Algebra · Variables and Equations · Lesson 2

Solving Equations with Counters and a Balance

An equation is just like the balance scale a trader uses at the market: both sides weigh the same. In this lesson you will use bags, counters, integer tiles and shapes on a balance to solve equations with your hands, say each move in words, and write the same moves in symbols.

NaCCA B7.2.3.1.2 ⏱ about 45 min

By the end you will be able to

✔Model a linear equation with bags (unknowns) and counters on a balance.
✔Solve an equation by doing the same thing to both sides so the balance stays level.
✔Use positive and negative counters (integer tiles) and zero pairs to solve equations such as $x + 5 = 2$ .
✔Describe each step of a solution in words and in symbols.

Key words

Balance: A scale that is level when both sides have equal mass; an equation is like a level balance.
Counter: A small object standing for 1. A bag or cup with an unknown number of counters stands for $x$ .
Integer tiles: Counters in two colours: yellow for +1 and red for −1.
Zero pair: One +1 and one −1 together. Their total is 0, so they can be added or removed freely.
Solution: The value of the unknown that makes both sides equal, e.g. $x = 4$ for $x + 3 = 7$ .

Curriculum reference

Content standard: B7.2.3.1 Demonstrate an understanding of linear equations of the form x + a = b (where a and b are integers) by modelling problems as a linear equation and solving the problems concretely, pictorially, and symbolically.

Indicator: Model and solve linear equations using concrete materials (e.g., counters and integer tiles) and describe the process orally and symbolically.

Starter

The tomato seller's scale

Auntie Esi sells tomatoes at Kejetia market in Kumasi. On her scale, a closed bag of tomatoes and three 1 kg weights balance seven 1 kg weights. She does not open the bag, yet she knows straight away that the tomatoes weigh 4 kg. She simply took 3 kg off each side in her mind. That idea — keep the scale level by doing the same to both sides — is the key to solving every equation.

1. An equation is a balance

Look at the equation $x + 3 = 7$ . The equals sign tells us that the left side, $x + 3$ , has exactly the same value as the right side, 7. Picture a balance scale: on the left pan is a bag holding an unknown number of counters plus 3 loose counters; on the right pan are 7 loose counters. The scale is level.

If you take something off one pan only, that side becomes lighter and the scale tips. If you take the same amount off both pans, it stays level. This gives us the golden rule of equations.

Our aim is always to get the bag (the unknown) alone on one side. Then whatever is on the other side tells us what is in the bag.

2. Bags and counters

Let us solve $x + 3 = 7$ with real objects. Use a small bag or an empty sachet for $x$ and bottle tops for the counters. Step through the animation and say each move aloud.

▶ Notice that 3 counters leave BOTH pans at the same time, so the scale never tips. The bag ends up alone, balancing 4 counters.

The indicator asks you to describe the process orally (in words) and symbolically (in algebra). Both describe the same moves:

What I do with the objects (orally)	What I write (symbolically)
A bag and 3 counters balance 7 counters.	$x + 3 = 7$
I take 3 counters away from each side.	$x + 3 - 3 = 7 - 3$
The bag alone balances 4 counters.	$x = 4$
I check: 4 counters and 3 more make 7.	$4 + 3 = 7$ ✓

3. Bags on both sides

Sometimes there are bags on both pans. All bags hold the same number of counters, so we may remove one bag from each side just as we removed counters. Take $3 x + 1 = x + 5$ : three bags and 1 counter balance one bag and 5 counters.

▶ Watch the scale stay level after every move. Bags are removed first, then counters, and finally we share equally.

1Remove the same number of bags from both sides so that bags are left on one side only.
2Remove the same number of counters from both sides so that counters are left on the other side only.
3If several bags remain, share the counters equally among them.
4Check by putting your answer back into the original equation.

4. Integer tiles and zero pairs

Bags and counters work well while everything is positive. What about $x + 5 = 2$ ? You cannot take 5 counters from a side that has only 2. For this we use integer tiles: a yellow tile is +1 and a red tile is −1.

A yellow and a red tile together make a zero pair: $(+ 1) + (- 1) = 0$ . Adding or removing zero pairs does not change the value of a side, so we can use them whenever we need extra tiles.

Equation	Move with tiles	Result
$x - 2 = 3$	Add 2 yellow tiles to each side. On the left they form 2 zero pairs with the red tiles and vanish.	$x = 5$
$x + 5 = 2$	Add 3 zero pairs to the right (now 5 yellow, 3 red). Remove 5 yellow from each side.	$x = - 3$
$x + 4 = - 1$	Add 4 red tiles to each side. On the left they cancel the 4 yellow; the right has 5 red.	$x = - 5$

5. Shapes on a balance

Balance puzzles can also use shapes of unknown mass. Each shape of the same kind has the same mass. Treat each shape like a bag: remove the same things from both pans, then share.

For example, three identical cubes and a 10 g mass balance one cube and a 50 g mass. Removing one cube from each pan leaves two cubes and 10 g balancing 50 g. Removing 10 g from each pan leaves two cubes balancing 40 g, so one cube is 20 g.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

One bag and counters

Model and solve

x + 3 = 7

with a bag and counters. Describe each step.

Show the solution

Step 1Model: a bag and 3 counters on the left, 7 counters on the right.
Step 2Say: "I take 3 counters off each side." Write: $x + 3 - 3 = 7 - 3$ .
Step 3Say: "The bag balances 4 counters." Write: $x = 4$ .
Step 4Check: 4 + 3 = 7 ✓

Answer: $x = 4$

Example 2

Bags on both sides

Solve

3 x + 1 = x + 5

using bags and counters.

▶ Each move is done to both pans, so the scale stays level until one bag is left alone.

Show the solution

Step 1Remove one bag from each side: $2 x + 1 = 5$ .
Step 2Remove 1 counter from each side: $2 x = 4$ .
Step 3Share 4 counters equally between 2 bags: $x = 2$ .
Step 4Check: left 3 × 2 + 1 = 7; right 2 + 5 = 7 ✓

Answer: $x = 2$

Example 3

Removing two bags

Solve

3 x + 4 = 2 x + 8

Show the solution

Step 1Remove 2 bags from each side: $x + 4 = 8$ .
Step 2Think of 8 as 4 + 4, so $x + 4 = 4 + 4$ .
Step 3Remove 4 counters from each side: $x = 4$ .
Step 4Check: 3 × 4 + 4 = 16 and 2 × 4 + 8 = 16 ✓

Answer: $x = 4$

Example 4

Using zero pairs

Use integer tiles to solve

x + 5 = 2

Show the solution

Step 1Model: an x-bag and 5 yellow tiles on the left; 2 yellow tiles on the right.
Step 2We need 5 yellow on the right, so add 3 zero pairs there: now 5 yellow and 3 red (still worth 2).
Step 3Remove 5 yellow tiles from each side: the left is $x$ , the right is 3 red tiles.
Step 4So $x = - 3$ . Check: −3 + 5 = 2 ✓

Answer: $x = - 3$

Example 5

Cubes on a balance

Three identical cubes and a 10 g mass balance one cube and a 50 g mass. Find the mass of one cube.

Show the solution

Step 1Let one cube have mass $x$ g: $3 x + 10 = x + 50$ .
Step 2Remove one cube from each side: $2 x + 10 = 50$ .
Step 3Remove 10 g from each side: $2 x = 40$ .
Step 4Share: $x = 20$ . Check: 3 × 20 + 10 = 70 and 20 + 50 = 70 ✓

Answer: Each cube has a mass of 20 g

Summary

An equation is like a level balance: both sides have the same value.
Golden rule: whatever you do to one side, do exactly the same to the other side.
Get the unknown alone: remove bags from both sides first, then counters, then share equally.
A yellow (+1) and red (−1) tile make a zero pair worth 0; add zero pairs when you need more tiles.
Describe every move in words and in symbols, and always check your answer in the original equation.

The tomato seller's scale

1. An equation is a balance

2. Bags and counters

3. Bags on both sides

4. Integer tiles and zero pairs

5. Shapes on a balance

Worked examples

One bag and counters

Bags on both sides

Removing two bags

Using zero pairs

Cubes on a balance

Summary

Practice: 15 questions