# Counting squares

Purpose

This counting collections activity engages students in finding the number of squares of any size on a chessboard.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
NA4-8: Generalise properties of multiplication and division with whole numbers.
NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.
NA5-2: Use prime numbers, common factors and multiples, and powers (including square roots).
NA5-9: Relate tables, graphs, and equations to linear and simple quadratic relationships found in number and spatial patterns.
Description of Mathematics

When a count forms a pattern, students can be encouraged to generalise. A generalised pattern supports students to reason algebraically (develop a rule) for the further counts in the pattern.

Counting collections promotes number sense, and is an essential foundation for students to be successful mathematicians. Recent literature (e.g. Boaler, 2008) suggests that flexible grouping practices best supports equitable opportunities for student learning.

It is important to share the mathematical focus of the task with students. Provide them with opportunities to find patterns within their count, to work collaboratively, to record their patterns and strategies and to share and justify their mathematical thinking using a variety of representations. The progression in the sophistication of students’ thinking when asked to count a collection of objects goes from counting in ones, to counting in groups, reasoning additively to reasoning multiplicatively. When the count forms a pattern (e.g. how many squares or the handshake problem), students should be encouraged to generalise and reason algebraically (develop a rule) for the nth count in the pattern. Consider the mathematical language your students are likely to use when grouping and counting and the language you want to develop.

Student agency is promoted if students have choice over their own counting and recording methods. Rather than suggesting a particular solution or counting method to the students teachers can use enabling prompts to support students who require assistance. Extending prompts can be offered when students have completed the task to build more sophisticated strategies and understandings.

Required Resource Materials
• Hundreds boards, including photocopies for students to manipulate
• Chess or draughts boards
Activity

The wondering for this mathematical inquiry is:

• I wonder how many squares there are on a hundreds board? OR
• I wonder how many squares there are on a chess board?
(Note this inquiry involves finding how many squares of all sizes e.g. 2x2, 3x3…)

Considerations when planning for the task introduction include:

• Do the students understand the context of the task?
• What key mathematical ideas and language do teachers need to clarify with students in order for them to engage with the task?

### Student Exploration of the Task

1. While the students are exploring the task, listen to their mathematical thinking and language particularly around noticing patterns.

2. Encourage students to record their thinking in a systematic way that illuminates possible patterns and encourages generalisations.

3. Ask students to formulate and check possible generalisations.

4. If students are struggling to get started, enabling prompts can be offered but only once students have had sufficient time to struggle with the problem themselves.
Examples include:
What other sizes of squares can you see?
Remember that squares can overlap.

5. Extending Prompts can be offered if students finish quickly or find the task easy. They encourage higher order thinking and generalisation within the same task.
Examples include:
What patterns have you noticed in your count?
Record the patterns so that you can share them with others.
Use your generalisation to find the squares on a different sized board.
How would your generalisation change if the board was made of triangles rather than squares?

### Student Representations

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