From Two Number Patterns to a Graph | Grade 5 Math | LevelKite

You already know two things. You can build a number pattern from a rule, such as add 2, that tells you how to get each next number. You can also plot a point on the coordinate plane, the grid made by an x-axis that runs across and a y-axis that runs up. In this lesson you put the two together, so you can see two patterns in one picture.

Here are two patterns. Each number in a pattern is called a term.

Pattern A starts at $0$ and follows the rule add 1: $0, 1, 2, 3, 4$
Pattern B starts at $0$ and follows the rule add 2: $0, 2, 4, 6, 8$

Read the patterns side by side. The first term of Pattern A goes with the first term of Pattern B. The second term goes with the second term, and so on. Terms in the same position are called corresponding terms. In the table below, corresponding terms share a row.

Two patterns side by side. Corresponding terms share a row.

Position	Pattern A (rule: add 1)	Pattern B (rule: add 2)
1st	$0$	$0$
2nd	$1$	$2$
3rd	$2$	$4$
4th	$3$	$6$
5th	$4$	$8$

Key concept

Each pair of corresponding terms is one point

An ordered pair is two numbers written in a set order, such as $(1, 2)$ . The first number is the x-coordinate. It tells how far to go right from the origin, the point $(0, 0)$ where the axes cross. The second number is the y-coordinate. It tells how far to go up.

To join two patterns, choose one pattern to give the x-coordinates (the x-pattern) and the other to give the y-coordinates (the y-pattern). Each pair of corresponding terms then makes one ordered pair, and each ordered pair is one point on the coordinate plane. The graph shows both patterns in one picture.

(x, y) = (\text{term from the x-pattern}, \text{term from the y-pattern})

Worked example

Form ordered pairs from two patterns

Pattern A is $0, 1, 2, 3, 4$ (start at $0$ , add $1$ ). Pattern B is $0, 2, 4, 6, 8$ (start at $0$ , add $2$ ). Use Pattern A for the x-coordinates and Pattern B for the y-coordinates. What ordered pairs do the two patterns make?

Line up the patterns by position: first term with first term, second with second, and so on. Only terms in the same position are corresponding terms, so only they can share a pair.
Pair the first terms. Pattern A gives the x-coordinate, so its term, 0, is written first. Pattern B gives the y-coordinate, so its term, 0, is written second.
$(0, 0)$
Pair the second terms in the same order. The second term of Pattern A is 1 and the second term of Pattern B is 2, so the x-coordinate is 1 and the y-coordinate is 2.
$(1, 2)$
Keep the same order for the rest. The third terms are 2 and 4, the fourth terms are 3 and 6, and the fifth terms are 4 and 8. The Pattern A term goes first each time because it is the x-coordinate.
$(2, 4), \quad (3, 6), \quad (4, 8)$

Answer

The two patterns make the ordered pairs $(0, 0)$ , $(1, 2)$ , $(2, 4)$ , $(3, 6)$ and $(4, 8)$ .

The pair

(3, 6)

and the swapped pair

(6, 3)

are two different points.

Worked example

Graph the ordered pairs from two patterns

The x-pattern starts at $0$ and follows the rule add 2, so it is $0, 2, 4, 6$ . The y-pattern starts at $1$ and follows the rule add 1, so it is $1, 2, 3, 4$ . Graph the ordered pairs that these two patterns make.

Form the ordered pairs from corresponding terms. Write the term from the x-pattern first and the term from the y-pattern second, because the x-coordinate always comes first.
$(0, 1), \quad (2, 2), \quad (4, 3), \quad (6, 4)$
Set up the graph. The x-axis counts the x-pattern and the y-axis counts the y-pattern. No term is below 0, so you only count right and up from the origin. That part of the coordinate plane is called the first quadrant.
Plot the first two pairs, starting from the origin each time. For (0, 1), the x-coordinate is 0, so do not move right. Go up 1 because the y-coordinate is 1. This point sits on the y-axis. For (2, 2), go 2 right because the x-coordinate is 2, then 2 up because the y-coordinate is 2.
Plot the last two pairs the same way. For (4, 3), go 4 right and 3 up. For (6, 4), go 6 right and 4 up.
Check the graph below against your pairs. Four pairs make four points, and each label on the graph matches one of the pairs you formed.

Answer

The graph shows the four points $(0, 1)$ , $(2, 2)$ , $(4, 3)$ and $(6, 4)$ , one point for each pair of corresponding terms.

Each point pairs an x-pattern term from

0, 2, 4, 6

with the y-pattern term from

1, 2, 3, 4

in the same position.

Look again at the graph above. From $(0, 1)$ to $(2, 2)$ you move $2$ right and $1$ up. From $(2, 2)$ to $(4, 3)$ you make the same move, and again from $(4, 3)$ to $(6, 4)$ .

That move comes straight from the two rules. The x-pattern follows add 2, so each point is $2$ farther right than the one before. The y-pattern follows add 1, so each point is $1$ higher than the one before.

Points that take the same step every time line up. They all sit along one straight line. If you laid a ruler from the first point to the last, its edge would pass through the points in between.

Sometimes you get the two rules and only the start of a table. That is enough to finish the job. Each rule tells you how to keep its pattern going, so you can fill in the missing terms yourself.

Every row you complete is a new ordered pair, and every new ordered pair is a new point. So when you finish the table, you can finish the graph too. If the rules have not changed, the new points should take the same step as the old ones.

A table to complete. The x column follows the rule add 3 and the y column follows the rule add 2.

x (rule: add 3)	y (rule: add 2)	Ordered pair
$0$	$2$	$(0, 2)$
$3$	$4$	$(3, 4)$
?	?	?
?	?	?

Worked example

Use the rules to finish a table and a graph

In the table above, the x column holds the x-pattern. It starts at $0$ and follows the rule add 3. The y column holds the y-pattern. It starts at $2$ and follows the rule add 2. Only the first two rows are filled in. Complete the table, graph every row as a point, and describe how the points line up.

Continue the x-pattern. The rule add 3 means each term is 3 more than the term above it. The last term in the x column is 3, so add 3 to get the next term, then add 3 again.
$3 + 3 = 6 \quad \text{and} \quad 6 + 3 = 9$
Continue the y-pattern the same way with the rule add 2. The last term in the y column is 4, so add 2 to get the next term, then add 2 again.
$4 + 2 = 6 \quad \text{and} \quad 6 + 2 = 8$
Write the ordered pair for each new row. Numbers in the same row are corresponding terms, so each row makes one pair, with the x-coordinate first.
$(6, 6) \quad \text{and} \quad (9, 8)$
Complete the graph, with the x-pattern along the x-axis and the y-pattern up the y-axis. Every row is one point. Plot (0, 2) and (3, 4) from the rows you were given. Then plot (6, 6) by going 6 right and 6 up, and (9, 8) by going 9 right and 8 up. The graph below shows all four points.
Describe how the points line up. From each point to the next, you go 3 right and 2 up, for the new points as well as the given ones. The 3 comes from the rule add 3 and the 2 comes from the rule add 2. The same step every time puts all four points along one straight line.

Answer

The completed table gives the ordered pairs $(0, 2)$ , $(3, 4)$ , $(6, 6)$ and $(9, 8)$ . On the graph, each point is $3$ right and $2$ up from the one before it, so the four points line up along one straight line.

From the given points

(0, 2)

and

(3, 4)

to the new points

(6, 6)

and

(9, 8)

, every step is

3

right and

2

up.

The takeaway: two number patterns can become one picture. Line the patterns up, pair the corresponding terms as $(x, y)$ with the term from the x-pattern first, and plot each pair as a point. Each add rule moves the points by the same amount every time, so every point is the same step from the one before, and the points line up along one straight line.

Next, you will graph tables of real-world data. You will decide what each axis stands for and use the graph to answer questions about the situation.