Number Lines

We will spend 15 minutes to complete the activity in this section.

What is it?

A number line is a straight line marked by points representing real numbers. Numbers increase in value as you move to the right, and decrease in value as you move to the left. Depending on the types of questions you are exploring with your students, number lines may include negative numbers, fractions, or decimals.

When adding a positive number, you are increasing in value, so you move along the number line to the right:

Starting at 5, we demonstrate "add positive 7" by jumping 7 spots to the right.

When subtracting, you move along the number line to the left:

Starting at 5, we demonstrate "subtract positive 3" by jumping 3 spots to the left.

Skip counting can also be demonstrated with a number line:

Skip counting by threes on a number line.

Lastly, addition or subtraction strategies can also be modelled with number lines. For instance, we can use an open number line (not every number is marked) to help us determine 26 + 37:

We can break "37" into a couple of more manageable pieces in order to help us add, as modelled on the number line. From 26, we can add 4 to get to 30, add 30 to get to 60, and then add the remaining 3 to get to 63 (37 = 4 + 30 + 3).

It should be noted that when working with open number lines, the absence of each individual number can sometimes lead to wonky scales. Like the example above (where the jump of 30 is larger than the jump of 4), always try and be as proportional as possible in order to avoid misconceptions.

How does it support working memory needs?

Number lines allow students to not only visualize adding, subtracting, or skip counting, but they also provide students with a way of keeping track of the steps in a multi-step problem. Students can do this by either drawing directly on the number line with a pencil, or by tracing their steps with a finger. In the last example above, students with working memory needs might not be able to mentally store the intermediate results of 30 or 60 while applying that particular strategy. Working with a number line would allow that student to complete the calculation without having to hold all the pieces in their head.

Your Task: Constant Difference on the Number Line

There are many mental math addition/subtraction strategies that can be modelled on a number line, but today we'll just look at one: constant difference.

Part 1:

Using either a piece of paper, a white board, or the Mathies Number Line tool/app (scroll down to Number Line), model the following statements on a number line. You may wish to draw each statement on a new number line.

HINT: If drawing by hand, an open line might be preferable to save time. Try to keep the scale the same on each number line, though.

65-29

62-26

71-35

63-27

66-30

Reflection:

What do you notice about the each of the subtraction statements?
How are the number lines of each of the statements similar?
Which of the above subtraction problems do you find the easiest?

In the first subtraction statement, the difference (or distance) between 65 and 29 on your number line is the same difference (or distance) as all the other statements. In essence, we are just shifting that difference back and forth on the number line.

Most people find 66-30 an easier subtraction than 63-27. By using a number line and the concept of constant difference, students can shift the difference of an unfriendly subtraction statement, to one with friendlier numbers (typically those that create a subtrahend (what is being subtracted) divisible by 10), making the operation more manageable.

Part 2:

Model the following statements on a number line, and THEN use the concept of constant difference to shift the distance to friendlier numbers on the same number line.

53-18

71-46

192-74