Using integer tiles
These exercises and activities are for students to use independently of the teacher to develop and practice number properties
Number Framework domain and stage:
Addition and subtraction, advanced additive to advanced multiplicative, stages 6 - 7
Number, level 5
Numeracy Project book reference:
These activities are an alternative to the teaching episodes based on hills and dales, Book 5, page 50.
Prior knowledge. Students should be able to:
- Explain the meaning of a negative number
- Place integers in order of size
- Explain that negative numbers and positive numbers are opposites
During these activities, students will meet:
- a representation of positive and negative numbers using tiles
- addition of positive and negative numbers
- subtraction of positive and negative numbers
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.
Question 7 is critical. It checks whether or not students understand how the integer tiles operate.
This activity gets 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.
This activity extends the previous exercise by steadily increasing the size of the numbers used.
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.
This exercise, like exercise 2 for adding, builds the size of the numbers so the use of the integer tiles for subtraction is more and more time-consuming and difficult.
Mixed 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
Two ways of getting the same result. This activity look at the equivalence of subtracting and adding the opposite.
Magic squares using integers
Practice exercises with answers:
Other resources:Learning Objects, Number level 4: Integer Addition, Integer Subtraction, Integer Addition and Subtraction.
Integer Cover Up (Material Master S3)
Figure It Out Number Year 7-8 Book pg 11 Integer Slam
Figure It Out Number Year 7-8 Book pg 12 Integer Links