5th Grade · 5.OA.B.3

Numerical patterns in 5th grade

Numerical patterns are lists of numbers that follow a rule, such as "start at 1 and add 3": 1, 4, 7, 10. In 5th grade, your child builds two patterns from two rules, compares them number by number and plots the pairs on a graph.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

What it means, in plain words

Each number in a pattern is called a term. Numerical patterns in 5th grade come in pairs. Children make two lists from two rules, then compare corresponding terms, which are the terms in the same position in each list: the first with the first, the second with the second.

Take "start at 0, add 2" and "start at 0, add 8". The lists are 0, 2, 4, 6 and 0, 8, 16, 24. Each term in the second list is 4 times the matching term in the first. Children also learn to say why: both patterns start at 0, and the second adds 4 times as much at every step.

Next, children pair corresponding terms into ordered pairs. An ordered pair is two numbers in parentheses, such as $(2, 8)$ , that mark a point on a coordinate plane (a grid with one number line across the bottom and one up the side). The first number says how far to go across and the second says how far to go up. When both rules add, the plotted points fall on a straight line.

Your child should be able to:

Write the first few terms of a pattern from its rule
Compare corresponding terms in two patterns and describe how they are related
Explain the relationship using the two rules
Turn corresponding terms into ordered pairs and plot them on a coordinate plane

Example

Pattern A starts at 0 and adds 7. Pattern B starts at 0 and adds 14. How do the corresponding terms compare?

Pattern A: 0, 7, 14, 21, 28.
Pattern B: 0, 14, 28, 42, 56.
Compare terms in the same position: 7 and 14, 14 and 28, 21 and 42.
Each term in Pattern B is 2 times the matching term in Pattern A, because Pattern B adds twice as much at every step.

Example

Pattern A starts at 0 and adds 1. Pattern B starts at 0 and adds 5. What are the first four ordered pairs, and what does the graph look like?

Pattern A: 0, 1, 2, 3. Pattern B: 0, 5, 10, 15.
Pair the terms in the same position: $(0, 0)$ , $(1, 5)$ , $(2, 10)$ , $(3, 15)$ .
Each point is 1 step across and 5 steps up from the one before.
The four ordered pairs are $(0, 0)$ , $(1, 5)$ , $(2, 10)$ and $(3, 15)$ , and they fall on a straight line.

Where children usually get stuck

For Pattern A: 0, 5, 10, 15 and Pattern B: 0, 10, 20, 30, your child says Pattern B is always 5 more.
The child compares only the first pair after 0, sees 5 and 10, and does not check the rest of the list.
What helps. Ask your child to test the rule on every pair. 15 and 30 are 15 apart, so "5 more" fails, but "2 times" works for every pair.
Your child plots the ordered pair $(2, 10)$ at 10 across and 2 up.
The order inside an ordered pair is a convention. Nothing in the numbers shows which one goes across.
What helps. Teach the phrase "walk before you climb": go across first, then up.
Your child sees that one pattern is 3 times the other but can only say "it just is".
Spotting a relationship in a table is easier than linking it back to the rules. The reason comes from the rules, not from the table.
What helps. Ask how much each rule adds per step. If one pattern adds 3 times as much each step and both start at 0, its total stays 3 times as big.

Try four practice questions

Real 5th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up

A student wants to compare Pattern A and Pattern B in the table shown. The student's work is shown under the table.

Term	Pattern A	Pattern B
1	4	7
2	9	12
3	14	17

Line 1	$\displaystyle \text{Row 1: } 7 - 4 = 3$
Line 2	$\displaystyle \text{Row 2: } 12 - 9 = 3$
Line 3	$\displaystyle \text{Row 3: } 17 - 14 = 3$
Conclusion	So Pattern A is 3 more than Pattern B in this table.

Select the statement that best describes the student's work.

The student's work is correct.

The student's same-row subtractions are correct, but the conclusion should say Pattern B is 3 more than Pattern A.

The student's work has an arithmetic mistake: 12 − 9 should be 2, not 3.

The student subtracted consecutive terms within Pattern A instead of same-row terms; Pattern B is 3 more than Pattern A.

Show the answer and how to solve it

The correct statement is the option saying the same-row subtractions are correct but the conclusion should say Pattern B is 3 more than Pattern A.

1Compare terms in the same row, because each row gives corresponding terms from Pattern A and Pattern B.
2Subtract Pattern A from Pattern B in each row to find the true difference.
3The same-row differences are all 3, so Pattern B is 3 more than Pattern A in this table.
4Now check the student's work. Each line subtracts the Pattern B term minus the Pattern A term in the same row ; those computations are correct. However, the conclusion says Pattern A is 3 more than Pattern B in this table, which reverses the relationship.
5Pattern B is larger in every row, so the conclusion has the direction backwards. The correct description is the option that says the same-row subtractions are right but the conclusion should say Pattern B is 3 more than Pattern A.

Common mistake. A common mistake is correctly finding same-row differences but writing the conclusion in the wrong direction, naming the smaller pattern as the one that is greater.

The idea underneath. A more-than relationship between two patterns comes from comparing corresponding terms in the same row, not from how one pattern changes down the table.

2On level
Pattern A and Pattern B are related, and Pattern B is 3 times Pattern A. A term in Pattern B is 18, and a student is finding the matching term in Pattern A — the student's work is shown below.
$\displaystyle 18 \times 3$
$\displaystyle =$
$\displaystyle 54$
$\displaystyle \text{Pattern A}$
$\displaystyle =$
$\displaystyle 54$
Select the statement that best describes the work shown.
A
In the work shown, the student multiplied when the problem required division.
B
In the work shown, the student divided when the problem required multiplication.
C
In the work shown, the student added or subtracted when the relationship needed multiplication or division.
D
The student's work is correct.
Show the answer and how to solve it
The best description is "In the work shown, the student multiplied when the problem required division." The matching Pattern A term is 6, not 54.
1. 1Look at the student's work below.
2. 2The rule says Pattern B is 3 times Pattern A. To go from Pattern A to Pattern B, multiply by 3. To go backward from Pattern B to Pattern A, divide by 3.
3. 3Find the matching Pattern A term.
4. 4The shown work multiplies 18 by 3 and gets 54, so the best description is "In the work shown, the student multiplied when the problem required division."
Common mistake. A common mistake is thinking "3 times" always means multiply. When the given term is in Pattern B and you need Pattern A, you must divide by 3.
The idea underneath. A pattern rule tells how to move from one pattern to the other; when you work backward, you use the opposite operation.
3On level
Marcus wants to list the first 5 terms for Pattern A and Pattern B. The table below shows his work.
Term Pattern A start at 4, add 4 Pattern B start at 4, multiply by 2
1 4 4
2 8 8
3 12 12
4 16 24
5 20 48
Select the statement that best describes the work.
A
Pattern A is wrong because it should start with 8 instead of 4.
B
Pattern B is wrong because term 3 should be 16, and terms 4 and 5 continue from that error.
C
Pattern B is correct because it adds 8 each time.
D
Both patterns are wrong because the rules were switched.
Show the answer and how to solve it
The correct statement is that Pattern B is wrong because term 3 should be 16, and terms 4 and 5 continue from that error. Pattern A is correct.
1. 1Check each pattern against its own rule in the table.
2. 2Pattern A starts at 4 and adds 4 each time, so its 5 terms should be these.
3. 3Pattern B starts at 4 and multiplies by 2 each time, so its 5 terms should be these.
4. 4Pattern A matches its rule perfectly. Pattern B does not. Term 3 in the table shows 12, but multiplying 8 by 2 gives 16. Terms 4 and 5 continue multiplying from the wrong value 12, giving 24 and 48 instead of 32 and 64.
Common mistake. A common mistake is checking only the first two rows, where both patterns look correct. The multiply-by-2 error in Pattern B first appears at term 3.
The idea underneath. A pattern table is correct only if every term in each column follows that column's own rule from the correct starting number.
4Stretch
Pattern A starts at 2 and multiplies by 2 each time. Pattern B starts at 2 and multiplies by 3 each time.
Enter the first 4 terms for each pattern in the table. Use the starting number as term 1.
Term Pattern A Pattern B
1
2
3
4
Show the answer and how to solve it
The first four terms are Pattern A: 2, 4, 8, 16 and Pattern B: 2, 6, 18, 54.
1. 1Term 1 is the starting number for each pattern.
2. 2Build Pattern A by multiplying by 2 each time.
3. 3Build Pattern B by multiplying by 3 each time.
4. 4Both patterns start at the same number and both multiply, but Pattern A uses factor 2 and Pattern B uses factor 3, so Pattern B grows much faster. Write the terms into the table in order.
Common mistake. A common mistake is applying the multiply-by-3 rule to Pattern A or the multiply-by-2 rule to Pattern B. Because both patterns share the same starting number, it is easy to mix up which factor belongs to which column.
The idea underneath. A numerical pattern is built one step at a time: each new term comes from the term right before it by repeating the same rule.

$\displaystyle 18 \times 3$	$\displaystyle =$	$\displaystyle 54$
$\displaystyle \text{Pattern A}$	$\displaystyle =$	$\displaystyle 54$