This unit provides an opportunity for students to use all the arithmetic operations, investigate the effect of combinations of arithmetic operations, communicate mathematical thinking using brackets and other mathematical signs and symbols correctly, and use their calculators correctly.
- Write and calculate arithmetic expressions precisely using the order of operations.
- Recognise the importance of the order of operations when using a calculator.
This unit is important for the development of a sound basis of algebra at Level 5 and beyond, during which students begin to study more sophisticated algebra. See Algebra Information for further information.
Algebraic manipulation is a generalisation of arithmetic manipulation. In order for the rules of algebraic manipulation to not become foreboding, student need to realise that they are already familiar with these "new" rules through their varied mathematical experiences. This unit places great stress on these rules or ‘conventions’. It not only helps students to see the need for them, but also gives them practice in using them. They see, first, how to first express what they want to say in words, before moving to the use of 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 available to students can encourage more sophisticated strategy use.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
- explicitly teaching students the conventions of order of operations. Acronyms like BEDMAS (Brackets-Exponents-Multiplication and Division-Addition and Subtraction) can help students to remember the order to enact operations.
- using frequent and systematic recording methods and tools to support students' thinking. Particularly useful symbols include 4/4 for ‘four divided by four’ and 4 – 4 = 0
- using materials and diagrams to model thinking and expressions
- strategically grouping students in order to encourage peer-learning
- using calculators to ease the cognitive load associated with working with larger whole number so students focus more on relationship among numbers rather than on finding answers. For example, 44/4 = 11, 4 x 4 = 16, 4! = 24, etc. become useful building blocks in the Four Fours problem.
The context for this unit is playing with operations on number. Introducing the cultural relevance of the investigations may motivate students. The origins of the four fours problem are in puzzle books that were fashionable in the late 1800s and early 1900s. Encourage students to investigate puzzles and games in their culture. Māori embraced board games bought by European settlers, partly because they had their own version of them, called mengamenga and mū tōrere. These games were played with stones on a board. Students might investigate the Rubik’s Cube, Soma cube, or Tower of Hanoi, as examples of different kinds of puzzles.
Te reo Māori kupu such as raupapa paheko (order of operations, BEDMAS), taiapa (brackets), pū (exponents), whakawehe (divide, division), whakarea (multiply, multiplication), tāpiri (add, addition), and tango (subtract, subtraction) could be introduced in this unit and used throughout other mathematical learning.
- A large chart to display students' work for about two weeks
- Four dice per group of four
- One calculator per group
- Four large-sized dice
Session 1
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.
- Set the context for the unit and introduce the game.
We are going to play a game called Four Fours. The origins of the four fours problem are in puzzle books that were fashionable in the late 1800s and early 1900s.
Do you know of any puzzles and games in your culture? Do they involve mathematics? Māori embraced board games bought by European settlers, partly because they had their own version of them, called mengamenga and mū tōrere. These games were played with stones on a board.
If time allows, you could ask students to bring in interesting games that reflect their culture, and explore these as a class. Alternatively, a member of your local community might be able to come in and teach your class games such as mengamenga and mū tōrere
- 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.
Write 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. - Begin the gameplay. Ask the four students to roll their dice. Write the resulting 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. - Continue playing the game until one team has reached 12 points. Roam and ensure all students understand how to play the game correctly.
- Using some of the examples from the game, discuss how to write the ways of 'making 12' using 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.
To emphasise this, you could display different possible notations for the expressions generated through the game, and get students to discuss and justify, in pairs, which expression is correct. - Direct students, in groups of four or six, (each group divided into two teams) to play a similar game: TARGET 24. Students write down the numbers thrown each time, along with the methods of getting 24 in words. Roam and observe 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)
Session 2
This session makes the transition from writing instructions in words to writing them with 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 games, discuss how to write some of the students’ ways of making 24 in mathematical notation. Stress the use of brackets for making students' intentions clear. For example, “add 5 and 1 then multiply by 4” can be clearly written as (5 + 1) x 4, but not as 5 + 1 x 4 which could be 9.
- Discuss and clarify other conventions as necessary. At this point, you may choose to introduce or revisit BEDMAS.
Brackets are used first.
Multiplications and divisions are performed next, working in order from left to right.
Additions and subtractions 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 useful examples and use them to end the session with a class discussion of the conventions.
Session 3
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.
Session 4
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 shared chart showing how to calculate many numbers (e.g. all up to 100 and some beyond). Students could also work on this independently, or in small groups. If appropriate, you might encourage students to capture their thinking in a variety of different ways (e.g. video, oral, digital presentation, physical movement, using materials). Students can add contributions daily, to be checked or challenged by classmates. Alternative solutions can also be recorded.
For students who require additional support, this task could be done (or started) with increased teacher guidance. Consider grouping students together with mixed levels of mathematical knowledge and confidence to encourage tuakana-teina (peer learning).
- 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 and 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.
- Draw on any interesting questions that arise to 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.
Session 5
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. Emphasise that this should be done with the use of 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.
Dear parents and whānau,
Your child will explain to you the Four Fours Challenge. We are having difficulty finding how to make
__________ __________ ________
(insert some numbers that have not yet been found by the class).
Could your family help us please? If you can’t get those numbers are there some numbers bigger than 100 that you can get? What is the biggest number that you can get?