Using integer tiles

Achievement objectives

NA4-2: Understand addition and subtraction of fractions, decimals, and integers.

Description of mathematics

• Addition and subtraction, AM (stage 7)

Required materials

See Materials that come with this resource to download:

• Using integer tiles activity (.pdf)
• Using integer tiles activity (.doc)

Activity

Prior knowledge

• Explain the meaning of a negative number.
• Place integers in order of size.
• Explain that negative numbers and positive numbers are opposites.

Background

Integer tiles are one way of using a material representation to introduce the concept of adding and subtracting integers. This model can be useful for working on the type of problem that many students find harder, that is problems like ^-4 + ^-5. However, in themselves, the tiles are rather abstract, and students need to be able to recognise the level of abstraction if they are to be successful in using them. This abstraction is spelled out in more detail in the next paragraph.

Having a red tile and a blue tile on their own, even if they are labelled "1" and "-1" does not mean that students recognise these things as being different. Indeed for some, if you asked "how much you have got in total", the answer "2" is perfectly reasonable, as they are seeing and counting the objects, rather than identifying that meaning has been given to the colour, or the numbers written on the objects. For the tiles to be of value, students must already have some understanding of integers (so the negative sign conveys meaning) and have the concept of the integers being the opposites of the whole numbers (so negative two is equal and opposite to positive two). This allows the underlying concept of the tiles to make sense and allows students to see that the negative one tile can cancel out the positive one tile.

Also remember that the ultimate aim of introducing the tiles is to scaffold learning, so students can work on the numbers themselves, without reference to the tiles, so these should be removed as soon as students have developed an understanding of the principle behind the model.

Comments on the exercise

Exercise 0

This exercise asks students to represent integers using drawings of tiles. Question 7 is critical. It checks whether or not students understand how the integer tiles operate.

Exercise 1

This exercise asks students to solve addition problems with integers. Students to represent additions using the tiles, and if necessary, use them to help answer the problems. The last question is again the most important in the exercise – have the students developed an understanding of the principle behind the tiles as a model for addition?

Exercise 2

This exercise asks students to solve addition problems with integers. This activity extends the previous exercise by steadily increasing the size of the numbers used. The problems include 2 digit integers.

Exercise 3

This exercise asks students to solve subtraction problems with integers. Representing subtractions with the tiles is harder than representing additions, as this involves taking tiles away. Here, the model becomes cumbersome. For example, with the problem ^-3 – 4: When represented with tiles, 4 positive one (red) tiles are being taken away from 3 negative one (blue) tiles, but as there are no positive tiles to take away, this problem is hard to do unless a "whole load of zeros" are thrown into the mix. That is, by adding 4 positive and 4 negative tiles to the table, 4 positive tiles can now be removed, leaving seven negative tiles on the table. Readers are invited to consider whether or not the use of the tiles is simplifying the problem for students.

Exercise 4

This exercise asks students to solve subtraction problems with integers. The problems include 2 digit integers.

Exercise 5

This exercise asks students to solve a mix of addition and subtraction problems. This activity is designed to help students decide which strategy is the most useful for answering questions across the range to which they have been introduced.

Exercise 6

This exercise asks students to explore the equivalence of subtracting and adding the opposite.

Exercise 7

This exercise asks students to solve magic squares puzzles using integers.

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