At a glance. One lesson of about 60 minutes. It builds directly on the quotient law (Lesson 3) and the power of a power (Lesson 4). Students walk a ladder of powers below the first power, state a0 and a−n, and use the laws to write answers with positive exponents. Scientific notation (Lesson 6) uses powers of 10 with negative exponents.
Calculator: no calculator. Nothing here needs a calculator. The values are small fractions that a student can work out, and the pattern is more convincing by hand.
What students will learn
Know
- a0 = 1 for any a that is not 0.
- a−n = 1/an, and (a/b)−n = (b/a)n.
- The laws of indices hold for zero and negative exponents.
Understand
- The definitions are forced by the pattern and by the quotient law.
- A negative exponent says how many times to divide, not that the number is negative.
- A power of a positive number stays positive and never reaches 0.
Do
- Evaluate zero and negative powers of whole numbers and fractions.
- Simplify products and quotients of powers, giving positive exponents.
- Write a number as a power with a negative exponent.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: a0 and a−n, with fractions (questions 1 to 7).
- Criterion B. Investigating patterns: the ladder. Ask students to continue the pattern for three different bases and state the rule they find for the rung at exponent 0.
- Criterion C. Communicating: justifying a0 = 1 from the quotient law, and explaining Khalid's slip (questions 1 and 9).
- Criterion D. Applying mathematics in real-life contexts: halving and doubling, such as a colony that doubles each hour read backwards in time (the what-if) and a medicine dose that halves (question 7).
The one thing to do with the figure
Ask for a prediction first. “What are 100 and 10−1?” Take guesses, then show the ladder and point at the ÷ 10 between the rungs.
Then set exponent 0 for three different bases and ask what the same rung tells you. Finally move to negative exponents and ask whether the value ever gets to 0 or below.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the ladder, predictions first. Include base 10 and base 2. |
| 10 min | The quotient-law argument for a0; key idea and the warning. |
| 10 min | Worked examples, with a negative exponent and with the laws; the faded 73 × 7−5. |
| 5 min | Question 1, the check. If many choose A, go back to the ladder and the divide step. |
| 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 | 75; 212; 4. |
|---|---|
| Worked examples | 0.008; 1/729. |
| Faded example | 1/49. |
| 1. A number to the power zero | C, 1. A thinks of nothing left; B is 61; D is not positive. |
| 2. A negative power as a decimal | 0.001. 1/103 = 1/1000. |
| 3. Zero powers added | 2. 1 + 1 = 2. |
| 4. Product with a negative exponent | 64. 2−6 = 1/64. |
| 5. A negative power of a fraction | 8. (1/2)−3 = 23. |
| 6. Ordering powers | A. 0.125, 0.5, 1, 2. |
| 7. Finding a negative exponent | −5. 30 ÷ 6 = 5 halvings, so 2−5. |
| 8. Adding negative powers | 0.75. 1/2 + 1/4 = 3/4. |
| 9. What a negative exponent does | B, 1/16. A is Khalid's slip; C puts the sign on the number; D multiplies 4 by 2. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Thought a zero power is 0, the base, or negative. | Use the ladder: 36, 6, then what? Each step divides by 6. |
| −1000 or 0.0001 in question 2 | Made the number negative, or miscounted the places. | Write 103 = 1000 and then 1/1000 on a place-value chart. |
| 0 or 13 in question 3 | Took a zero power as 0, or added the bases. | Evaluate each zero power separately, then add. |
| 2, 8 or 512 in question 4 | Used only one factor, or took the base. | Add the exponents on paper: 3 + (−9). |
| 0.125 or −1.5 in question 5 | Did not flip the fraction, or multiplied by the exponent. | Say: negative exponent, flip the fraction. |
| B, C or D in question 6 | Ordered by the size of the exponent's digits, or did not turn the powers into decimals. | Convert each to a decimal, then order. |
| 5, 30 or 6 in question 7 | Dropped the sign, gave the hours, or gave the hours for one halving. | Count the halvings first: 30 ÷ 6. |
| 0.125 or 0.25 in question 8 | Added exponents, or used one term. | Write each term as a decimal. |
Questions worth asking
- Why must a0 be 1? Because a3 ÷ a3 = a0 and anything divided by itself is 1. Listen for “divided by itself”.
- Can a power of 2 ever be negative? No: powers of 2, positive or negative exponent, are all positive. Listen for “fraction, not negative”.
- What is 00? Undefined or debated; do not settle it in class, but note why a0 = 1 needs a not 0.
- What is (−2)−3? 1/(−2)3 = −1/8; the base can be negative even if the exponent's effect is a reciprocal.
Differentiation
A way in for anyone stuck
- Give a ladder with the base filled in and the division arrows drawn; students fill the rungs below 21.
- Use base 2 and base 10 before 3, 5 and 7.
- Write each negative power as a fraction before simplifying anything else.
Stretch
- Question 8, then: evaluate (2/5)−2. (25/4.)
- Write 0.0001 as a power of 10. (10−4.)
Students learning in English as an additional language
“Reciprocal” means one over a number: the reciprocal of 4 is 1/4. “Negative exponent” does not mean a negative answer. Say “one over” each time a negative exponent is read.
| Word | Plain meaning | Example |
|---|---|---|
| zero exponent | an exponent of 0, which gives 1 | 60 = 1 |
| negative exponent | an exponent below 0, which gives a reciprocal | 2−3 = 1/8 |
| reciprocal | one divided by a number | the reciprocal of 4 is 1/4 |
| positive exponent | an exponent above 0 | 3 in 53 |
| flip | swap the top and the bottom of a fraction | 2/3 flips to 3/2 |
| simplify | write in a shorter or simpler form | 95 × 9−8 = 1/729 |
Sentence starters for explaining aloud or in writing:
- Each step down divides by …, so the next rung is …
- A negative exponent means …
- The answer stays positive because …
Two things not to say
“A negative exponent makes it negative.” It makes it a reciprocal; the number stays positive.
“Anything to the power 0 is 0.” It is 1. Say why with the quotient law, not only the rule.
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.
- Negative powerEvaluate 4−2.
1/16. - Zero powerEvaluate 1000.
1.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tank loses half its water each hour. Write the fraction left after 3 hours as a power of 2 with a negative exponent.
2−3 = 1/8. - Earlier work (Lesson 4)Simplify (3a2)2.
9a4.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that a−1 is the reciprocal of a.
a × a−1 = a1 + (−1) = a0 = 1. - LettersSimplify 5x−2 × 4x5.
20x3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Write a ladder on the board and ask students to fill in the rungs below the first power.