Index laws: a week of ready-to-teach maths

Five lessons for Year 9 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus copy cards to match, power line-up cards, a 16-card law loop, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair, and cut them before Day 1: the copy cards are for Day 1, the line-up cards for Day 4 and the law loop for Day 5.
4. Calculators are for checking only, and only from Day 4.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

An exponent counts copies: a⁵ is five copies of a multiplied together. Three laws follow from counting. Multiplying powers of the same base adds the exponents (a³ × a⁴ = a⁷); dividing subtracts them (a⁷ ÷ a⁴ = a³); and raising a power to a power multiplies them ((a³)² = a⁶). An exponent outside a bracket reaches every factor inside, the number as well as the letters: (2x²)³ = 8x⁶. Keeping the laws true for every integer exponent forces a⁰ = 1 and a⁻ⁿ = 1/aⁿ: a negative exponent means a reciprocal, never a negative number. With letters, the numbers in front multiply or divide as numbers while the exponents follow the laws.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these expressions

Exponents are written raised, as in 3m². A letter on its own has exponent 1, so a means a¹. Answers are written with the number first and the letters in alphabetical order, as in 10a³b⁴. From Day 4, a final answer is given with positive exponents, as in 3/a² rather than 3a⁻², unless a question asks otherwise. Every letter stands for a number that is not zero.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Days 1 to 3 carry the three laws from numbers to letters and coefficients; Day 4 pushes the exponents down through zero into the negatives; and Day 5 uses every law at once and checks answers by putting a number in. It builds on Year 8, where students met the product, quotient and power laws and the zero exponent with numbers. It prepares scientific notation for very large and very small quantities later in Year 9, expanding and factorising, and the exponential growth and decay of Year 10.

Materials for the week (one trip)

• From the classroom: scissors, envelopes for the cards, pencils, calculators for checking from Day 4, this pack printed.
• Cut-out sheet 1, one per pair: 12 copy cards (Day 1) and 12 power line-up cards (Day 4).
• Cut-out sheet 2, one per class: the 16-card law loop (Day 5), one card per student or pair. No other equipment to buy.

Words for the word wall

base, exponent, index, power, index form, expanded form, coefficient, product law, quotient law, power law, zero exponent, negative exponent, reciprocal, simplify, positive exponents, substitute, counterexample

Dear families

This week in maths, Year 9 is learning the index laws: the rules for multiplying, dividing and raising powers such as x⁵. They all come from one idea, that an exponent counts how many copies are multiplied. We use them with letters as well as numbers, and we find out what a zero or a negative exponent means.

Try this at home

• Ask: what is 2³ × 2²? (2⁵ = 32: three twos and two more make five twos.)
• Ask: is (2x)² the same as 2x²? (No: (2x)² = 4x², because the square reaches the 2 as well.)
• Ask: what is 10⁰? (1. Any number that is not zero, to the power 0, is 1.)
• Ask: is 2⁻³ a negative number? (No: 2⁻³ = 1/2³ = 1/8. A negative exponent means “one over”.)

My index-law week

Fill one row a day. Write a power you simplified, the law you used, and how you checked it.

Printed from the free seegongsik Index laws teaching pack · seegongsik.com/au/y9/algebra/AC9M9A01/pack

The product law with letters

Year 8 multiplied powers of numbers by adding exponents. The same counting works for letters, and a number in front of a letter multiplies as a number: 3m² × 4m⁵ = 12m⁷.

We are learning to

• write a product of copies in index form, and an index form as copies,
• multiply terms with coefficients and several letters: multiply the numbers, add the exponents of each letter,
• test a claimed simplification by putting in a number.

Success criteria

• I can simplify 5a²b × 2ab³ and explain each exponent by counting copies.
• I can show that a³ × a⁴ is not a¹² by putting a = 2 into both sides.

You need

The worksheet; the 12 copy cards (cut-out sheet 1), one set per pair. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the card match; start the second with the three board products.

Watch for these ideas

• Multiplying the exponents: a³ × a⁴ is a⁷, not a¹². Count the copies.
• Adding the coefficients: 3m × 3m = 9m², not 6m². Numbers in front multiply.
• Mixing up y² × y² = y⁴ with y² + y² = 2y².

Answers

• Part A: m × m × m × m = m⁴; 3 × p × p × q = 3p²q; a × b × a × b × a = a³b²; 2 × k × 5 × k = 10k².
• Part B: x⁴ × x⁶ = x¹⁰; a × a⁵ = a⁶; 3m² × 4m⁵ = 12m⁷; −2p³ × 6p = −12p⁴; 5a²b × 2ab³ = 10a³b⁴; 4x³y² × (−3x²y⁴) = −12x⁵y⁶.
• Part C, at 2: 1 false, a⁷ (128 and 4096); 2 true (192 and 192); 3 false, y⁴ (16 and 8); 4 true (−4 and −4); 5 true (36 and 36); 6 false, b⁶ (64 and 32).
• Part D: 2³ × 3³ = 8 × 27 = 216 = 6³, because three 2s and three 3s pair up into three 6s; so aⁿbⁿ = (ab)ⁿ. 2³ × 3² = 72 has no single power: the bases and exponents both differ.

The product law with letters

A. Copies to index form

Write each product in index form, number first.

B. Multiply

Multiply the numbers, then add the exponents of each letter. Show one line of working.

C. True or false? Test it at 2

Put 2 in for the letter on both sides. If the two sides differ, the claim is false: write the correct right-hand side.

D. Stretch: different bases

Work out 2³ × 3³ and 6³. Explain with copies why they are equal, and write a rule for aⁿ × bⁿ. Can 2³ × 3² be written as one power? Why not?

The quotient law with letters

Dividing cancels copies that the top and the bottom share, so the exponents subtract. The numbers in front divide as numbers: 20a⁷ ÷ 4a² = 5a⁵.

We are learning to

• divide terms with coefficients and several letters: divide the numbers, subtract the exponents of each letter,
• use the product and quotient laws together in one fraction,
• find a missing exponent by reasoning about the law.

Success criteria

• I can simplify 12p⁴q⁶ ÷ 3pq² and explain the exponents by cancelling.
• I can say why the y’s disappear in 36x⁵y² ÷ 9x²y².

You need

The worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second by multiplying back one Part A answer.

Watch for these ideas

• Dividing the exponents: a⁶ ÷ a² is a⁴, not a³. Cancel copies.
• Subtracting the coefficients: 20a⁷ ÷ 4a² has 5 in front, not 16.
• Leaving a y when every y cancels: y² ÷ y² = 1, so the y simply goes.

Answers

• Part A: x⁸ ÷ x³ = x⁵; m⁶ ÷ m = m⁵; 20a⁷ ÷ 4a² = 5a⁵; 18y⁵ ÷ (−3y³) = −6y²; 12p⁴q⁶ ÷ 3pq² = 4p³q⁴; 36x⁵y² ÷ 9x²y² = 4x³.
• Part B: (6a³ × 2a⁴) ÷ 4a² = 12a⁷ ÷ 4a² = 3a⁵; (x⁵y³ × x²y) ÷ x⁴y² = x⁷y⁴ ÷ x⁴y² = x³y²; 10m⁶ ÷ (2m × 5m²) = 10m⁶ ÷ 10m³ = m³.
• Part C: box 8, x⁸ ÷ x³ = x⁵; box 7, 2⁷ × 2⁴ = 2¹¹; box 7, a⁹ ÷ a⁷ = a²; box 3, y⁶ × y³ ÷ y⁴ = y⁵.
• Part D: x³ ÷ x³ = 1 (every copy cancels), and the law gives x⁰, so x⁰ = 1. x³ ÷ x⁵ leaves two x’s on the bottom, 1/x², and the law gives x⁻²: the idea of Day 4.

The quotient law with letters

A. Divide

Divide the numbers, then subtract the exponents of each letter. Check one by multiplying back.

B. Two laws in one

Simplify the brackets first, then divide.

C. The missing exponent

Find the number in the box. Write the equation the exponents make.

D. Stretch: when the bottom has more copies

Write x³ ÷ x³ and x³ ÷ x⁵ as fractions of copies and cancel. What is left in each? What does the quotient law say each one equals? What must x⁰ and x⁻² mean for the law and the cancelling to agree?

Powers of powers and brackets

Raising a power to a power makes copies of copies, so the exponents multiply. An exponent outside a bracket reaches every factor inside: the number in front is raised too, and so is each letter.

We are learning to

• raise a power to a power by multiplying the exponents,
• raise a bracket of factors to a power, including the number and its sign,
• spot and fix the classic slips, and use two laws in one expression.

Success criteria

• I can simplify (2p³)⁴ = 16p¹² and explain the 16 and the 12.
• I can explain why (−2y²)³ is negative but (−x)² is not.

You need

The worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the three board powers.

Watch for these ideas

• Adding instead of multiplying: (x²)³ = x⁶, not x⁵.
• Leaving the number unpowered: (3x)² = 9x², not 3x².
• Dropping the sign: (−2y²)³ = −8y⁶, because three negatives multiply to a negative.

Answers

• Part A: (a⁴)³ = a¹²; (m²)⁵ = m¹⁰; (3x)² = 9x²; (2p³)⁴ = 16p¹²; (−2y²)³ = −8y⁶; (a²b³)⁴ = a⁸b¹².
• Part B: 1 wrong, (x²)³ = x⁶; 2 wrong, (3a)² = 9a²; 3 right; 4 wrong, (−x)² = x²; 5 right.
• Part C: (x³)² × x⁴ = x⁶ × x⁴ = x¹⁰; (2a²)³ ÷ 4a = 8a⁶ ÷ 4a = 2a⁵; (m⁴)³ ÷ (m²)⁵ = m¹² ÷ m¹⁰ = m²; (3x²y)² × 2xy³ = 9x⁴y² × 2xy³ = 18x⁵y⁵.
• Part D: (x²/3)³ = x⁶/27, as the 3 is cubed too. At x = 1, y = 1: (x + y)² = 4 but x² + y² = 2, so a bracket of a sum does not share its exponent: only a bracket of factors does.

Powers of powers and brackets

A. Raise to the power

Raise the number to the power, and multiply each letter’s exponent by the power.

B. Right or wrong?

Tick the right ones. For a wrong one, write the correct answer and what went wrong.

C. Two laws in one

Do the powers first and write that line, then multiply or divide.

D. Stretch: a fraction and a sum in brackets

Simplify (x²/3)³. Then test (x + y)² and x² + y² at x = 1 and y = 1. Does a bracket of a sum share its exponent the way a bracket of factors does?

Zero and negative exponents

Each step down a staircase of powers divides by the base. Keep stepping past the first power and the laws force a⁰ = 1 and a⁻ⁿ = 1/aⁿ: a negative exponent means a reciprocal, not a negative number.

We are learning to

• explain why a⁰ = 1 and a⁻ⁿ = 1/aⁿ, from the staircase and from the quotient law,
• evaluate zero and negative powers exactly, including negative bases,
• write an expression with positive exponents, keeping track of which factor the exponent belongs to.

Success criteria

• I can evaluate (−2)⁻³ exactly and say why it is negative while 2⁻³ is not.
• I can rewrite 4y⁻² as 4/y² and explain why the 4 stays on top.

You need

The worksheet; the 12 line-up cards (cut-out sheet 1), one set per pair; calculators for checking Part B only.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the line-up; start the second with the staircase for 3 again.

Watch for these ideas

• Reading a negative exponent as a negative number: 2⁻³ = 1/8, not −8.
• Thinking a zero exponent gives 0: 9⁰ = 1.
• Moving the number with the letter: 4y⁻² = 4/y², but (2m)⁻¹ = 1/(2m), because there the bracket holds the 2.

Answers

• Part A: 5³ = 125, 5² = 25, 5¹ = 5, 5⁰ = 1, 5⁻¹ = 1/5, 5⁻² = 1/25.
• Part B: 2⁻⁴ = 1/16; 6⁻² = 1/36; 10⁻³ = 1/1000; 9⁰ = 1; (−3)⁻² = 1/9; (−2)⁻³ = −1/8; 6⁻¹ + 3⁻¹ = 1/2; 5⁰ + 5⁻¹ = 6/5.
• Part C: x⁻⁵ = 1/x⁵; 4y⁻² = 4/y²; a³b⁻² = a³/b²; (2m)⁻¹ = 1/(2m); 1/k⁻³ = k³; 6p⁻¹q = 6q/p.
• Line-up: 10⁻² (1/100) < 2⁻⁵ (1/32) < 3⁻³ (1/27) < 5⁻² (1/25) < 2⁻³ (1/8) < 4⁻¹ (1/4) < 2⁻¹ (1/2) < 7⁰ (1) < (1/2)⁻¹ (2) < 2² (4) < (1/3)⁻² (9) < 10¹ (10).
• Part D: (1/2)⁻³ = 2³ = 8: a negative exponent turns the fraction upside down. A positive base raised to any integer power is positive, because it is copies of a positive number or one over them.

Zero and negative exponents

A. The staircase for 5

Each step down divides by 5. Write each value as a whole number or a fraction.

B. Evaluate exactly

Write each answer as a whole number or a fraction. Check two with a calculator.

C. Positive exponents only

Rewrite with positive exponents. Decide first which factor each exponent belongs to.

D. Stretch: a fraction to a negative power

Work out (1/2)⁻³ by going down a staircase for 1/2, or by the rule a⁻ⁿ = 1/aⁿ. What happens to a fraction under a negative exponent? Can a positive number to any power ever be negative? Explain.

All the laws together

Students simplify with every law at once, give answers with positive exponents, find and fix four mistakes, and check a claim by putting in x = 2. The week ends with the law loop and the self-check quiz.

We are learning to

• choose and combine the laws to simplify, giving the answer with positive exponents,
• find a mistake in worked simplifications and explain it,
• check a simplification by substituting a number into both sides.

Success criteria

• I can simplify (x²y⁻³)⁻¹ to y³/x² and say which law each step uses.
• I can test a claim at x = 2 and explain what a failed test proves.

You need

The worksheet; the 16 law loop cards (cut-out sheet 2); the unit’s self-check quiz for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part C; start the second with the law loop and the quiz.

Watch for these ideas

• Moving the number with a negative exponent: 3x⁻² = 3/x², not 1/(3x²).
• Stopping at a negative exponent when positive ones are asked for: x⁻² is right, 1/x² is finished.
• Trusting one test: a claim that works at x = 2 may still be false; one that fails is false for sure.

Answers

• Part A: x⁴ × x⁻⁶ = x⁻² = 1/x²; 12a³ ÷ 4a⁵ = 3a⁻² = 3/a²; (y⁻²)³ = y⁻⁶ = 1/y⁶; (3m⁻¹)² = 9m⁻² = 9/m²; 5p⁻² × 2p⁷ = 10p⁵; (x²y⁻³)⁻¹ = x⁻²y³ = y³/x².
• Part B: Jai: 3x⁻² = 3/x², not 1/(3x²) (the −2 belongs to x only, so the 3 stays on top); Mia: (2a³)⁴ = 16a¹², not 8a¹² (the 4 reaches the 2 as well: 2⁴ = 16); Sam: a⁶ ÷ a² = a⁴, not a³ (subtract the exponents, do not divide them); Lena: (−4)⁰ = 1, not −1 (any non-zero base to the power 0 is 1).
• Part C, at x = 2: 1 1/4 and 1/4, true; 2 4 and 4, true; 3 1/4 and 1, false: (2x)⁻¹ = 1/(2x).
• Part D: 2ⁿ = 1/32 = 2⁻⁵, so n = −5. 4ⁿ = (2²)ⁿ = 2²ⁿ and 2²ⁿ = 2⁻⁶, so 2n = −6 and n = −3.

All the laws together

A. Simplify

Use the laws, then give each answer with positive exponents.

B. Find and fix the mistake

Each student made one mistake. Write the right answer and what went wrong.

C. Check it at x = 2

Work out both sides at x = 2. Are they equal? If not, the claim is false.

D. Stretch: find the exponent

Find n if 2ⁿ = 1/32. Then find n if 4ⁿ = 2⁻⁶. (Hint: write 4 as a power of 2.)

Copy cards and power line-up cards

Copy cards (Day 1): 12 cards

Lay the cards face down. Turn two over; keep them if one is a product of copies and the other is its index form.

Power line-up (Day 4): 12 cards

Write the value on the back of each card, then lay the cards in a line, smallest on the left.

Teacher note: the keys are with the mini-check answers, so this sheet can go to pairs whole.

The law loop

For Day 5, one set per class. Hand out the 16 cards. The student holding x⁷ starts by reading its question; whoever holds the answer says “I have ...” and reads the next question, until the loop comes back to the start.

Teacher note: the order of the loop is with the mini-check answers.

What we know: Index laws

Work on your own. No calculator. Give answers with positive exponents. Show your working.

1. Simplify 4a³ × 5a⁶. ______________________
2. Simplify 24x⁸y³ ÷ 6x²y. ______________________
3. Simplify (3m⁴)². ______________________
4. Evaluate 4⁻² and 7⁰, and then 7⁰ + 2⁻¹. ______________________
5. Write 5x⁻³ and (2y)⁻² with positive exponents. ______________________
6. Simplify x³ × x⁻⁷. ______________________
7. Ty wrote (2a³)² = 2a⁶. Explain the mistake and give the right answer. ______________________
8. Simplify a⁶ ÷ a², then check your answer by putting a = 2 into both sides. ______________________

Answers and marking guide

Answers with working

1. 4a³ × 5a⁶: 4 × 5 = 20 and 3 + 6 = 9, so 20a⁹.
2. 24 ÷ 6 = 4; x: 8 − 2 = 6; y: 3 − 1 = 2; so 4x⁶y².
3. 3² = 9 and 4 × 2 = 8, so 9m⁸.
4. 4⁻² = 1/16; 7⁰ = 1; 7⁰ + 2⁻¹ = 1 + 1/2 = 3/2.
5. 5x⁻³ = 5/x³ (the −3 belongs to x only); (2y)⁻² = 1/(2y)² = 1/(4y²).
6. Exponents 3 + (−7) = −4, so x⁻⁴ = 1/x⁴.
7. The 2 was not squared: the exponent outside the bracket reaches every factor, so (2a³)² = 4a⁶.
8. a⁶ ÷ a² = a⁴; at a = 2: 64 ÷ 4 = 16 and 2⁴ = 16, so they agree.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working, and writes final answers with positive exponents.

Card keys

Copy cards (Day 1): a × a × a × b = a³b; 2 × x × x × y × y × y = 2x²y³; m × n × m × n = m²n²; 5 × p × p × p × p = 5p⁴; (−k) × k = −k²; 3 × a × 3 × a = 9a². Power line-up (Day 4), smallest first: 10⁻² (1/100), 2⁻⁵ (1/32), 3⁻³ (1/27), 5⁻² (1/25), 2⁻³ (1/8), 4⁻¹ (1/4), 2⁻¹ (1/2), 7⁰ (1), (1/2)⁻¹ (2), 2² (4), (1/3)⁻² (9), 10¹ (10). Law loop (Day 5): x⁷ → a⁶ → y⁶ → 12m³ → 8p³ → 5k⁴ → 7 → 1/4 → 1/n³ → 1/b² → 1 → 9a⁴ → 2/x² → 4y² → 25 → m⁶n³ and back to x⁷.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: the product law, the quotient law, powers and brackets, zero and negative exponents, and all the laws together.