C. Algebra · Patterns, Expressions, Equations, Inequalities, Coding and Modelling

Representing Patterns: Tables, Graphs, Expressions and Equations

C1.2 A1About 40 minutes

Learning goals

I can create repeating, growing and shrinking patterns and show them in different ways.
I can show a pattern with a table of values and a graph.
I can write an algebraic expression and an equation for a linear growing pattern.
I can translate a pattern from one representation to another.

Words to know

representation
A way of showing something: pictures, words, a table, a graph, an expression or an equation.

table of values
A table that pairs each term number with its term value.

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

term value (t)
The number in that position, like how many toothpicks are in Figure 3.

algebraic expression
Numbers, variables and operations with no equal sign, like 3n + 1.

equation
A statement with an equal sign, like t = 3n + 1. It says the two sides have the same value.

Remember from Grade 5

In Grade 5 you showed growing and shrinking patterns with tables of values and graphs, and you wrote term-number rules in words, like chairs = term number × 2 + 2.

In Grade 6 you will write those rules the way mathematicians do: as an algebraic expression like 2n + 2 and an equation like t = 2n + 2. You can do this for any linear growing pattern.

Let's start with toothpicks

Leo builds squares in a row with toothpicks. Squares that touch share a toothpick.

Toothpick squares in a row. 1 square uses 4 toothpicks, 2 squares use 7 toothpicks and 3 squares use 10 toothpicks.

Representation 1: Words

Start at 4 and add 3 each time. Each new square needs only 3 more toothpicks, because it shares a side with the square before it. The jump is always 3, so the pattern is linear.

Representation 2: A table of values

Term number (n): squares	Term value (t): toothpicks
1	4
2	7
3	10
4	13
5	16

A table of values for the toothpick squares.

Representation 3: A graph

The term number goes across. The term value goes up. We don't join the dots, because there is no such thing as 2½ squares in this pattern.

A graph of the toothpick pattern. The points (1, 4), (2, 7), (3, 10), (4, 13) and (5, 16) rise in a straight line.

Representation 4: An expression and an equation

Here's how to find the rule for a linear growing pattern:

Find the jump. Here it is 3. So the rule starts with 3 × n, written 3n.
Compare. For n = 1, 3n is 3, but the real value is 4. For n = 2, 3n is 6, but the real value is 7. The real value is always 1 more.
Write the rule. The expression is 3n + 1. The equation is t = 3n + 1 (the term value equals 3 times the term number, plus 1).
Check with another term. n = 5: 3 × 5 + 1 = 16 ✔ (the table says 16).

It matches the picture too: there is 1 toothpick at the left end, and every square adds 3 more.

Writing multiplication in algebra

3n means 3 × n. The number is written in front of the letter, and the × sign is left out. So 3n + 1 means multiply n by 3, then add 1.

Use any letter you like for the variable, as long as you say what it stands for. In this unit, n is usually the term number and t the term value.

Representation	The toothpick pattern
Words	Start at 4 and add 3 each time
Table	1 → 4, 2 → 7, 3 → 10, 4 → 13, ...
Graph	Points rising in a straight line, 3 units up for each step across
Expression	3n + 1
Equation	t = 3n + 1

Five ways to show the same pattern. Translating means changing from one representation to another.

Worked example

Example 1: From an equation to a table and a graph

Make a table of values for t = 5n + 2 for n = 1 to 5. Describe its graph.

n = 1: 5 × 1 + 2 = 7. n = 2: 5 × 2 + 2 = 12. n = 3: 5 × 3 + 2 = 17.
n = 4: 5 × 4 + 2 = 22. n = 5: 5 × 5 + 2 = 27.
Each value is 5 more than the one before, so the graph points go up 5 for every 1 across.

Table: 1 → 7, 2 → 12, 3 → 17, 4 → 22, 5 → 27. The graph is 5 points rising in a straight line.

Worked example

Example 2: From a graph to an equation

A graph shows the points (1, 6), (2, 10), (3, 14), (4, 18). Write an expression and an equation for the pattern.

The term values are 6, 10, 14, 18. The jump is +4 every time, so the pattern is linear and the rule starts with 4n.
Compare: for n = 1, 4n = 4, but the value is 6. That's 2 more.
Check n = 3: 4 × 3 + 2 = 14 ✔.

Expression: 4n + 2. Equation: t = 4n + 2.

Worked example

Example 3: A shrinking pattern in a table and a graph

A bag has 60 hockey pucks for practice. The coach hands out 8 pucks to each team that arrives. Make a table for the first 5 terms and describe the graph.

Term 1 is 60 (the full bag). Then subtract 8 each time: 60, 52, 44, 36, 28.
Table: 1 → 60, 2 → 52, 3 → 44, 4 → 36, 5 → 28.
On the graph, each point is 8 units lower than the one before.

The points go down in a straight line, because the same amount (8) is taken away each time. (Grade 6 writes expressions only for linear growing patterns, so a table and graph are enough here.)

Worked example

Example 4: Translating a repeating pattern

Show the pattern 🔺🔵🔵🔺🔵🔵🔺🔵🔵 in three other ways.

Letters: the core is one triangle then two circles, so write ABB ABB ABB.
Actions: clap, snap, snap, clap, snap, snap, clap, snap, snap.
Table: term numbers 1, 4 and 7 are 🔺. Term numbers 2, 3, 5, 6, 8 and 9 are 🔵.

All of these show the same core of 3 elements (one, then two the same), so they are the same pattern in different representations.

Worked example

Example 5: From words to every representation

A snow fort has 5 blocks on day 1. Each day the builders add 4 blocks. Show the pattern as a table, an expression and an equation.

Table: 1 → 5, 2 → 9, 3 → 13, 4 → 17.
The jump is 4, so start with 4n. For n = 1, 4n = 4, and the value is 5: that's 1 more.
Check n = 4: 4 × 4 + 1 = 17 ✔.

Expression: 4n + 1. Equation: t = 4n + 1, where n is the day and t is the number of blocks.

Check your rule

The number in front of n is the jump.
The number you add (or subtract) is the fix-up that makes term 1 correct.
Always test your equation on two rows of the table. If one doesn't match, change the fix-up and try again. That is good math, not a failure!

Just for fun

Riddle: Why do Leo's toothpick squares never argue?

Because every new square shares a side with the one before it. (That's why each one needs only 3 more toothpicks!)

Hands-on activity

Toothpick houses

You need: about 40 toothpicks (or cotton swabs or uncooked spaghetti pieces), grid paper, a pencil.

Build a row of houses. Each house is a square with a two-toothpick roof on top. Houses that touch share a wall.
Build 1, 2, 3 and 4 houses. Count the toothpicks each time and make a table of values.
Graph your table on grid paper.
Find the jump and write an expression and an equation for the pattern.
Use your equation to predict how many toothpicks 10 houses need. Build it if you have enough toothpicks!

What to look for: 6, 11, 16, 21 toothpicks; the equation t = 5n + 1; for 10 houses, 5 × 10 + 1 = 51 toothpicks.

Grown-up note: toothpicks have sharp points. Use cotton swabs or spaghetti with younger children nearby.

Your turn

Make a table of values for t = 2n + 3, for n = 1 to 4.

Write an expression for the pattern 5, 9, 13, 17, ...

Write an equation for 10, 20, 30, 40, ...

A graph shows the points (1, 3), (2, 8), (3, 13). Write an equation for the pattern.

Make a table for a shrinking pattern that starts at 45 and subtracts 5 each time, for terms 1 to 5.

Translate 🍁🍁❄️🍁🍁❄️ into letters and into actions.

Write an equation for: start at 6 and add 6 each time.

More practice: warm-up

What does 4n mean?

Use t = 3n + 1 to find t when n = 10.

A table shows 1 → 5, 2 → 10, 3 → 15. What is the next row?

In t = 2n + 7, which letter stands for the term number?

On a pattern graph, does the term number go across or up?

More practice: on your own

Write an expression for 9, 15, 21, 27, ...

Make a table for t = 7n − 4, for n = 1 to 4.

Which equation matches 8, 11, 14, 17: t = 3n + 5, t = 5n + 3 or t = n + 7?

Sasha writes t = n + 5 for the pattern 2, 7, 12, 17. Is she right? If not, fix it.

A snow fort has 12 blocks on day 1 and gains 6 blocks each day. Write an equation. How many blocks on day 5?

Challenge

Explain where each part of t = 5n + 1 comes from in the toothpick houses.

A table shows only some rows: 2 → 11, 4 → 19, 6 → 27. Write an equation for the linear pattern.

Pattern A is t = 2n + 10. Pattern B is t = 4n + 2. For which term number do they have the same value? What happens after that?

Check your understanding

Quiz

Which expression matches 4, 7, 10, 13, ...?

For t = 2n + 5, what is t when n = 3?

Which pattern matches t = 6n?

On a graph, the points go down in a straight line. The pattern is ...

In t = 4n + 3, what is the jump from one term to the next?

Which equation means start at 9 and add 2 each time?

For parents and teachers

Big idea: Grade 6 students create and translate repeating, growing and shrinking patterns using tables of values, graphs, and, for linear growing patterns, algebraic expressions and equations (C1.2). Writing a rule like t = 3n + 1 is the new step this year.

Common difficulties:

Using the first term as the number in front of n (writing t = 4n + 3 for 4, 7, 10). The number in front of n is the jump.
Forgetting to check. Have your child test the equation on two rows of the table.
Joining the dots on a graph. For figures like squares or houses, we leave the points unjoined because there is no Figure 2½.

How to help: Build patterns with toothpicks, cotton swabs, LEGO or coins, and ask for the table, the graph and the equation. Then reverse it: give an equation like t = 2n + 1 and ask your child to build Figures 1 to 3. Mathematical processes (A1) are woven in: your child represents, connects and checks their thinking.