Grade by Grade
C. Algebra · Patterns, Integers, Equations, Inequalities, Coding and Modelling

Pattern Rules: Predictions, Missing Elements and Unknown Values

C1.3 A1About 45 minutes
Learning goals
  • I can find a pattern rule and use it to extend a pattern.
  • I can make predictions about a pattern and justify them.
  • I can find missing elements in repeating, growing and shrinking patterns with whole numbers and decimals.
  • I can use an equation to solve for an unknown term number or term value in a linear pattern.

Words to know

pattern rule
A description of how a pattern works. A term-to-term rule says how to get the next term (add 0.7). A general rule (an equation) gives any term from its term number (t = 4n + 2).
extend
Continue a pattern to find more terms.
prediction
A statement about a term you haven't calculated one by one, based on the pattern rule.
justify
Explain why your answer makes sense, using the rule.
missing element
A term that has been left out of a pattern.
solve
Find the value of the variable that makes an equation true.

Remember from Grade 6

In Grade 6 you used pattern rules to extend patterns, make predictions and find missing terms. You also solved for an unknown term number: for t = 4n + 4, you found that 100 is term 24.

In Grade 7 you'll do the same with more decimal patterns and more shrinking patterns, and you'll write and solve the equation step by step.

Let's start with a mystery

A scarf pattern has rows of beads. Some numbers were smudged:

3.2, ?, 4.6, ?, 6.0

The pattern is linear. What are the missing numbers? There are two jumps between 3.2 and 4.6.

33.544.555.56+0.7+0.7+0.7+0.73.2, 3.9, 4.6, 5.3, 6.0
From 3.2 to 6.0 in four equal jumps of 0.7. The missing terms are 3.9 and 5.3.

Finding missing elements

In a linear pattern, count the jumps between two terms you know. Divide the change by the number of jumps.

From 3.2 to 4.6 the change is 1.4, in 2 jumps. 1.4 ÷ 2 = 0.7. So the missing terms are 3.2 + 0.7 = 3.9 and 4.6 + 0.7 = 5.3. Check: 5.3 + 0.7 = 6.0 ✓.

In a non-linear pattern, look for multiplying or dividing. 243, ?, 27, 9: each term is the one before ÷ 3, so the missing term is 81.

In a repeating pattern, find the core. Then any position is easy: divide the position by the length of the core and look at the remainder.

Solving for unknown values

The equation of a linear pattern lets you jump straight to any term, or work backwards to find a term number.

  • Unknown term value: for t = 4n + 2, term 50 is 4 × 50 + 2 = 202.
  • Unknown term number: which term is 98? Write 4n + 2 = 98 and solve. Subtract 2: 4n = 96. Divide by 4: n = 24.
  • Is a number in the pattern? Is 200 a term? 4n + 2 = 200 gives 4n = 198 and n = 49.5. Term numbers must be whole numbers, so 200 is not in the pattern.
Building the term:n = 249698× 4+ 2Undoing it (go backwards):9896n = 24− 2÷ 44n + 2 = 98 → 4n = 96 → n = 24
To build a term of t = 4n + 2, multiply by 4 and then add 2. To find which term is 98, undo the steps in reverse order: subtract 2, then divide by 4.

Undo in reverse order

To solve 4n + 2 = 98, the pattern multiplied by 4 then added 2. Undo it backwards: subtract 2 first, then divide by 4. Always check by putting your answer back in: 4 × 24 + 2 = 98 ✓.

Worked example

Example 1: Which term is it?

A pattern has the equation t = 50 − 1.5n. Which term has the value 5?

  1. Write the equation: 50 − 1.5n = 5.
  2. The pattern took away 1.5n from 50 and got 5. So 1.5n must be 50 − 5 = 45.
  3. n = 45 ÷ 1.5 = 30.
  4. Check: 50 − 1.5 × 30 = 50 − 45 = 5 ✓.

Term 30 has the value 5.

Worked example

Example 2: Missing terms with decimals

Find the missing terms: 7.25, ?, ?, 9.5.

  1. Between 7.25 and 9.5 there are 3 jumps.
  2. Change: 9.5 − 7.25 = 2.25. Each jump: 2.25 ÷ 3 = 0.75.
  3. 7.25 + 0.75 = 8, and 8 + 0.75 = 8.75. Check: 8.75 + 0.75 = 9.5 ✓.

8 and 8.75.

Worked example

Example 3: Predict and justify

Maya has $35 and saves $12.50 each week for a new pair of skates that cost $160 (a made-up price). Predict when she'll have enough. Justify your prediction.

  1. Equation: s = 12.5w + 35, where w is the number of weeks.
  2. Solve 12.5w + 35 = 160. Subtract 35: 12.5w = 125. Divide by 12.5: w = 10.
  3. Justify: after 10 weeks she has 12.5 × 10 + 35 = 125 + 35 = $160 ✓. After 9 weeks she has only $147.50.

Maya will have exactly enough after 10 weeks, assuming she saves $12.50 every week.

Worked example

Example 4: Is it in the pattern?

Is 250 a term of 9, 16, 23, 30, ...? Is 247?

  1. Constant rate 7, initial value 9 − 7 = 2. Equation: t = 7n + 2.
  2. 250: 7n + 2 = 250, so 7n = 248. 248 ÷ 7 = 35.43... That is not a whole number, so 250 is not a term.
  3. 247: 7n + 2 = 247, so 7n = 245, and n = 35. Check: 7 × 35 + 2 = 247 ✓.

250 is not in the pattern. 247 is term 35.

Worked example

Example 5: Far along a repeating pattern

Ice-rink lights flash in the order white, white, red, blue, then repeat. What colour is the 100th flash? The 102nd?

  1. The core has 4 flashes.
  2. 100 ÷ 4 = 25 with no remainder, so flash 100 is the last flash of a core: blue.
  3. 102 ÷ 4 = 25 R2, so flash 102 is the 2nd flash of a core: white.

Flash 100 is blue. Flash 102 is white.

Just for fun

Why was 200 left out of the pattern t = 4n + 2?

It would have to be term 49.5, and nobody has ever met half a term!

Hands-on activity

Smudged pattern game

You need: paper, a pencil and a grown-up or sibling to play with. A calculator is handy for checking.

  1. Player 1 secretly writes a linear pattern with decimals (for example: start at 2.4 and add 0.6) and lists its first 6 terms.
  2. Player 1 rewrites the list, replacing two terms with question marks, but never both of the first two.
  3. Player 2 finds the missing terms and the equation, explaining their steps out loud.
  4. Player 1 then names a big number. Player 2 uses the equation to decide whether it is in the pattern and which term it is.
  5. Switch roles.

Math talk: "How many jumps were there between the numbers you knew? How did you check?"

Your turn

1

Give the rule and the next two terms: 2.5, 3.75, 5, 6.25, ...

2

Find the missing terms: 14, ?, 26, 32, ?

3

Find the missing terms: 1.8, ?, ?, 3.0

4

For t = 5n + 3, which term has the value 103?

5

Is 75 a term of t = 4n + 1? Explain.

6

What is the 30th letter in A B C A B C ...?

7

Find the missing term: 400, 200, 100, ?, 25

More practice: warm-up

1

Next term: 10, 20, 30, 40, ...

2

Missing term: 5, ?, 15, 20

3

Missing term: 0.2, 0.4, ?, 0.8

4

For t = 2n + 1, what is term 10?

5

What is the 9th letter in A B A B A B ...?

More practice: on your own

1

For t = 3.5n + 4, which term has the value 46?

2

Find the missing terms: 10.4, ?, ?, ?, 13.6

3

A snow sculpture is 150 cm tall and melts 4.5 cm each day. Predict its height after 12 days. On which day will it first be less than 60 cm?

4

Is 88 a term of 6, 10, 14, 18, ...? If so, which term?

5

A pattern goes 1, 3, 9, 27, .... Predict the 7th term and justify.

Challenge

1

The pattern 0.5, 1, 1.5, 0.5, 1, 1.5, ... repeats. What is the sum of the first 20 terms?

2

Pattern A is t = 8n + 5 and pattern B is t = 5n + 41. Which term number gives the same value in both? What is the value?

3

Find the missing terms in 3, ?, 48, ? in two different ways: once as a linear pattern and once as a pattern that multiplies.

Check your understanding

Quiz

1.

Find the missing term: 2.6, 3.4, ?, 5.0

2.

For t = 6n + 4, which term has the value 64?

3.

To solve 5n + 7 = 52, what should you do first?

4.

Is 100 a term of t = 3n + 2?

5.

What is the 50th item in 🍁 ❄️ ❄️ ⛸️ 🍁 ❄️ ❄️ ⛸️ ...?

6.

Find the missing terms: 2, ?, ?, 8 (linear)

For parents and teachers

Big idea: Grade 7 students determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in repeating, growing and shrinking patterns with whole numbers and decimal numbers, and they use algebraic representations of pattern rules to solve for unknown values in linear growing patterns (C1.3).

Common difficulties:

  • Counting terms instead of jumps. Between term 1 and term 4 there are 3 jumps, not 4.
  • Undoing in the wrong order (dividing before subtracting). Say: "Undo the last thing first."
  • Remainders in repeating patterns. A remainder of 0 means the last item in the core.
  • Forgetting that term numbers must be whole numbers, so 200 is not in t = 4n + 2.

How to help: Play the smudged pattern game, and ask "How do you know?" after every answer. Praise checking an answer by substituting it back. This lesson builds A1 skills: reasoning, justifying predictions and persevering with multi-step problems.