At a glance. One lesson of about 60 minutes. It uses powers (Year 7, Unit 1, lesson 2) and powers of negative numbers (Year 8, Unit 3, lesson 5). Students join and cancel runs of factors in the figure, state the two laws, and use them with coefficients. Zero and negative exponents (Lesson 5) are the natural continuation of the quotient law.
Calculator: no calculator. Nothing here needs a calculator. The laws turn a long product into one power, and the values asked for are small enough to work out by hand.
What students will learn
Know
- In a power, the base is repeated as a factor exponent times.
- am × an = am+n and am ÷ an = am−n, for the same base.
- A number with no written exponent has exponent 1: 7 = 71.
Understand
- Adding and subtracting exponents is counting factors, not a trick.
- The base does not change when powers are multiplied or divided.
- The laws need the same base; rewrite as a common base where possible.
Do
- Write a product or quotient of powers of the same base as one power.
- Simplify expressions with numbers and letters in front of the powers.
- Rewrite a number as a power of another so that the bases match.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the two laws, with and without coefficients (questions 1 to 7).
- Criterion B. Investigating patterns: the figure, which also suits a full-lesson investigation of the product rule. Ask students to tabulate b, m, n and the exponent of the product and say the rule they find.
- Criterion C. Communicating: explaining why 23 × 24 is not 47, using “factors” (question 9).
- Criterion D. Applying mathematics in real-life contexts: file sizes counted in powers of 2 (the hook and question 8).
The one thing to do with the figure
Ask for a prediction first. “23 × 24: is it 27, 212 or 47?” Take votes, then use the figure with base 2, first exponent 3 and second exponent 4.
Then switch to Divide and ask which boxes cancel. Ask students to predict the number of boxes left before you move the slider.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, predictions first. Include a base of 10 and a base with exponent 1. |
| 10 min | Key idea and the expanded working; the warning on multiplying bases. |
| 10 min | Worked examples with and without coefficients; the faded 20m7 ÷ 5m2. |
| 5 min | Question 1, the check. If many choose B, C or D, return to the runs of boxes. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 1000; −8; 5/9. |
|---|---|
| Worked examples | 64; 15a6. |
| Faded example | 4m5. |
| 1. Product of powers of the same base | A, 57. B multiplies the exponents; C multiplies the bases; D adds the bases. |
| 2. A missing exponent | 4. 6 + 4 = 10. |
| 3. Quotient of powers evaluated | 8. 29−6 = 23. |
| 4. Product with coefficients | 11. 3 + 8 = 11. |
| 5. Quotient with coefficients | 3. 18 ÷ 6 = 3. |
| 6. Spotting the odd one out | C. 2 + 5 = 7, not 10; the others all give exponent 10. |
| 7. Product and quotient together | 3. 9 + 4 − 10 = 3. |
| 8. Powers of 2 in context | 1024. 210 = 32 × 32. |
| 9. Different bases made the same | B. A gives up too soon; C multiplies bases; D keeps the wrong base. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Multiplied the exponents, or multiplied or added the bases. | Write each power out as factors, count the factors of 5 in total. |
| 16 or 60 in question 2 | Added or multiplied the exponents on the right. | Ask: what must be added to 6 to get 10? |
| 3 or 1.5 in question 3 | Stopped at the exponent, or divided the exponents. | Ask for the value: what is 23? Subtract, never divide, the exponents. |
| 24 or 5 in question 4 | Multiplied or subtracted the exponents of p. | Write p3 and p8 as factors and count. |
| 12 or 24 in question 5 | Subtracted or added the numbers in front. | The numbers in front are an ordinary division: what is 18 ÷ 6? |
| A, B or D in question 6 | Misread NOT, or did a sum of exponents wrongly. | Compute the exponent for each choice and compare with 10. |
| 23 or 5 in question 7 | Added the divisor's exponent, or subtracted where a product was. | Do it in two steps: write the product as one power first. |
| 10 or 1.5 in question 8 | Gave the exponent or divided the exponents. | Ask: is the answer a number of photos or an exponent? |
Questions worth asking
- Why do the exponents add when the powers multiply? Each exponent counts factors, and multiplying joins the runs. Listen for “factors in a row”.
- Why can't 23 × 34 be written as a single power? The factors are different, so there is no one base. Listen for “not the same base”.
- What is a × a3? a4, because a = a1.
- What happens if the two exponents are equal when dividing? Everything cancels and the answer is 1. This leads to a0 in Lesson 5.
Differentiation
A way in for anyone stuck
- Give a card with the two laws and one example each; let students write a power out as factors for the first three questions.
- Start with base 2 and exponents under 5 before bases of 7 or 10.
- Separate the coefficient step from the power step with two columns on the page.
Stretch
- Question 9, then: write 9 × 81 as one power of 3. (32 × 34 = 36 = 729.)
- Solve 2x × 2x = 210 by adding the exponents. (x = 5.)
Students learning in English as an additional language
“Power”, “exponent” and “index” are used for the small raised number; the plural of index is indices. “Base” is the number being multiplied by itself. Say “two to the power of three”.
| Word | Plain meaning | Example |
|---|---|---|
| power | a number multiplied by itself a number of times | 23 = 2 × 2 × 2 |
| base | the number that is multiplied by itself | 2 in 23 |
| exponent | the number of factors of the base; also called the index | 3 in 23 |
| factor | a number that is multiplied | each 2 in 2 × 2 × 2 |
| coefficient | the number in front of a letter or power | 15 in 15a6 |
| indices | more than one index, or the laws about exponents | the laws of indices |
Sentence starters for explaining aloud or in writing:
- I added the exponents because …
- The base stays the same because …
- These cannot be combined because the bases are …
Two things not to say
“Add the powers.” Say “add the exponents” and show the factors; adding powers is a different operation.
“Multiply the bases and add the exponents.” This is the commonest wrong rule; the base is a factor, not a quantity to be combined.
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.
- ProductWrite 63 × 62 as a single power.
65. - QuotientWrite 107 ÷ 104 as a single power.
103.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA disk holds 216 blocks and each folder uses 28 blocks. How many folders fit?
28 = 256. - Earlier work (Lesson 1)Write 3/8 as a decimal.
0.375.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that 24 × 24 × 24 = 212.
Add the exponents: 4 + 4 + 4 = 12. This is the lesson for the next law. - LettersSimplify (x5 × x3) ÷ x4.
x5+3−4 = x4.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The three sliders and the two buttons work with the keyboard (tab to each, then the arrow keys or Enter) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Write each power out as factors on squared paper and cross out the cancelled ones.