At a glance. One lesson of about 60 minutes. Powers of negative integers, even and odd powers, (−k)ⁿ against −kⁿ, and the order of operations with integers.
Calculator: scientific calculator. A scientific calculator is used once, deliberately: typing −3² shows that the machine follows the convention. Everything else is mental or written.
What students will learn
Know
- (−k)ⁿ is positive for even n and negative for odd n.
- −kⁿ = −(kⁿ), always negative for k ≠ 0.
- The order of operations applies unchanged to integers.
Understand
- Notation decides meaning: the power belongs to what it touches.
- Even and odd powers follow from pairing negative factors.
Do
- Evaluate powers of negative integers.
- Evaluate expressions with several operations and negative numbers.
- Use a scientific calculator to check, typing brackets when they are meant.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: questions 2 to 8.
- Criterion B. Investigating patterns: this lesson is the natural place for a criterion B investigation into the sign of (−a)ⁿ. Students choose their own bases and powers, find the even/odd rule, test it on new cases and justify it by pairing factors. The figure supports the pattern-finding; do not show it during the task itself.
- Criterion C. Communicating: writing −(3²) and (−3)² correctly and explaining the difference in a sentence (exit ticket).
- Criterion D. Applying mathematics: the calculator note, judging whether a calculator's output matches what was meant.
The one thing to do with the figure
Ask for predictions first: “In which rows will the two bars point the same way?” Let students commit, then set k = 2 and read the rows.
Then move k to 3 and ask whether the answer changed. It does not: only the parity of the power matters. Ask for a reason using the Lesson 4 idea of counting negative factors.
A possible order
| 5 min | Warm-up: a power, a product of negatives, and the order of operations. |
|---|---|
| 10 min | What the power touches: (−3)² against −3², with predictions. |
| 10 min | The figure: even and odd powers; the reason from pairing factors. |
| 10 min | Order of operations with integers; the faded example in pairs. |
| 5 min | The calculator note: every student types both versions. |
| 5 min | Question 1, the check. |
| 15 min | Your turn and exit ticket (question 9). |
The answers
| Warm-up | 8; 25; 11. |
|---|---|
| Worked examples | −17; −10; 4. |
| Faded example | 10 − 16 = −6. |
| 1. What the power touches | B. A is the key misconception of the lesson; C or D mean squaring is being read as doubling. |
| 2. Even power | 16. |
| 3. Odd power of −1 | −1. |
| 4. Higher power | −32. |
| 5. Order of operations | −17. |
| 6. Power, then multiply | 17: 9 − (−8). |
| 7. Large power | 1. |
| 8. Brackets compared | 162: 81 − (−81). |
| 9. Calculator convention | A. B is the lesson's key misconception, now in calculator form. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1; 25 for −5² | The power is applied to the minus sign too. | Ask “what is the power touching?” and rewrite with explicit brackets: −(5²). |
| −8 or 8 in question 2 | Squaring read as doubling. | Write the power as a product before calculating. |
| 32 in question 4 | The digits are right; whether the power is even or odd was ignored. | Count the negative factors. |
| −9 in question 5 | Worked left to right, ignoring the order of operations. | Underline the multiplication first. |
| 1 or −17 in question 6 | 1: lost the sign of 4 × (−2). −17: treated (−3)² as −9, then added the −8 instead of subtracting it. | One operation per line. |
| 81 in question 8 | Found one value, not the gap. | Ask for both values on a number line. |
Questions worth asking
- Is (−k)ⁿ ever equal to −kⁿ? Yes, whenever n is odd (and when k = 0).
- Can a square ever be negative? Not for any number met so far. (−k)² and k² are both positive.
- Why does a calculator square before it applies the minus sign? Because it follows the same order of operations we do.
Differentiation
A way in for anyone stuck
- Write every power as a product before calculating: (−2)⁴ = (−2)(−2)(−2)(−2).
- Colour the part the power touches.
- One operation per line for every order-of-operations question.
Stretch
- Is (−2)⁶ greater or smaller than (−3)⁵? (64 against −243.)
- Find a positive integer n for which (−2)ⁿ + 2ⁿ = 0, and explain which n work. (Any odd n.)
Students learning in English as an additional language
“Power”, “index” and “exponent” all name the small raised number. Use one consistently and show the others once.
| Word | Plain meaning | Example |
|---|---|---|
| power (exponent) | how many times the base is written as a factor | in 2⁵ the base 2 is a factor 5 times: 2 × 2 × 2 × 2 × 2 = 32 |
| base | the number being multiplied | in (−2)⁵ the base is −2 |
| even / odd | divides exactly by 2 / does not | 4 is even, 5 is odd |
| brackets | show what belongs together and is done first | (−3)² |
| order of operations | the agreed order for calculating | brackets, powers, × and ÷, + and − |
Sentence starters for explaining aloud or in writing:
- The power is touching …, so I …
- There are … negative factors, which is even / odd, so the answer is …
- First I worked out … because …
Two things not to say
“A negative squared is negative.” It depends entirely on the brackets; say what the power touches.
“The calculator is always right.” It does exactly what was typed. The exit ticket is about precisely that.
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.
- PowersWork out (−5)², −5² and (−5)³.
25, −25 and −125. - OrderWork out 7 + (−2) × 6.
−5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Quiz scoreA quiz gives −3 points for each wrong answer and 10 for each right one. Lena has 4 wrong and 2 right. Write one calculation for her score and work it out.
4 × (−3) + 2 × 10 = −12 + 20 = 8 points. - SquaresA square has area (−6)² written by mistake for 6². Does the area change? Explain.
No: (−6)² = 36 = 6². The side length is 6 either way.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningWithout calculating, is (−7)¹¹ × (−7)² positive or negative?
Negative: 13 negative factors, an odd number. - From Lesson 4Work out (−3) × (−4) × (−1)².
12.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Students need a scientific calculator for the calculator note only. A phone's calculator app in landscape mode usually has the x² key.
“Show me the steps” gives every row for k = 2, and two rows for k = 3, in words, which suits a printed handout.