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

Growing and Shrinking Patterns in Tables and Graphs

C1.2 A1About 40 minutes

Learning goals

I can make a growing or shrinking pattern from a rule.
I can show a pattern with pictures, a table of values and a graph.
I can translate a pattern from one representation to another.
I can tell from a graph whether a pattern grows or shrinks.

Words to know

representation
A way to show a pattern: pictures, words, numbers, a table or a graph.

translate
Change a pattern from one representation to another, like from a table to a graph.

table of values
A table with term numbers in one column and term values in the other.

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

term value
The number in that position, like the number of toothpicks in Figure 3.

graph
A picture of a pattern with points. The term number goes across and the term value goes up.

Remember from Grade 4

In Grade 4 you showed growing patterns with tables of values and graphs. The term number goes across the bottom of the graph, and the term value goes up the side. We leave the points unjoined, because there is no Figure 2½.

In Grade 5 you will also show shrinking patterns, and you will translate any pattern from pictures to a table to a graph and back again.

Let's start with toothpicks

Ava builds a row of triangles with toothpicks. Neighbouring triangles share a toothpick.

Toothpick triangles in a row. 1 triangle uses 3 toothpicks, 2 triangles use 5 toothpicks and 3 triangles use 7 toothpicks.

Each new triangle needs only 2 more toothpicks, because it shares one side with the triangle before it. The rule is start at 3 and add 2 each time.

Representation 1: A table of values

Term number (triangles)	Term value (toothpicks)
1	3
2	5
3	7
4	9
5	11

The toothpick pattern in a table. The term values go up by 2.

Representation 2: A graph

For each row of the table, go across to the term number, then up to the term value, and draw a point.

Graph of the toothpick pattern: points at (1, 3), (2, 5), (3, 7), (4, 9) and (5, 11). The points climb in a straight line.

The points go up to the right: it is a growing pattern. They line up in a straight line because the pattern adds the same amount each time.

A shrinking pattern

Kai has 36 hockey cards. He gives 4 cards to friends at the end of each week. How many does he have left after each week?

Week (term number)	Cards left (term value)
1	32
2	28
3	24
4	20
5	16

Kai's cards: start at 32 and subtract 4 each week.

Graph of Kai's cards: points at (1, 32), (2, 28), (3, 24), (4, 20) and (5, 16). The points go down in a straight line.

What a graph tells you

Points that go up to the right show a growing pattern.
Points that go down to the right show a shrinking pattern.
Points in a straight line mean the pattern adds or subtracts the same amount each time.
Points that bend upward more and more mean the jumps are getting bigger, like a doubling pattern: 1, 2, 4, 8, 16.

Worked example

Example 1: From a graph to a table and a rule

A graph has points at (1, 5), (2, 9), (3, 13), (4, 17). Make a table, write the rule and find term 6.

Each point is (term number, term value). Table: 1 → 5, 2 → 9, 3 → 13, 4 → 17.
The values go up by 4 each time: 5 + 4 = 9, 9 + 4 = 13, 13 + 4 = 17.
Rule: start at 5 and add 4 each time.
Term 5: 17 + 4 = 21. Term 6: 21 + 4 = 25.

Term 6 is 25. The points go up in a straight line, so it is a growing pattern.

Worked example

Example 2: From a rule to a table and a graph

Make a shrinking pattern with the rule start at 50 and subtract 6 each time. Show 5 terms in a table, and tell what the graph looks like.

Term 1: 50.
Term 2: 50 − 6 = 44. Term 3: 44 − 6 = 38.
Term 4: 38 − 6 = 32. Term 5: 32 − 6 = 26.
Graph: plot (1, 50), (2, 44), (3, 38), (4, 32), (5, 26).

Table: 1 → 50, 2 → 44, 3 → 38, 4 → 32, 5 → 26. The points go down in a straight line.

Worked example

Example 3: From a table to pictures

A tile pattern has this table: 1 → 4, 2 → 8, 3 → 12. Describe what Figure 4 could look like.

The values go up by 4 each time. Figure 4 has 12 + 4 = 16 tiles.
One way to build it: Figure 1 is a row of 4 tiles. Each new figure adds another row of 4.
So Figure 4 is 4 rows of 4 tiles, a 4 by 4 square.

Figure 4 has 16 tiles, for example 4 rows of 4. (Other arrangements with 16 tiles that follow the rule are fine too.)

Just for fun

Engineers love Ava's triangles too. A triangle made of sticks can't be squished out of shape the way a square can, which is why so many bridges and roof frames are built from triangles. Toothpicks: tiny but mighty!

Hands-on activity

Build it, table it, graph it

You need: about 25 toothpicks, cotton swabs or uncooked spaghetti pieces, paper (grid paper if you have it), a pencil and a ruler.

Build Ava's triangle pattern with 1, 2, 3, 4 and 5 triangles. Count the toothpicks each time.
Make a table of values with two columns: triangles and toothpicks.
Draw a graph. Number the bottom 0 to 6 and the side by 2s from 0 to 12. Plot a point for each row of your table.
Now make a shrinking pattern: put 5 triangles together, then take one triangle away at a time. Make a table for 5, 4, 3, 2, 1 triangles. What do the points on a graph do?
Explain to your grown-up how the table and the graph show the same pattern.

What to look for: 3, 5, 7, 9, 11 toothpicks; in step 4 the toothpicks go 11, 9, 7, 5, 3, and the graph points go down.

Grown-up note: toothpick ends are sharp, and spaghetti can snap into small pieces. Supervise younger children.

Your turn

Make a table with 5 terms for the rule start at 4 and add 3 each time.

Make a table with 5 terms for the rule start at 40 and subtract 5 each time.

A table shows 1 → 6, 2 → 12, 3 → 18, 4 → 24. Write the rule and find term 5.

A graph has points at (1, 2), (2, 4), (3, 6), (4, 8). Is the pattern growing or shrinking? Write the rule.

A graph has points at (1, 20), (2, 17), (3, 14), (4, 11). Write the rule and find term 5.

Why don't we join the points on the toothpick graph with a line?

How many toothpicks does Ava need for 6 triangles?

More practice: warm-up

In a table of values, what does the left column usually show?

The points on a graph go down to the right. Is the pattern growing or shrinking?

A table shows 3, 6, 9, 12. What is the next term value?

To plot the point (3, 10), which way do you go first?

Make a table for start at 10 and add 10 each time (4 terms).

More practice: on your own

A table shows 1 → 45, 2 → 41, 3 → 37. Find term 5.

A graph has points at (1, 7), (2, 12), (3, 17). What is the term value at term 5?

Pattern A is 60, 54, 48, 42. Pattern B is 6, 12, 18, 24. Which graph goes down? Explain.

Figure 1 has 2 tiles. Each figure adds 4 tiles. Make a table up to Figure 4.

Kai starts with 36 cards and gives away 4 each week. After which week will he have 0 cards?

Challenge

Pattern A: start at 5 and add 5. Pattern B: start at 35 and subtract 5. At which term number do they have the same value? What would the graphs show?

The pattern 1, 2, 4, 8, 16 doubles each time. Why don't its points make a straight line on a graph?

A growing pattern adds the same amount each time. Its graph has points at (2, 11) and (4, 19). Find term 1 and term 3.

Check your understanding

Quiz

A table shows 1 → 8, 2 → 13, 3 → 18. What is the rule?

The points on a graph go down in a straight line. What kind of pattern is it?

What is term 4 of start at 30 and subtract 7?

Which point belongs on the graph of 3, 6, 9, 12?

Ava's toothpick triangles use 3, 5, 7, 9, ... toothpicks. How many for 7 triangles?

Why are the points on a pattern graph usually not joined?

For parents and teachers

Big idea: Grade 5 students create and translate growing and shrinking patterns using several representations, including tables of values and graphs (C1.2). Translating means moving between pictures, words, a table and a graph and seeing that they all show the same pattern.

Common difficulties:

Plotting points backwards (up first, then across). Say "across the hall, then up the stairs."
Uneven scales. If the side of the graph counts by 5s, every line must be 5 more than the one below.
Starting a shrinking table at the wrong number. Read the story carefully: is term 1 the starting amount, or the amount after one week?

How to help: Give your child grid paper and let them build, table and graph a pattern from toothpicks, coins or blocks. Ask "How can you tell from the graph that the pattern is shrinking?" Social-emotional learning (A1): praise neat, careful work and patience with plotting, which takes practice.