Algeblocks With Integers By Reda Berry familybriannasarah@yahoo.com Math Institute

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. − +

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 + −

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

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

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

Adding Integers: 5 + -8= Step 2. Make a zero if you can: 5 + -5 = 0 Adding Integers: 5 + -8= Step 2. Make a zero if you can: 5 + -5 = 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. − +

Adding Integers Step 3. Read the answer from the mat 5 + -8 = -3 Now have them work out finding the sum of – 5 + - 8 = -13. 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

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. − - + - +

Example 1: -3 - -1 Step 2. Add zeros, if necessary. Step 3. Take away (remove) -1 Step 4. Answer on Mat. -3 - -1 = -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 - - - -

Example 2: 4 - -1 Step 2. Add zeros, if necessary. Step 3. remove -1 Answer on Mat. 4 - -1 = 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. - - - -

Example 3: -2 - -3 Step 2. Add zeros, if necessary. Step 3. remove -3. Answer on Mat. -2 - -3 = 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. - - - -

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.

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. + −

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. + −

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. + −

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

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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Resources: Johnston, Anita M., Algeblocks, South-Western Publishing Co., Cincinnati, Ohio 1994 ISBN 0-538-62493-0

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