Exponential relations: a week of ready-to-teach maths

Five lessons for Year 10 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 twelve table cards to sort, twelve rule-and-graph cards to match, 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. Cut the 12 table cards before Day 1 and the 12 rule-and-graph cards before Day 3, and keep each set in an envelope.
4. Calculators are used every day. On Day 5 a spreadsheet or a graphing app on a laptop or tablet does what the descriptor calls a digital tool; every task also works with a calculator.
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

A linear pattern grows by adding the same amount each step; an exponential one grows by multiplying by the same factor. That single difference shows in every form of the relation: in a table the ratios of neighbouring values stay the same, in the rule y = a × b^x the start value a and the factor b stand out, and on a graph the curve bends ever more steeply (or flattens towards zero when b is less than 1) while it never touches the x-axis. Some exponential equations can be solved exactly by writing both sides as powers of the same base: 2^x = 8 means 2^x = 2³, so x = 3. Others, such as 2^x = 10, have answers that are not tidy, and a table, a graph or a digital tool finds them to any accuracy we need.

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

A rule is written y = a × b^x: a is the value when x = 0 and b is the factor for each step of 1 in x. A fraction base goes in brackets, (1/2)^x. Answers to equations that do not come out exactly are given to two decimal places, and answers about real things end with a sentence.

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

Day 1 separates multiplying from adding; Day 2 draws the curves; Day 3 moves between table, graph and rule; Day 4 solves the equations that come out exactly; and Day 5 solves the ones that do not, in real situations. It builds on Year 9, where students used the exponent laws with integer exponents and graphed quadratic functions, solving equations graphically and numerically with digital tools. It prepares the modelling of growth and decay that follows in Year 10, and the logarithms of senior mathematics.

Materials for the week (one trip)

• From the classroom: calculators, scissors, envelopes for the cards, pencils, this pack printed.
• Cut-out sheet 1, one per pair: 12 table cards (Day 1).
• Cut-out sheet 2, one per pair: 12 rule-and-graph cards (Day 3).
• If you have them: laptops or tablets with a spreadsheet or graphing app for Day 5 (one per pair is plenty). No other equipment to buy.

Words for the word wall

linear, exponential, constant difference, constant ratio, factor, base, exponent, start value, growth, decay, curve, asymptote, y-intercept, power, digital tool

Dear families

This week in maths, Year 10 is studying exponential relations: patterns that grow by multiplying, like savings with interest, a doubling colony of cells, or a bouncing ball that loses a share of its height each bounce.

Try this at home

• Ask: would you rather have $1000 a day for 30 days, or 1 cent doubled every day for 30 days? (The doubling: over $5 million on the last day.)
• Ask: 3, 6, 12, 24, what comes next? And 3, 6, 9, 12? (48, by doubling; 15, by adding 3.)
• Ask: what power of 2 is 32? (2⁵ = 32.)
• Ask: $1000 grows by 5% a year. Roughly how long until it doubles? (About 15 years.)

My growth week

Fill one row a day. Write something that grows or shrinks, whether it adds or multiplies, and by how much each step.

Adding or multiplying?

A linear pattern adds the same amount each step; an exponential pattern multiplies by the same factor. Students grow both from one start, name and continue four patterns, and sort twelve table cards.

We are learning to

• tell a linear pattern from an exponential one by its differences and ratios,
• write the rule of each: y = a + dx, or y = a × b^x,
• explain why multiplying overtakes adding in the end.

Success criteria

• I can say why 3, 6, 12, 24 is exponential and 3, 6, 9, 12 is linear.
• I can write y = 5 × 3^x for the pattern 5, 15, 45, 135.

You need

Calculators; the table cards (cut-out sheet 1), one set per pair; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; sort the cards and do Part D next time.

Watch for these ideas

• “Exponential means growing fast.” It means the same factor each step; a factor below 1 shrinks: 100, 50, 25.
• Checking one step only: 1000, 1100, 1210 adds 100 and then 110. Check every step.
• Reading the factor as an amount added: in 5, 15, 45 the factor is 3; nothing is added.

Answers

• Part A: y = 3 + 2x gives 3, 5, 7, 9, 11, 13 (add 2); y = 3 × 2^x gives 3, 6, 12, 24, 48, 96 (multiply by 2).
• Part B: 1. exponential, next 405, y = 5 × 3^x; 2. linear, next 45, y = 5 + 10x; 3. exponential, next 1, y = 81 × (1/3)^x; 4. exponential, next 1464.1, y = 1000 × 1.1^x.
• Part C: the card key is with the mini-check answers.
• Part D: at x = 9, 550 against 2⁹ = 512; at x = 10, 600 against 2¹⁰ = 1024. The doubling overtakes at x = 10 and stays ahead for good.

Adding or multiplying?

A. One start, two patterns

Fill both rows. The first three columns are done for you.

B. Name it, continue it, write the rule

C. Sort the cards

Sort the twelve table cards into linear and exponential, then write the letters.

D. Stretch: the race

The equation becomes a curve

Every rule y = b^x with b positive makes a curve through (0, 1) that never touches the x-axis. Students tabulate three rules for negative x as well, plot all three on one grid, and name what the curves share and how they differ.

We are learning to

• tabulate an exponential rule for negative, zero and positive x,
• plot the points and draw one smooth curve through them,
• describe what every such curve shares: the point (0, 1), y always positive, and the x-axis approached but never reached.

Success criteria

• I can explain why 2⁻² = 1/4 and plot it.
• I can say why every curve y = b^x passes through (0, 1).

You need

Calculators; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; do Parts B to D next time.

Watch for these ideas

• “2⁻² = −4.” A negative exponent divides: 2⁻² = 1/2² = 1/4, still positive.
• Joining the points with straight segments: the curve bends smoothly between them.
• “The curve meets the x-axis on the left.” It gets ever closer, but halving never reaches 0.

Answers

• Part A: y = 2ˣ: 1/4, 1/2, 1, 2, 4, 8; y = 3ˣ: 1/9, 1/3, 1, 3, 9, 27 (only 3 to the power 3 = 27 is off the grid).
• Part B: 8, 4, 2, 1, 1/2, 1/4: the curve of y = 2^x reflected in the y-axis.
• Part C: 1. (0, 1): any base to the power 0 is 1. 2. No: y is always positive. 3. y = 3^x is steeper for x > 0 and lower for x < 0. 4. y = (1/2)^x falls as x grows.
• Part D: at x = 3, 2³ = 8 and 3² = 9; at x = 10, 1024 against 100. They meet at x = 2 and x = 4, and after x = 4 the exponential stays ahead.

The equation becomes a curve

A. Two growing curves

Fill both rows as fractions or whole numbers. The x = 0 column is done for you.

B. A shrinking curve

Plot all three rules on this grid and draw a smooth curve through each. One value of Part A will not fit; leave it off.

C. What the curves share

D. Stretch: power against square

Table, graph and rule

In y = a × b^x the start value a is y at x = 0, and the factor b is the ratio from one x to the next. Students read a and b from tables and from two points of a graph, and match six rules to their graphs.

We are learning to

• read the start value and the factor from a table,
• find the rule from the points of a graph at x = 0 and x = 1,
• link a rule to its graph: where it starts, and whether it grows or decays.

Success criteria

• I can write the rule for the table 80, 40, 20, 10.
• I can tell y = 3 × 2^x from y = 2 × 3^x on a graph, though both pass through (1, 6).

You need

The rule-and-graph cards (cut-out sheet 2), one set per pair; calculators; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; start the next with Part B.

Watch for these ideas

• Taking a from x = 1: a is the value at x = 0.
• Finding b by subtracting: in 80, 40, 20 the factor is 40 ÷ 80 = 1/2, not 40.
• “Both pass through (1, 6), so they match.” Look at x = 0: one starts at 3, the other at 2.

Answers

• Part A: a = 5, b = 2, so y = 5 × 2^x; a = 2, b = 3, so y = 2 × 3^x; a = 80, b = 0.5, so y = 80 × (1/2)^x; a = 500, b = 1.2, so y = 500 × 1.2^x.
• Part B: (0, 3) and (1, 4.5) give y = 3 × 1.5^x; (0, 8) and (1, 2) give y = 8 × (1/4)^x.
• Part C: the card key is with the mini-check answers.
• Part D: b = 12 ÷ 6 = 2; one step back from x = 1 gives a = 6 ÷ 2 = 3, so y = 3 × 2^x.

Table, graph and rule

A. From a table

Find the start value a (at x = 0) and the factor b, then write the rule y = a × b^x. The first is done for you.

B. From a graph

Each graph shows its points at x = 0 and x = 1. Write each rule, and say whether it grows or decays.

C. Match the cards

Match each rule card to its graph card. Write the letters.

D. Stretch: a late start

Solving by matching the base

When two powers of the same base are equal, their exponents are equal. Students write numbers as powers of 2, 3, 5 and 10, solve exponential equations exactly, and change the base when the two sides start from different bases.

We are learning to

• write whole numbers and fractions as powers of a base, with negative exponents for fractions,
• solve an exponential equation by matching the bases and setting the exponents equal,
• change both sides to a common base when the bases differ.

Success criteria

• I can solve 5^x = 1/25 and check it.
• I can solve 4^x = 32 by writing both sides as powers of 2.

You need

Calculators, for checking only; the worksheet, one per student.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part B; start the next with Part C.

Watch for these ideas

• Dividing instead of counting: 2^x = 8 is not x = 4; three 2s multiply to 8, so x = 3.
• Missing the negative exponent: 1/25 = 5⁻².
• Changing one side only: in 4^x = 32 both sides become powers of 2: 2^2x = 2⁵.

Answers

• Part A: 8 = 2³, 32 = 2⁵, 1 = 2⁰, 1/16 = 2⁻⁴, 81 = 3⁴, 1/27 = 3⁻³, 125 = 5³, 1/25 = 5⁻².
• Part B: 2^x = 32, x = 5; 3^x = 81, x = 4; 5^x = 1/25, x = −2; 10^x = 1/1000, x = −3; 2^{x + 1} = 16, x = 3; 3^2x = 81, x = 2.
• Part C: 4^x = 32 becomes 2^2x = 2⁵, x = 5/2 = 2.5; 9^x = 27 becomes 3^2x = 3³, x = 3/2 = 1.5; 8^x = 4 becomes 2^3x = 2², x = 2/3 = 0.6667; 27^x = 9 becomes 3^3x = 3², x = 2/3 = 0.6667.
• Part D: 1. 2^x × 2^{x + 1} = 2^{2x + 1} = 32 = 2⁵, so x = 2. 2. No solution: every power of 2 is positive.

Solving by matching the base

A. Powers of a base

Write each number as a power of the base shown. The first is done for you.

B. Equal bases, equal exponents

Solve, then check on a calculator. The first is done for you.

C. Change the base

Write both sides as powers of the same smaller base, then solve. Answers may be fractions.

D. Stretch

When the bases will not match

Most exponential equations do not come out exactly: 2^x = 10 has no tidy answer. Students trap it in a table, finish with a digital tool, and answer three growth and decay questions in context. The self-check quiz ends the week.

We are learning to

• narrow down the solution of an exponential equation with a table of values,
• use a digital tool (a graphing app, a spreadsheet or a calculator) to solve to two decimal places,
• answer a growth or decay question in words, with the whole number the situation needs.

Success criteria

• I can show that the solution of 2^x = 10 lies between 3.3 and 3.4.
• I can say in which whole year $1000 at 5% a year has doubled.

You need

Calculators; laptops or tablets with a spreadsheet or graphing app if you have them; the worksheet; the self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions? Stop after Part A; do Parts B to D and the quiz next time.

Watch for these ideas

• Rounding the time down: if it has doubled after 14.2 years, that happens during year 15.
• Stopping at a rough graph reading: a table or a tool narrows the answer to the accuracy the question asks.
• Treating 5% growth as adding $50 a year: the interest is 5% of a balance that keeps growing.

Answers

• Part A: 2³ ≈ 8.00, 2^3.1 ≈ 8.57, 2^3.2 ≈ 9.19, 2^3.3 ≈ 9.85, 2^3.4 ≈ 10.56, 2^3.5 ≈ 11.31; 10 lies between x = 3.3 and x = 3.4. A tool gives x ≈ 3.32.
• Part B: 1.05^t = 2 gives t ≈ 14.21. After 14 years the balance is $1,979.93; after 15 it is $2,078.93. It has doubled by the end of year 15.
• Part C: 2⁹ = 512 and 2¹⁰ = 1024, so the colony first passes 1000 cells after 10 hours.
• Part D: 2 × 0.6ⁿ: after 5 bounces 15.6 cm, after 6 9.3 cm. Bounce 6 is the first under 10 cm.

When the bases will not match

A. Trapping 2^x = 10

Fill the table to two decimal places. The first entry is done for you.

B. Doubling money

C. A doubling colony

D. Stretch: a bouncing ball

Linear or exponential?

For Day 1, one set per pair. Each card is a pattern at x = 0, 1, 2 and 3. Sort the cards into linear (the same difference each step) and exponential (the same factor each step). Check every step, not just the first.

Teacher note: the key is with the mini-check answers.

Rules and their graphs

For Day 3, one set per pair. Six cards show a rule; six show a graph with its points at x = 0 and x = 1. Match each rule to its graph. Some graphs share their point at x = 1: look at where each one starts.

Teacher note: the key is with the mini-check answers.

What we know: Exponential relations

Work on your own. A calculator is allowed. Show your working.

1. For each pattern at x = 0, 1, 2, 3, say linear or exponential and write the rule:   (a) 4, 12, 36, 108   (b) 4, 12, 20, 28
2. Complete y = 3^x for x = −2, −1, 0, 1, 2. Where does the graph cross the y-axis? ____________
3. Write the rule y = a × b^x for the pattern 6, 3, 1.5, 0.75 at x = 0, 1, 2, 3. ____________
4. Solve 2^x = 64. ____________
5. Solve 3^{x − 1} = 27. ____________
6. The graph of y = 5^x, circle one:   crosses the x-axis at x = 0  ·  passes through (0, 1)  ·  passes through (0, 5)
7. Solve 4^x = 2. ____________
8. $500 is saved at 8% a year, compounded yearly. In which whole year does it first pass $1000? Show how you know. ____________

Answers and marking guide

Answers with working

1. (a) exponential, factor 3: y = 4 × 3^x. (b) linear, difference 8: y = 4 + 8x.
2. 1/9, 1/3, 1, 3, 9; it crosses the y-axis at (0, 1).
3. a = 6 and b = 0.5: y = 6 × (1/2)^x.
4. 64 = 2⁶, so x = 6.
5. 27 = 3³, so x − 1 = 3 and x = 4.
6. passes through (0, 1): any base to the power 0 is 1.
7. 2^2x = 2¹, so 2x = 1 and x = 1/2 = 0.5.
8. 1.08^t = 2: after 9 years $999.50, after 10 years $1,079.46, so it first passes $1000 in year 10.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working, gives exact answers where the bases match, and ends Question 8 with a sentence.

Card keys

Table cards (Day 1): linear B, E, H, J, K; exponential A, C, D, F, G, I, L. Rule and graph cards (Day 3): S and M, P and V, Q and X, N and T, W and R, U and O.

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: adding or multiplying, the curves, table, graph and rule, solving exactly, and solving with tools in real situations.