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

Showing Patterns with Tables of Values and Graphs

C1.2 A1About 40 minutes

Learning goals

I can create repeating and growing patterns.
I can show a growing pattern with pictures, a table of values and a graph.
I can translate a pattern from one representation to another.
I can read a graph of a pattern and describe the pattern.

Words to know

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

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

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

term value
The number of things in a term, like the number of chairs in Figure 3.

graph
A picture that shows numbers as points on a grid.

axis
One of the two number lines on the side and bottom of a graph. More than one axis: axes.

Remember from Grade 3

In Grade 3 you showed patterns with shapes, numbers and tables of values, and you translated patterns, like turning a pattern of shapes into letters. In Grade 4 you will add a new tool: a graph. A graph lets you see how a pattern grows.

Let's start with a story

The Grade 4 class is having a winter potluck. They push square tables together in a row. One chair fits on each open side of a table.

1 table seats 4 chairs. 2 tables in a row seat 6. 3 tables in a row seat 8. The dots are chairs.

Count the chairs: 4, 6, 8. Each new table adds 2 chairs (one on the top, one on the bottom). The pattern rule is start at 4 and add 2 each time.

Pictures take a long time to draw. Let's show the same pattern in faster ways.

Representation 1: A table of values

A table of values has two columns. The left column is the term number (here, the number of tables). The right column is the term value (the number of chairs).

Number of tables (term number)	Number of chairs (term value)
1	4
2	6
3	8
4	10
5	12

The table goes down by +1 on the left and +2 on the right.

Representation 2: A graph

To make a graph:

Draw a line across the bottom (the horizontal axis). Label it with the term numbers: 1, 2, 3, ...
Draw a line up the side (the vertical axis). Label it with the values. Count by 2s here, since the numbers go up to 12.
For each row in the table, find the term number on the bottom. Go straight up to its value. Draw a point.

For term 1, go to 1 on the bottom, then up to 4. For term 2, go to 2, then up to 6. And so on.

Graph of the chairs pattern. Each point shows one row of the table: (1 table, 4 chairs), (2, 6), (3, 8), (4, 10), (5, 12).

Why we don't join the dots

There is no such thing as 2½ tables in this pattern, so there is nothing between the points. We leave the points unjoined. Notice how the points go up in a straight line. That happens when a pattern grows by the same amount every time.

What does a repeating pattern look like on a graph?

The pattern 2, 5, 2, 5, 2, 5 repeats. Its graph goes up, down, up, down. It never climbs higher.

Graph of the repeating pattern 2, 5, 2, 5, 2, 5. The points zigzag up and down.

Worked example

Example 1: From a rule to a table

Create a growing pattern with the rule start at 3 and add 5 each time. Show the first 5 terms in a table of values.

Term 1 is the start: 3.
Term 2: 3 + 5 = 8. Term 3: 8 + 5 = 13.
Term 4: 13 + 5 = 18. Term 5: 18 + 5 = 23.
Put the term numbers 1 to 5 in the left column and 3, 8, 13, 18, 23 in the right column.

Term 1 → 3, term 2 → 8, term 3 → 13, term 4 → 18, term 5 → 23.

Worked example

Example 2: From a graph to a table

A graph has points at (1, 5), (2, 10), (3, 15) and (4, 20). The first number is the term number and the second is the value. What is the pattern?

Write the points as a table: 1 → 5, 2 → 10, 3 → 15, 4 → 20.
Find the jumps: +5, +5, +5.
The points go up in a straight line, which also tells us the jumps are the same.

The pattern is 5, 10, 15, 20: start at 5 and add 5 each time. The next term would be 25.

Worked example

Example 3: Translate a repeating pattern

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

The core is 🍁❄️❄️. Give each different element its own letter: 🍁 = A, ❄️ = B.
Letters: A B B A B B A B B.
Actions: 🍁 = clap, ❄️ = snap. clap, snap, snap, clap, snap, snap, clap, snap, snap.
Numbers: 🍁 = 1, ❄️ = 2. 1, 2, 2, 1, 2, 2, 1, 2, 2.

All four show the same pattern, because the core always has 1 of the first element and 2 of the second, in the same order.

Worked example

Example 4: Translate the toothpick pattern

In the last lesson, the toothpick squares used 4, 7, 10 and 13 sticks. Show this pattern in a table and say what its graph looks like.

Table: term 1 → 4, term 2 → 7, term 3 → 10, term 4 → 13.
On a graph, the points are (1, 4), (2, 7), (3, 10), (4, 13).
Each point is 3 higher than the one before, because each jump is +3.

The graph shows 4 points going up in a straight line, each one 3 higher than the last.

Just for fun

What did the table of values say to the graph?

"You make a lot of good points!"

Hands-on activity

Build it, table it, graph it

You need: building blocks, coins or cereal pieces, paper (graph paper if you have it), a pencil and a ruler.

Make Figure 1 with 2 blocks. Make Figure 2 with 5 blocks. Make Figure 3 with 8 blocks. Keep adding 3 blocks each time to build Figure 4 and Figure 5.
Make a table of values: term number on the left, number of blocks on the right.
Draw a graph. Label the bottom axis 1 to 5 (term number). Label the side axis 0, 2, 4, ... up to 16 (number of blocks). Plot one point for each row.
Do the points go up in a straight line?
Now make up your own growing pattern, and ask your grown-up to make the table and graph. Check their work!

What to look for: the table reads 2, 5, 8, 11, 14, and the points rise in a straight line.

Grown-up note: keep small items like these away from babies and toddlers.

Your turn

Make a table of values for start at 6, add 4 each time, for terms 1 to 5.

In the chairs pattern, how many chairs fit around 6 tables in a row?

A table of values says 1 → 9, 2 → 18, 3 → 27. Describe the pattern.

Translate A B C A B C into shapes using ● ■ ▲.

To plot the point for term 3, value 12, where do you start and which way do you go?

A pattern's graph zigzags up and down. Is it repeating or growing?

Look at the chairs graph. Why are the points not joined with a line?

More practice: warm-up

In a table of values, which column shows the term number?

Fill in the table: 1 → 2, 2 → 4, 3 → 6, 4 → ?

Translate clap, stomp, clap, stomp into letters.

A point is at (2, 7). What is the term number? What is the value?

What is the rule for 10, 20, 30, 40?

More practice: on your own

A triangle needs 3 toothpicks. Triangles in a row share sides: 1 triangle uses 3 toothpicks, 2 use 5, 3 use 7. Make a table up to 5 triangles.

A graph shows points at (1, 3), (2, 7), (3, 11), (4, 15). Describe the pattern and find term 5.

Translate 1, 1, 2, 1, 1, 2 into two other representations.

Make a table for start at 40, subtract 5 each time for 5 terms. What do the points do on a graph?

Sam's table: 1 → 5, 2 → 8, 3 → 12, 4 → 14. Sam says it grows by 3 each time. Find his mistake.

Challenge

Two patterns: A starts at 2 and adds 4. B starts at 10 and adds 2. Make a table for terms 1 to 5 for each. At which term are they equal?

A repeating pattern goes 4, 1, 1, 4, 1, 1, ... What is the value of term 10? Explain.

Create a growing pattern whose graph goes up by 3 each time and starts at 7. Write the first 4 terms and the rule.

Check your understanding

Quiz

In a table of values for the chairs pattern, what does the right column show?

A table shows 1 → 7, 2 → 14, 3 → 21. What is term 4?

Which translates ★ ● ● ★ ● ● correctly?

To plot (4, 10), you...

The points on a graph go up in a straight line. What does that tell you?

Triangles in a row use 3, 5, 7, 9 toothpicks. How many for 6 triangles?

For parents and teachers

Big idea: The Ontario curriculum (C1.2) asks Grade 4 students to create and translate repeating and growing patterns using various representations, including tables of values and graphs. Graphing is new in Grade 4 algebra. The key idea is that each row of a table becomes one point on a graph.

Common difficulties:

Mixing up the axes. Say "Walk across first, then climb up": term number along the bottom, value up the side.
Uneven scales on the side axis (0, 2, 5, 10). Use graph paper and count by the same amount every line.
Joining the dots. It is good to notice the straight line, but explain that 2½ tables doesn't make sense.
Translating with the wrong number of elements (A B A B for ★ ● ●). Point to each element as you say its letter.

How to help: Build patterns with Lego, coins or cereal. Ask your child to make a table and a graph on graph paper. Encourage neat, careful work (Strand A: organizing and representing) and asking "Does my graph match my table?"