At a glance. One lesson of about 60 minutes. It uses the geometric sum formula from lesson 4. Students decide when a sum to infinity exists, find it, and use it for recurring decimals and a bouncing ball. The infinite sum is presented as a limit value, not as an addition that is ever completed.
Calculator: no calculator. No calculator: ratios are chosen to give exact results, and the reasoning is about size.
What students will learn
Know
- S∞ = a ÷ (1 − r) when |r| < 1.
- The sums do not settle when |r| is 1 or more.
- A recurring decimal is a geometric series with ratio 0.1, 0.01 and so on.
Understand
- Why rn shrinks when |r| < 1.
- A sum to infinity is the value the sums approach, not a finished addition.
- A negative ratio makes the sums alternate around the value.
Do
- Decide whether a sum to infinity exists.
- Find the sum to infinity, or a missing ratio or term.
- Write a recurring decimal as a fraction.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the condition and the value (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe what the sums do as the ratio changes.
- Criterion C. Communicating: explaining in a sentence why the sum is a ceiling.
- Criterion D. Applying mathematics in real-life contexts: the corridor (hook), a bouncing ball (worked example 2) and a growing plant (question 4).
The one thing to do with the figure
Ask for the prediction first. Then set the ratio to 0.5 and move the number of terms from 1 to 12, asking how big each extra step is. Then set the ratio to 1 and then 1.1 and ask what is different.
Ask: “What does the ceiling depend on, and what does the speed of approach depend on?” The ceiling depends on a and r; a ratio closer to 1 approaches more slowly.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the corridor; collect predictions and reasons. |
| 12 min | Figure, key idea and warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C, ask what the sums do when each term is 5. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 15; 0.375; 7/8. |
|---|---|
| Hook | The walks are 15, 7.5, 3.75, … with a = 15 and r = 0.5, so the total approaches 30 m and never passes it. |
| Worked examples | 24; 40 m; 3/11. |
| Faded example | 13.5. |
| 1. When a sum to infinity exists | B. A has ratio 2.5; C has ratio 1; D has ratio −2. |
| 2. Sum to infinity | 67.5. 45 ÷ (2⁄3). |
| 3. Decimal ratio | 25. 5 ÷ 0.2. |
| 4. Finding the ratio | 0.75. 1 − r = 0.25. |
| 5. Recurring decimal with a digit that does not repeat | 43. 36⁄90 + 7⁄90. |
| 6. A sum from k = 1 | 2. 1.2 ÷ 0.6. |
| 7. Finding a term | 4.8. r = 0.6. |
| 8. Reading partial sums | D. A is the last sum shown; B uses the sums as terms; C uses the last sum as the first term. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Chose a ratio of 1 or more in size. | Write each ratio and ask: do the terms shrink? |
| 135 or 33.75 in question 2 | Left out the 1 minus, or used 1 plus r. | Check with 1 + 1/2 + 1/4 + …: the sum to infinity should be 2. |
| 20 or 6.25 in question 3 | Used the second term as a, or divided by r. | Name a and r before substituting. |
| 0.25 or 4 in question 4 | Gave 1 − r, or its reciprocal. | Ask: what does the equation 60 = 15 ÷ (1 − r) solve for first? |
| 7 or 106 in question 5 | Left out the 0.4 that does not repeat, or started the repeating part at 0.7. | Write 0.4 + 0.07 + 0.007 + … and ask which part is geometric. |
| 5 or 1.2 in question 6 | Started at k = 0, or gave only the first term. | Write the first three terms and read off a and r. |
| 12 or 0.6 in question 7 | Gave the rest of the sum, or the ratio. | Ask: what is the second term, in terms of a and r? |
| A, B or C in question 8 | Gave the last sum shown, used the sums as the terms, or used the last sum as the first term. | Find the gaps between sums: they are the terms. |
Questions worth asking
- Why does the sum to infinity need |r| to be less than 1? Listen for: otherwise the terms do not shrink.
- Does the ball ever stop in the model? Listen for: the model never reaches the ceiling, but the total distance is finite.
- What is the sum to infinity when r is negative? Listen for: the sums alternate above and below it.
- Is 0.999… equal to 1? Listen for: yes, as the value the sums approach.
Differentiation
A way in for anyone stuck
- Give a table of partial sums for ratio 0.5 and ask where they are heading.
- Provide the formula with blanks for a and r, and the check |r| < 1 as a first line.
- Give the frame: “The first term is … and the ratio is …, so S∞ = …÷ … = ….”
Stretch
- Find the values of x for which 1 + x + x2 + … has a sum to infinity, and write the sum in terms of x.
- A ball is dropped from 12 m and each rebound is 3⁄4 of the previous height. Find the total distance it travels.
Students learning in English as an additional language
“Infinity” means going on without end. “Ceiling” here means a value that something rises towards but does not pass. “Settle” means come closer and closer to one value. “Recurring” means repeating forever.
| Word | Plain meaning | Example |
|---|---|---|
| sum to infinity | the value the partial sums of a series approach | S∞ |
| partial sum | the sum of the first n terms | S6 |
| converge | get closer and closer to one value | sums near a limit |
| recurring decimal | a decimal whose digits repeat for ever | 0.333… |
| ceiling | a value the sums stay below | the line in the figure |
Sentence starters for explaining aloud or in writing:
- The ratio is …, so the terms ….
- The sums approach … because …
- There is no sum to infinity because …
Two things not to say
“The sum reaches the value.” Say instead: the sums get as close as we like to it.
“Any infinite sum is infinity.” Say instead: it depends on the ratio.
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.
- FindFind the sum to infinity of 20 + 10 + 5 + …
40. - DecideDoes 2 + 6 + 18 + … have a sum to infinity? Why?
No: the ratio is 3.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextThe tip of a pendulum in a museum in Quito swings 40 cm on its first swing, and each swing is 90% of the one before. What total distance does the tip travel in the long run?
40 ÷ 0.1 = 400 cm. - Earlier work (lesson 4)Find 1 + 3 + 9 + 27 + 81.
121.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningCan a series with ratio 0.5 and positive first term have a sum to infinity less than its first term? Explain.
No: S = a ÷ 0.5 = 2a, which is more than a. - Working backwardsA series has sum to infinity 9 and ratio 2/3. Find a.
a = 9 × 1/3 = 3.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed.