It's All in the Numbers

# 6. Number Properties

Suggested Learning Intentions

• To explore a range of number properties and use these properties to arrange numbers into sets
• To recognise that some pairs of sets can have common numbers, and some pairs of sets do not have common numbers

Sample Success Criteria

• I can identify properties of numbers
• I can organise numbers into sets with common properties
• I can explain my thinking using manipulatives

The purpose of this activity is review some concepts students may have been previously exposed to, either as part of the activities in this sequence or in previous years. Should some concepts be unfamiliar to some students, you can use this activity to teach these specific concepts ‘in the moment’.

Present students with a series of cards, with assorted numbers on them. For example:

Ask students to use these cards to make different sets of numbers, using these cards only.

Depending on the prior knowledge of your students regarding number properties, you could ask them to produce sets of:

• Even numbers, or odd numbers (two different sets)
• Factors of 24, or 60 (two different sets)
• Multiples of 3, or 7 (two different sets)
• Prime Numbers
• Composite Numbers
• Square Numbers
• Triangular Numbers

Use this opportunity to teach any specific concepts that individual or small groups of students may be unfamiliar with.

Other questions can be posed to further gauge student understanding of these concepts, including:

• Which numbers may appear in more than one set? (This question can be used to gauge understanding of common factors and multiples).
• Which sets couldn’t have any numbers in common? (For example, evens and odds, or primes and composites).
• Find a number that would fit into as many, or as few of the sets as possible (for example, the 12 card would fit into the set of even cards, the factors of 24, the factors of 60, the multiples of 3, and the composite numbers set).
• Write your own description of a number property, such as ‘factors of 50’ or ‘composite numbers greater than 20’ and use the cards to create this set.

Show students a grid such as this one below:

Select one number for discussion with the class, such as 25. Invite students to explain which squares they believe 25 could be placed in. For example, 25 could be placed in the bottom right corner, because it is both an odd number and a square number. Facilitate similar discussions using other numbers.

Ask students to identify nine different numbers that could be used to complete the grid.

Enable students requiring further support by providing a 2 x 2 grid to begin with, and/or a set of numbers to select from to fill the grid with (such as the cards used in the initial task).

Question students to draw out their understanding of the different sets, and the relationships that exist between them. Sample questions include:

• Are there any squares in the grid that can only be filled by one number? Why is this? A clear explanation of why 2 must go in the top left corner (because 2 is the only even prime) will help consolidate student understanding of prime numbers.
• If the headings were placed in different parts of the grid, would you still be able to fill it entirely? Students should recognise that even and odd numbers must go on the same ‘side’ of the table, as they have no common values. The same is true for primes and squares (remember 1 is not a prime!)

A sample solution is provided below. The 2 and the 5 are highlighted in red to indicate that they are the only possible numbers that can go in this part of the table.

Extend students by challenging them to construct their own grid. This could be undertaken in different ways:

• Construct a grid with the headings included and the spaces blank and ask a classmate to fill it out.
• Construct a grid where the numbers are included but the headings are blank and ask a classmate to work out what the headings should be.
• Construct a grid that can be filled with consecutive numbers. A sample solution is provided above, using the numbers 1 - 9. Note the use of ‘Factors of 168’ which reflects the challenge present in what appears a simple task.

Areas for further exploration

Present students with the ‘Factors and Multiples’ puzzle on NRICH, published by the University of Cambridge.

Read through the Teachers’ Resources section on NRICH to know how to present this task to students, or use the resources provided, such as the printable worksheets. Note that the website has sample solutions provided by students, so it is best not to direct students to the website themselves.

This puzzle is like the problems presented earlier in this stage but is larger and more complex. The puzzle includes a 5 x 5 grid and students are provided with 10 different headings and 25 numbers with which to complete the grid.

Encourage students to reflect on the strategies they have used in the previous activities to help determine any headings that need to go on the ‘same side’ of the table.

Enable students requiring further support by providing prompts such as:

• Can you think of any numbers that are even and odd? What does this tell you about where the ODD NUMBER and EVEN NUMBER headings should go?
• Can you think of any numbers that are less than 20 AND more than 20?

This is a particularly challenging puzzle and students will require a lot of time to engage with it.

This stage exposes students to a range of mathematical skills and concepts which you may wish to review and consolidate with students. Consider the progress made by students when determining which skills and concepts you select to review.

Two suggested concepts for review are presented below.

1. Square numbers can be found by multiplying a value by itself

You may wish to emphasise the following concepts:

• square numbers can be located on the diagonal of a common multiplication grid, found in many classrooms
• square numbers can be represented visually by an array of square dots
• working ‘backwards’ from a square number is called ‘finding the square root’ of the number.

2. All numbers have a set of factors and a set of multiples

You may wish to emphasise the following concepts:

• there are a range of approaches we can use to find factors of numbers; some are more systematic than others
• 1 is a factor of all whole numbers
• a number is both a factor and a multiple of itself
• multiples of a number can be found by ‘counting by’ that number.

You could support students to consolidate their understanding of factors and multiples by playing the ‘Factors and Multiples Game’ on nRich, published by the University of Cambridge. This game uses a 1-100 number grid and can be played in different ways.

Students can play competitively in pairs; the first player chooses a starting number and crosses it off the grid. The second player then chooses a factor or multiple of that number and crosses it off. Play alternates, with each player selecting a factor or multiple of the last number crossed off, until one player cannot choose a number, at which point the other player is the winner.

Alternatively, players can play collaboratively, and try and form the longest chain possible of connected factors and multiples, using the rules of the game. This chain could be submitted to help assess student understanding of factors and multiples.

The game can be played either online or on printable grids provided by nRich. The Teachers Resources section provides further advice as to how you can best use this with students.

University of Cambridge, n.d. Factors and Multiples Game. [Online]
Available at: nrich.maths.org/factorsandmultiples
[Accessed 15 March 2022].

University of Cambridge, n.d. Factors and Multiples Puzzle. [Online]
Available at: nrich.maths.org/5448
[Accessed 15 March 2022].