Samia Henaine designed the prompt for pupils in grades 2 and 3 to test a statement about the number line. On seeing the prompt for the first time, pupils often claim that it is false because there are only four numbers: 0, 1, 2, and 3. 

However, a number line represents more numbers than just those that are marked. Indeed, it is always possible to find a number in the interval between any two points. 

There is not a single correct set of '5 numbers' for the interval between zero and three; several different choices can all be valid. If pupils use the whole numbers in their examples, then the fifth number will be a fraction or a decimal (for example, 0, 1, 1.5, 2, and 3). 

This will lead to unequal intervals on the number line, although the spaces between the whole numbers should be the same. Pupils might draw their number lines to scale or might estimate the position of fractions and decimals.

One line of inquiry that develops from the prompt involves the search for equal intervals. If there are five numbers (including zero and three) on the number line, then the interval value is 0.75 and the five numbers are 0, 0.75. 1.5, 2.25. and 3.

Question, notice, and wonder

In the first phase of the inquiry, Samia anticipates pupils making these contributions:

Question

• Do the five numbers have to include 0 and 3?

• Can we choose any five numbers, or does it have to be a special set?

• What if we wanted six numbers instead of five? 

Notice

• There are only two numbers actually marked on the line: 0 and 3.

• 0, 1, 2 and 3 are only four numbers, not five.

• We can put a number like 1.5 halfway between the two whole numbers. 

Wonder

• I wonder how many different ways we could choose five numbers on this line.

• I wonder if there's a limit to how many numbers we could fit on this line. 

Learning

The teacher aims to co-construct an understanding of the position of numbers on a number line by drawing out the knowledge pupils already hold and then drawing upon it to develop lines of inquiry. Whether pupils require a more structured explanation of, for example, the position of decimals and fractions emerges from their responses to the prompt. 

Pupils create examples by placing three more numbers on the number line between zero and three. They ensure each one is in the correct position in relation to the others and then justify their number line by comparing and contrasting it with those of their peers.

One line of inquiry that develops from the prompt involves the search for equal intervals (see Lines of inquiry below). If there are five numbers (including 0 and 3) on the number line, then there would be four intervals or gaps between them. The interval value is 0.75 and the five numbers are 0, 0.75. 1.5, 2.25. and 3.

Terminology

An interval is the set of all real numbers that lie between two given points. The interval in the prompt is 0 ≤ n ≤ 3 (where n is any real number) or [0, 3]. It is a closed interval because zero and three are included. During the inquiry, 'intervals' is used synonymously with jumps, spaces, or gaps between the marks on a number line.

Extending the inquiry

Samia designed the Intervals inquiry as the first of two inquiries. The teacher can use the Two number lines inquiry, which is aimed at pupils in grades 4 and 5, to extend ideas and concepts.

November 2026