At a glance. One lesson of about 60 minutes. Students build a pattern in a diagram, record it as a table and in words, find any later term by counting steps, and use the gaps to tell linear patterns from other growing ones.
Calculator: no calculator. Nothing here needs one. The skill is seeing the step and counting steps, and the numbers are small enough to do in the head or on paper.
What students will learn
Know
- The words term, sequence, pattern and core.
- A linear pattern adds or subtracts the same amount each time.
- A pattern can be shown as a diagram, a table, words or a rule.
Understand
- Seeing what stays the same and what grows is how a pattern becomes a rule.
- The number of steps from the first term is one less than the term number.
- Growing is not the same as linear: the gaps decide.
Do
- Continue a diagram or number pattern and describe it in words.
- Find a term well beyond those drawn, by counting steps.
- Decide whether a pattern is linear from the gaps.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: continuing patterns, finding a later term and testing for a constant gap (questions 1 to 7).
- Criterion B. Investigating patterns: the first figure. Ask students to record the table, spot the constant gap and predict the 8th and then the 20th total before it is drawn.
- Criterion C. Communicating: describing the same pattern in a diagram, a table and a sentence, and explaining why the doubling pattern is not linear (question 9).
- Criterion D. Applying mathematics in real-life contexts: bamboo frames, tile borders and matchsticks, where counting every piece would take too long (the worked and faded examples).
The one thing to do with the figure
Ask for a prediction before the slider moves. One frame is 4 canes and two are 7. “How many for five?” Take the guesses from the room, then slide to 5 and read 16.
Then go to the second figure and ask only one question per pattern: is the gap the same every time? Make students say the gaps aloud for patterns 3 and 4 before the colour of the gap text confirms it.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea; the second figure. Students find the gaps and decide linear or not. |
| 10 min | Worked example on the 20th term; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose B or C, return to the table in the first figure. |
| 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 | 19; 8; 6, 12, 18, 24. |
|---|---|
| Worked example | 61 canes for 20 frames: 4 + 19 × 3. |
| Faded example | 49 tiles: 5 + 11 × 4. |
| 1. Describing a pattern | A, start at 7 and add 4. B swaps the start and the step; C uses the second term as the step; D multiplies where the pattern adds. |
| 2. Continuing a pattern | 29. 5 + 6 × 4 = 29. |
| 3. The core of a repeating pattern | 20. 10 cores, 2 dots each. |
| 4. A term far along | 4. 40 − 6 × 6 = 4. |
| 5. The first term past a target | 41. 38 is below 40, then 38 + 3 = 41. |
| 6. Linear or not | C. A and B have constant gaps (+4 and −3); D has a constant gap of 0. Only C has gaps 1, 2, 4. |
| 7. Building from a diagram | 21. 3 + 9 × 2 = 21. |
| 8. Which term number | 20. 95 ÷ 5 = 19 steps, so the 20th term. |
| 9. Why not linear | B. A confuses growing with linear; C looks at the start, not the step; D mixes up having a rule with being linear. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B or C in question 1 | Swapped the start and the step, or used the second term as the step. | Ask for the first term and then the gap between neighbours, written as two separate numbers. |
| 28, 35 or 33 in question 2 | Multiplied the number of tiles by the step, or added the step one time too many. | Draw the first three tiles and count the pins added at each step. Ask how many tiles come after the first. |
| 10 in question 3 | Chose the wrong number of loops or dots in the core. | Have them write out the first 12 symbols and circle every core. |
| 10 in question 4 | Stopped one step early, giving the 6th term. | Number every term: 1st, 2nd and so on. The 7th is 6 steps from the 1st. |
| 38 in question 5 | Chose the last term below the target. | Ask whether 38 is greater than 40. |
| B or D in question 6 | B: thought a pattern that goes down cannot be linear. D: thought a gap of 0 is not a pattern. | Write the gaps under each pattern and compare them. |
| 30 or 20 in question 7 | Treated every triangle as new, or forgot the first needs 3. | Draw 4 triangles and count the sticks each addition needs. |
| 19 in question 8 | Gave the number of steps rather than the term number. | Ask what the term number of the first term is, and how many steps it is from itself. |
Questions worth asking
- Why does a second frame need only 3 canes? Because it shares a side with the first. Listen for “shared” or “it already has one wall”.
- How many steps are there from the 1st term to the 12th? 11. The term number is one more than the number of steps. This is the idea behind the position rule in lesson 3.
- Can a pattern go down and still be linear? Yes: 75, 68, 61 goes down by 7 every time. Linear means the same gap, up or down.
- Can you make a pattern that grows but is not linear? Doubling, or adding 1 more each time. Ask them to show you the gaps.
Differentiation
A way in for anyone stuck
- Give each student a strip of squared paper and ask them to draw the frames for 1, 2, 3 and 4 squares. Count the canes together.
- Fill in the table with the new canes next to the total, so the step is visible before they use it.
- Start with the pattern the figure opens on (add 4) before showing one that grows by more each time.
Stretch
- Frames are now built two high, in columns. One column of 2 squares needs 7 canes. Draw 2 and 3 columns and find how many canes n columns need. (7, 12, 17: each column adds 5, so 5n + 2.)
- A pattern starts at 1 and its gaps are 2, 4, 6, 8, and so on. What is the 10th term? (The nine gaps before it run 2, 4, 6, up to 18 and add to 90, so the 10th term is 91.)
Students learning in English as an additional language
Pattern vocabulary is new. Link each word to the picture: point at the frame for term, the repeating part for core, and the gap for step.
| Word | Plain meaning | Example |
|---|---|---|
| pattern | numbers or shapes that follow a rule | 4, 7, 10, 13 |
| term | one number in a pattern | 7 is the 2nd term of 4, 7, 10 |
| sequence | a list of terms in order | 5, 9, 13, 17 |
| core | the shortest part of a repeating pattern | dot, dot, loop |
| linear | the same amount is added or subtracted each time | 75, 68, 61 |
| gap | the difference between neighbouring terms | the gaps in 2, 5, 8 are 3 and 3 |
Sentence starters for explaining aloud or in writing:
- The first term is … and each term is … more than the one before.
- The gaps are …, so the pattern is / is not linear.
- To find the …th term I add … a total of … times.
Two things not to say
“Just add the step each time.” Fine for the next term, but for the 20th it hides the idea that the step is added 19 times.
“It grows, so it is linear.” Growth is not the test. Always check the gaps.
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.
- Next termsWrite the next three terms of 8, 13, 18, 23, …
28, 33, 38. - Linear or notIs 100, 90, 80, 70 linear? What is the gap?
Yes: the gap is −10, so subtract 10 each time.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cup-seller in Marrakech stacks cups. One cup is 9 cm tall and each extra cup adds 2 cm. How tall is a stack of 11 cups?
29 cm. 9 + 10 × 2. - Earlier work, order of operationsWork out 4 + 3 × 6 − 2.
20. Multiply first: 18, then 4 + 18 − 2.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningIs 50 a term of 4, 10, 16, 22, …? Explain.
No. The terms are 4 more than a multiple of 6 (or: 50 − 4 = 46, and 46 ÷ 6 is not whole). - Earlier work, powersWrite the first four terms of the pattern that starts at 1 and multiplies by 3 each time. Is it linear?
1, 3, 9, 27. The gaps are 2, 6, 18, so it is not linear.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Build the frames from matchsticks or drawing straws on the desk and count.