1. CPM Educational Program
How many of you are familiar with algebra tiles and the area model? For factoring? Completing the Square? Long Division?
CPM Educational Program

2. CPM Educational Program
This may seem like simple arithmetic but it is crucial to developing a factoring process later on.
CPM Educational Program

3. CPM Educational Program
For shorter workshops quickly share how this looks in class but skip doing depending on participants backgrounds.
CPM Educational Program

4. Perimeter Area Name
P = 4 A = 1 un sq Unit Tile
P = 2x A = 1(x) = x x Tile
P = 4x A = x(x) = x sq x sq Tile 1 1 x 1 x x

5. CPM Educational Program
Algebra Tiles
Definitely do as it models sum to product. x^2+6x+8=(x+2)(x+4)
It’s important they make a sketch of the rectangle the made with the tiles.
CPM Educational Program

6. CPM Educational Program
Again depending on participants background only do a couple of these. Always draw a sketch.
CPM Educational Program

7. CPM Educational Program
This can be a challenge but let them try on their own to model it with tiles and draw a sketch. The actual product is the goal but secondary to the activity.
Product to Sum
CPM Educational Program

8. CPM Educational Program
Could skip, check time.
CPM Educational Program

9. CPM Educational Program
Explain the necessity for a general drawing versus drawing each tile piece as numbers get larger.
CPM Educational Program

10. CPM Educational Program
Diagrams like the one in problem 3-66 are referred to as generic rectangles. Generic rectangles allow you to use an area model to multiply expressions without using the algebra tiles. Using this model, you can multiply with values that are difficult to represent with tiles.
Multiply and simplify the following expressions using either generic rectangle or the Distributive Property.
The term generic rectangle is introduced.
CPM Educational Program

11. The Generic Rectangle Challenge
Have participants do these. They appear to have no value beyond finding area and dimensions so be sure to point out your really find the entire area as a sum and product. Always right out your answer as sum=product.
CPM Educational Program

12. CPM Educational Program
Looking at patterns to help find a process for getting the outer dimensions which is area as a product when no outside dimension is given. They may notice a greatest common divisor typically works.
CPM Educational Program

13. CPM Educational Program
Finding the product without using algebra tiles. Part (b) we know we need two numbers that multiply to make 24x^2 but which pair should be use and why?
CPM Educational Program

14. Using algebra tiles, factor
Here is where the diamond problem can help. This is an algorithm that students can easily see why it works from their previous work with generic rectangles and diamond problems.
CPM Educational Program

15. Completing the Square Use your tiles to make a square out of x2+8x+10.
So y=x2+8x+10 is the same as y=(x+4)2 - 6
Try y=x2+4x+9
How would you do y=x2+5x+2?
Powerful but may need to skip depending on time. Interested participants can figure it out from the handout.
CPM Educational Program

16. CPM Educational Program
The next slide shows a multiplication problem that can be used to recognize patterns in the generic rectangle.
CPM Educational Program

17. Complete 114-119 in your teams.
If you always multiplied polynomial written in descending power where would the products highest term be located. What patterns do you notice.
Keep in mind that if you are missing a power you need to put a placeholder into your table or the degrees won’t line up in a diagonal. This works for any size polynomials.
Complete in your teams.
CPM Educational Program

18. Thank you for you time. Please visit us at cpm.org
For more information please contact
If you have more questions or are curious about the activities in this packet please stop by our booth.
CPM Educational Program