Orders Please

Teacher Notes

Programme Overview

Before Viewing

Work through some simple calculations mentally with the class, considering each and establishing whether there is more than one possible answer.

Encourage students to describe to the class the order they used to achieve their answers.

Invite students to make up simple examples that can be calculated in different orders to give different answers.

Establish that expressions can be ambiguous when several operations are involved.

During or After Viewing

Ask students to consider some of the expressions Lisa fed into the Poke-It Calculator to establish the order that the machine used to work them out and the function of the brackets. We saw the following in the programme:

4 ÷ 4 + 4 = 5
4 + 4 ÷ 4 = 5
4 — 4 ÷ 4 = 3
4 x 4 + 4 ÷ 4 = 17
(4 + 4) ÷ 4 = 2
4 + (4 + 4) ÷ 4 = 6

Discuss strategies for tackling complex expressions like the last one Lisa made in the programme:

((4 — 4 ÷ 4) + ((4 + 4) ÷ 4)) x 4 + (4 ÷ 4) = 21

Students can make up some more like this, using multiple brackets.

The class could complete Lisa’s set of numbers, using only the basic operators, brackets and the number 4, by finding expressions which give the following answers:

7, 9, 10, 11, 13, 14, 15, 18, 19

If a number can be made in more than one way, which is the simplest?

(Alternatively, as an activity or homework prior to viewing the video, you could ask students to finds ways of making the numbers 1 to 20.)

Discuss questions like: What could you do if the multiplication on your calculator wasn’t working? Which buttons can you manage without and which are essential?

Explore other notational conventions by looking at expressions like:

2(3 + 4)
¹⁸/₂ + 4
18 + ⁴/₂

Comments

Keep the numbers simple so that students can readily monitor what is happening and appreciate the need for clarity.

Understanding the order of operations is obviously of particular importance when using a calculator. Generally, ‘scientific’ calculators do multiplication and division first, whereas simpler models work from left to right. Students can explore what happens on their own calculator when they enter some of the expressions they have met. Can they describe the order that it is using to perform operations? Do they need to adapt the way they enter calculations? How? Experiment with the use of bracket or memory keys.

Most students should be able to make a start on solving Lisa’s problem. A well-known puzzle is to make all the numbers from 1 to 100 using exactly four 4s — which is more challenging and requires *4 to be used as well. For forming larger numbers, these facts may also be useful:

4! = 24
4 ÷ 0.4 = 10

The Worksheets

Worksheet 1: Order It Now!

1. The Poke-It Calculator used the convention: brackets first, then multiplication and division, and finally addition and subtraction.

2. The question could be extended as follows:

On your own calculator, use these expressions, and others of your own, to explore the order in which your calculator performs operations. Think about how you have to enter each expression to achieve the correct result. You can record the keys you press and the answers you get.

Students might attempt the questions mentally first, as this will help them to recognise what is happening when they try them on their calculator. Emphasise the importance of understanding the order of precedence that their particular calculator is using. They should appreciate that more complex calculations can be performed using the same procedures as are needed here.

3. Once students have found out how to place the brackets, they could explore the use of brackets with a calculator. Ask students how they can get the required answers using their own calculator. They could record key strokes to describe their work.

By building on the strategies developed for smaller numbers, the work can be extended to more complex calculations. These could include larger integers, decimals or fractions. Students should recognise that they can check the order of operations by using small simple numbers.

4. Here are some solutions:

123 — 4 — 5 — 6 — 7 + 8 — 9 = 100
(1 x 2 x 3) — (4 x 5) + (6 x 7) + (8 x 9) = 100
(1 x (2 + 3) x 4 x 5) +6 — 7 — 8 + 9 = 100

(From Mathematical Activities by Brian Bolt (CUP).)

Worksheet 2: The Wizard’s Challenge

This problem is harder than Lisa’s and more open-ended. All students should be able to find some results and express them using their understanding of order of operations and brackets. Collaboration, pooling results and comparing solutions can make the activity more accessible and enjoyable, and require students to be precise in expressing their meaning, so that others can follow their reasoning.

Worksheet 3: Mail Order

This activity provides a real context in which to explore combining numbers to achieve different totals. The initial target of finding expressions for numbers between 20 and 40 should allow most students to make some progress. All can be made, and several have alternative answers which can lead to consideration of the ‘best’ solution.

A 50p book could contain 5 x 3p and 5 x 7p stamps or 12 x 3p and 2 x 7p stamps.

Most students should be able to discover that a choice of two even denominations is not sensible, and explain why. Similarly, denominations with common factors restrict the range of possible totals. They should also appreciate why too large or too small denominations may give fewer practical solutions.

Tables of charges and information about available denominations can be found at post offices or at: http://www.royalmail.co.uk/athome/default.htm.

Alternatively, students could try an activity based on money. Suppose you are waiting for a bus and you don’t know the fare. You don’t want to hold up the bus driver and you want to have the exact change ready. What coins do you get out to cover all possible fares?

Further Ideas

(1) Students could devise games in which the object is to combine sets of numbers to get a result as near as possible to a given target, using the four operations and brackets. They should record their solutions, so that opportunities arise to examine the correct order of operations and use of brackets. A calculator with a random number generator may be useful; alternatively students could make up sets of numbered cards.

Students may be familiar with the numbers game on the Channel 4 television quiz Countdown. They can play this game on the programme’s website at: http://www.channel4.co.uk/entertainment/countdown/intro_game.html.

(2) A challenge often set by teachers in the twentieth century was to invite students to use the digits of the year to make the numbers 1 to 20 or 1 to 100. Students could consider how soon this could be reintroduced to our classrooms. Will their generation ‘miss out’?

Other questions around this activity could be explored: for example, the relative difficulty of the task for different years or decades in the last century. Were students’ minds sometimes unfairly taxed by their teachers?