C. Algebra · Patterns, Equations, Inequalities, Coding and ModellingSolving Equations with Whole Numbers up to 100
C2.3 A1About 40 minutes
Learning goals- I can write an equation for a story problem.
- I can solve equations with adding, subtracting, multiplying and dividing, using whole numbers up to 100.
- I can use different strategies: a related fact, the opposite operation, a balance, or guess and check.
- I can verify a solution by substituting it back into the equation.
Words to know
equation
A math sentence with an equal sign. Both sides have the same value.
solve
Find the value of the variable that makes the equation true.
solution
The number that makes the equation true.
inverse (opposite) operation
An operation that undoes another: subtracting undoes adding, and dividing undoes multiplying.
verify
Check a solution by putting it back into the equation.
Remember from Grade 4
In Grade 4 you solved equations like n + 12 = 30 and 6 × v = 42, with numbers up to 50. You thought of a related fact or used the opposite operation, and you verified your answer.
In Grade 5 the numbers go up to 100, the stories get bigger, and you'll even solve some equations with two steps.
Let's start with a story
The class is collecting 100 pairs of mittens for a winter clothing drive. They have 63 pairs so far. How many more pairs, m, do they need?
63 + m = 100
Think: 63 plus what makes 100? Count up: 63 + 7 = 70, then 70 + 30 = 100. That's 7 + 30 = 37. So m = 37.
Verify: 63 + 37 = 100 ✔
An equation is a balance
The equal sign means both sides have the same value, like a level balance. If you do the same thing to both sides, it stays level.
To get n alone, take 38 away from both sides: n = 92 − 38 = 54. Verify: 54 + 38 = 92 ✔
Strategies for solving
| Equation | Strategy | Solution |
|---|
| n + 38 = 92 | Opposite operation: 92 − 38 | n = 54 |
| 100 − m = 37 | Related fact: 100 − 63 = 37, or count up from 37 to 100 | m = 63 |
| 7 × b = 84 | Opposite operation: 84 ÷ 7 | b = 12 |
| q ÷ 4 = 15 | Opposite operation: 15 × 4 | q = 60 |
| k × 8 = 96 | Guess and check: 10 × 8 = 80 (too low), 12 × 8 = 96 ✔ | k = 12 |
Worked exampleExample 1: An addition equation
Solve n + 38 = 92 and verify.
- Undo adding 38 by subtracting 38: n = 92 − 38.
- 92 − 38: 92 − 40 = 52, then add back 2: 54.
n = 54. Verify: 54 + 38 = 92 ✔
Worked exampleExample 2: The variable is subtracted
- Ask: 100 take away what leaves 37?
- The missing part is 100 − 37 = 63.
m = 63. Verify: 100 − 63 = 37 ✔
Worked exampleExample 3: A multiplication story
A teacher buys 84 pencils. They come in boxes of 7. How many boxes did she buy?
- Let b be the number of boxes. Equation: 7 × b = 84.
- Undo multiplying by dividing: b = 84 ÷ 7.
- Think: 7 × 10 = 70, and 7 × 2 = 14. 70 + 14 = 84, so 7 × 12 = 84.
b = 12 boxes. Verify: 7 × 12 = 84 ✔
Worked exampleExample 4: A division equation
- Ask: what number shared into 4 equal groups gives 15 in each?
- Undo dividing by multiplying: q = 15 × 4 = 60.
q = 60. Verify: 60 ÷ 4 = 15 ✔
Two-step equations
Some equations have two operations. Picture the balance below.
Worked exampleExample 5: A two-step equation
- Undo the last step first. The + 5 happens last, so take 5 from both sides: 3 × n = 45.
- Now undo the multiplying: n = 45 ÷ 3 = 15.
n = 15. Verify: 3 × 15 + 5 = 45 + 5 = 50 ✔
Solve, then verify
- Write what the variable means (b = number of boxes).
- Write the equation.
- Solve it with a strategy that makes sense to you.
- Verify by putting your answer back into the equation.
- Answer the question in a sentence, with units.
If your check doesn't work, that's useful information, not a failure. Try another strategy.
Just for funWhy do the two sides of an equation never argue?
Because they are always equal!
Hands-on activityMystery bag balance
You need: a paper bag or envelope, about 100 dried beans, pasta pieces or paper clips, paper and a pencil.
- Your grown-up secretly puts some beans in the bag (fewer than 60). They put 35 more beans next to it and tell you: the bag and the loose beans have 82 beans altogether.
- Write the equation: b + 35 = 82.
- Solve it and write your solution. Then open the bag and count to verify.
- Next round: your grown-up makes 4 equal piles that total 72 beans. Write and solve 4 × p = 72. Count one pile to verify.
- Swap jobs: you make up an equation with the beans for your grown-up to solve.
What to look for: b = 47 in step 3 and p = 18 in step 4 (if your grown-up used those numbers). Your child writes the equation before solving and checks by counting.
Grown-up note: keep beans and small items away from babies and toddlers.
Your turn
n = 80 − 45 = 35. Verify: 35 + 45 = 80 ✔
Undo subtracting by adding: m = 46 + 27 = 73. Verify: 73 − 27 = 46 ✔
k = 90 − 52 = 38. Verify: 90 − 38 = 52 ✔
a = 72 ÷ 6 = 12. Verify: 6 × 12 = 72 ✔
d = 13 × 5 = 65. Verify: 65 ÷ 5 = 13 ✔
6The class has 63 pairs of mittens and wants 100. Write and solve an equation.
63 + m = 100, so m = 37 more pairs.
7Is n = 9 the solution of 8 × n = 81? Explain.
No. 8 × 9 = 72, not 81. (81 is not a multiple of 8, so there is no whole-number solution.)
More practice: warm-up
5Verify: is n = 16 the solution of n + 34 = 50?
More practice: on your own
p = 94 − 57 = 37. Verify: 57 + 37 = 94 ✔
t = 96 ÷ 4 = 24. (4 × 20 = 80 and 4 × 4 = 16, so 4 × 24 = 96 ✔)
3A hockey team scored 91 goals this season. They scored 48 at home. Write and solve an equation for the goals scored away, a.
48 + a = 91, so a = 91 − 48 = 43 away goals. Verify: 48 + 43 = 91 ✔
472 students go skating. They are split equally into 6 groups. Write and solve an equation for the number in each group, g.
6 × g = 72 (or 72 ÷ 6 = g), so g = 12 students. Verify: 6 × 12 = 72 ✔
r = 9 × 8 = 72. Verify: 72 ÷ 8 = 9 ✔
Challenge
Take 6 from both sides: 5 × n = 75. Divide: n = 75 ÷ 5 = 15. Verify: 75 + 6 = 81 ✔
2Mira has some money. She earns $4 for each lawn she rakes. After raking some lawns she has $70, including the $22 she started with. How many lawns did she rake?
22 + 4 × L = 70. Take 22 from both sides: 4 × L = 48. L = 48 ÷ 4 = 12 lawns. Verify: 22 + 48 = 70 ✔
Undo the doubling first: n + 9 = 60 ÷ 2 = 30. Then n = 30 − 9 = 21. Verify: 2 × (21 + 9) = 2 × 30 = 60 ✔
Quiz
1. What is the solution of n + 29 = 75?
2. What is the solution of 8 × k = 64?
3. What is the solution of p ÷ 3 = 25?
4. Which equation matches: Some students are in 4 equal teams. There are 92 students. How many are on each team?
5. Which is the solution of 3 × n + 4 = 31?
6. How do you verify a solution?
Big idea: Grade 5 students solve equations that involve whole numbers up to 100 in various contexts and verify the solutions (C2.3). They may use known facts, the opposite operation, a balance model, or guess and check. Two-step equations appear only as a stretch in this lesson.
Common difficulties:
- Using the same operation instead of the opposite one (for n − 27 = 46, subtracting 27 again). Ask: "What would undo taking away 27?"
- For 100 − m = 37, adding 100 + 37. Act it out: "100 take away something leaves 37."
- Skipping the check. Make verifying a habit; it catches most mistakes.
How to help: Turn everyday moments into equations: "We need 100 cups for the party and have 64. How many more?" Have your child write the equation, solve, verify and answer in a sentence. Social-emotional learning (A1): celebrate the check, even when it shows a mistake.