Keep Your Balance

Teacher Notes

Programme Overview

Before Viewing

Introduce the idea of a balance, and explore the conditions necessary to maintain it

An approach consistent with the programme content would be to represent the ‘unknown’ in a box and the ‘knowns’ as weights, which could be shown as small circles.

Develop the idea that balance is maintained provided that the same operations are performed on both sides. Focus on this idea, rather than finding the contents of the box, at this stage.

You could use diagrams in your discussion. For example:

This box balances these weights.

What if I do this?

How can I get the scales to balance again? What do I need to do on the right-hand side?

What if I do this? What could I put on the right-hand side?

Invite students to suggest other modifications and solutions.

Once students have grasped the idea that a balance can always be maintained by performing the same operation on both sides, explore examples where combinations of operations lead to discovery of the contents of the box. For example:

If each box has the same weight, how can we find the contents of one box?

What if I now do this?

Get students to explore questions like: ‘If I put 4kg in the box, can you give me some examples of scales that will balance?’

Emphasise the mathematical equivalents of the terms you use. For example, ‘remove a box’ = ‘subtract a box’; ‘double the weights’ = ‘multiply the weights by 2’.

During or After Viewing

Work through the examples given in the programme, focusing on the use of letters to represent unknowns. The equations were:

A + 5 = 8
W + A + 3 = 15
F + 7 = 5
E + F + 3A = 32
6x — 5 = 7

Discuss the necessary steps and how these are written down algebraically.

Link the idea of the physical balance to the idea of balancing the two sides of an abstract equation.

Develop the concept of inverse operations and how we decide what needs to be done to both sides to reach a solution.

Invite students to make up further examples using the context of the programme, and discuss their solutions. Does the answer have to be a whole number?

Explore simple word problems that can be solved by identifying an unknown and writing an equation. For example:

Jane is 4 years older than Petra, who is 3 years older than Luke. How old will they each be when the sum of their ages is 70?

Comments

Students need to appreciate that the strategies they use to solve simple problems with algebraic methods will also be applicable to more complex problems.

The drawing of balances can soon become tedious, and the more succinct representation using equations should be a welcome alternative, allowing a wider range of problem types to be solved.

The Turkey Problem

The following balancing example could be presented for students to work on.

Can you help a fat grocer with a weighty problem on Christmas Eve?

‘Twas the night before Christmas; he’d shut up the shop.
He’d had nothing to eat since his lunchtime lamb chop;
And all that was left of his produce and goods
Was a turkey, some onions, and three Christmas puds.

‘My diet must be balanced!’ he said to himself
As he took all the leftovers down from the shelf.
And the gleam of the scales inspired him to see
Just how balanced his diet could possibly be.

Three puds and the turkey he put on one tray;
On the other, 12 onions: the same they did weigh.
Next, on the left pan, he put just the hen.
With three puds and six onions it balanced again.

It was then that he saw he was in a right fix,
Because onions and puds are a terrible mix.
So he left out the puds, and used onions alone.
How many will balance the bird on its own?

Adam Hart-Davis

The Worksheets

Worksheet 1: Balancing Act

No particular method of finding and recording solutions is specified here, to allow for differentiation within or between groups of students.

The equations represented by the balances can be solved in the order presented. Many students should be able to write down and solve algebraic expressions. The diagrams suggest a simple representation that students can reproduce if they need more concrete practice with the concept of a balance.

The statements can be represented either algebraically or as balances. You might consider here what to do with the kilogram units when translating into algebra. Students will need to choose an appropriate order of solution. The negative value of ‘air’ presents students with the same problem as ‘fire’ did in the programme.

Students can share and solve one another’s suggestions for custard (they could be presented as balances if appropriate), which should generate plenty of equations. A greater range of equation types are possible if students can work directly with algebra and include subtraction and division, as well as larger numbers.

The work can be extended, with students choosing their own sets of values and creating sets of consistent equations for others in the group to solve. Do their sets of equations give sufficient information? Do they give unique solutions? What order of solution do they suggest? Can they be solved in a different order? What if the variables can take fractional values? Some students may prefer to work with black and yellow bile, phlegm and blood!

Worksheet 2: Amazing Balances

The activity aims to give practice in solving equations, substituting values, and reasoning. It includes a wider range of equation types which cannot easily be represented by balances using weights. It is assumed that students will use algebraic methods to find the solution and that they will recognise that the letters stand for the ‘elements’ given in the programme.

The route ends with the equation: C — 7A = 8.

Any of the equations found on the correct route can be used to find or check values. The remaining ‘balances’ are not true.

Different solutions can be compared.

The correct values are:

A = 4
C = 36
E = 20
F = —6
W = 12

Students could explore what happens if custard had a different value — for example, 22, 50 or 30.

Students could design their own maze of balances for their own choice of values. They need to be careful not to create dead ends with their false balances.

Worksheet 3: Wizard Training!

This activity aims to provide a context for students to translate instructions into algebra and to develop notions of algebraic proof. Everyone should be able to substitution different values.

To start with you could go over the first puzzle with the students; while calling out the statements you could write the equivalent algebraic expressions on a flipchart or pad of paper which is hidden from the class. For example:

Think of a number.	N
Add 5.	N + 5
Take away 3.	N + 5 — 3 = N + 2
Take away the number you first thought of.	N + 2 — N = 2

Some students may struggle to express their work algebraically but nevertheless appreciate what is happening and be able to express this verbally. Many students should be able to give explanations and proofs using equations. Some students may find flow charts more accessible than equations for monitoring and describing the puzzles.

Creation of their own puzzles should allow students to develop their understanding of substitution and inverses.

The Challenge will work with anyone aged under 100. The answer, in base 10, should always be the house number followed by the age.

Further Ideas

Students can meet and develop their skills in solving equations using an activity which is often known as ‘Pyramids’.

The pyramids are built by adding pairs of numbers. Initially a pyramid of base 3 is sufficient. Students explore whether they can complete a pyramid if some of the numbers are missing. Different sizes of pyramid can be explored with starting numbers given in different locations. Students can readily make up their own examples and generate their own variations and questions.

The use of a letter to stand for an unknown in the pyramid leads to the construction and solution of simple equations from the information given. This can be extended for more able students to include two unknowns and solution sets, fractional and decimal solutions, and the use of other operations (—, x, ÷) to build pyramids. Some examples are given below: