This unit provides an opportunity for students to use all the arithmetic operations, investigate the effect of combinations of arithmetic operations, communicate with each other using brackets and other mathematical signs and symbols correctly, and use their calculators correctly.
- to write and calculate arithmetic expressions precisely usign the order of operations.
- to realise the importance of the order of operations on a calculator
The students will begin to study algebra proper at Level 5. This unit is important for the development of a sound basis for the algebra at that Level and beyond. (See Algebra Information.)
Algebraic manipulation is a generalisation of arithmetic manipulation. In order for the rules of algebraic manipulation to not become forbidding, student need to realise that they are just the rules that they have always used with numbers. Hence they need to be aware of what those rules are. This unit places great stress on these ‘conventions’. It not only helps students to see the need for them but also gives them practice in using them. They see how to first express what they want to say in words and then in symbols.
The importance of the order of operations is stressed by the use of calculators. It is not good enough simply to type numbers in punctuated by operations. The right answer can only be obtained if care is taken in the order that these entities are put into the machine.
Links to Numeracy
This unit provides an opportunity to focus on the strategies students are using to solve number problems, particularly in the domains of addition and subtraction and multiplication and division.
As students play the target game there will be ample opportunity for students to consider the different strategies they are using to combine the numbers. Encourage students to focus on this by comparing and contrasting their own strategies with those of others. As students explain their strategies, both to each other and to the teacher, there will be an opportunity for students to learn about the strategies of others. If students are not grouped according to strategy stage, they will be exposed to a greater range of strategies for the same combinations of numbers.
For example if the dice showed 6, 4, 2 and 1 and the target was 24, useful combinations of numbers are
6 + 4 = 10 6 X 2 = 12 6 + 2 = 8
These combinations can then be combined in a variety of ways to reach the target, for example:
10 + 12 + 2 12 + 8 + 4 10 + 10 + 8 – 4
Which combinations of numbers make tidy numbers to work with?
Are there any other groups of numbers you can see that would be useful? Why?
Which numbers will be useful with the operation of multiplication? Why?
Making dice with larger numbers would encourage more sophisticated strategy use and may be appropriate for some students.
a large cardboard chart to display students' work for about two weeks
four dice per group of four
one calculator per group
four large-sized dice
order of operations, notation, conventions, brackets, combinations, ambiguous, ambiguity
This session introduces a game that uses the four arithmetic operations. The aim of this game is to use numbers in interesting and creative ways. The students write their operations in words.
- We will start by playing a simple version together and then you will be able to play a harder version of the game in your groups. The teacher writes TARGET 12 in a box on the board, gives a large die to each of four selected students, and divides the class into two teams.
- The four students all roll their dice and the teacher writes the numbers on the board.
Can anyone make 12 by combining some, or all, of these numbers? Each number can only be used once.
If no one can make 12, then the four students roll their dice again. If they can make 12, then selected students explain their methods.
For example, if 5, 4, 1, 2 are rolled, someone may correctly suggest “add 5 and 1 then multiply by 2”. Write this in words on the board.
If 4, 4, 5, 3 are thrown; someone may suggest “just multiply 4 by 3”. Someone else may suggest “add 4, 3 and 5”. Write this in words on the board.
If a student suggests a correct way of making 12, then their team wins a point. The other team can challenge it and if the challenge is upheld then the challengers take the point instead. Teams take turns in suggesting a method. When no one can suggest another way, the dice are rolled again. Play continues until one team has reached 12 points.
- Using some of the examples from the game, discuss how to write the ways of making 12 in mathematical notation. Stress the use of brackets for making intentions clear. For example, “add 5 and 1 then multiply by 2” has to be written as (5 + 1) x 2 and not as 5 + 1 x 2 which could be misinterpreted as 7.
- Groups of four or six students (each divided into two teams) now play a similar game with TARGET 24. Students write down the numbers thrown each time, along with the methods of getting 24 in words. Teacher observes students working to collect good examples for subsequent discussion.
- End the session with a discussion of any patterns that the students noticed (e.g. 2, 3, 4 and 6 can all be used with multiplication, because they are factors of 24)
This session makes the transition from writing instructions in words to writing them in mathematical symbols. People need to be able to communicate exactly what calculations they have in mind. This is done by agreed rules (conventions) and by using brackets.
- Using some of the examples from the earlier game, discuss how to write some of the students’ ways of making 24 in mathematical notation. Firstly stress the use of brackets for making their intentions clear. For example, “add 5 and 1 then multiply by 4” can be clearly written as (5 + 1) x 4 and not as 5 + 1 x 4 which could be 9.
- Discuss and clarify other conventions as necessary.
Brackets are cleared first.
Multiplications and divisions are performed next, working in order from left to right.
Additions and subtractions are follow next, working in order from left to right.
Example: 3 + 4 x 5 –10 + (6 + 10)/8
Step 1 3 + 4 x 5 –10 + 16/8
Step 2 3 + 20 –10 + 16/8
Step 3 3 + 20 –10 + 2
Step 4 23 –10 + 2
Step 5 13 +2
Step 6 15
Recommend the use of brackets wherever ambiguity may arise.
- Play the game TARGET 20 in groups with four dice, this time recording the calculations in both words and mathematical symbols. Collect good examples and use them to end the session with a class discussion of the conventions.
This session aims to raise students’ awareness of the fact that when using calculators the order of operations is important. If they are not done in the correct order, the answer will be wrong.
- Students should at least know that they need to be careful that their calculator is doing what they intend. For example, does pressing the button sequence give you (5 + 2) x 3 or not?
5 + 2 x 3 =
You see it could give you either the correct answer to 5 + (2 x 3) (= 11) or the correct answer to
(5 + 2) x 3 (= 21). If this sequence of buttons gives you the wrong answer, how do you enter it to
make it correct?
- Each student now tests what their own calculator does on problems that mix addition, subtraction, multiplication and division, and write a short report with examples to explain how to use their calculator properly. A good strategy is to start with simple whole number examples where the answers can be easily found, and progress to more complicated numbers. Final examples should be written in words, mathematical symbols and calculator button sequences and could include expressions like (5.1 + 2.2) x 3, 5.1 + (2.2 x 3), 24.6 ÷ 3.4 ÷ 2, 24.6 ÷ (3.4 ÷ 2), (24.6 ÷ 3.4) ÷ 2, 48 ÷ (3 x 4), (48 ÷ 3) x 4, 6 + 18 ÷ 3, 100 – (75 – 20), (100 – 75) – 20, etc.
FOUR FOURS CHALLENGE
The class is challenged to make as many numbers as they can using AT MOST four fours and any mathematical sign. For example 4 x 4 + 4 + 4 = 24, 4 – 4 + 4 – 4 = 0, 44 + 4/4 = 45, 4 + 4 = 8. The teacher will need to negotiate with the class what signs are allowed (e.g. brackets, +, x, -, ÷, etc.), This activity may extend over several days or weeks, as students gradually build up a wall chart showing how to do many numbers (e.g. all up to 100 and some beyond). Students can add contributions daily, to be checked or challenged by classmates. Alternative solutions can also be recorded.
- Introduce the problem and illustrate with a simple case, such as 4 + 4 + 4 + 4 = 16. Ask for other suggestions from the class and record these in words (if necessary) and symbols.
- Challenge the class to see which of the numbers from 1 to 100 they can make. Divide the class into groups to let students make a start on the problem. Make sure the groups record their answers in words as well as symbols when necessary.
- Some suggestions will need brackets inside brackets. This may need to be discussed with the class.
- Many interesting questions will arise that will extend students’ knowledge of numbers and operations. For example, students who write 4 – 4 – 4 + 4 may wonder whether this is equal to zero (it is) because they need to go into negative numbers to work it out.
- Add selected solutions to the wall chart.
- Finish the session with a progress report from the groups and set them the challenge of seeing how many other numbers can be done.
Conduct this session when good progress has been made on completing the four fours chart, e.g. when most of the numbers from 1 to 100 have been completed. Select examples from the chart to illustrate the following points:
The order of operations matters e.g. working out 4 x 4 – 4 ÷ 4 you can get many answer
((4 x 4) – 4) ÷ 4 = 3
4 x 4 – 4 ÷ 4 = 15
4 x (4 – 4 ÷ 4) = 12
4 x (4 – 4) ÷ 4 = 0
People need to be able to communicate exactly what calculations they intend to be done. This is done by agreed rules (conventions) and by using brackets.
- Discuss and clarify the use of brackets and conventions as necessary. Promote the use of brackets wherever ambiguity may arise.
- Link with calculator use as before.