Two number lines inquiry

The prompt

Mathematical inquiry processes: Explore and generate examples; test cases; conjecture and generalise; reason. Conceptual field of inquiry: Number lines; equal intervals; ordering integers, decimals, and fractions; correspondence and infinity.

Samia Henaine designed the prompt for pupils in grades 4 and 5 to test a statement about two number lines. On seeing the prompt for the first time, pupils often claim that it is true because the distance between two and three is greater than the distance between a third and a half.

However, the length of an interval is not the same thing as how many numbers fit inside it. We can use fractions and decimals to generate new numbers in an interval that looks 'full'. Between any two numbers, however close together they are, we can always find another number - either by finding the halfway point or by writing a number with more decimal places.

Thus, the set of numbers in each interval is infinite and, therefore, the sizes of the sets are equal. Students come to realise that the statement is false.

Question, notice, and wonder

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

Question

Are the two number lines drawn to scale?
How can we tell how many numbers there are in each gap on the number line?

Notice

There are nine numbers between 2 and 3 and only two between a third and a half.
The distance between 2 and 3 looks a lot bigger then the distance between a third and a half so the top number line represents more numbers.
We can always cut a gap on the number line in half and find a new number in the middle.
Even the small gap between a third and a half has room for more fractions.

Wonder

I wonder if a bigger gap on the number line always contains more numbers.
I wonder if a gap can get so small that there is no room left for new numbers.

Learning

The teacher aims to co-construct an understanding of number lines 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 concepts such as fractions and decimals emerges from their responses to the prompt.

Pupils create examples by placing as many numbers as they can on the number lines. 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 can develop from the prompt involves the search for equally-spaced fractions between a third and a half (see Lines of inquiry below).

As students change the prompt and investigate other number lines, they begin to conjecture that there are an infinite set of numbers in any interval. To explain why the size of the set of numbers between two and three is equal to the size of the set of numbers between a third and a half, the teacher might introduce the idea of a one-to-one correspondence (also in Lines of inquiry).

Modelling inquiry

Samia designed the Two number lines inquiry as the second of two inquiries. The teacher can use the Intervals inquiry, which is aimed at pupils in grades 2 and 3, as a preliminary step through which to model the inquiry process before giving students more independence in the second inquiry.

November 2026

Samia Henaine

Samia is an internationally-renown PYP educator and instructional coach. Her website Math Bridges contains more inquiry prompts and other resources.

Samia regularly writes articles about classroom inquiry here.

Lines of inquiry

The slides contain lines of inquiry based on three regulatory cards. In a structured inquiry, the teacher directs the class to follow the sequence of actions on the cards in order. Pupils can use the cards to review the course of the inquiry and develop their understanding of mathematical inquiry.

For the meaning of each card in the context of the inquiry, see the descriptions below.

Pupils try to list as many numbers as they can inside each interval, starting with more obvious ones (such as 2.5, or a number halfway between a third and and a half) and then squeezing in more by adding decimal places or finding midpoints. They then compare and contrast their number lines with those of a partner.

The teacher might structure the search for equally-spaced fractions between a third and a half by using equivalent fractions (see Teacher scaffolds below). If pupils develop a general method to find any number of equally-spaced fractions, then they will understand that the number of fractions is infinite because the process will always work for the next case.

If pupils claim that they cannot find any more numbers on the number lines in the prompt or they continue to maintain that there are more numbers between the integers than the fractions, they might explore other cases by changing the endpoints of each interval.

Are there more fractions between, for example, a quarter and a half than a third and a half? How many numbers are there between 20 and 30 compared to two and three?

Even though the length of the interval has increased in both cases, pupils might begin to understand the one-to-one correspondence between the number lines.

In answer to the second question above, if 21 corresponds to 2.1, then 22 corresponds to 2.2, 23 to 2.3, and so on. Tenths on the first line correspond to hundredths on the second: 20.1 corresponds to 2.01, 20.2 to 2.02, and 20.3 to 2.03. As the process continues, each number on one line has a corresponding number on the other line. Hence, the size of the sets of numbers on each number line are equal.

Pupils test whether they can ever get stuck or run out of new numbers by repeatedly halving the smallest gap on each number line. New numbers are recorded in their correct position and the process is repeated.

Pupils use their two number lines to discuss and reason about whether the original statement makes sense given that you can always find another number regardless of how small the gap.

Even though some pupils conclude that there is an equal (infinite) set of numbers between two and three and between a third and a half, others might continue to argue that one infinity is greater than the other.

Teacher scaffolds

If pupils get stuck generating numbers between a half and a third, the teacher can offer three scaffolds to prompt thinking. The scaffolds involve, in order of accessibility, decimals, equivalent fractions, and midpoints.

Convert to decimals

Since ⅓ ≈ 0.333 and ½ = 0.5, pupils can read off values like 0.4, 0.45, or 0.49 almost by inspection. By adding more decimal places, pupils will always be able to find another number on the number line.

2. Equivalent fractions

The teacher can encourage pupils to express a third and a half with a common denominator. When they use sixths, however, they realise that the denominator is 'too small' because there cannot be another sixth between two-sixths and three-sixths.

This prompts pupils to try a 'larger' denominator. Twelfths gives four-twelfths and six-twelfths, with five-twelfths fitting neatly in between. The general point of the scaffold is that a common denominator (after sixths) always creates room for more fractions.

If pupils use the smaller/larger terminology, the teacher should challenge the potential misconception that a larger denominator means a bigger fraction. A class discussion might start with the question: Is a twelfth greater than a sixth?

3. Repeated halving or midpoints

Finding the midpoint between a third and a half gives five-twelfths; finding the midpoint between a third and five-twelfths (or five-twelfths and a half) gives another new fraction. This process can be repeated indefinitely, and connects to the idea that a gap can always be halved without ever closing.

Resources

Prompt sheet

PowerPoint

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