At a glance. One lesson of about 60 minutes. It follows Lesson 3 (product and quotient laws) and Year 8, Unit 3, lesson 5 (powers of negative numbers). Students draw a power of a power as rows of factors, regroup a product to a power, and compare a sum to a power. Zero and negative indices follow in Lesson 5.
Calculator: no calculator. Nothing here needs a calculator. Every value is small, and the point is the structure of the factors, not the arithmetic.
What students will learn
Know
- (am)n = amn.
- (ab)n = anbn and (a/b)n = an/bn.
- (−a)n is positive for even n and negative for odd n, and −an is the negative of an.
Understand
- Exponents add when runs are joined and multiply when a run is repeated.
- A power of a product distributes over every factor; a power of a sum does not distribute.
- Brackets decide what the exponent touches.
Do
- Simplify a power of a power with numbers and letters.
- Raise a product or quotient, including a coefficient, to a power.
- Evaluate powers of negative numbers with and without brackets.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the three laws and negative bases (questions 1 to 7).
- Criterion B. Investigating patterns: the second figure. Ask students to find states where (a + b)n equals an + bn and explain why only n = 1 works.
- Criterion C. Communicating: explaining why every factor takes the power, using “factor” and “bracket” (question 7).
- Criterion D. Applying mathematics in real-life contexts: the area of a square tile from its side as a power of 2 (question 2) and the volume of a cube (extra questions).
The one thing to do with the figure
Ask for a prediction first. “(23)2: is it 25 or 26?” Many will say 25, from Lesson 3. Let the figure show two rows of three.
In the second figure, ask students to find a state where the product agrees with the powers (always) and one where the sum does (only with exponent 1).
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions. Return to Lesson 3's law for the contrast. |
| 10 min | The second figure and the warning on sums. |
| 10 min | Negative bases; the worked examples and the faded (5y2)3. |
| 5 min | Question 1, the check. If many choose A, go back to rows of factors. |
| 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 | 510; 9; 19. |
|---|---|
| Worked examples | 16x12; 6561. |
| Faded example | 125y6. |
| 1. A power of a power | B, 46. A adds; C is 32; D squares the base. |
| 2. A power of a power | 8. (24)2 = 28. |
| 3. A negative base with brackets | 16. Four negative factors make a positive. |
| 4. A power of a quotient with a coefficient | 6. (a2)3 = a6. |
| 5. A sum to a power | 40. 81 − 41 = 40. |
| 6. Which equals minus sixteen | B. A and C are 16; D is −32. |
| 7. Every factor takes the power | 15. a = 9 and b = 6. |
| 8. Matching bases across a power | 4. 46 = 212 and 3 × 4 = 12. |
| 9. Which is not equal | C. It is 28; the rest are all 212. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Added the exponents, raised the exponent, or squared the base. | Draw two rows of three boxes, then count. |
| 6 or 16 in question 2 | Added the exponents, or raised one to the other. | Ask how many rows and how many boxes in a row. |
| −16 or −8 in question 3 | Left the sign outside the bracket, or multiplied base by exponent. | Write the four factors out and multiply in pairs. |
| 5 or 8 in question 4 | Added the exponents or raised one to the other. | Cube the 3, and multiply the exponent of a by 3. |
| 0 or −23 in question 5 | Split the sum or squared by doubling. | Evaluate both brackets in full, side by side. |
| A, C or D in question 6 | Dropped the brackets' effect on the sign. | Read each as: what does the power touch? |
| 11 or 8 in question 7 | Left a letter without the power, or added every exponent. | Underline each factor in the bracket and apply the power to all. |
| 2 or 12 in question 8 | Did not match the bases first. | Write 4 as 22 before using the law. |
Questions worth asking
- Why does a power of a power multiply and a product add? A product joins runs, a power of a power repeats a run. Listen for “rows”.
- Is (a + b)2 ever equal to a2 + b2? Only if 2ab = 0, which needs a or b to be 0. Listen for “cross term” if students expand.
- Why is (−2)4 positive but (−2)5 negative? Pairs of negatives cancel; an odd one out stays. Listen for “pairs”.
- What is (−1)100? 1, because 100 is even.
Differentiation
A way in for anyone stuck
- Provide rows of boxes drawn on squared paper; students count factors rather than apply a rule.
- Teach the power of a power with base 2 first; then the product with two small numbers.
- Give a sentence frame: “There are … rows of … factors, so … factors in all.”
Stretch
- Question 8, then: solve 8k = 212. (8 = 23, so 3k = 12 and k = 4.)
- Expand (a + b)2 and find the term that the false rule leaves out. (2ab.)
Students learning in English as an additional language
“Raised to the power of” names the exponent; “squared” and “cubed” are special names for 2 and 3. “Bracket” is a parenthesis. Say “the whole bracket is raised”.
| Word | Plain meaning | Example |
|---|---|---|
| power of a power | a power that is itself raised to a power | (23)2 |
| bracket | a pair of curved lines that group terms | (−2)4 |
| negative base | a base that is below zero | −2 in (−2)4 |
| even power | a power whose exponent is even | 4 in (−2)4 |
| odd power | a power whose exponent is odd | 5 in (−2)5 |
| expand | write out a power or a product in full | 23 = 2 × 2 × 2 |
Sentence starters for explaining aloud or in writing:
- There are … rows of … factors, so …
- Every factor in the bracket takes the power, so …
- The power touches only …, so the sign is …
Two things not to say
“Multiply the little numbers.” That rule works only for a power of a power; students will use it on a product.
“Square both.” That is only true of a product; (a + b)2 is a sum and cannot be split.
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.
- Power of a powerWrite (25)3 as one power.
215. - Negative baseEvaluate (−3)3.
−27.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cube has side 102 mm. Find its volume as one power of 10.
(102)3 = 106 mm3. - Earlier work (Lesson 3)Simplify 12x8 ÷ 3x2.
4x6.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that (ab)3 = a3b3 by writing out the factors.
(ab)(ab)(ab) = a × a × a × b × b × b. - Matching basesFind k: 27k = 36.
27 = 33, so 3k = 6 and k = 2.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All six sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Draw rows of boxes on squared paper; counting boxes shows both laws.