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Level Four > Number and Algebra

Using integer tiles

Achievement Objectives:

Achievement Objective: NA4-2: Understand addition and subtraction of fractions, decimals, and integers.
AO elaboration and other teaching resources

Purpose: 

These exercises and activities are for students to use independently of the teacher to develop and practice number properties

Specific Learning Outcomes: 
solve problems that involve addition of positive and negative numbers
solve problems that involve subtraction of positive and negative numbers
Description of mathematics: 

Addition and subtraction, AM (Stage 7)

Required Resource Materials: 
Practice exercises with answers (PDF or Word)
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 Exercises

Exercise 0
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
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
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
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
Asks students to solve subtraction problems with integers.  The problems include 2 digit integers.

Exercise 5
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
Asks students to explore the equivalence of subtracting and adding the opposite.

Exercise 7
Asks students to solve magic squares puzzles using integers.

AttachmentSize
IntegersPracticeSheet.pdf140.78 KB
IntegersPracticeSheet.doc120 KB