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

Creating and Translating Patterns

C1.2 A1About 45 minutes

Learning goals

I can create repeating, growing and shrinking patterns with whole numbers and decimals.
I can show the same pattern with objects or pictures, words, a table, a graph, an expression and an equation.
I can translate a pattern from one representation to another.

Words to know

representation
A way to show a pattern: objects, a picture, words, a table of values, a graph, an expression or an equation.

translate
To change a pattern from one representation to another, for example from a picture to an equation.

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

algebraic expression
A combination of numbers, variables and operations, like 2n + 2. It has no equal sign.

equation
A statement that two things are equal, like c = 2n + 2.

variable
A letter, like n or c, that stands for a number that can change.

Remember from Grade 6 and the last lesson

In Grade 6 you showed patterns with tables of values, graphs, expressions and equations. In the last lesson you learned that a linear pattern has a constant rate and an initial value.

Today you'll create your own patterns with whole numbers and decimals, and translate them: change one representation into another. Each representation shows the pattern in a different way, so being able to switch between them is a powerful skill.

Let's start with a party

The class is setting up square tables in a row for a Family Day lunch. One chair goes on each side of a table that is open. How many chairs do you need for 25 tables? You could draw 25 tables... or you could find the rule.

Tables in a row. 1 table has 4 chairs, 2 tables have 6 chairs and 3 tables have 8 chairs.

Six ways to show the same pattern

Objects or pictures: the drawing above.
Words: There is always one chair at each end (2 chairs). Each table adds one chair on top and one on the bottom (2 more).
Table of values: see below.
Graph: points in a straight line.
Expression: 2n + 2, where n is the number of tables.
Equation: c = 2n + 2, where c is the number of chairs.

For 25 tables: c = 2 × 25 + 2 = 52 chairs.

tables (n)	1	2	3	4	5
chairs (c)	4	6	8	10	12

The chairs go up by 2 each time. Going back one jump from 4 gives the initial value 2 (the two end chairs).

Graph of c = 2n + 2. The points line up in a straight line. Going back to n = 0 gives 2, the two end chairs.

Where is the rule hiding?

Notice how the same two numbers show up everywhere:

In the picture: 2 end chairs, and 2 more chairs for each table.
In the table: the values go up by 2, and going back one step gives 2.
In the graph: the points rise by 2 for each step, and the line starts at 2.
In the equation: c = 2n + 2.

Patterns that are not linear can also be shown in a table or graph. Their points make a curve, not a straight line. And repeating patterns can be translated too: clap, snap, snap can be written as letters A B B or numbers 1 2 2.

Start from what you know

To translate a pattern into an equation, find the constant rate (the jump) and the initial value (term 0). To translate an equation into a table, substitute n = 0, 1, 2, 3, ... and calculate.

Worked example

Example 1: From an equation to other representations

Show the pattern t = 2.5n + 1 with a table (n = 0 to 4), words and a graph description.

Table: n = 0 gives 1. n = 1 gives 3.5. n = 2 gives 6. n = 3 gives 8.5. n = 4 gives 11.
Words: start at 1 and add 2.5 each term.
Graph: the first point is (0, 1) on the vertical axis. Each step right, the points rise 2.5. They line up in a straight line.

Table: 1, 3.5, 6, 8.5, 11. Words: start at 1, add 2.5. Graph: straight line, starting at 1, rising 2.5 per step.

Worked example

Example 2: A shrinking pattern from words

A camping water jug holds 20 L. A family uses 1.5 L each hour. Translate this into a table (h = 0 to 4) and an equation.

The initial value is 20 (before any hours pass). The pattern decreases by 1.5 each hour.
Table: h = 0: 20. h = 1: 18.5. h = 2: 17. h = 3: 15.5. h = 4: 14.
Equation: w = 20 − 1.5h, where w is the litres of water left and h is the number of hours.

w = 20 − 1.5h. Its graph goes down in a straight line.

Worked example

Example 3: Translating a repeating pattern

Translate stomp, clap, clap, stomp, clap, clap, ... into letters, numbers and pictures.

The core is stomp, clap, clap. The first action is A and the new action is B.
Letters: A B B A B B ... Numbers: 1 2 2 1 2 2 ... Pictures: 🍁 ❄️ ❄️ 🍁 ❄️ ❄️ ...
All four show the same structure: one thing, then two of another thing.

A B B (core of 3). Any representation with the same structure is a correct translation.

Worked example

Example 4: Create a pattern to match

Create a linear pattern with a constant rate of 0.25 and an initial value of 3. Give a real-life story for it.

Equation: t = 0.25n + 3.
Table: n = 0: 3. n = 1: 3.25. n = 2: 3.5. n = 3: 3.75. n = 4: 4.
Story: A bus ride costs $3.00 to get on plus $0.25 for each stop you ride. (A made-up fare, just for practice.)

3, 3.25, 3.5, 3.75, 4, ... One story: a $3 fare plus $0.25 for each stop.

Just for fun

Translating a pattern is like changing outfits: clap, snap, snap can dress up as A B B, or as 1 2 2, or as 🍁 ❄️ ❄️. Same pattern underneath, different clothes on top!

Hands-on activity

Build it, then translate it

You need: 6 sticky notes or square pieces of paper (the tables), about 20 coins or buttons (the chairs), paper and a pencil. Keep small objects away from young children.

Build the table-and-chair pattern for 1, 2, 3 and 4 tables. Count the chairs each time.
Now create your own pattern with the same materials. For example: tables in a row, but two chairs at each end.
Show your pattern in all six ways: build it, draw it, describe it in words, make a table, sketch a graph on paper, and write the equation.
Give a grown-up only your table. Can they find your equation? Then give them only your equation. Can they build figure 3?

(For two chairs at each end: c = 2n + 4.)

Your turn

Make a table of values for t = 3n + 2 for n = 0 to 4.

A table has n = 0, 1, 2, 3 and t = 6, 8.5, 11, 13.5. Write the equation.

Describe t = 0.5n + 4 in words.

Toothpick triangles in a row: 1 triangle uses 3 toothpicks, 2 triangles use 5, 3 triangles use 7. Write the equation. How many toothpicks for 10 triangles?

Translate jump, jump, spin, jump, jump, spin, ... into letters.

A 50 cm ribbon has 2.5 cm pieces cut off it, one at a time. Make a table for 0 to 3 cuts and write the equation.

Describe the graph of t = 4n + 3.

More practice: warm-up

In t = 5n + 1, what is t when n = 2?

Describe t = n + 10 in words.

Translate 🍁 ❄️ 🍁 ❄️ 🍁 ❄️ into letters.

Complete the table for t = 2n: n = 0, 1, 2, 3.

Write the equation for a linear pattern with initial value 7 and constant rate 3.

More practice: on your own

A class orders pizzas for $12.50 each plus a $5 delivery fee (made-up prices). Write an equation for the cost c of p pizzas. Make a table for 1 to 4 pizzas.

The terms for n = 1, 2, 3, 4 are 0.3, 0.55, 0.8, 1.05. Write the equation.

A graph has the points (0, 12), (1, 10.5), (2, 9), (3, 7.5). Describe the pattern in words and write the equation.

Create a table (n = 0 to 4) for a pattern with initial value 1.2 and constant rate 0.6.

The square numbers 1, 4, 9, 16, ... make a pattern. What is the next term? Would its graph be a straight line?

Challenge

Write a real-life story that matches t = 4n + 6. Then make its table for n = 0 to 3.

A tablet battery shows 100% at the start, 92.5% after 1 hour and 85% after 2 hours. Write the equation. If the pattern continues, after how many hours will it show 25%?

Sam's table for t = 3n + 1 (n = 0 to 3) reads 1, 4, 7, 11. Find his mistake and explain how the graph would show it.

Check your understanding

Quiz

Which equation matches the table n: 0, 1, 2, 3 and t: 5, 9, 13, 17?

What is term 4 of t = 1.5n + 2?

Which words describe t = 30 − 2n?

Which is a correct translation of red, blue, blue, red, blue, blue, ...?

Toothpick squares: 1 square uses 4, 2 squares use 7, 3 squares use 10. Which equation fits?

The graph of a pattern is a curve, not a straight line. What do you know?

For parents and teachers

Big idea: Grade 7 students create and translate repeating, growing and shrinking patterns involving whole numbers and decimals, using various representations, including algebraic expressions and equations for linear growing patterns (C1.2). Being able to move between a picture, words, a table, a graph and an equation helps them see that it's all the same pattern.

Common difficulties:

Using term 1 instead of term 0 for the number added in the equation. For 4, 6, 8 chairs, the equation is c = 2n + 2, not c = 2n + 4.
Mixing up the constant rate and the initial value (writing t = 2n + 2.5 instead of t = 2.5n + 2). Ask: "Which number is the jump?"
Decimal place value when adding 0.25 or 1.5 repeatedly. Encourage lining up the decimal points.

How to help: Do the build-and-translate activity together, and swap roles: you give a table, your child finds the equation. Ask your child to make up stories for equations. This lesson builds A1 skills: representing ideas in different ways and communicating them clearly.