Multiplying with Algebra Tiles
Name: _______________________________
Multiplying with Algebra Tiles Multiplication can be thought of performing repeated addition. We can model this algebra tiles. For example: 3 × 2 translates to “add 3 groups of 2 tiles”
Therefore we have a total of 6 tiles.
3 × −2 translates to “add 3 groups of 2 red (grey) tiles”
Therefore we have a total of -6 tiles.
When we start with a negative number, we use a process similar to subtraction. −3 × 2 translates to “remove 3 groups of 2 tiles”. Since we are removing tiles, we need to add enough zero pairs so that we can remove 3 groups of 2 tiles. Then remove 3 groups of (+)2 tiles.
We are left with -6 tiles.
−3 × −2 translates to “remove 3 groups of 2 red (grey) tiles” Again, since we are removing tiles, we need to have enough zero pairs so that we can remove 3 groups of –2 tiles (this time we need to remove the red/grey tiles.
Therefore we are left with 6 tiles. Use Algebra Tiles to model the following multiplication problems. Be sure to sketch your answers. 1. 6 × 3
2. 4 × 2
3. −5 × 3
4. −5 × (−3)
5. 3 × (−4)
6. −2 × 6
7. −4 × (−4)
8. Based on your observations, a. What is the product of two positive numbers?
b. What is the product of a positive number and a negative number?
c. What is the product of two negative numbers?
d. What is the product of a negative number and a positive number?
In all multiplication problems you can model the problem by building a rectangle. Then the result of the multiplication is the area of the rectangle. In the following problems, start with the number of groups each quantity. Then draw a rectangle that has the same amount of algebra tiles. State the dimensions of the rectangle and the resulting algebraic expression. For instance: 2(𝑥 + 3) start with 2 groups of 𝑥 + 3
This makes a rectangle that is 2 by 𝑥 + 3 and the rectangle contains 2 – 𝑥 tiles and 6 unit tiles. So the area of the rectangle is 2𝑥 + 6
Use Algebra Tiles to model the following multiplication problems. Make sure to draw the resulting rectangle and state the dimensions and resulting algebraic expression. 9. 3(𝑥 + 5)
10. 4(𝑥 − 2)
11. 3(𝑥 2 + 3𝑥 + 5)
12. −2(−2𝑥 + 3)
13. −3(2𝑥 2 − 3𝑥 − 2)
In looking at the previous problems, the dimensions of the rectangle are the factors of the problems. So, if we want to determine 𝑥(𝑥 + 3), the dimension of the rectangle we will need are 𝑥 and 𝑥 + 3. 𝑥(𝑥 + 3) x
+ 3
x
This rectangle contains 1 – 𝑥 2 and 3 – 𝑥 tiles. So the area of the rectangle is 𝑥 2 + 3𝑥 Use Algebra Tiles to model the following multiplication problems. Make sure to draw the resulting rectangle and state the dimensions and resulting algebraic expression. 14. 𝑥(𝑥 + 4)
15. 𝑥(2𝑥 + 3)
16. 3𝑥(𝑥 + 4)
17. 2𝑥(3𝑥 + 2)
18. (𝑥 + 4)(𝑥 + 2)
19. (2𝑥 + 1)(𝑥 + 4)
20. (2𝑥 + 5)(3𝑥 + 2)
All of the above problems we used positive factors. What happens when we have negative factors? Remember from our previous work, a negative means we need to take away. So, in this case we need to take away some of the area of the rectangle. For instance: 𝑥(𝑥 − 3) x
- 3 We can take the area away by laying the red (grey) tiles over the 𝑥 2 tile. This can be read two different ways (think back to your subtractions). You can read it as 𝑥 2 − 3𝑥 or 𝑥 2 + (−3𝑥).
x
Use Algebra Tiles to model the following multiplication problems. Make sure to draw the resulting rectangle and state the dimensions and resulting algebraic expression. 21. 𝑥(𝑥 − 2)
22. 2𝑥(𝑥 − 3)
23. 𝑥(3𝑥 − 4)
24. (𝑥 + 3)(𝑥 − 5)
25. (2𝑥 − 1)(𝑥 + 3)
26. (𝑥 − 3)(𝑥 − 2) – Remember, multiplying two negatives equals a positive.
Multiplying with Algebra Tiles Multiplication can be thought of performing repeated addition. We can model this algebra tiles. For example: 3 × 2 translates to “add 3 groups of 2 tiles”
Therefore we have a total of 6 tiles.
3 × −2 translates to “add 3 groups of 2 red (grey) tiles”
Therefore we have a total of -6 tiles.
When we start with a negative number, we use a process similar to subtraction. −3 × 2 translates to “remove 3 groups of 2 tiles”. Since we are removing tiles, we need to add enough zero pairs so that we can remove 3 groups of 2 tiles. Then remove 3 groups of (+)2 tiles.
We are left with -6 tiles.
−3 × −2 translates to “remove 3 groups of 2 red (grey) tiles” Again, since we are removing tiles, we need to have enough zero pairs so that we can remove 3 groups of –2 tiles (this time we need to remove the red/grey tiles.
Therefore we are left with 6 tiles. Use Algebra Tiles to model the following multiplication problems. Be sure to sketch your answers. 1. 6 × 3
2. 4 × 2
3. −5 × 3
4. −5 × (−3)
5. 3 × (−4)
6. −2 × 6
7. −4 × (−4)
8. Based on your observations, a. What is the product of two positive numbers?
b. What is the product of a positive number and a negative number?
c. What is the product of two negative numbers?
d. What is the product of a negative number and a positive number?
In all multiplication problems you can model the problem by building a rectangle. Then the result of the multiplication is the area of the rectangle. In the following problems, start with the number of groups each quantity. Then draw a rectangle that has the same amount of algebra tiles. State the dimensions of the rectangle and the resulting algebraic expression. For instance: 2(𝑥 + 3) start with 2 groups of 𝑥 + 3
This makes a rectangle that is 2 by 𝑥 + 3 and the rectangle contains 2 – 𝑥 tiles and 6 unit tiles. So the area of the rectangle is 2𝑥 + 6
Use Algebra Tiles to model the following multiplication problems. Make sure to draw the resulting rectangle and state the dimensions and resulting algebraic expression. 9. 3(𝑥 + 5)
10. 4(𝑥 − 2)
11. 3(𝑥 2 + 3𝑥 + 5)
12. −2(−2𝑥 + 3)
13. −3(2𝑥 2 − 3𝑥 − 2)
In looking at the previous problems, the dimensions of the rectangle are the factors of the problems. So, if we want to determine 𝑥(𝑥 + 3), the dimension of the rectangle we will need are 𝑥 and 𝑥 + 3. 𝑥(𝑥 + 3) x
+ 3
x
This rectangle contains 1 – 𝑥 2 and 3 – 𝑥 tiles. So the area of the rectangle is 𝑥 2 + 3𝑥 Use Algebra Tiles to model the following multiplication problems. Make sure to draw the resulting rectangle and state the dimensions and resulting algebraic expression. 14. 𝑥(𝑥 + 4)
15. 𝑥(2𝑥 + 3)
16. 3𝑥(𝑥 + 4)
17. 2𝑥(3𝑥 + 2)
18. (𝑥 + 4)(𝑥 + 2)
19. (2𝑥 + 1)(𝑥 + 4)
20. (2𝑥 + 5)(3𝑥 + 2)
All of the above problems we used positive factors. What happens when we have negative factors? Remember from our previous work, a negative means we need to take away. So, in this case we need to take away some of the area of the rectangle. For instance: 𝑥(𝑥 − 3) x
- 3 We can take the area away by laying the red (grey) tiles over the 𝑥 2 tile. This can be read two different ways (think back to your subtractions). You can read it as 𝑥 2 − 3𝑥 or 𝑥 2 + (−3𝑥).
x
Use Algebra Tiles to model the following multiplication problems. Make sure to draw the resulting rectangle and state the dimensions and resulting algebraic expression. 21. 𝑥(𝑥 − 2)
22. 2𝑥(𝑥 − 3)
23. 𝑥(3𝑥 − 4)
24. (𝑥 + 3)(𝑥 − 5)
25. (2𝑥 − 1)(𝑥 + 3)
26. (𝑥 − 3)(𝑥 − 2) – Remember, multiplying two negatives equals a positive.
