Introducing Algebra

Years 6 - 8

Summary

The idea of each plug having a value - usually 1 - is built into almost all of the Calculating Changes activities. In this activity that idea is extended based on learning from Task 130, Protons & Anti-Protons, in which each blue plug is worth positive 1 and each yellow plug is worth negative one. Therefore any collection of equal numbers of blue and yellow plugs is worth zero. But can we use the red plugs for anything?

Well, they could represent the value of a mystery collection of yellow and blue plugs. Aha! Now we can represent algebraic expressions and equations and explore that 'text book stuff' in a visual, concrete way. Suitable for threading.

Materials

  • One Poly Plug set per pair

Acknowledgement

This activity was contributed by Tony Wright, Bellarine Secondary College.

Preparation

This activity builds on experience with Protons & Anti-Protons from the Mathematics Task Centre. In this task, a plug blue side up represents +1 and a plug yellow side up represents -1. The task also introduces two key concepts of integer arithmetic which are the basis of the Introducing Algebra activity.

  1. Any positive or negative number can be represented by an infinite number of collections. For example, each of the three collections shown is worth 3 Protons (or +3) because equal numbers of positive and negative units sum to zero.

  1. Adding a negative gives the same answer as subtracting a positive.

 

Content

  • algebra, equivalent expressions
  • algebra, factorisation
  • algebra, like & unlike terms
  • algebra, linear
  • conservation of number
  • equations: creating/solving
  • multiplication - array model
  • negative numbers
  • order of operations
  • problem solving
  • properties of number
  • properties of zero
  • recording - written
  • using brackets
  • visual and kinaesthetic representation of number

Procedure

So if we know what a number is, we can make it. We can also represent it with any collection that suits our problem at the time. But what happens if we don't know what the number is, for example, if someone has put some plugs in a box?

Well, we could represent the unknown number in the box by drawing the box, or by using something, say a red plug, to stand for it. Then we could try to find out more about how the value in the box combines with numbers we do know.

In this approach, algebraic representation becomes just another What happens if...? adventure of the sort mathematicians love to play with. Now the procedure is to explore as a threaded activity - a little bit often - to help the children grow confident with their manipulation and recording of algebraic expressions. Not because algebraic manipulation is the main objective, but because the challenges explored are interesting and a mathematician's work begins with an interesting problem.

What are the learning features of this activity likely to make it interesting for children?

Algebraic Expressions

The preparation activities grow confidence in representing any particular positive or negative number. For example:
  • Make a collection of yellow and blue plugs with a value of  -7.
  • Make the same value another way.
  • Make the same value so it includes exactly three  -1 plugs.
  • Make and record 5 different collections with a value of  -7.

The journal recording (apart from pictures) of any of these collections is the number -7. But how do we record the hidden number of plugs, which is being represented by a red plug? Well, as long as we don't actually use a number, it doesn't matter what we choose. We could draw a box, but mathematicians usually choose a letter of the alphabet. So,

  • How about x?
(If you want to get really sophisticated down the track, perhaps a sticker could be put on one side of the red plug to represent  -x.)

Now we can make all sorts of collections that include red plugs.

  • The red plugs represent a box with an unknown number of yellow and blue plugs, and
  • If there is more than one red plug in the same problem, each one represents the same unknown number.

For example:
This collection represents 6x - 12 because adding the negative has the same effect as subtracting the positive.

Make this collection and record how you can sort it into equal groups to represent:

  • 6(x - 2) meaning 6 rows of (x - 2)
  • 3(2x - 4) meaning 3 rows of (2x - 4)
  • 2(3x - 6) meaning 2 rows of (3x - 6)
That's what factorising is all about. Finding ways to make a total collection into rectangles.

Expanding is asking the reverse question, for example:

We have 5 rows of (x plus 1), ie: 5(x + 1), what is value of the total collection?

Isn't this more fun and more meaningful than drill and kill exercises from a text book?

Linear Equations

Now suppose the person who hid the plugs in the box still doesn't tell us the value of the box, but does give us a clue that will help us solve the problem. For example, suppose we are told that 5x - 6 = 4. Our challenge is to find the value of the plugs hidden in the box. This value is represented by a red plug when we make the equation and by x when we record it.
So, what is the value of x if 5x - 6 = 4?
  • Start by making the equation. What starts equal has to stay equal...

  • Keep it equal by doing the same thing to both sides. In this case adding a number that makes the total collection of known plugs zero. Then all the value on the right hand side must come from the red plugs...

  • So five things with the same value are worth 10, so we can work out the value of each thing (red plug, mystery box)...

  • And we get the answer x = 2...
  • Substitute that value back into the original collection to check that it works.
So, the person hid plugs with a value of 2 inside the mystery box.

As with the algebraic expressions section above, children tackling one or two of these equation solving challenges a day (perhaps at a work station), sometimes invented by the teacher, sometimes invented by the children, for three or four days each week over two or three weeks, leads to more involvement, more understanding and more confidence with tacking text-type questions.

  • What happens if there are x plugs on both sides, eg: 3x - 6 = 4 - 2x?

Extensions

  • Other 'mystery box' activities that lead into our complement this activity are Number Shapes and Trial, Record & Improve (Free Tour) and Box Hunt, Six Plus and What's My Rule? (Members).
  • Maths300 members could enrich the activity with Lesson 19, Backtracking, Lesson 34, What's My Rule?, Lesson 84, Number Charts, Lesson 94, Trial, Record & Improve, Lesson 156, Chart Strategies, and Lesson 160, Algebra Charts.

Calculating Changes ... is a division of ... Mathematics Centre