C. Algebra · Algebra: Patterns, Rational Numbers, Equations, Inequalities, Coding and Modelling

Pattern Rules: Predict, Find Missing Terms and Solve for Unknowns

C1.3 A1About 45 minutes

Learning goals

I can find recursive and explicit pattern rules for growing and shrinking patterns with rational numbers.
I can use a pattern rule to extend a pattern and to make and justify predictions.
I can find missing terms in a pattern.
I can use an equation to find which term has a given value.

Words to know

pattern rule
A description of how a pattern works.

recursive rule
A rule that says how to get each term from the term before it, like "start at 9 and subtract 2".

explicit rule
A rule that gives any term from its term number, like t = 2n + 1. It is also called the general term.

justify
To give reasons, with math, that show your answer is right.

unknown value
A number you need to find, often shown with a variable.

Remember from Grade 7

In Grade 7 you used pattern rules to extend patterns, find missing terms and solve for unknown values. This year the patterns can use any rational numbers, and you'll use equations to answer questions like "Which term has the value 300?" and then justify your predictions.

Let's start with a food drive

A school food drive collects 35 kg of food on day 1. After that, the total grows by 12.5 kg each day.

The principal asks: Will they reach 300 kg by day 20? You could list 20 terms... or you could use a rule. By the end of this lesson, you'll answer in a few lines.

Cardboard boxes of donated cans, pasta, rice and cereal are lined up on tables in a school gym. — A food drive grows a little every day as donations come in. If the same amount is added each day, the total makes a linear pattern you can write a rule for. AI-generated image

Two kinds of rules

A recursive rule tells you how to get from one term to the next: "Start at 35 and add 12.5." It's great for the next few terms.
An explicit rule uses the term number: t = 12.5n + 22.5. It's great for faraway terms. (The initial value is 35 − 12.5 = 22.5.)

With the explicit rule, day 20 is 12.5 × 20 + 22.5 = 250 + 22.5 = 272.5 kg. So no, they won't reach 300 kg by day 20.

Which day will they reach 300 kg? Solve 12.5n + 22.5 = 300: 12.5n = 277.5, so n = 22.2. Day 22 isn't quite enough (297.5 kg), so they reach 300 kg on day 23 (310 kg). That's a prediction you can justify: you showed the math.

A growing L-shaped pattern. Each figure adds one square to the tall arm and one to the bottom arm, so it grows by 2 squares each time.

Using the rule for the L-shapes

The numbers of squares are 3, 5, 7, ... The rate is 2 and the initial value is 1, so t = 2n + 1. You can see it in the picture: figure n has n + 1 squares going up and n squares going across: (n + 1) + n = 2n + 1.

Figure 50 has 2 × 50 + 1 = 101 squares.
Which figure has 75 squares? 2n + 1 = 75, so 2n = 74 and n = 37.
Could a figure have 80 squares? 2n + 1 = 80 gives n = 39.5. Figure numbers are whole numbers, so no. (Every figure has an odd number of squares.)

Finding missing terms

If two known terms have a gap between them, count the jumps, not the blanks. For 4.2, __, __, 7.8 there are 2 blanks but 3 jumps. Divide the total change by the number of jumps: (7.8 − 4.2) ÷ 3 = 1.2.

Between 4.2 and 7.8 there are 2 blanks but 3 jumps. Each jump is 3.6 ÷ 3 = 1.2, so the missing terms are 5.4 and 6.6.

Worked example

Example 1: Missing terms

Find the missing terms: 4.2, __, __, 7.8 (linear).

Total change: 7.8 − 4.2 = 3.6.
There are 3 jumps, so each jump is 3.6 ÷ 3 = 1.2.
4.2 + 1.2 = 5.4, 5.4 + 1.2 = 6.6. Check: 6.6 + 1.2 = 7.8 ✓.

5.4 and 6.6

Worked example

Example 2: An explicit rule with negatives

Find an explicit rule for −7, −4.5, −2, ... (term 1 is −7) and predict term 40.

Rate: +2.5. Initial value: −7 − 2.5 = −9.5.
Rule: t = 2.5n − 9.5.
Term 40: 2.5 × 40 − 9.5 = 100 − 9.5 = 90.5.

t = 2.5n − 9.5; term 40 is 90.5.

Worked example

Example 3: Which term?

Which term of t = 4n − 9 has the value 63?

Set the rule equal to 63: 4n − 9 = 63.
Add 9 to both sides: 4n = 72.
Divide both sides by 4: n = 18.
Verify: 4 × 18 − 9 = 72 − 9 = 63 ✓.

Term 18

Worked example

Example 4: Is it a term at all?

Is 100 a term of t = 3n + 4? Is 101?

3n + 4 = 100 → 3n = 96 → n = 32. That's a whole number, so 100 is term 32.
3n + 4 = 101 → 3n = 97 → n = 32.33... That's not a whole number.

100 is a term (term 32). 101 is not a term.

Worked example

Example 5: A non-linear rule

Find a rule and predict the 6th term: 64, 16, 4, 1, ...

The differences change (−48, −12, −3), so it isn't linear.
Each term is 1/4 of the one before (divide by 4).
Term 5: 1/4. Term 6: 1/16.

Rule: start at 64 and divide by 4 each time. Term 6 is 1/16.

Hands-on activity

Toothpick triangles

You need: about 40 toothpicks (or cotton swabs, pasta or matchsticks with the heads removed by an adult).

Make 1 triangle (3 toothpicks). Then build a row of 2 triangles that share a side (5 toothpicks), then 3 triangles (7 toothpicks).
Write a recursive rule and an explicit rule for the number of toothpicks.
Predict the toothpicks for 25 triangles, then explain your prediction.
Predict how many triangles you can build with 100 toothpicks.
Build 6 triangles to test your rule.

What to look for: recursive: "start at 3, add 2"; explicit: t = 2n + 1. 25 triangles: 51 toothpicks. With 100 toothpicks: 2n + 1 ≤ 100, so 49 triangles (using 99 toothpicks). 6 triangles use 13 toothpicks.

Just for fun

Why is the explicit rule everyone's favourite?

It can tell you about term 1000 without having to say hi to terms 1 to 999 first.

Your turn

Find the missing terms: 12, __, 5, __ (linear).

Find an explicit rule for 0.5, 1.25, 2, 2.75, ... and find term 20.

Which term of t = 6n − 5 has the value 115?

Is 50 a term of t = 4n + 3? Justify.

Find a rule and the 6th term: 2, 6, 18, 54, ...

Find the 4 missing terms: −6, __, __, __, __, 9 (linear).

In the L-shape pattern, how many squares are in figure 10?

More practice: warm-up

Next term: 1.2, 1.5, 1.8, ...

Missing term: 10, __, 20 (linear).

Find term 4 of t = 5n − 2.

Write a recursive rule for 9, 7, 5, 3, ...

Next term: 48, 24, 12, ... (halving).

More practice: on your own

Find an explicit rule for 3/8, 1/2, 5/8, ... and find term 14.

For t = −2.5n + 40, which term is 0? Which term is −10?

Find the missing terms: −4.5, __, __, __, 1.5 (linear).

A 750 g jar of honey loses 35 g each day. Write a rule. After how many days will there first be less than 100 g left?

Is 0.4 a term of 4.4, 3.9, 3.4, ...? If so, which one?

Challenge

A pattern has the rule t = 1.5n + k, and term 8 is 9.5. Find k and term 30.

In 1, 4, 9, 16, ... (square numbers), which term is 400? Is 500 a term? Justify.

Pattern A: t = 3n + 7. Pattern B: t = 5n − 13. Is there a term number where they're equal? For which term numbers is B greater? Justify.

Check your understanding

Quiz

What is the missing middle term of 2.6, __, 4.2 (linear)?

Which term of t = 7n + 1 has the value 64?

Which explicit rule matches 5, 3.5, 2, 0.5, ... (term 1 is 5)?

Is 45 a term of t = 2n + 1?

What is the next term of 81, 27, 9, 3, ...?

A recursive rule tells you...

For parents and teachers

Big idea: C1.3 asks Grade 8 students to determine pattern rules, extend patterns, make and justify predictions, find missing elements in patterns with rational numbers, and use algebraic rules to solve for unknown values.

Common difficulties:

Counting blanks instead of jumps when finding missing terms.
Using only recursive thinking (listing every term). Encourage the explicit rule for faraway terms.
Accepting a non-whole term number. If n = 22.2, the value isn't exactly a term. In context, think about what that means (the food drive reaches 300 kg on day 23).
Not justifying. A prediction needs reasons: the rule, the calculation and a check.

How to help: Do the toothpick activity together and ask your child to predict first, then build. Ask "How do you know?" often. This lesson builds A1 skills: reasoning and proving, and perseverance.