours.”
Comparison: “Alice has 4 cookies; Ralph has five times as many.”
Combinations: “Cindy has five different shirts and four different pairs of pants; how many different shirt/pants outfits can she make?”
Area: “Ricky buys a rectangular rug 5 feet long and 4 feet wide.”
Multiplication is typically introduced as repeated addition or “skip counting”, but students need to develop progressively more sophisticated understandings of multiplication as they move through elementary school and into the middle grades.
“Groups-of” and “Area” conceptions of multiplication are flexible models that can be easily extended from whole numbers to rational numbers.
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA Publishing, 1998, Chapter 5.

12. 3.6: Represent Multiplication Using an Area Model
Use your base ten blocks to represent
3 x 6 = 18
The area model is closely related to the “groups of” and “array” meanings of multiplication (the applet even references “3 groups of 6”), but it has two important distinctions:
It’s a measurement.
It always uses square units.
National Library of Virtual Manipulatives – Rectangle Multiplication

13. 3.6: Represent Multiplication Using an Area Model
Or did yours look like this?
Rotating the rectangle doesn’t change its area.
Commutative Property:
The area model provides a nice illustration of the commutative property:
Changing the orientation of the rectangle doesn’t change the length of its sides (the factors) or its area (the product).
National Library of Virtual Manipulatives – Rectangle Multiplication

14. 3.6: Represent Multiplication Using an Area Model
Use your base ten blocks to represent
5 x 14 = 70
What is the area of the red inner rectangle?
The area model is closely related to the “groups of” and “array” meanings of multiplication (the applet even references “3 groups of 6”), but it has two important distinctions:
It’s a measurement.
It always uses square units.
What is the area of the blue inner rectangle?
National Library of Virtual Manipulatives – Rectangle Multiplication

15. 5.19: Distributive Property of Multiplication
3.6: Represent Multiplication Using an Area Model
How could students record the area of the 5 x 14 rectangle?
5 x 4 = 20 14
x 5
5 x 10 → 50
5 x 4 → + 20 70
5 x 10 = 50
The area model is closely related to the “groups of” and “array” meanings of multiplication (the applet even references “3 groups of 6”), but it has two important distinctions:
It’s a measurement.
It always uses square units.

16. 5.19: Distributive Property of Multiplication Over Addition
Understanding the Standard: “The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products (e.g., 3(4 + 5) = 3 x x 5, 5 x (3 + 7) = (5 x 3) + (5 x 7); or (2 x 3) + (2 x 5) = 2 x (3 + 5).”
Essential Knowledge & Skills:
“Investigate and recognize the distributive property of whole
numbers, limited to multiplication over addition, using
diagrams and manipulatives.”
“Investigate and recognize an equation that represents the
distributive property, when given several whole number
equations, limited to multiplication over addition.”
The area model is closely related to the “groups of” and “array” meanings of multiplication (the applet even references “3 groups of 6”), but it has two important distinctions:
It’s a measurement.
It always uses square units.
National Library of Virtual Manipulatives – Rectangle Multiplication

17. 5.19: Distributive Property of Multiplication Over Addition
Use base ten blocks to build a 12 x 23 rectangle.
The traditional multi-digit multiplication algorithm finds the sum of the areas of two inner rectangles.
Most participants will be familiar with only the traditional algorithm, so that point of this slide (and the next two) is to connect the area model to the traditional algorithm.
*If you download the free trial version of the NLVM software ( you can show other examples using this applet without having access to the Internet.
National Library of Virtual Manipulatives – Rectangle Multiplication

18. 5.19: Distributive Property of Multiplication Over Addition
The partial products algorithm finds the sum of the areas of four inner rectangles.
Look familiar?
F.irst
O.uter
I.nner
L.ast
Connect to the previous slide.
Specifically ask participants to discuss at their tables all of the connections that they can see. Have them share out with the whole group.
Acknowledging the connection to FOIL when it comes up (it’s almost guaranteed that someone will bring it up, but we should be sure to if no one else does) is a powerful way to increase participant “buy in” for using the area model and partial products as a foundational instructional model.
National Library of Virtual Manipulatives – Rectangle Multiplication

19. Strengths of the Area Model of Multiplication
Illustrates the inherent connections between multiplication and division:
Factors, divisors, and quotients are represented by the
lengths of the rectangle’s sides.
Products and dividends are represented by the area of
the rectangle.
Versatile:
Can be used with whole numbers and decimals (through
hundredths).
Rotating the rectangle illustrates commutative property.
Forms the basis for future modeling: distributive
property; factoring with Algebra Tiles; and Completing
the Square to solve quadratic equations.
The last bullet is the key point.

20. 4.16b: Associative Property for Multiplication
Use your base ten blocks to build a rectangular solid
2cm by 3cm by 4cm
Base: 3cm by 4cm; Height: 2cm
Volume: 2 x (3 x 4) = 24 cm3
The associative property states that the grouping of the factors doesn’t affect the product.
Possible confusion: There are actually three different possible orientations of this rectangular solid, but the associative property only deals with two of them at a time. We may need to address this by showing how it would work with the missing orientation from our sample problem.
e.g., (3 x 2) x 4 and 3 x (2 x 4)
*Participants may confuse this idea with the commutative property, but we are not talking here about 2 x 3 x 4 = 2 x 4 x 3.
Associative Property: The grouping of the factors does not affect the product.
Base: 2cm by 3cm; Height: 4cm
Volume: (2 x 3) x 4 = 24 cm3
National Library of Virtual Manipulatives – Space Blocks

21. Expressions and Equations

22. A Look At Expressions and Equations
A manipulative, like algebra tiles, creates a concrete foundation for the abstract, symbolic representations students begin to wrestle with in middle school.
Tell participants that they will use the algebra tiles on their tables as we work with expressions and equations. 22

23. What do these tiles represent?
1 unit
Area = 1 square unit
Tile Bin
Unknown length, x units
Area = x square units
1 unit
x units
Note to presenters: It would be good to mention that with an interactive white board works nicely with this type of exploration. It might also be good to tell the audience that no special program was used to create the representations of the tiles.
We want to start by establishing the value of the algebra tiles. (~10 minutes) 
Many students will relate these tile to the base ten blocks and want to name them very concretely with a number value.
We start with the yellow tile which does represent one unit.
We declare that it is one unit tall and one unit wide, so it has an area of one square unit.
Next, we move to the green tile. It is one unit tall. However, its width can vary.
The tiles shown above are about 2.5 units long. The manipulatives our participants will have are about 5.1 units long.
The width is unknown, so we name it with a variable, x. The green tile is one unit by x units, so it has an area of x square units.
(The Multiplicative Identity can be mentioned here.)
After this, we take a look at the blue tile. It is x units tall and x units wide.
(Many students will still go back to the yellow tiles as a unit of measurement.)
The blue tile has an area of x2 square units.
Finally, discuss that the red tiles are used to denote negative quantities.
x units
Area = x2 square units
The red tiles denote negative quantities.

24. Modeling expressions x + 5 5 + x Tile Bin
Our next idea is to focus us modeling algebraic expressions. (~20 minutes)
Now that we have established the value of each tile, have participants model each expression using tiles.
a) x + 5 (one green tile and 5 yellow tiles)
b) 5 + x (5 yellow tiles and 1 green tile)
These two expressions represent the same group of tiles… Commutative Property of Addition 24

25. Modeling expressions x - 1 Tile Bin c) x - 1
Start with x and take away one.  (Put up one green tile.)
We don't have one to take away. We only have an x.
This is another picture of x where we can take one away. (Still have one green tile. Add one yellow tile and one red tile.)
We added a zero pair. (The red tile and the yellow tile combine to make the zero pair… Additive Inverse)
After we take one away, or subtract one, this is what we are left with. (Remove the yellow tile representing one, leaving one green and one red tile.)
This process with the tiles is helpful in building understanding that subtracting one is the same as adding negative one. 25

26. Modeling expressions x + 2 2x Tile Bin
Compare the expressions x + 2 and 2x (Have participants model each expression at their tables.)
Should they look the same?
x is an x with 2 ones added onto the end. (one green tile and two yellow tiles)
2x means you have an x two times. (two green tiles) 26

27. Modeling expressions x2 + 3x + 2 Tile Bin
Now, we are ready to move to simplifying algebraic expressions. (~20 minutes)
(combining like terms and applying the Distributive Property)
a) Build x2 + 3x + 2
(one blue tile, three green tiles, and two yellow tiles)
This expression has three terms and each term looks quite different. 27

28. Simplifying expressions
x2 + x - 2x2 + 2x - 1
Tile Bin
Build x2 + x - 2x2 + 2x – 1
(From left to right, pull in one blue tile, one green, two red large square tiles, two green, and one small red square tile.)
Some of these terms do look “like.”
Let’s rearrange, using the Commutative Property of Addition.
(Rearrange so like tiles are together.)
We see one zero pair with our x2 and -x2 (Additive Inverse). We see three x tiles.
It is easy to see that the expression simplifies to x2 + 3x - 1.
zero pair
Simplified expression
-x2 + 3x - 1 28

29. Simplifying expressions
Tile Bin
What would the model of this expression look like?
Invite a volunteer to come up and draw what they have modeled. Ask if anyone has a different model to share (and compare).
It is likely that most participants will make two sets of tiles that each contain 2 greens and 3 yellows. It is possible that someone may think about the area model. This should lead nicely into the next slide.
This expression is the first one we have seen with parentheses. One would use the Distributive Property to simplify the expression.
Simplified expression
4x + 6 29

30. Two methods of illustrating the Distributive Property:
Example: 2(2x + 3)
One way to model the Distributive Property is to copy and paste.
The copy and paste method reinforces the idea that you will be getting the whole quantity within the parentheses multiple times.
Another method to model the Distributive Property is to use an area model for multiplication and build an array.
The area model connects to the earlier work we did with multi-digit multiplication, and it carries over into Algebra I when multiplying and factoring polynomials is introduced.
Both of these methods have value for the students.

31. Solving Equations How does this concept progress as we move through middle school?
6th grade:
The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
7th grade:
7.14 The student will
solve one- and two-step linear equations in one variable; and
solve practical problems requiring the solution of one- and two-step linear equations.
8th grade:
The student will
a) solve multistep linear equations in one variable on one and two sides of the
equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
Finally, we can work with equations.
6th grade one-step equations remember that these students have not learned about integer operations.
7th grade up to two-step equations we can now apply operations with integers.
8th grade up to four-step equations simplifying expressions within an equation and working with variables on both sides of the equation make their debut.

32. Solving Equations Tile Bin
We should start with a discussion of the word equation.
It is a statement that two expressions are equal.
This idea is mirrored in the equation mat.
Each rectangle should show the model of an expression. The two expressions are equivalent.

33. Solving Equations x + 3 = 5 Tile Bin
The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
x + 3 = 5
Tile Bin
Let’s start with some examples from 6th grade SOL.
x + 3 = 5
What would that look like on the equation mat?
(Pull in one green and three yellow on left and five yellow on the right.)
We want to isolate the x term.
How can we achieve that goal?
Take three yellows off each side
Add two negatives to each side (Note this is not a 6th grade response.)

34. Solving Equations x + 3 = 5 x + 3 = 5 ̵ 3 ̵ 3 x = 2 x + 3 = 5 ̵ 3 ̵ 3
The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
x + 3 = 5
x + 3 = 5
̵ 3 ̵ 3
x = 2
x + 3 = 5
̵ 3 ̵ 3
Here we start with a pictorial representation of what we just built with the tiles.
How can we solve for x ?
You can take three away from each side. You can see a pictorial representation as well as the symbolic representation of that step.
Finally, we are left with the green tile that represents x alone on the left and two yellow tiles on the right. The solution is x = 2.
Note: the “condensed symbolic representation” column allows you to rework the problem in a more stand-alone symbolic version.
x = 2

35. Solving Equations 2x = 8 Tile Bin
The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
2x = 8
Tile Bin
Let’s try another 6th grade example, 2x = 8. Model this on your equation mats. What does it look like?
We have two green tiles on the left side, and eight yellow tiles on the right.
If we want to split the green tiles to determine what each one is worth individually, how would we separate the yellow tiles to distribute them fairly into 2 equal groups?
(Pull the yellow tiles one at a time into two separate groups, distributing one at a time to show the fair sharing idea.
Discuss that the picture still shows 2x = 8, it has just been reorganized.)
How many x’s do we want? Can you tell me what 1x is worth? Delete all but one section by editing.

36. Solving Equations 3 = x - 1 Tile Bin
7.14 The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
3 = x - 1
Tile Bin
3 = x - 1 (This is a 6th grade problem that would be challenging with the tiles. It would
require a discussion of zero pairs. Although integer operations and the Additive
Inverse Property are 7th grade curriculum, we could approach problems like this
to help create a bridge to the 7th grade content.)
To model with tiles:
Put three yellow tiles on the left and one green tile on the right.
We don’t have any unit tiles to remove (or subtract), but we could add a zero pair (one yellow and one red unit tile that combine to equal zero). Then, we could remove the yellow unit tile (take away one) and this would leave x and negative one, or x plus -1 on the right.
This example would cause sixth grade students to consider what might happen and would be an early example for seventh graders as they formulate their own rules for operating with integers.

37. Solving Equations 2x + 3 = 13 Tile Bin
7.14 The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
2x + 3 = 13
Tile Bin
7th grade examples: 2x + 3 = 13
2x + 3 = 13
Let’s first model the equation on the equation mat.
(Pull in two green tiles and three yellow tiles on the left, and thirteen yellow tiles on the right.)
We know that the two boxes must be equivalent. What can you do to solve the equation?
What would be your first step? (First step is to subtract three tiles from each side, leaving two greens on left and 10 yellows on right.) What would you do next?
Have participants model and discuss at their tables. Go to next slide for whole group discussion.

38. Solving Equations 2x + 3 = 13 2x + 3 = 13 ̵ 3 ̵ 3 2x + 3 = 13 ̵ 3 ̵ 3
7.14 The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
2x + 3 = 13
x = 5
2x = 10
2x + 3 = 13
̵ ̵ 3
2x + 3 = 13
̵ ̵ 3
2x = 10
2x + 3 = 13
This chart shows the progression of steps we followed. On the left, you see a pictorial representation of what we did with the tiles.
In the middle column, you see the more abstract, symbolic representation of the our moves with the tiles.
On the right, you see the condensed version that one would expect students to work toward as they become efficient.
To recap this solution, remove three from each side (second row of chart) and then practice fair sharing (third row).
x = 5

39. Solving Equations 0 = 4 – 2x Tile Bin
7.14 The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
0 = 4 – 2x
Tile Bin
Another 7th grade example: 0 = 4 – 2x
What would that model look like?
Have participants model and solve, keeping track of the steps to solve. Use next slide for whole group discussion of solution.
Invite volunteer to draw pictorial representation on chart paper before moving to next slide.

40. Solving Equations 0 = 4 – 2x 0 = 4 – 2x ̵ 4 ̵ 4 0 = 4 – 2x ̵ 4 ̵ 4
7.14 The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
0 = 4 – 2x
0 = 4 – 2x
̵ 4 ̵ 4
-4 = -2x
2 = x
0 = 4 – 2x
̵ 4 ̵ 4
-4 = -2x
-2 = -x
In the first row, we see the yellow tiles and representing 4 and the two red tiles representing -2x.
Students could put up 2 zero pairs and take away two x’s, leaving the 2 negative x tiles.
We want to isolate the x term, so the first step is to add four red tiles to each side. You can easily see the -4 on the left side of the equation mat. On the right side of the equation, this gives us 4 zero pairs and the -2x.
In the third row, you see the separation into two equal groups. But we are interested in the value of one x, so we can remove all but one of those groupings. This leaves us with -2 = -x.
We can think about it this way.
If the opposite of two is equal to the opposite of x, then 2 is equal to x.
In the third column, what looks different? (Division by negative 2)
Presenters have the option of solving additional equations that fit within the realm of the 7th grade curriculum. These equations could be solved on chart paper, with participants recording their pictorial and symbolic representations to foster discussion:
x + 2 = 1 (Students could add -2 to each side and then simplify each side OR add a zero pair to the right side in order to then take away two from each side)
3 = x – 4 (Students could add four to each side and then simplify each side)
-3x = 9 (Fair sharing and opposites in either order)
-9 = -5x +1 (Add -1 to each side, then fair sharing and opposites in either order OR add one zero pair to -9 in order to take away positive one from each side, then division and opposites or division by negative 5)
2 = x

41. Solving Equations 3x + 5 – x = 11 Tile Bin 8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
Tile Bin
3x + 5 – x = 11
Now we’re ready to try some examples from 8th grade.
Model 3x + 5 – x = 11 on your equation mat. What does that look like? What would you do first?
Solve at your tables and keep track of your steps.
Go to next slide for whole group discussion of solution.

42. Solving Equations 3x + 5 – x = 11 3x + 5 – x = 11 2x + 5 = 11
The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
3x + 5 – x = 11
2x + 5 = 11
3x + 5 – x = 11
2x + 5 = 11
2x = 6
x = 3
2x + 5 = 11
2x = 6
We see the first model has the red tile. Where does it go? (talk here about zero pair/ combining like terms)
After we combine like terms, what remains is a two-step equation. First we subtract five from each side. In the third row of the chart, you see those five tiles have been pulled out of the boxes on the equation mat. In the second step, we divide both sides by two. We can see the solution is three.
Again, the third column shows the more efficient method of recording work that students will work toward.
x = 3

43. Solving Equations x + 2 = 2(2x + 1) Tile Bin 8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
Tile Bin
x + 2 = 2(2x + 1)
Another 8th grade example: x + 2 = 2(2x + 1)
There is a lot going on here. What would we tackle first? (Simplify the expression on the right using the Distributive Property.)
Have participants work at tables to go through the steps to find the solution. Use next slide for whole group discussion.

44. Solving Equations x + 2 = 2(2x + 1) x + 2 = 4x + 2 x + 2 = 2(2x + 1)
The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
x + 2 = 2(2x + 1)
x + 2 = 4x + 2
x + 2 = 2(2x + 1)
x + 2 = 4x + 2
-x x
2 = 3x + 2
0 = 3x
0 = x
x + 2 = 4x + 2
-x x
2 = 3x + 2
0 = 3x
Things to bring out in the discussion:
First step is application of the Distributive Property. Here we have modeled the “copy and paste” method.
What would happen if the 4x were subtracted from both sides instead of the x?
Presenters can continue with additional equations that fall in the realm of 8th grade curriculum on chart paper:
6 = 2(x - 3) + x (Distribute and combine like terms to create a two-step equation.)
** 3x - 1 = x + 7 (Variables on both sides of the equation. In this example you can simply
remove an x from each side to create a two-step equation.)
** 3x - 1 = -x + 7 (Now, we need to extend our operations with integers to the variables.)
**We thought this might be a spot where the group could split in half.
Each half could work through one of these two, slightly different equations.
We could then talk about the similarities and differences as a whole group.
0 = x

45. Modeling Multiplication and Division of Fractions45 45

46. So what’s new about fractions in Grades 6-8?
SOL 6.4
The student will demonstrate multiple representations of multiplication and division of fractions. 46

47. Thinking About Multiplication
The expression…
We read it…
It means…
It looks like…
Make the connection to the earlier discussion of the variety of meanings of multiplication (slide 8) and think about which make sense for fractions.
Repeated Addition: makes sense if fraction is multiplied by a whole number (2 ∙ 1/3 = 1/3 + 1/3), but not for fraction times fraction
Groups-Of: makes sense if fraction is multiplied by whole number (2 groups or sets of 1/3)
Rectangular Array: makes sense for fraction multiplied by whole number (2 rows with 1/3 apple in each row)
Rate: “Dave walked for 2 hours at a rate of 1/3 mile per hour” or “Dave walked ½ hour at a rate of 1/3 mile per hour”
Comparison: “Susie has 2 cookies, and Ralph has 1/3 as many cookies as Susie”
Combinations:
Area: “I need to cover a bulletin board that measures ½ yard long and 1/3 yard wide; how many square yards of fabric are needed?” 47 47

48. Thinking About Multiplication
The expression…
We read it…
It means…
It looks like…
2 times 3
two groups of three
2 times
two groups of one-third
times
one-half group of one-third
Make the connection to the earlier discussion of the variety of meanings of multiplication (slide 8) and think about which make sense for fractions.
Repeated Addition: makes sense if fraction is multiplied by a whole number (2 ∙ 1/3 = 1/3 + 1/3), but not for fraction times fraction
Groups-Of: makes sense if fraction is multiplied by whole number (2 groups or sets of 1/3)
Rectangular Array: makes sense for fraction multiplied by whole number (2 rows with 1/3 apple in each row)
Rate: “Dave walked for 2 hours at a rate of 1/3 mile per hour” or “Dave walked ½ hour at a rate of 1/3 mile per hour”
Comparison: “Susie has 2 cookies, and Ralph has 1/3 as many cookies as Susie”
Combinations:
Area: “I need to cover a bulletin board that measures ½ yard long and 1/3 yard wide; how many square yards of fabric are needed?”

49. Making sense of multiplication of fractions using paper folding and area models
Enhanced Scope and Sequence, 2004, pages
Take participants through the enhanced scope and sequence activity. Spend no more than 15 minutes on this…
Participants will need blank paper and something to write with. 49

50. Making sense of multiplication of fractions using paper folding and area models
Enhanced Scope and Sequence, 2004, pages
Take participants through the enhanced scope and sequence activity. Spend no more than 15 minutes on this…
Participants will need blank paper and something to write with. 50

51. Making sense of multiplication of fractions using paper folding and area models
Enhanced Scope and Sequence, 2004, pages
Take participants through the enhanced scope and sequence activity. Spend no more than 15 minutes on this…
Participants will need blank paper and something to write with. 51

52. The Importance of Context
Builds meaning for operations
Develops understanding of and helps illustrate the relationships among operations
Allows for a variety of approaches to solving a problem

53. Contexts for Modeling Multiplication of Fractions
The Andersons had pizza for dinner, and there was one-half of a pizza left over. Their three boys each ate
one-third of the leftovers for a late night snack.
How much of the original pizza did each boy get for snack?
Ask participants to draw a picture to solve the problem. One possible pictorial solution is on the following slide.

54. One-third of one-half of a pizza is equal to one-sixth of a pizza.
Which meaning of multiplication does this model fit?
We start with one-half of one pizza. That is, in effect, the “whole” amount we have to work with.
Each boy gets one-third of that amount, so we divide the half into three equal portions (dividing into three equal shares is the same as finding one-third of the amount).
Since the question refers to the share of the original pizza, we have to determine what each of those three pieces is in relation to the original whole. The dotted segments extend across to show that each boy had 1/6 of the original pizza.

55. Another Context for Multiplication of Fractions
Andrea and Allison are partners in a relay race. Each girl will run half the total distance. On race day, Andrea stops for water after running of her half of the race.
What portion of the race had Andrea run when she stopped for water?

56. Students need experiences with problems that lend themselves to a linear model.

57. Another Context for Multiplication of Fractions
Mrs. Jones has 24 gold stickers that she bought to put on perfect test papers. She took of the stickers out of the package, and then she used of that half on the papers.
What fraction of the 24 stickers did she use on the perfect test papers?

58. Problems involving discrete items may be represented with set models.
One-third of one-half of the 24 stickers is of the 24 stickers.
What meaning(s) of multiplication does this model fit?
Problems involving discrete items
may be represented with set models.

59. What’s the relationship between multiplying and dividing?
Multiplication and division are inverse relations
One operation undoes the other
Division by a number yields the same result as multiplication by its reciprocal (inverse). For example:
This slide serves as a transition between multiplication and division, and should be reached by the 30 min. mark for this segment… 59 59

60. Meanings of Division For 20 ÷ 5 = 4…
Divvy Up (Partitive): “Sally has 20 cookies. How many
cookies can she give to each of her five friends, if she gives each friend the same number of cookies?
- Known number of groups, unknown group size
Measure Out (Quotitive): “Sally has 20 minutes left on
her cell phone plan this month. How many more 5-minute calls can she make this month?
- Known group size, unknown number of groups
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA Publishing, 1998.

61. Sometimes, Always, Never?
When we multiply, the product is larger than the number we start with.
When we divide, the quotient is smaller than the number we start with.
These questions are posed to get participants to consider what really happens during the operations of multiplication and division, and to recognize that the ways the two numbers in each expression are used is different (one is the multiplier, the other is a quantity or amount).. 61 61

62. “I thought times makes it bigger...”
When moving beyond whole numbers to situations involving fractions and mixed numbers as factors, divisors, and dividends, students can easily become confused. Helping them match problems to everyday situations can help them better understand what it means to multiply and divide with fractions. However, repeated addition and array meanings of multiplication, as well as a divvy up meaning of division, no longer make as much sense as they did when describing whole number operations.
Using a Groups-Of interpretation of multiplication and a Measure Out interpretation of division can help:
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power, LEA Publishing, 1998.

63. “Groups of” and “Measure Out”
1/4 x 8: “I have one-fourth of a box of 8 doughnuts.”
8 x 1/4: “There are eight quarts of soda on the table. How many whole gallons of soda are there?”
1/2 x 1/3: “The gas tank on my scooter holds 1/3 of a gallon of gas. If I have 1/2 a tank left, what fraction of a gallon of gas do I have in my tank?”
1¼ x 4: “Red Bull comes in packs of four cans. If I have 1¼ packs of Red Bull, how many cans do I have?”
3½ x 2½: “If a cross country race course is 2½ miles long, how many miles have I run after 3½ laps?
3/4 ÷ 2: “How much of a 2-hour movie can you watch in 3/4 of an hour?” *This type may be easier to describe using divvy up.
2 ÷ 3/4: “How many 3/4-of-an-hour videos can you watch in 2 hours?”
3/4 ÷ 1/8: “How many 1/8-sized (of the original pie) pieces of pie can you serve from 3/4 of a pie?”
2½ ÷ 1/3: “A brownie recipe calls for 1/3 of a cup of oil per batch. How many batches can you make if you have 2½ cups of oil left?”

64. Thinking About Division
The expression…
We read it…
It means…
It looks like…
20 ÷ 5

65. Thinking About Division
The expression…
We read it…
It means…
It looks like…
20 ÷ 5
20 divided by 5
20 divided into groups of 5;
20 divided into 5 equal groups…
How many 5’s are in 20?
20 divided by
20 divided into groups of …
How many ’s are in 20? 65 65

66. Thinking About Division
The expression…
We read it…
It means…
It looks like…
one-half divided by one-third
divided into groups of …
How many ’s are in ? ?
Is the quotient more than one or less than one? How do you know?

67. Contexts for Division of Fractions
The Andersons had half of a pizza left after dinner. Their son’s typical serving size is pizza. How many of these servings will he eat if he finishes the pizza?

68. pizza divided into pizza servings = 1 servings
If 1/3 pizza is one serving, how many servings are in the ½ pizza we are starting with in this problem?
Since 1/3 is less than 1/2, our solution will be more than one serving.
Since the ½ pizza we have is less than 2/3 (or two servings), the solution will be less than 2 servings.
We can portion off one full serving, leaving 1/6 of the original pizza. That sixth is ½ of a serving. We have a total of 1 ½ servings in our half pizza, that are each 1/3 pizza in size.
serving 68

69. Another Context for Division of Fractions
Marcy is baking brownies. Her recipe calls for
cup cocoa for each batch of brownies. Once she gets started, Marcy realizes she only has cup cocoa. If Marcy uses all of the cocoa, how many batches of brownies can she bake?

70. 1 cup cup 0 cups Three batches (or cup) Two batches (or cup) 1 batches
One batch (or cup)
How many thirds are in one half?
Measurement situations can be represented with a linear model, even if length isn’t being measured. 70

71. Another Context for Division of Fractions
Mrs. Smith had of a sheet cake left over after her party. She decides to divide the rest of the cake into portions that equal of the original cake.
How many cake portions can Mrs. Smith make from her left-over cake?

72. What could it look like?
The blue amount represents the leftover ½ sheet cake.
The green shaded amount shows 1/3 of the original cake.
You can see that one complete 1/3 cake portion exists and then ½ of another portion.
Another way to think: common denominators.
We know that ½ is 3-sixths and 1/3 is 2-sixths. How many 2/6 are in 3/6? There is one 2/6 and then ½ of another 2/6, for a total of 1 ½.
With this diagram you can see that the “little bit more” is really one sixth… so why isn’t “one sixth” a part of the answer? The answer is 1½ … where is that one half in the model? 72

73. What does it look like numerically?
Connect the discussion regarding the pictorial representation to the numerical solution. Toggle back and forth between the two slides pointing out where the common denominator 6 comes from , as well as where the 1 ½ comes from. 73

74. What is the role of common denominators in dividing fractions?
Ensures division of the same size units
Assist with the description of parts of the whole
Connect concrete to abstract by giving participants some division problems to solve using the common denominator method. 74

75. the traditional algorithm?
What about
the traditional algorithm?
If the traditional “invert and multiply” algorithm is taught, it is important that students have the opportunity to consider why it works.
Representations of a pictorial nature provide a visual for finding the reciprocal amount in a given situation.
The common denominator method is a different, valid algorithm. Again, it is important that students have the opportunity to consider why it works.
Invariably, the traditional algorithm will come up even if the teacher has not introduced it himself.
Teachers need to be ready to discuss it.
Remember that one focus of the 2009 standards is to model multiple representations for division of fractions. Time considering the algorithm is worthwhile but it’s not the only way to divide fractions.
The next slides attempt to give teachers some ideas about how to help students think about that traditional algorithm. 75 75

76. the traditional algorithm?
What about
the traditional algorithm?
Build understanding:
Think about 20 ÷ .
How many one-half’s are in 20?
How many one-half’s are in each of the 20 individual wholes?
Experiences with fraction divisors having a numerator of one illustrate the fact that within each unit, the divisor can be taken out the reciprocal number of times. 76

77. the traditional algorithm?
What about
the traditional algorithm?
Later, think about divisors with numerators > 1.
Think about 1 ÷
How many times could we take from 1?
We can take it out once, and we’d have left. We
could only take half of another from the remaining
portion. That’s a total of
In each unit, there are sets of .
This is a difficult idea to understand and could take several days in itself. The gist is that we should always be teaching for understanding, and we want students to understand the procedures they are using to solve problems.
Multiplying by the reciprocal is not the only method for dividing fractions. 77 77

78. Multiple Representations
Instructional programs from pre-k through grade 12 should enable all students to –
Create and use representations to organize, record and communicate mathematical ideas;
Select, apply, and translate among mathematical representations to solve problems;
Use representations to model and interpret physical, social, and mathematical phenomena.
Participants will fold a sheet of paper into four quadrants, then represent a given problem contextually, graphically, and numerically. Assign each table a different expression to work with.
¾ ÷1/2
2/3 ÷ ¼
5/6 ÷ 2/6
2/3 ÷ 1/6
1/6 ÷ 3/6
12/3÷ 1/2
It is important that everyone come back together and have some debriefing.
from Principles and Standards for School Mathematics (NCTM, 2000), p. 67. 78

79. Using multiple representations to express understanding
Given problem
Check your solution
Contextual situation
Solve numerically
Solve graphically
Participants will fold a sheet of paper into four quadrants, then represent a given problem contextually, graphically, and numerically. Assign each table a different expression to work with.
¾ ÷1/2
2/3 ÷ ¼
5/6 ÷ 2/6
2/3 ÷ 1/6
1/6 ÷ 3/6
12/3÷ 1/2
It is important that everyone come back together and have some debriefing. 79

80. understanding of division of
Using multiple
representations
to express
understanding of division of
fractions 80

81. Mean: Fair Share and Balance Point

82. Mean: Fair Share 2009 5.16: The student will
a) describe mean, median, and mode as measures of center;
b) describe mean as fair share;
c) find the mean, median, mode, and range of a set of data; and
d) describe the range of a set of data as a measure of variation.
Understanding the Standard: “Mean represents a fair share concept of the data. Dividing the data constitutes a fair share. This is done by equally dividing the data points. This should be demonstrated visually and with manipulatives.”
The blue text represents additional changes in the Standard.
The red text represents the change that we are addressing today.
*This convention is also used on Slide 76 for SOL
Understanding the Standard is copied directly from the 2009 Curriculum Framework.

83. Understanding the Mean
Each person at the table should:
Grab one handful of snap cubes.
Count them and write the number on a sticky note.
Snap the cubes together to form a train.
We will be asking participants to engage in two separate activities – this one uses snap cubes to demonstrate mean as fair share, and the sticky note activity demonstrates mean as balance point.
Michael has suggested that we should take the group up through Slide 79 (the example where the mean doesn’t work out neatly to a whole number) BEFORE we conduct the second activity. The instructions for that activity are in the notes for Slide 79.

84. Understanding the Mean
Work together at your table to answer
the following question:
If you redistributed all of the cubes from your handfuls so that everyone had the same amount (so that they were “shared fairly”), how many cubes would each person receive?
“Teach” this lesson similarly to the way you would with students.

85. Understanding the Mean
What was your answer?
- How did you handle “leftovers”?
- Add up all of the numbers from the
original handfuls and divide the sum
by the number of people at the table.
- Did you get the same result?
Obviously, the answer represents the mean – but make sure they understand it represents a “fair share” as the underlying concept of the mean.
- What does your answer represent?

86. Understanding the Mean
Take your sticky note and place it on
the wall, so they are ordered…
Horizontally: Low to high, left to right; leave one space if there is a missing number.
Vertically: If your number is already on the wall, place your sticky note in the next open
space above that number.
Building a line plot for the activity at the end of Slide 79.

87. Understanding the Mean
How did we display our data?
From the Curriculum Framework. c

88. Understanding the Mean
Looking at our line plot, how can we
describe our data set? How can we
use our line plot to:
- Find the range?
- Find the mode?
- Find the median?
“Teach” again to make sure they can find the range, mode, and median before moving to the next slide about using a line plot to find the mean (the balance point).
- Find the mean?

89. Mean: Balance Point 2009 6.15: The student will
a) describe mean as balance point; and
b) decide which measure of center is appropriate
for a given purpose.
Understanding the Standard: “Mean can be defined as the point on a number line where the data distribution is balanced. This means that the sum of the distances from the mean of all
the points above the mean is equal to the sum of the distances of all the data points below the mean.”
Understanding the Standard and Essential Knowledge & Skills info is taken directly from the Curriculum Framework.
*For the record, the last bullet on this slide is not the only bullet in the Framework:
• Find the mean for a set of data.
• Describe the three measures of center and a situation in which each would best represent a set of data.
• Identify and draw a number line that demonstrates the concept of mean as balance point for a set of data.
Essential Knowledge & Skills:
Identify and draw a number line that demonstrates the
concept of mean as balance point for a set of data.

90. Where is the balance point for this data set?X X X X X X
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 90

91. Where is the balance point for this data set?X X X X X X
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 91

92. Where is the balance point for this data set?X X X X X X
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 92

93. Where is the balance point for this data set?X X X X X X
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 93

94. Where is the balance point for this data set?X X X X X X
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean. 94

95. Where is the balance point for this data set?X X X X X X
Now refer back to the original line plot. We know that 3 is the balance point or mean. Talk about the sum of the distances above the mean being the same as the sum of distances from the mean below the mean. 95

96. Where is the balance point for this data set?
Move 2 Steps
Move 2 Steps
Move 2 Steps
Move 2 Steps
We can manipulate the data points to help us see where the balance point would be.
Even though we don’t yet know the balance point, as long as we make balanced /equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean.
*Note, in case it comes up: This works the way it would with a fulcrum in physics (Force times Distance). That means it’s possible to move different numbers of data points on either side. For example, you could move two points one space each on the left and one point two spaces on the right without affecting the balance.
And, not that you’d want to during our presentation, you could also make moves away from the center without affecting the mean.
4 is the Balance Point 96

97. The Mean is the Balance Point
We can confirm this by calculating:
= 36
36 ÷ 9 = 4
The Mean is the Balance Point 97

98. Where is the balance point for this data set?
If we could “zoom in” on the space between 10 and 11, we could continue this process to arrive at a decimal value for the balance point.
Move 1 Step
The Balance Point is between 10 and 11 (closer to 10).
Move 2 Steps
Move 1 Step
Move 2 Steps
Sticky Note Activity:
Work with the whole group to use this strategy to find the mean number of cubes in one handful based on our data set.
If it doesn’t work out to be whole number, discuss how we could find the exact decimal value of the mean if we could “zoom in” and how we could estimate the mean based on the modified line plot.
*You may want to have a calculator handy to find the actual mean of your whole group data set. 98

99. Mean: Balance Point When demonstrating finding the balance point:
CHOOSE YOUR DEMONSTRATION DATA SETS INTENTIONALLY.
Use a line plot to represent the data set.
Begin with the extreme data points.
Balance the moves, moving one data point from each side an equal number of steps toward the center.
Continue until the data is distributed symmetrically or until there are only two values left on the line plot.
By “intentionally” we mean that teachers should begin with data sets that have whole number means, only progressing to rational number means once students understand the concept and process.
When introducing decimal means, teachers should begin with “neat” data sets.
- For example: {2, 3, 4, 5, 6, 7} would have a mean of 4.5 (27/6 = 4.5), which would be easy to see once the line plot was transformed so that there were three 4’s and three 5’s.

100. Assessing Higher-Level Thinking
Key Points for & 6.15:
Students still need to be able to calculate the mean by summing up and dividing, but they also need to understand:
- why it’s calculated this way (“fair share”);
- how the mean compares to the median and the
mode for describing the center of a data set; and
- when each measure of center might be used
to represent a data set.
Emphasize the shift to higher-order thinking from the 2001 to 2009 Standards.

101. Mean: Fair Share & Balance Point
“Students need to understand that the mean ‘evens out’ or ‘balances’ a set of data and that the median identifies the ‘middle’ of a data set. They should compare the utility of the mean and the median as measures of center for different data sets. …students often fail to apprehend many subtle aspects of the mean as a measure of center. Thus, the teacher has an important role in providing experiences that help students construct a solid understanding of the mean and its relation to other measures of center.”
- NCTM Principles & Standards for School Mathematics, p. 250
Connection between the new SOL and the NCTM Standards emphasizing that the teacher must provide learning experiences that will promote students’ conceptual understanding.
The days of “just add ‘em up and divide” are behind us :o)

102. Inequalities
102
102

103. Inequalities
SOL 6.20
The student will graph inequalities on a number line.
SOL 7.15
The student will
a) solve one-step inequalities in one variable; and
graph solutions to inequalities on the number line.
SOL 8.15
b) solve two-step linear inequalities and graph the results on a number line
Why spend the time talking about inequalities at grade 6?
103
103

104. Inequalities
What does inequality mean in the world of mathematics? mathematical sentence comparing two unequal expressions How are they used in everyday life?
to solve a problem or describe a relationship for which there is more than one solution
104
104

105. Equations vs. Inequalities
x = 2 x > 2
How are they alike?
How are they different?
So, what about x > 2?
Engage participants in dialogue comparing x = 2 with x > 2. Have participants suggest real life connections to the two statements, then use white boards or paper to show graphical representations of the two. So, what about x>=2? What have mathematicians done to make sure people can understand when they mean x>2 and when they mean x>=2?
105
105

106. Equations vs. Inequalities
x = 2 x > 2 2 2
Engage participants in dialogue comparing x = 2 with x > 2. Have participants suggest real life connections to the two statements, then use white boards or paper to show graphical representations of the two. So, what about x>=2? What have mathematicians done to make sure people can understand when they mean x>2 and when they mean x>=2? 2
106
106

107. Open or Closed? x > 16 -5 > y m > 12 n < 341 -3 < j 1616 -5 12
341
How do you read these? What do they really mean in real life terms? It’s more than just putting dots and arrows, isn’t it? -3
and, which way should the ray go?
107
107

108. Equations vs. Inequalities
x + 2 = 8 x + 2 < 8
How are they alike?
How are they different?
So, what about x + 2 < 8?
Engage participants in dialogue comparing x +2 = 8 with x+2 > 8. What is the difference between using the = or the <? What is the difference between using the < and the <? How could we represent these in real life terms? (i.e. (x+2 =8) Antonio has $8 to ride go-karts and play games. If the go-karts cost $2, how much can he spend on games? (x+2<8) Antonio has $8 to ride go-karts and play games at the fair. If the go-kart costs $2, what is the most he can spend on games?) Have participants use white boards or paper to show graphical representations of the two. So, what about x+2>=8?
108
108

109. Equations vs. Inequalities
x + 2 = 8 x + 2 < 8
How are they alike?
Both statements include the terms: x, 2 and 8
The solution set for both statements involves 6.
How are they different?
The solution set for x + 2 = 8 only includes 6. The solution set for x + 2 < 8 does includes all real numbers less than 6. What about x + 2 < 8?
Engage participants in dialogue comparing x +2 = 8 with x+2 > 8. What is the difference between using the = or the <? What is the difference between using the < and the <? How could we represent these in real life terms? (i.e. (x+2 =8) Antonio has $8 to ride go-karts and play games. If the go-karts cost $2, how much can he spend on games? (x+2<8) Antonio has $8 to ride go-karts and play games at the fair. If the go-kart costs $2, what is the most he can spend on games?) Have participants use white boards or paper to show graphical representations of the two. So, what about x+2>=8?
The solution set for this inequality includes 6 and all real numbers less than 6.
109
109

110. Equations vs. Inequalities
x+ 2 = 8 x+ 2 < 8 6 6
Engage participants in dialogue comparing x = 2 with x > 2. Have participants suggest real life connections to the two statements, then use white boards or paper to show graphical representations of the two. So, what about x>=2? What have mathematicians done to make sure people can understand when they mean x>2 and when they mean x>=2? 6
110
110

111. Inequality Match
With your tablemates, find as many matches as possible in the set of cards.
111
111

112. X > 5
X is greater
than 5
SAMPLE MATCH
112

113. Operations with Integers
113

114. Operations with Integers
a: The student will
a) model addition, subtraction, multiplication and division of integers; and
b) add, subtract, multiply, and divide integers.
Is this really a “new” SOL?
: The student will formulate rules for and solve practical problems involving basic operations (addition, subtraction, multiplication, and division) with integers.
This 2009 SOL has flown under the radar because at first glance the content doesn’t not appear to have changed – it’s still about operating on integers in 7th Grade.
However, the new verb “model” should not be underestimated.
“Model”
114

115. Assessing Higher-Level Thinking
7.3a: The student will model addition, subtraction, multiplication, and division of integers.
These “chip” models must be read from left to right: FOLLOW THE ARROWS.
= 1
= -1
3 + (-7) = -4
What operation does this model?
115

116. Assessing Higher-Level Thinking
7.3a: The student will model addition, subtraction, multiplication, and division of integers.
This one doesn’t seem to “do” very much, but the arrow is showing a move from 3 groups of (-4) to a product of (-12).
= 1
= -1
What operation does this model?
3 • (-4) = -12
116

117. Assessing Higher-Level Thinking
7.3a: The student will model addition, subtraction, multiplication, and division of integers.
= -12
5 + (-17) = -12
Most follow the arrows – begin at the zero.
This model could be representing:
5 + (-17) = -12 OR
= -12
The slide animations will display both number sentences.
What operation does this model?
117

118. Assessing Higher-Level Thinking
7.3a: The student will model addition, subtraction, multiplication, and division of integers.
3 • (-5) = -15
Could also be shown using “hops”.
What operation does this model?
118

119. Another Example of Assessing Higher-Level Thinking
7.5c: The student will describe how changing one measured attribute of a rectangular prism affects its volume and surface area.
Describe how the volume of the rectangular prism shown (height = 8 in.) would be affected if the height was increased by a scale factor of ½ or 2.
8 in.
Emphasize the vocabulary: “scale factor”.
3 in.
5 in.
119

120. Tying it All Together
Improved vertical alignment of content with increased cognitive demand.
Key conceptual models can be extended across grade levels.
Refer to the Curriculum Framework.
Pay attention to the changes in the verbs.
120

121. Exit Slip
1. Aha...
2. Can’t wait to share…
3. HELP!
121