An equation says two things are equal. Find the value that makes it true, then check it.
Shops, recipes and sports tables all hide an equation: a total that has to match. Finding the missing number, and checking it, is what solving means.
An expression is a calculation with a letter in it, such as 2x+3. An equation puts an = sign between two expressions: it says they are equal, like a balance scale with the same weight on each side. The solution is the value of the letter that makes it true.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Which value balances?
2x+3=11
Each bag on the left holds x blocks, and there are two bags and 3 loose blocks. The right side has 11 blocks. How many blocks must be in each bag for the scale to balance?
TryTry it on the board and compare the result with your prediction.
Find the balance
Change x with the slider, or the − and + buttons. Watch both sides and stop when the scale balances.
Try it yourselfYour edits are saved in this browser.
Find the value of x that balances the scale.
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Check a solution
To check a value, put it in place of the letter and work out each side.
Worked exampleBefore each step, say what you think comes next.
2x+3=11Is x=4 a solution? Start with the equation.
2(4)+3Put 4 in place of x on the left. The brackets show that 2x means 2 times x.
=8+3=11Work out the left side.
11=11The left side equals the right side.
So x=4 is a solution.If the two sides were different, it would not be.
Read Sam's working one line at a time. Where does a number come from that should not?
Look for the = sign.
Is x=5 a solution of 3x−2=13? Start by working out the left side, 3x−2, when x=5.
Sam decides that x=4 is not a solution of 2x+3=11, but it balanced the scale. Click the line where his working goes wrong.
ModifyChange the problem on purpose and see what happens.
New scales
Each task loads a new scale. Find the value of x that balances it.
Need a hint?
Each bag holds the same number of blocks. How many blocks are on the right, shared between four bags?
Now both sides change when x changes. The solution still makes the two sides equal.
Need a hint?
Watch both sides change as you move x.
Another hint
Try each value from 0 upwards and compare the two totals.
Try every value. What happens between x=3 and x=4?
Try it yourselfYour edits are saved in this browser.
Balance 4x=20.
Use the board. This ticks when you get there.
Balance x+7=3x+1.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Check a value
Work out 5x−4 when x=3.
Which value of x solves 3x+2=20? Try 5, 6 and 7.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Your own equation
Make an equation that a friend could solve. Check it by substitution before you press Check answer.
Stretch: write one with x on both sides.
Need a hint?
Start with what x=6 gives you: 2x is 12.
Another hint
Then add the same number to both sides: 2x+1=13.
Write your own equation whose solution is x=6. Use x at least once.
For teachers
Students treat an equation as a balance, find the value that balances it, and check a value by substitution (AC9M7A03, node 1). About 45 minutes.
I can check whether a value is a solution by putting it in place of the letter and working out both sides.
I can tell an expression from an equation.
I can explain that a solution makes both sides of an equation equal.
Before class
Do the Run and Modify boards yourself first. The surprise is the last Modify task: 2x=7 never balances with a whole number, and that is the point. Be ready to ask "what would x have to be?" and accept 3.5 without teaching a method yet: solving step by step is the next two lessons.
Running the lesson
Predict: students lock in a value before touching the board. Ask two students with different predictions to explain them.
Run: the board shows each side worked out for the current x. Point at the line under each pan: that is substitution, done for them.
Investigate: walk through the worked example together, one step at a time. Before each click, ask what comes next.
Say 2x aloud as 'two times x', never 'two x', for the whole lesson. It prevents the 24-for-8 slip in the second task.
Modify: the last task has no whole-number answer. Let students try every value before discussing.
Make: share a few student equations and have the class check them by substitution.
When students get stuck
A student says x = 4 gives 2x = 24. Ask: What does 2x mean? Say it as a times sentence. Fix: 2x means 2 × x. Write the multiplication sign in, or use brackets: 2(4) = 8.
The student moves the slider at random and stops when it balances, without noticing why. Ask: Before you move it, which way will it tip? Why? Fix: Ask them to read both values under the pans aloud before each move.
A student writes an expression, such as 2x + 3, for the Make task. Ask: What is your expression equal to? Where is the = sign? Fix: An equation needs two sides. Work out 2x + 3 when x = 6, then write = and that number.
Stretch and support
Stretch: Write an equation with x on both sides whose solution is 6, then one whose solution is not a whole number.
Support: Use the board for the Investigate questions too: set up 3x − 2 and 13 by hand on paper as bags and blocks, then count.
Where it leads
Next: solving by backtracking, then the balance method (do the same to both sides). Both lean on the habit built here: check every answer by substitution.
Where it leads in class: missing totals in recipes or budgets ("4 tickets and a $5 booking fee cost $41"), and formulas in science where one value is unknown.
What each step is for
which value balances?: Committing to a value before trying it makes the balance a test of their thinking, not a guessing game.
find the balance: The solution is the one value that balances the scale: every other value tips it one way or the other.
check a solution: Substitution is how you check a solution: replace the letter with the value, work out each side, and see if they are equal.
new scales: The idea holds for any equation, even with the letter on both sides, and some equations have no whole-number solution at all.
check a value: Substitution is a habit: every value gets checked by working out both sides.
your own equation: Building an equation from its solution shows that the solution is the value that makes both sides equal.
Now you can
Check whether a value is a solution by putting it in place of the letter and working out both sides.
Tell an expression from an equation.
Explain that a solution makes both sides of an equation equal.