4. Using Symbols to Recognise Unknowns

• A container and 10 balls
• Plastic cups
• Counters
• Envelopes
• Whiteboard and whiteboard pens
• Sticky Triangles
• Information for teachers about the use of pronumerals

1. How Many Balls?

The purpose of this activity is to encourage students to identify ‘how many balls are in the box’ using only the clues provided.

Put four balls into a container without the students seeing you do this. Ask the following questions:

• How many balls are in this container?
• You are all giving different answers. Why is this?
• If we want to be certain about how many balls are in the box without me giving you the number, what do you need?
• If I give you the following clue, could you work out with certainty how many balls are in the container?

CLUE: If I put two more balls into the container, I would have six balls. How many balls do I have?

Ask students to work out the problem, then ask some students to share their strategies. Ask:

How would I write this problem using a mathematical equation?

You might get this response: 'blank' + 2 = 6

The use of a blank or a square to represent an unknown is a pre-cursor to the development of using pronumerals. 

Explain that they were using the knowns in this equation to work out the unknown, which is ‘how many balls were in the box?’ This is known as algebra.

Ask: Rather than just leaving a blank or using a box for an unknown, how could we represent this instead?

Explain that we can use symbols (pronumerals) to represent the unknown. For example, ‘x’ represents one unknown number and ‘y’ represents another unknown number. If a letter appears more than once in an expression it represents the same number. For example, with the above equation, it can now be read as:

Ask: How could we work out what x is?

Demonstrate that if we take 2 away from the first side, we need to also take away 2 from the second side so that the equation remains balanced. For example:

By seeing the relationship between the numbers and understanding the need for equivalence, equations can be manipulated to work out unknown numbers.

1. How Many Balls Now?

Ask students work in pairs to replicate the How Many Balls? activity. Give each pair of students two cups and 20 counters.

Tell the students to take it in turns to hide some counters in a cup and provide their partner with a clue. In turn the partner must calculate the number of counters in the cup based on this clue, using a similar clue to the one used as an example in the Get started activity.

Throughout the activity roam around and ask the following questions:

• What clues are you using?
• Are some clues easier to work out than others? Why?
• How are you working out the unknown?

Enable students by encouraging them to model each clue with their own counters. For example:

‘If I add three balls to the box and now, I have a total of 10 balls can you tell me how many balls were in the box in the beginning?’

Encourage students to model the equation by starting with ten counters and then separating three counters from the ten, thus modelling the difference of seven.

Extend students by encouraging them to:

• Use different operations in their clues. For example, ‘If I multiplied the number of balls, I have by 3 I would have a total of 12 balls. How many balls do I have?’
• Use more than one operation in their clue. For example, ‘If I divide the number of balls, I have by 3 and then take away 2 balls I would have 3 balls left. How many balls do I have?’
• Record each clue as an equation, using a pronumeral to represent the unknown quantity.

2. What are Their Ages?

Present the following scenario to the class:

Last night my niece Emily slept over at our house. My daughter Ali and Emily were comparing ages. That’s when we all realised that Ali is twice as old as Emily and that the sum of their ages is 18. Can you work out each of their ages?

Before students are sent off to work on this problem, ask them if they understand what the question is asking them. Also check to see if there are any words that they do not understand. Once any issues have been cleared up, encourage the students to work with a partner to find a solution to the problem.

While students are working together roam around and observe how they are approaching the problem. Ask the following questions if needed:

• How might you work out this problem?
• What is the relationship between the two ages?
• How could you represent the unknown ages of the two girls?
• Can you write an equation and use a pronumeral to represent the unknown?
• What functions/operations will you use?
• What will you do first?

Note: Either girl’s age can be represented by a pronumeral and the other age represented in relation to that age. The equations will look different, but the outcome is still the same. For example:

If Emily’s age is x, Ali’s age is twice x, and the equation will be:

If Ali’s age is y, Emily’s is half of y, and the equation will be

Enable students by assisting them through the problem step by step, asking the following questions:

• Whose age would you like to start with?
• How can you represent this age using a pronumeral?
• Do you see a relationship between the two ages? What is that relationship? (One is twice as old as the other)
• How will you represent the age of the other girl using this pronumeral?
• It says that the sum of their ages is 18; write this as an equation?
• How will you work it out now?

Extend students by encouraging them to create a similar riddle about a family member or friend and their age using a different relationship other than double or half someone’s age. For example:

• One person is 5 years older.
• One person is 13 years younger.
• One person is three times as old as the other.
• One person is a one-fifth the age of another.

Have students write an algebraic equation for the problem and then ask a friend to try to solve it.

Areas for further exploration

1. Sticky Triangles

In the Sticky Triangles activity, students explore a growing pattern of triangles. A triangle is made out of matches. Six more matches are added to create 4 triangles. Then another row is added and so on.

Ask: How many matchsticks are needed every time you add a row?

Encourage students to record their results in a table. Ask them to look for a pattern and write an algebraic equation to represent this pattern. Encourage them to use a symbol (pronumeral) to represent the unknown.

The preceding tasks and stages have encouraged students to explore the concepts of patterns and equivalence, and to explore using symbols (pronumerals) to represent unknowns. To consolidate student understanding, the below tasks allow students to represent the unknowns in both equivalent equations and algebraic equations used to represent patterns. Students also learn that functions/operations are applied when trying to work out the value of an unknown.

Present students with some written problems and ask the students to write an algebraic equation for each one, using a symbol (pronumeral) to represent the unknown in each situation. Use a mixture of types of unknowns, such as result unknown, start unknown, change unknown, whole unknown, part unknown, difference unknown etc (Cognitively Guided Instruction, Carpenter, P et al, Heinemann, 1999). Examples include:

• Jon had some football cards. Jane gave him 13 more. Then he had 29 football cards. How many football cards did he have to begin with?
• Martha has 75 apples. 48 are red and the rest are green. How many green apples does Martha have?
• Frank has 14 skateboards. Pete has 8 more than Frank. How many skateboards does Pete have?

Now ask the students to use their understanding of equivalence and functions to find the value of the unknown (pronumeral).

Present students with a pattern. Ask them to find the rule and write an algebraic equation to represent the rule. Then ask them to work out what the twentieth term would be (adapted from Van de Walle et al, 2014).

• Complete the table above
• How many triangles are needed for step 10? Step 20? Step 100? Explain your reasoning.
• Write a rule (in words and symbols) that gives the total number of pieces to build any step number. Use a symbol (pronumeral) to represent the unknown.

For both situations above ask students the following questions:

• How did you work out the unknown number?
• How did you use your knowledge of balanced equations to help you out?
• What functions (operations) did you use?
• What patterns did you notice? Did this help you work it out?

Responses to these questions can assist in assessing the students’ understanding of algebra.

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