2.2.2 Measurement Model

Number & Operations for Teachers

Measurement Model

Corresponding measurement models for the operations ^-2 + 4 = 2 and 2 – 4 = ^-2 are shown in Figure 2.7. In both operations, an arrow with its tail at the origin points to the first term in the expression. The second term is represented by an arrow arranged head-to-tail with the first. The head of the second arrow is positioned over the indicated sum or difference.

Figure 2.7: Measurement models for ^-2 + 4 = 2 and 2 – 4 = ^-2

In order to accommodate addition and subtraction of both positive and negative numbers, the following conventions are used in the measurement model.

· To add a positive, face right and step forward (i.e., to the right) along the number line

· To subtract a positive, face left and step forward (i.e., to the left) along the number line

· To add a negative, face right and step backward (i.e., to the left) along the number line

· To subtract a negative, face left and step backward (i.e., to the right) along the number line

Example 2.1

Using both a set model and a measurement model, represent the operation 2 – ^-2 = 4.

Solution 2.1

- = - =

2 - ^-2 = 2 + 0 + 0 - ^-2 = 4

By adding 0, or , twice to the first term, it is possible to “take away” two red disks, or ^-2, represented numerically as

[2] – [ ^-2 ] = [ 2 + (^-1 + 1) + (^-1 + 1)] – [^-1 + ^-1] = 4

^-2

___________________

0 1 2 3 4 5

At the point 2, an arrow head is drawn facing left, indicating subtraction. Backing to the right two units (dashed line) represents subtracting ^-2 ® 4.

Example 2.2

Using both a set model and a measurement model, represent the operation 2 + ^-3 = ^-1.

Solution 2.2

+ = =

[2] + [ ^-3 ] = [(^-1 + 1) + (^-1 + 1) + ] + ^-1] = ^-1

^-3

___________________

^-1 0 1 2 3 4

At the point 2, an arrow head is drawn facing right to indicate addition. Facing to the left, move forward three units to ^-1.

Fact families for addition and subtraction of integers are similar to those for addition and subtraction of whole numbers. Both operations are closed on the set of integers.

Example 2.3a

Write a fact family for 2 + ^-5 = ^-3

Solution 2.3a

2 + ^-5 = ^-3

^-3 – ^-5 = 2

^-5 + 2 = ^-3

^-3 – 2 = ^-5

Example 2.3b

Write a fact family for 2 - 5 = ^-3

Solution 2.3b

2 – 5 = ^-3

5 + ^-3 = 2

2 – ^-3 = 5

^-3 + 5 = 2

Tech Resources

Investigate Number Line Bars – Integers at the

National Library of Virtual Manipulatives for Interactive Mathematics

http://matti.usu.edu/nlvm/nav/topic_t_1.html