At a glance. One lesson of about 60 minutes. Students find percentages of amounts mentally from 10%, 5% and 1%, compare percentages of different amounts, and check a calculator answer against an estimate.
Calculator: scientific calculator. A scientific calculator is used once, for an awkward percentage such as 37% of 2450. The estimate from 10% is done by hand, because a calculator only does what is typed.
What students will learn
Know
- 10% is a tenth, 5% is half of 10% and 1% is a hundredth.
- p% of A = A × p ÷ 100, or A times the decimal (35% is 0.35).
- p% of A and A% of p are equal.
Understand
- A percentage is a share of its own whole, so the size of the whole matters.
- Any percentage can be built from 10%, 5% and 1%.
- An estimate from benchmarks catches a slipped decimal place on a calculator.
Do
- Find a percentage of an amount mentally by building from 10%, 5% and 1%.
- Use the decimal multiplier on a scientific calculator, and check it with an estimate.
- Compare percentages of different amounts.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: percentages of amounts by benchmarks and by multiplier (questions 1 to 6).
- Criterion B. Investigating patterns: the second figure. Ask students to find pairs of percentages and amounts that give equal results, then say what they notice (p% of A = A% of p).
- Criterion C. Communicating: explaining why 70 − 20 does not find 20% of 70, using the words share, tenth and hundredth (question 9).
- Criterion D. Applying mathematics in real-life contexts: cloth prices, school travel, rice milling and seating (the hook and questions 5, 7 and 8).
The one thing to do with the figure
Ask for a prediction before the first slide. “10% of 360 is 36. What is 35%? Which blocks would you add?” Take 126 or 108 + 18 from the room and let the figure and the read-outs settle it.
Then try 47% of 200, which needs the 1% slivers, and 100%, where ten blocks make the whole. In the second figure, find two settings that give equal results with different percentages and ask why.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea; the second figure, searching for equal amounts. |
| 10 min | Worked example on 35% of 360; students finish the faded example in pairs. |
| 5 min | Question 1, the check. If many choose A or C, go back to the first figure and halve the 10% block. |
| 15 min | Your turn, Levels 1 to 3. Question 7 is the calculator question. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 25%; 30%; 36. |
|---|---|
| Worked example | 126 naira. |
| Faded example | 108 rupees. |
| 1. Finding 5% | B. A gives 20%; C gives 2%; D multiplies instead of taking a share. |
| 2. 10% of an amount | 45. 450 ÷ 10. |
| 3. 25% of an amount | 22. 88 ÷ 4. |
| 4. 15% of an amount | 42. 28 + 14. |
| 5. 4% of an amount | 50. 4 × 12.5. |
| 6. Comparing percentages of amounts | B. A compares the percentages and ignores the wholes; D: both are quick from 10%. |
| 7. Percentage on a calculator | 906.5. 2450 × 0.37. |
| 8. Percentage then remainder | 156. 240 − 84, or 65% of 240. |
| 9. Percentage is a share, not a subtraction | B. A subtracts the percentage number; C copies it; D stops at 10%. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Confused 5% with a fifth or a fiftieth, or multiplied. | Draw ten blocks and shade 5%: it is half of one block. |
| 4.5 or 4500 in question 2 | Divided by 100 or multiplied by 10. | Ask how many tenths are in a whole, and how many hundredths. |
| 440 in question 2 | Took 10 away from 450. | Ask what 10% of 450 is, and why 10 off would not be 10%. |
| 63 in question 3; 265 in question 4; 1246 in question 5 | Subtracted the percentage number from the amount. | Ask what 10% of the amount is, and why 10 off would not be 10%. |
| 28 in question 4 | Stopped at 10%. | Write 15% = 10% + 5% before calculating. |
| 12.5 in question 5 | Found 1% and stopped. | Ask how many 1% pieces make 4%. |
| A in question 6 | Compared the percentages and ignored the wholes. | Use the second figure: put 40% on 60 and 15% on 180 and read the shaded lengths. |
| 90.65 or 9065 in question 7 | Slipped a decimal place in 0.37. | Estimate first with 40% of 2450. Which answers are near it? |
| 84 in question 8 | Stopped at the guests. | Underline the question word: the rest. |
Questions worth asking
- How would you find 15%, 35% and 95% of an amount using only 10%, 5% and halving? 10% + 5%; 3 × 10% + 5%; 100% − 5%. Listen for the 100% − 5% shortcut.
- Why does the estimate come before the calculator? Because the calculator will give an answer to whatever is typed, including a slipped decimal point.
- Which is quicker: 4% of 150, or 150% of 4? 150% of 4 is 1.5 × 4 = 6. They are equal because p% of A = A% of p.
- If 30% of an amount is 45, how could you find the amount? 10% is 15, so 100% is 150. This is next lesson's idea of working backwards.
Differentiation
A way in for anyone stuck
- Give each student ten strips of paper, one for each 10%, to fold and label.
- Keep to amounts that divide by 10 cleanly (multiples of 20 or 40) until the building is secure.
- Write the pieces in a column and add them, naming each piece: 10%, 10%, 10%, 5%.
Stretch
- Question 7, then: find 37% of 2450 without a calculator, to within 10. (40% is 980, 3% is 73.5, so 906.5.)
- Show that 12.5% of an amount is an eighth of it, and use it to find 12.5% of 360. (360 ÷ 8 = 45.)
Students learning in English as an additional language
“Of” means multiply in “35% of 360”. Practise the phrase with three different amounts before the questions.
| Word | Plain meaning | Example |
|---|---|---|
| percentage of | a share, in hundredths, of an amount | 35% of 360 is 126 |
| amount | the whole quantity that a share is taken from | 360 naira |
| benchmark | a value known by heart, such as 10%, to build others from | 10% of 360 is 36 |
| multiplier | the number you multiply by to find a share | 35% has multiplier 0.35 |
| estimate | a quick answer that is close to the real one | 30% of 3000 is about 900 |
| tenth | one of ten equal parts | 10% is a tenth |
Sentence starters for explaining aloud or in writing:
- 10% of … is …, so …% is …
- My estimate is … and the calculator says …, so …
- …% of … is greater than …% of … because …
Two things not to say
“Just take off the percentage.” Taking off is for a reduction, which is the idea of lesson 5; here the share is what is wanted.
“Of means times, so multiply by the percentage number.” Multiply by the hundredths: 35% is 0.35, not 35.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- 10% and 5%Find 10% of 80 and 5% of 80.
8 and 4. - BuildFind 30% of 150.
45.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA school of 640 students has 15% walking to school. How many walk?
96. 10% is 64 and 5% is 32. - Earlier work (Lesson 2 of this unit)What is 3/8 of 240? Change 3/8 to a percentage first.
90. 3/8 = 37.5%: 30% is 72, 5% is 12, 2.5% is 6.
3Challenge
Reasoning, generalising and working backwards.
- GeneralizingIs 4% of 150 the same as 25% of 24? Show why without a calculator.
Both are 6: 4 × 150 = 600 and 25 × 24 = 600, and both are divided by 100. - Working backwards10% of a number minus 5% of it is 18. What is the number?
5% of it is 18, so 1% is 3.6 and the number is 360.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Draw a bar of ten equal boxes on the board and label them with tenths of the amount.