Relationships Between Corresponding Terms | Grade 5 Math | LevelKite

You already know how to make two number patterns from two rules and line them up in a table. In this lesson, every rule gives a starting number and an amount to add each time. Now you will read across the table to compare the two patterns.

Each number in a pattern is a term. Corresponding terms are the terms in the same position in two patterns: the 1st term with the 1st term, the 2nd with the 2nd, and so on. In the tables in this lesson, corresponding terms sit in the same row.

Start with two patterns. Pattern A starts at $0$ and adds $3$ each time. Pattern B starts at $0$ and adds $6$ each time.

Pattern A and Pattern B, side by side

Position	Pattern A: start at 0, add 3	Pattern B: start at 0, add 6
1st	$0$	$0$
2nd	$3$	$6$
3rd	$6$	$12$
4th	$9$	$18$
5th	$12$	$24$

Read each row from left to right and compare the two terms.

2nd terms: $3$ and $6$ . In words, six is twice three, which means 2 times three. As a number sentence, $6 = 2 \times 3$ .
3rd terms: $6$ and $12$ . Twelve is twice six: $12 = 2 \times 6$ .
4th terms: $9$ and $18$ . Eighteen is twice nine: $18 = 2 \times 9$ .
5th terms: $12$ and $24$ . Twenty-four is twice twelve: $24 = 2 \times 12$ .
1st terms: $0$ and $0$ . This row fits too, because $2 \times 0 = 0$ .

A relationship between two patterns is one statement that is true for every pair of corresponding terms. Here it is: each term of Pattern B is twice the corresponding term of Pattern A. You can say it the other way too: each term of Pattern A is half of the corresponding term of Pattern B.

One row is not enough to be sure. Test your statement on every row you have.

The table shows only five pairs. The two rules tell you why the relationship holds for every pair, even pairs the table does not show.

Each time a pattern adds, call that one step. Both patterns start at $0$ . On each step, Pattern A adds $3$ and Pattern B adds $6$ . Each $6$ is the same as $3 + 3$ . So on every step, B adds twice as much as A.

Corresponding terms are in the same position, so they come after the same number of steps. Both patterns began at $0$ , so each term is just the total added so far. B's total is always twice A's total. That is why B's term is always twice A's term.

See it for the 4th terms, $9$ and $18$ . Each pattern has taken three steps, so A has added $3 + 3 + 3$ and B has added $6 + 6 + 6$ . Split each $6$ into $3 + 3$ . Then regroup the six threes into two groups of $3 + 3 + 3$ :

18 = 6 + 6 + 6 = (3 + 3) + (3 + 3) + (3 + 3) = 2 \times (3 + 3 + 3) = 2 \times 9

Key concept

The two rules decide the relationship

Corresponding terms are linked because of the two rules. Each rule in this lesson has two parts: the starting number and the amount added on each step.

Suppose both patterns start at $0$ , and one pattern adds $2$ times as much as the other on each step. Then after the same number of steps it has added $2$ times as much in total. So every one of its terms is $2$ times the corresponding term of the other pattern. The same reasoning works for $3$ times, $4$ times, and so on.

Both parts of the rules matter. With other starting numbers or other amounts added, the relationship can be a different one. So test every row of your table, then use the rules to explain what you see.

Worked example

Compare the terms, then explain

Pattern C starts at $0$ and adds $2$ each time. Pattern D starts at $0$ and adds $8$ each time. How does each term of Pattern D compare with the corresponding term of Pattern C? Explain why.

Write the first five terms of each pattern from its rule. This gives you pairs to compare.
$\text{C: } 0,\ 2,\ 4,\ 6,\ 8 \qquad \text{D: } 0,\ 8,\ 16,\ 24,\ 32$
Match the terms by position. The 1st terms are 0 and 0. They fit many statements, because any number times 0 is 0. So start with a pair that is not zero. The 2nd terms are 2 and 8, and 8 is 4 times 2.
$8 = 4 \times 2$
Test "4 times" on every other pair, because a relationship must be true in every row. It fits each one, and it fits the 1st terms too.
$16 = 4 \times 4 \qquad 24 = 4 \times 6 \qquad 32 = 4 \times 8 \qquad 0 = 4 \times 0$
Explain it from the rules. Both patterns start at 0. On each step C adds 2 and D adds 8, and 8 is four 2s. So D adds 4 times as much on every step. After the same number of steps, D's term is 4 times C's term.
$8 = 2 + 2 + 2 + 2 = 4 \times 2$

Answer

Each term of Pattern D is $4$ times the corresponding term of Pattern C. This is true for every pair, because both patterns start at $0$ and D adds $4$ times as much as C on each step.

Now use the same thinking in word problems. In these stories, two amounts grow at the same time, and each amount follows its own rule.

Here is a plan:

Find the two rules in the story. For each amount, ask: what does it start at, and how much is added each time?
Put the two patterns in a table with one row for each step of time, such as each night or each week.
Read across the rows. Each row holds a pair of corresponding terms. Find the relationship and test it on every row.
Explain the relationship from the rules. Then use it to answer the question.

Worked example

Two readers, two rules

Mia and Dev each start a new book on the same day. Mia reads $4$ pages every night. Dev reads $12$ pages every night. (a) Make a table of the total pages each has read, from the start (night $0$ ) through night $4$ . How do Dev's pages compare with Mia's? (b) If they keep reading this way, how many pages has Dev read when Mia has read $28$ pages?

Find the two rules. Both books are new, so both totals start at 0 pages. Each night Mia adds 4 pages and Dev adds 12 pages.
$\text{Mia: start at } 0,\ \text{add } 4 \qquad \text{Dev: start at } 0,\ \text{add } 12$
Make the table with one row for each night. Night 0 is the start, before any reading, so both columns begin at 0. Go down Mia's column adding 4 and down Dev's column adding 12.
$\text{Mia: } 0,\ 4,\ 8,\ 12,\ 16 \qquad \text{Dev: } 0,\ 12,\ 24,\ 36,\ 48$
Read across each row. The two numbers in a row are corresponding terms. In every row, Dev's pages are 3 times Mia's pages.
$0 = 3 \times 0 \quad 12 = 3 \times 4 \quad 24 = 3 \times 8 \quad 36 = 3 \times 12 \quad 48 = 3 \times 16$
Explain it from the rules. Both start at 0. Each night Dev adds 12 pages, and 12 is three 4s. So Dev adds 3 times as much as Mia every night. His total stays 3 times her total on every night, not only on nights 0 to 4.
$12 = 4 + 4 + 4 = 3 \times 4$
Answer part (b) with the relationship. It holds on every night, so when Mia has read 28 pages, Dev has read 3 times as many. You can check by extending the table: night 7 gives 28 pages for Mia and 84 pages for Dev.
$3 \times 28 = (3 \times 20) + (3 \times 8) = 60 + 24 = 84$

Answer

On every night, Dev has read $3$ times as many pages as Mia. When Mia has read $28$ pages, Dev has read $84$ pages.

Pages read: the table from Mia and Dev's problem

Night	Mia's pages (add 4)	Dev's pages (add 12)	Dev compared with Mia
0	$0$	$0$	$0 = 3 \times 0$
1	$4$	$12$	$12 = 3 \times 4$
2	$8$	$24$	$24 = 3 \times 8$
3	$12$	$36$	$36 = 3 \times 12$
4	$16$	$48$	$48 = 3 \times 16$

Worked example

Same amount added, different starts

Kai has $10$ stickers and Zoe has $5$ stickers. Each week, their teacher gives each of them $3$ more stickers. (a) Make a table from the start (week $0$ ) through week $4$ . How does Kai's number of stickers compare with Zoe's? (b) If this keeps going, how many stickers does Kai have when Zoe has $26$ ?

Find the two rules. This time the starting numbers are different, and the amount added is the same.
$\text{Kai: start at } 10,\ \text{add } 3 \qquad \text{Zoe: start at } 5,\ \text{add } 3$
Make the table with one row for each week. Week 0 is the start, so the columns begin at 10 and 5. Then add 3 down each column.
$\text{Kai: } 10,\ 13,\ 16,\ 19,\ 22 \qquad \text{Zoe: } 5,\ 8,\ 11,\ 14,\ 17$
Read across and test. Week 0 looks like "twice", since 10 is 2 times 5. Week 1 does not fit: 2 times 8 is 16, not 13. Try subtracting instead. Kai has 5 more than Zoe in every row.
$10 - 5 = 5 \quad 13 - 8 = 5 \quad 16 - 11 = 5 \quad 19 - 14 = 5 \quad 22 - 17 = 5$
Explain it from the rules. Kai starts 5 ahead. Each week both add the same amount, 3, so the gap between them cannot change. Kai stays 5 ahead every week.
$(10 + 3) - (5 + 3) = 13 - 8 = 5$
Answer part (b) with the relationship. It holds every week, so when Zoe has 26 stickers, Kai has 5 more.
$26 + 5 = 31$

Answer

Kai always has $5$ more stickers than Zoe. When Zoe has $26$ stickers, Kai has $31$ stickers.

Stickers: the table from Kai and Zoe's problem

Week	Kai's stickers (add 3)	Zoe's stickers (add 3)	Kai minus Zoe
0	$10$	$5$	$10 - 5 = 5$
1	$13$	$8$	$13 - 8 = 5$
2	$16$	$11$	$16 - 11 = 5$
3	$19$	$14$	$19 - 14 = 5$
4	$22$	$17$	$22 - 17 = 5$

Keep this big idea. Corresponding terms sit in the same position in two patterns, and you compare them by reading across a table. Find one relationship that is true in every row. Then use both parts of the two rules, the starting numbers and the amounts added, to explain why it holds for every pair. In a word problem, find the two rules in the story, build the table, and let the relationship help you answer the question.