C. Algebra · Algebra: Expressions, Equations and Coding

Generalizing Patterns with Algebraic Expressions

C1.2 C1.3 A1About 70 minutes

Learning goals

I can describe a growing or shrinking pattern with an algebraic expression for figure n.
I can use the constant difference and the starting value (figure 0) to write the expression for a linear pattern.
I can see a pattern in more than one way and write a different expression for each way.
I can recognize some non-linear patterns, such as n² + 1.

Words to know

generalize
To write one rule or expression that works for every case, not just the ones you checked.

figure number (n)
The position of a figure or term in a pattern: 1st, 2nd, 3rd, ...

first differences
The amounts you add (or subtract) to get from one term to the next. If they are all the same, the pattern is linear.

linear pattern
A pattern that grows or shrinks by the same amount each step. Its expression looks like an + b.

non-linear pattern
A pattern whose first differences are not all the same, such as 1, 4, 9, 16.

equivalent expressions
Expressions that give the same value for every value of the variable, such as 4n + 4 and 4(n + 1).

Remember from Grade 8

In Grade 8 you described linear patterns by their constant rate and initial value, used tables and graphs, and solved for missing terms. In Grade 9 you will turn patterns into expressions quickly, see one pattern in several ways, and decide whether different expressions really describe the same thing.

Let's start with toothpicks

These figures are made of toothpicks. How many toothpicks would figure 50 need? Nobody wants to build it to find out!

Each new square adds 3 toothpicks (a top, a bottom and one new side). Figure 1 has 4, figure 2 has 7, figure 3 has 10.

Figure number n	0	1	2	3	4
Toothpicks	1	4	7	10	13

The first differences are all +3. Working back one step from figure 1 gives “figure 0” = 1 (the single toothpick on the left).

From a table to an expression

For a linear pattern:

Find the first difference. That is the coefficient of n. Here it is 3, so the expression starts 3n.
Find the value at figure 0 (go back one step from figure 1). That is the constant. Here it is 4 − 3 = 1.
Write the expression: 3n + 1.
Check with a figure you know: figure 3 → 3(3) + 1 = 10 ✓.

Now figure 50 is easy: 3(50) + 1 = 151 toothpicks.

One pattern, many ways of seeing

People often see a pattern differently, and that's fine. Each way of seeing gives an expression.

“Figure 1 has 4. Each of the other (n − 1) squares adds 3.” → 4 + 3(n − 1)
“There are n tops, n bottoms and (n + 1) upright sticks.” → n + n + (n + 1)
“Each square uses 3, plus 1 extra at the start.” → 3n + 1

All three simplify to 3n + 1. Expressions like these that always give the same value are called equivalent. You'll learn to test and prove equivalence in the next lesson.

A square pool n tiles wide (here n = 3) has a border of tiles. Each way of seeing the border gives a different-looking expression, but all four give 16 when n = 3.

Worked example

Example 1: The toothpick pattern

Use the toothpick pattern (3n + 1). (a) How many toothpicks are in figure 50? (b) Which figure uses exactly 100 toothpicks?

(a) Substitute n = 50: 3(50) + 1 = 150 + 1 = 151.
(b) We need 3n + 1 = 100. Subtract 1: 3n = 99. Divide by 3: n = 33.
Check: 3(33) + 1 = 99 + 1 = 100 ✓.

(a) 151 toothpicks (b) figure 33

Worked example

Example 2: Test four expressions

Check that all four pool-border expressions give the same number of tiles when n = 3 and when n = 10.

n = 3: 4(3) + 4 = 16; 4(3 + 1) = 16; 2(3 + 2) + 2(3) = 10 + 6 = 16; (3 + 2)² − 3² = 25 − 9 = 16.
n = 10: 4(10) + 4 = 44; 4(11) = 44; 2(12) + 2(10) = 24 + 20 = 44; 12² − 10² = 144 − 100 = 44.
All four agree both times. That suggests they are equivalent. (The first three simplify to 4n + 4. You'll prove the fourth after you learn to multiply binomials.)

All give 16 when n = 3 and 44 when n = 10.

Worked example

Example 3: A shrinking pattern from a table

A snowbank is measured each morning during a warm spell. Write an expression for its height on day n.

Day 1: 84 cm. Day 2: 78 cm. Day 3: 72 cm. Day 4: 66 cm.

First differences: 78 − 84 = −6, and the same each day. The coefficient is −6.
Day 0: go back one step from day 1: 84 + 6 = 90.
Expression: 90 − 6n (or −6n + 90).
Check day 4: 90 − 6(4) = 90 − 24 = 66 ✓.

Height = 90 − 6n cm

Worked example

Example 4: A non-linear pattern

Find an expression for the pattern 2, 5, 10, 17, 26, ...

First differences: 3, 5, 7, 9. They are not constant, so the pattern is non-linear.
Compare with the square numbers 1, 4, 9, 16, 25 (that is, n²). Each term is exactly 1 more.
Expression: n² + 1. Check n = 4: 16 + 1 = 17 ✓.

n² + 1

Square tables pushed together in a row. Each table adds 2 chairs (one above it and one below it), and there is always 1 chair at each end: 2n + 2.

Avoid the figure-1 trap

Always test your expression on at least two figures you already know. One match can happen by luck; two or three matches make you much more confident. And remember: the constant is the value at figure 0, not figure 1.

Hands-on activity

Build and generalize

You need: toothpicks, dry spaghetti broken into equal pieces, or coins; paper and pencil.

Build triangles in a row that share sides: figure 1 is one triangle (3 pieces), figure 2 is two triangles sharing a side, and so on. Build figures 1 to 4.
Make a table of figure number and number of pieces.
Write an expression for figure n. Describe how you see the pattern.
Predict figure 10 with your expression, then build it to check.
Ask someone at home how they see the pattern. Write their expression too and check that it gives the same values.

Model answer: 3, 5, 7, 9 pieces → difference 2, figure 0 = 1 → 2n + 1. Figure 10 needs 21 pieces. Another way of seeing it: 3 + 2(n − 1), which also gives 21.

Just for fun

A pattern walks into a café. Then a slightly bigger pattern walks in. Then an even bigger one.

The server sighs: “I can see where this is going.”

Your turn

Write an expression for the pattern 6, 10, 14, 18, ... and find term 20.

Triangles in a row made of toothpicks use 3, 5, 7, ... toothpicks. Write an expression and find figure 25.

Square tables in a row seat 4, 6, 8, ... people (see the tables picture above). How many tables are needed for 30 people?

Write an expression for this table: n = 1, 2, 3, 4 gives 11, 8, 5, 2.

Is 1, 4, 9, 16, ... linear? Write an expression for it.

Ana writes 4 + 3(n − 1) for the toothpick pattern and Ben writes 3n + 1. Show that they are the same.

A pattern has expression 5n − 3. Write figures 1, 2 and 3.

More practice: warm-up

Write an expression for 3, 6, 9, 12, ...

Write an expression for 5, 7, 9, 11, ...

What is the figure-0 value of 8, 13, 18, ...?

Evaluate 10n + 5 for n = 7.

What is the constant first difference of 20, 17, 14, 11, ...?

More practice: on your own

A fence has n sections. Each section has 3 rails, and there is a post at each end of every section (neighbouring sections share a post). Write an expression for the total number of pieces (posts plus rails) and evaluate it for 12 sections.

Write an expression for this table: n = 0, 1, 2, 3 gives −4, −1, 2, 5.

A rectangular pool is n tiles long and 2 tiles wide. Write an expression for the number of border tiles around it. Check with n = 1.

A 120 cm snowbank melts 8 cm per day. Write an expression for its height after d days, and find when it is gone.

Write an expression for 3, 6, 11, 18, 27, ... and find term 10.

Challenge

An “L” pattern of square tiles: figure n has a row of n tiles and a column of n tiles that share the corner tile (figure 1 is a single tile). Write two different expressions and show they are equivalent.

Dot rectangles: figure n is a rectangle of dots n wide and n + 1 tall. Write an expression, list figures 1 to 4, and find which figure has 110 dots.

Explain why the constant first difference of a linear pattern becomes the coefficient of n.

Check your understanding

Quiz

Which expression describes 7, 11, 15, 19, ...?

The toothpick pattern is 3n + 1. How many toothpicks are in figure 10?

Which pattern is linear?

A square pool is n tiles wide. Which expression does NOT give the number of border tiles?

Tables in a row seat 2n + 2 people. How many tables are needed to seat 22 people?

In the expression 5n + 2 for a linear pattern, what does the 2 tell you?

For parents and teachers

Big idea: C1.2 asks students to create expressions that generalize relationships given in words, numbers and pictures, and C1.3 asks them to compare expressions to find equivalent ones. Pattern work does both: different ways of seeing the same pattern give different but equivalent expressions.

Common difficulties:

Using figure 1 as the constant. For 6, 10, 14 the expression is 4n + 2, not 4n + 6.
Assuming every pattern is linear. Check the first differences.
Testing only one figure. Encourage two or three checks.

How to help: Build a pattern with coins or toothpicks and ask “How do you see it growing?” Then share how you see it. Writing both expressions and checking that they match is exactly what this lesson is about.