C. Algebra · Patterns, Equations, Coding and Modelling

Creating and Translating Patterns: Shapes, Numbers and Tables

C1.2 A1About 40 minutes

Learning goals

I can create repeating patterns and patterns that grow or shrink by a repeating operation.
I can translate a pattern: show the same pattern with shapes, letters, actions or numbers.
I can show a growing or shrinking pattern in a table of values.
I can make a picture of a pattern from its table of values.

Words to know

translate a pattern
Show the same pattern in a different way, like turning shapes into letters or numbers.

representation
A way of showing math: with objects, pictures, numbers, words or a table.

table of values
A table with two columns. One column tells the position in the pattern, and the other tells the amount at that position.

term
One step or position in a pattern. Figure 1 is the 1st term, Figure 2 is the 2nd term, and so on.

figure
One picture in a picture pattern.

Remember from Grade 2

In Grade 2 you translated patterns. You turned a shape pattern into a number pattern, and a number pattern into shapes.

This year you will also use a new math tool called a table of values. Scientists, builders and computer coders use tables like this to keep track of patterns.

One pattern, many ways

The same pattern can be shown in many ways. Each way is a representation. Look at this pattern of actions:

clap, stomp, stomp, clap, stomp, stomp, …

The core is clap, stomp, stomp. Here is the very same pattern shown in other ways.

Representation	The pattern
Actions	clap, stomp, stomp, clap, stomp, stomp
Letters	A B B A B B
Shapes	▲ ■ ■ ▲ ■ ■
Numbers	1 2 2 1 2 2
Emoji	👏 🦶 🦶 👏 🦶 🦶

All five rows show an ABB pattern. Only the elements change. The structure stays the same.

Patterns of movement can be translated too. An arrow turning a quarter turn to the right each time points up, right, down, left, up, right, down, left. In letters, that is A B C D A B C D. With your body: face the window, then the door, then the wall, then the closet, and repeat!

Growing patterns and tables of values

Look at this growing pattern made of squares.

Figure 1 has 3 squares. Figure 2 has 5 squares. Figure 3 has 7 squares. Each figure has 2 more squares than the one before.

We can put this pattern in a table of values. The left column is the figure number (which term it is). The right column is the number of squares in that figure.

Figure number	Number of squares
1	3
2	5
3	7
4	9
5	11

A table of values for the square pattern. Going down the right column, add 2 each time.

Reading down the right column, the operation add 2 repeats: 3, 5, 7, 9, 11. We extended the table to Figures 4 and 5 without drawing them!

The pattern rule is: Start at 3. Add 2 each time.

Two columns, two jobs

The left column counts the terms: 1, 2, 3, 4, … It always goes up by 1.
The right column shows the amount in each term. That is where you look for the repeating operation.

Just for fun

What if the ABB pattern were a dance? Clap, stomp, stomp, clap, stomp, stomp... Try it fast. The whole house will know you are learning about patterns!

Worked example

Example 1: Translate a repeating pattern

Translate 🐻 🐻 🐰 🐻 🐻 🐰 into letters, actions and numbers.

Find the core: 🐻 🐻 🐰.
Give each different element a letter: 🐻 = A, 🐰 = B. So the core is A A B.
Actions: use one action for A and a different action for B. For example, stomp, stomp, jump.
Numbers: 🐻 = 1, 🐰 = 2. So the core is 1 1 2.

A A B A A B; stomp, stomp, jump, stomp, stomp, jump; 1 1 2 1 1 2. They all show the same AAB pattern.

Worked example

Example 2: Create a pattern from a rule

Leo has 10¢ in his piggy bank. Each day he adds a nickel (5¢). Make a table of values for the first 5 days.

Start: after Day 1 he has 10¢ + 5¢ = 15¢. (We count the nickel he puts in on Day 1.)
Each day, the operation add 5¢ repeats.
Day 2: 15¢ + 5¢ = 20¢. Day 3: 25¢. Day 4: 30¢. Day 5: 35¢.

Day 1: 15¢, Day 2: 20¢, Day 3: 25¢, Day 4: 30¢, Day 5: 35¢. The rule: start at 15¢, add 5¢ each day.

Day	Money in the piggy bank
1	15¢
2	20¢
3	25¢
4	30¢
5	35¢

Leo's table of values.

Leo's savings as a bar graph. Each bar is 5¢ taller than the one before, because add 5¢ repeats every day.

Worked example

Example 3: A shrinking pattern in a table

A box has 24 granola bars. The family eats 4 bars each day. Make a table of values. On which day is the box empty?

Day 1: 24 − 4 = 20 bars left.
Day 2: 20 − 4 = 16. Day 3: 16 − 4 = 12. Day 4: 12 − 4 = 8. Day 5: 8 − 4 = 4. Day 6: 4 − 4 = 0.
The operation subtract 4 repeats in the right column.

The box is empty at the end of Day 6. Bars left: 20, 16, 12, 8, 4, 0.

Worked example

Example 4: From a table to a picture

Draw the first 3 figures for this table: Figure 1 → 2 circles, Figure 2 → 4 circles, Figure 3 → 6 circles.

Figure 1: draw 2 circles.
Figure 2: draw 4 circles. That is 2 more.
Figure 3: draw 6 circles. That is 2 more again.
A neat way: draw each figure as a column of 2 circles, then 2 columns, then 3 columns.

Figure 1: ⚪⚪; Figure 2: ⚪⚪⚪⚪; Figure 3: ⚪⚪⚪⚪⚪⚪. The pattern starts at 2 and adds 2 each time.

Your turn

Translate ■ ● ● ■ ● ● into letters.

Translate A B C A B C into numbers.

Make a table of values: Figure 1 has 4 squares, and each figure has 4 more squares than the one before. Show Figures 1 to 4.

In the square pattern in this lesson (3, 5, 7, 9, 11), how many squares are in Figure 6?

Create a growing pattern with the rule Start at 1. Add 3. Write the first 5 numbers.

A table shows: Day 1 → 50, Day 2 → 40, Day 3 → 30. What operation repeats? What is the value for Day 4?

Show the pattern up, down, up, down with actions.

Hands-on activity

Build it, then table it

You need: about 30 small blocks (LEGO, cubes, dry pasta or coins), paper and a pencil.

Build Figure 1 with 2 blocks.
Build Figure 2 with 2 more blocks than Figure 1 (4 blocks). Build Figure 3 with 2 more again.
Draw a table of values with two columns: Figure number and Number of blocks. Fill it in for Figures 1, 2 and 3.
Without building, use the table to find the number of blocks in Figures 4 and 5. Then build them to check!
Now make up your own rule (like start at 1, add 3) and ask a grown-up to build it from your table.

Math talk: "Where in the table can you see the operation that repeats?"

Grown-up note: keep small items away from younger children who might put them in their mouths.

More practice: warm-up

Translate 🍎 🍌 🍎 🍌 into letters.

In a table of values, what does the left column usually go up by?

Rule: Start at 10. Add 10. Write the first 4 numbers.

Rule: Start at 20. Subtract 2. Write the first 4 numbers.

Translate 1 1 2 1 1 2 into shapes.

More practice: on your own

Make a table of values for spiders' legs: 1 spider has 8 legs. Show 1 to 4 spiders. What operation repeats?

A table shows Figure 1 → 5, Figure 2 → 8, Figure 3 → 11. Write the pattern rule and find Figure 5.

Translate the movement pattern slide right, slide right, jump into letters and into numbers.

Kai reads 15 pages each night. Make a table of values for the total pages read after 1, 2, 3 and 4 nights.

Two patterns: ■ ▲ ■ ▲ and 2 7 2 7. Do they have the same structure? Explain.

Challenge

A table of values has these numbers in the right column: 6, 12, 24, 48. What operation repeats? What is the next value?

Sam's table: Figure 1 → 7, Figure 2 → 10, Figure 3 → 14, Figure 4 → 16. One number does not fit. Which one? Fix it.

Create your own growing pattern that has 20 in Figure 4. Show it in a table.

Check your understanding

Quiz

Which shows ■ ★ ★ ■ ★ ★ in letters?

In a table of values, Figure 1 → 6, Figure 2 → 9, Figure 3 → 12. What is Figure 4?

What does the left column in a table of values usually show?

Rule: Start at 30. Subtract 5. Which numbers follow this rule?

Which pattern has the same structure as 1 2 2 1 2 2?

A tricycle has 3 wheels. In a table for 1, 2, 3, 4 tricycles, which values are right?

For parents and teachers

Big idea: A pattern's structure can be shown in many ways (C1.2). Grade 3 adds the table of values, a two-column table that pairs each term number with its value. It is an early step toward graphing and algebra in later grades.

Common difficulties:

Looking for the repeating operation in the wrong column. The left column always goes up by 1. The pattern's operation is in the right column.
Off-by-one errors. In Leo's piggy bank example, children may start the table at 10¢. Talk about what "after Day 1" means.
Translating with mixed-up letters. Each different element gets its own letter, and the same element always keeps its letter.

How to help: Make tables of values together from daily life: allowance saved each week, pages read each night, chairs and legs. Let your child build patterns with LEGO or pasta and record them in a table.