The equal sign and finding the unknown
What the equal sign really means, deciding if an equation is true or false, and finding a missing number
About three lessons of 35 to 40 minutes
The equal sign is a balance, not an arrow to an answer
Most sums you have seen look like 3 + 4 = ?, with the equal sign right before the answer. That can quietly teach the wrong idea, that equals means 'here comes the answer.' But look at 7 = 3 + 4. That is also true, because 3 + 4 really is 7. The equal sign means both sides weigh the same amount, like a balanced scale, not 'the answer is coming next.'
Once the equal sign means 'the same amount as' on both sides, you can do two new things: decide whether an equation is true or false, even a tricky-looking one like 4 + 1 = 5 + 2, and find a missing number anywhere in an equation, not just at the very end.
- 6 + 6 on one side, 11 on the otherfalse, because 6 + 6 is really 12, not 11
- 5 + 2 = 2 + 5true, both sides equal 7
- 8 + ? = 11the missing number is 3, because 8 + 3 = 11
- 6 + 6 = ? + 4the missing number is 8, because both sides must equal 12
What students will be able to do
Students will understand that the equal sign means both sides of an equation represent the same amount, judge whether addition and subtraction equations are true or false, and find the missing whole number in an equation involving three numbers.
- I can explain that the equal sign means both sides are the same amount.
- I can decide whether an equation is true or false by working out both sides.
- I can find a missing number in an equation, even when it is not at the very end.
Standards this unit teaches
- 1.OA.D.7Common Core (US)Meaning of the equal sign
Understand that the equal sign means both sides are the same amount, and judge whether simple equations are true or false.
- 1.OA.D.8Common Core (US)Find the unknown in an equation
Find the missing whole number in an addition or subtraction equation involving three numbers.
- AC9M1N04Australian Curriculum v9 (ACARA)Number sentences to solve problems (Year 1)
Solve one- and two-digit adding and subtracting problems by writing number sentences and using part-part-whole reasoning.
- AC9M1A02Australian Curriculum v9 (ACARA)Addition facts to 20 (Year 1)
Build fluency with addition facts to twenty and use them to work out the matching subtraction facts.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Equation
- a maths sentence with an equal sign showing two amounts are the same
- Equal sign
- the = symbol, meaning both sides of the equation are the same amount
- True equation
- an equation where both sides really do work out to the same amount
- False equation
- an equation where the two sides do not actually work out to the same amount
- Unknown
- a missing number in an equation, shown as a blank or a question mark
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. What the equal sign really means
ConcretePicture a balance scale. If both sides hold the same amount, the scale sits level, balanced. The equal sign works exactly the same way: whatever is written on the left and whatever is written on the right must be the same amount for the equal sign to be used honestly. It does not mean 'now write the answer,' it means both sides balance.
That is why 7 = 3 + 4 is a perfectly normal, true equation, even though the single number sits on the left this time and the sum sits on the right. Both sides equal 7, so the scale balances, and the equal sign is used correctly either way round.
- Does the equal sign mean 'the answer comes next', or 'both sides are the same amount'?
- Is 9 = 4 + 5 written correctly, even though the single number comes first?
2. Deciding if an equation is true or false
PictorialTo check an equation, work out both sides separately, then compare. Take 4 + 1 = 5 + 2. The left side, 4 + 1, is 5. The right side, 5 + 2, is 7. Five does not equal seven, so the scale does not balance, and the equation is false, even though it might look fine at a glance.
Now compare that with 5 + 2 = 2 + 5. The left side is 7, and the right side is also 7. Both sides match, so this equation is true. The only way to be sure is to actually work out both sides, never to guess from how the equation looks.
Is the equation 6 + 6 = 11 true or false?
- Work out the left side: 6 + 6 = 12.
- The right side is already given as 11.
- Compare: 12 does not equal 11.
Answer: False. 6 + 6 is really 12, not 11.
- Is 7 = 8 - 1 true or false?
- Is 4 + 1 = 5 + 2 true or false, and how do you know?
3. Finding a missing number in an equation
AbstractSometimes one number in an equation is hidden. To find it, work out whichever side you can, then figure out what the missing number needs to be so both sides balance. For 8 + ? = 11, the right side is already 11, so the missing number must make the left side reach 11 too: 8 + 3 = 11, so the missing number is 3.
The missing number is not always at the end. In 6 + 6 = ? + 4, work out the side you can: 6 + 6 = 12. Now the other side, ? + 4, must also equal 12. What plus 4 makes 12? Eight. So the missing number is 8.
Find the missing number: 5 = ? - 3.
- The left side is already known: 5.
- The right side, ? - 3, must also equal 5.
- What number, minus 3, leaves 5? Count on from 5: 5, 6, 7, 8, so 8 - 3 = 5.
Answer: The missing number is 8.
- Find the missing number in 9 = ? + 4.
- In 6 + 6 = ? + 4, why must the missing number make the right side equal 12?
Common misconceptions and how to address them
MisconceptionThe child insists 7 = 3 + 4 is written wrongly, or tries to 'fix' it by flipping it to 3 + 4 = 7, because equals is expected to come right before the answer.
Why it happens: Nearly every equation seen so far has the answer immediately after the equal sign, so that pattern gets mistaken for a rule rather than just a habit.
How to address it: Show a balance scale with 7 on one side and 3 blocks plus 4 blocks on the other. Both sides weigh the same, so the scale is level no matter which side the single number sits on.
MisconceptionFacing 4 + 5 = ? + 6, the child adds every number in the whole equation together, including the 6 on the right, instead of solving for the missing number so both sides balance.
Why it happens: Without a secure relational understanding of the equal sign, the equation looks like one long string of numbers to add rather than two separate, balanced sides.
How to address it: Cover one side of the equation at a time. Work out the visible side first (4 + 5 = 9), then ask what has to join the 6 on the other side to also reach 9, rather than adding straight across the equal sign.
MisconceptionWhen judging true or false, the child decides based on how the equation looks, such as assuming it is true because a number appears on both sides, without actually calculating both sides.
Why it happens: A quick visual check feels faster than calculating, but it does not verify the amounts actually match.
How to address it: Make a habit of covering the equal sign, working out the left amount and the right amount separately in writing, then comparing the two numbers before deciding true or false.
MisconceptionWhen finding a missing number, the child guesses randomly or always writes the same familiar number rather than working out what value makes both sides balance.
Why it happens: Without a clear method, an unfamiliar equation shape (with the blank in the middle or on the left) feels like there is nothing to calculate.
How to address it: Always work out the side without the blank first, then ask the direct question: what number is needed on the other side to reach that same total?
Guided practice (with answers)
1. Is 7 = 3 + 4 true or false?
Answer: True. 3 + 4 = 7, so both sides are the same amount.
2. Is 5 + 2 = 9 true or false?
Answer: False. 5 + 2 = 7, not 9.
3. Find the missing number: 8 + ? = 11.
Answer: 3, because 8 + 3 = 11.
4. Find the missing number: 5 = ? - 3.
Answer: 8, because 8 - 3 = 5.
5. Find the missing number: 6 + 6 = ? + 4.
Answer: 8, because 6 + 6 = 12, and 8 + 4 = 12.
6. Is 4 + 1 = 5 + 2 true or false?
Answer: False. 4 + 1 = 5, but 5 + 2 = 7, and 5 does not equal 7.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. The missing addend sheet practises finding an unknown number directly, and addition and subtraction sheets build the fact fluency needed to check both sides of an equation quickly.
Differentiation
- Start with the number sentence read aloud as a balance: 'this side and this side are the same amount.'
- Keep the missing number at the end of the equation at first, before moving it to the middle or the start.
- Use physical objects on two sides of a real or drawn balance scale before moving to written equations.
- Work out and write both sides separately every time, rather than trying to judge true or false by eye.
- Write your own true equation and your own false equation with the missing number in an unusual position, for a partner to check.
- Find a missing number in an equation with the blank on the left side, such as ? + 5 = 9 + 2.
- Explore equations with the same total made three different ways, such as 12 = 6 + 6, 12 = 8 + 4, and 12 = 20 - 8.
- Turn a true equation false by changing just one number, and explain what changed.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling the meaning of equals, true or false, and a missing number.
1. What does the equal sign mean?
Answer: Both sides of the equation are the same amount.
2. Is 8 = 3 + 4 true or false?
Answer: False. 3 + 4 = 7, not 8.
3. Find the missing number: 9 = ? + 4.
Answer: 5, because 5 + 4 = 9.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 the meaning of the equal sign (section 1), Lesson 2 deciding true or false (section 2), Lesson 3 finding a missing number in different positions (section 3) plus the exit ticket.
- This is a genuinely important unit, not a small technicality: research on early algebra (Carpenter, Franke and Levi) shows many students carry a purely 'operational' view of the equal sign, meaning 'here comes the answer,' well into upper primary unless it is directly challenged with equations like 7 = 3 + 4 and 4 + 5 = ? + 6. Building the balanced, 'relational' view here pays off in every later algebra unit.
- Language to keep saying: both sides are the same amount, cover one side and work it out first, and what makes this side match that side. These phrases pre-empt the misconceptions above.
- Curriculum note: US Grade 1 states the meaning of the equal sign (1.OA.D.7) and finding an unknown in an equation (1.OA.D.8) as two closely related standards. ACARA Year 1 covers writing number sentences to solve problems (AC9M1N04) and building addition and matching subtraction fact fluency (AC9M1A02), a close conceptual match without naming the equal sign's meaning as its own explicit descriptor.
- Present mode and print both work: use the Print button for a student worksheet, or project the page and use a real or drawn balance scale to teach the equal sign's meaning before moving to written equations.