1. Algeblocks With Integers
By Reda Berry
Math Institute

2. Algeblocks Basic Mat − +
One green block, , represents one unit or whole
number.
The algeblocks basic mat is used to model direction with Algeblocks. The top half which represents positive values and the bottom half which begins below the line represents negative values. − +

3. Representing Integers
2 blocks
placed “up”
represent 2
2 blocks
placed “down”
represent -2
What are some situations where negative numbers or opposites are used? Temperature, sea level, a gain of five yards, a profit of $2, a penalty of 10 yards, 7 steps down, and checking accounts
Have the students think about and decide which of the two numbers is greater? The two blocks placed on the positive side are greater.
Discuss which sign ( + or -) would be associated with each opposite situation. Positive numbers as “up” and Negatives as “down”
Using a directional approach with Algeblocks and mat helps your student understand relationship between positive and negative numbers
TEACHING: Like a thermometer, placing blocks, on top half of the mat indicates positive values and one bottom half indicate negative values. Note: The sign is often left off of a positive integer + −

4. Finding the opposite of 3:
Step 3. The
opposite 0f
3 is -3
Step 1. Place
3 blocks on the top
part of the mat.
Step 2. Take
the opposite by
moving the 3
blocks down 3 + + +
In this example, it was not necessary to add zeros since there are enough negative blocks to remove one negative block -3 - - - 4

5. Finding the opposite of -2 can be written --2
Step 3. The opposite
of -2 (written as - -2)
is 2
Step 1. Place
-2 on the mat.
Step 2. Finding the
opposite of -2 can
be written --2 + + +
In this example, it was not necessary to add zeros since there are enough negative blocks to remove one negative block 2 - - - 5

6. Adding Integers: 5 + -8 = Step 1. Place 5 and -8 on the mat.− +

7. Adding Integers: 5 + -8= Step 2. Make a zero if you can: 5 + -5 = 0
Adding Integers: = Step 2. Make a zero if you can: = 0. Remove the blocks that make the zero.
On the overhead, a large zero made from heavy paper to circle clocks will help students see you “making zero.” Then emphasize that equal numbers of blocks from opposite sides of the mat add to zero. − +

8. Adding Integers Step 3. Read the answer from the mat 5 + -8 = -3
Now have them work out finding the sum of – = Place on flipchart: At 4 a.m. the temperature was 0 degrees F. By 9 a.m. the temperature had increased by 7 degrees F (place 7 units on positive side). Just before 10 a.m. the temperature dropped 7 degrees F (place 7 units on negative side) when a cold front came through. What was the temperature at 10 a.m.? Answer: on the mat should be 7 units in the positive and 7 units in the negative, they are equal number of blocks from opposite sides of the mat and add to zero degrees. − +
Answer -3

9. Subtracting Integers What integer is represented on each mat?+ - + -
Although the first mat is obvious, all of the mats in the beginning activity are models of 1. Have students show possible models for -2, 3, -3. These kinds of models will be needed for subtraction. − - + - +

10. Example 1: -3 - -1 Step 2. Add zeros, if necessary. Step 3. Take away
(remove) -1
Step 4.
Answer on
Mat.
= -2
Step 1. Place
-3 on the mat. + + + +
In this example, it was not necessary to add zeros since there are enough negative blocks to remove one negative block - - - -

11. Example 2: 4 - -1 Step 2. Add zeros, if necessary. Step 3. remove -1
Answer on
Mat.
= 5
Step 1. Place
4 on the mat. + + + +
In this example, there are no negative blocks to remove in Step 3. It is necessary to represent 4 another way. This is done by adding zero in the form of +1 and -1. - - - -

12. Example 3: -2 - -3 Step 2. Add zeros, if necessary. Step 3. remove -3.
Answer on
Mat.
= 1
Step 1. Place
-2 on the mat. + + + +
In this example, there are not enough negative blocks to remove 3 negative blocks. It is necessary to represent -2 in another way. This is done by adding zero in the form of +1 and -1. - - - -

13. Algeblocks Quadrant Mat+ −
Patterns in Multiplication: Explain that the heavy lines around the center vertical column and the center horizontal row separate the grid into four “quadrants.” Explain how the quadrants are numbered: the upper right quadrant is Quadrant 1. Moving counterclockwise, the upper left quadrant is Quadrant 2. The lower left and lower right quadrants are, respectively, Quadrant 3 and Quadrant 4.

14. Example 1: Find the product of 2 and 3.
Step 4. Answer
is number of
cubes with sign
of quadrants.
2 x 3 = 6
Step 2. Place 3
in the positive part
of vertical axis. Begin
at left of center
Step 3. Form
rectangular area
in quadrant bounded
by cubes.
Step 1. Place 2
in positive part of
horizontal axis. Begin
at right of center. + −

15. Example 2: Find the product of -3 and 1.
Step 4. Answer
is number of
cubes with sign
of quadrants.
-3 x 1 = -3
Step 2. Place 1
in the positive part
of vertical axis. Begin
at left of center
Step 3. Form
rectangular area
in quadrant bounded
by cubes.
Step 1. Place 3
in negative part of
horizontal axis. Begin
at left of center. + −

16. Example 3: Find the product of -3 and -1.
Step 4. Answer
is number of
cubes with sign
of quadrants.
-3 x -1 = 3
Step 2. Place 1
in the negative part
of vertical axis.
Step 3. Form
rectangular area
in quadrant bounded
by cubes.
Step 1. Place 3
in negative part of
horizontal axis. Begin
at left of center. + −

17. divisor (2) in the positive part of the horizontal axis.
Example 1: Dividing integers: division is the inverse of multiplication. That is, because (2)(-3) = -6, -6/2 = -3
Step 2. Form a rectangle of 6 cubes in the negative quadrant (quadrant 4) bounded by the divisor. One side of the rectangle will be 2.
Step 3. Complete the
other dimension of the
rectangle. This is the
answer by cubes.
Step 1. Place the
divisor (2) in the positive
part of the horizontal
axis. + −
Step 1
Step 3
Step 2 17

18. In Summary: Algeblocks Basic Mat Algeblocks quadrant mat
Addition of integers
Subtraction of integers
Algeblocks quadrant mat
Multiplication of integers
Division of integers

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20. Resources:
Johnston, Anita M., Algeblocks, South-Western Publishing Co., Cincinnati, Ohio 1994
ISBN

21. + −