Arithmetic patterns in 3rd grade

What the standard says

What it means, in plain words

Where children usually get stuck

Try four practice questions

3rd Grade · 3.OA.D.9

Arithmetic patterns are rules that repeat in numbers, like the way every product in the 2s row of the multiplication table is even. In 3rd grade, your child finds patterns in the addition table and the multiplication table and explains why they always work.

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

3.OA.D.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.

An addition table and a multiplication table are grids with numbers along the top and down the side. Each box holds the sum or the product of its row number and its column number. Your child looks for what repeats: rows that grow by the same step each time, products that are always even and a grid that matches itself across its diagonal.

Patterns in the multiplication table come from the properties of multiplication. The table matches across its diagonal because $2 \times 9$ and $9 \times 2$ are the same. Every product in the 4s row is even, which means it splits into two equal parts. 4 groups of a number are 2 groups plus 2 more groups, such as $4 \times 7 = 14 + 14 = 28$ .

Arithmetic patterns in 3rd grade have two halves. Spotting the pattern is the first half. Saying why it always works, with groups, an array or a turned-around fact, is the second.

Your child should be able to:

Find a pattern in a row, a column or a diagonal of the addition or multiplication table
Explain why 4 times any number is even
Explain a pattern with groups, an array or a turned-around fact
Use a pattern to fill in a missing number in a table

Example

In the multiplication table, why is each product in the 6s row double the product above it in the 3s row?

Pick any column, such as 9: $3 \times 9 = 27$ and $6 \times 9 = 54$ , and 54 is double 27.
6 groups of 9 are 3 groups of 9 plus 3 more groups of 9.
So $6 \times 9 = 27 + 27 = 54$ .
The same split works in every column, so each product in the 6s row is always double the one in the 3s row.

Example

In the addition table, why does the sum go up by 2 each time you move one box down and one box right?

Moving down one box adds 1 to the first number. Moving right one box adds 1 to the second number.
For example, $3 + 4 = 7$ , and one box down and to the right is $4 + 5 = 9$ .
Both numbers grew by 1, so the sum grows by $1 + 1 = 2$ .
The sum goes up by 2 every time.

Your child says every product in the 3s row is odd, after checking 3, 9 and 15.
3 is odd, so the products feel like they should be odd too. Three examples fit, so the rule seemed proven.
What helps. Go along the whole row together: 3, 6, 9, 12. Then ask why the products take turns, odd then even.
Asked why the 4s row goes 4, 8, 12, 16, your child says "it just goes up by 4".
That describes the pattern but does not explain it. Each box in the row is one more group of 4 than the box before.
What helps. Build the row with blocks: one group of 4, then add another group of 4 each time. Ask what got added at each step.
Your child says $3 \times 5$ must be even, because two odd numbers make an even number.
Odd plus odd really is even, so the rule gets carried over to multiplication. But $3 \times 5$ is $5 + 5 + 5$ , three odd numbers added, and that is odd.
What helps. Lay out 3 rows of 5 coins and pair them up. One coin is always left over, so 15 is odd.

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Real 3rd 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
Look at the addition table below. What number is shown in all the red cells?
+ 0 1 2 3 4 5
0 0 1 2
$\displaystyle \textcolor{#ef4444}{3}$
4 5
1 1 2
$\displaystyle \textcolor{#ef4444}{3}$
4 5 6
2 2
$\displaystyle \textcolor{#ef4444}{3}$
4 5 6 7
3
$\displaystyle \textcolor{#ef4444}{3}$
4 5 6 7 8
4 4 5 6 7 8 9
5 5 6 7 8 9 10
A
3
B
1
C
6
D
5
Show the answer and how to solve it
The constant sum is 3. Every red cell on the highlighted diagonal shows 3, because $0+3$ , $1+2$ , $2+1$ , and $3+0$ all equal 3.
1. 1Look at the addition table. The red cells form a diagonal that runs from the top-right corner toward the bottom-left corner.
2. 2Each cell in the table shows the sum of its row addend and its column addend. For example, the red cell in row 1 and column 2 shows $1 + 2 = 3$ .
3. 3Read the number printed in each red cell. They all show 3.
4. 4Every red cell shows the same sum: 3.
Common mistake. A common mistake is reading a row or column header instead of the number inside a red cell. The header 1 is an addend. Each red cell shows 3.
The idea underneath. On an addition table, every cell on the same anti-diagonal shows the same sum, because each cell on that diagonal is just a different way of breaking the sum into two addends.
2On level
Look at the highlighted row in the multiplication table below. Which statement is always true for every product in this row?
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
10
12
14
16
18
3
0
3
6
9
12
15
18
21
24
27
4
0
4
8
12
16
20
24
28
32
36
5
0
5
10
15
20
25
30
35
40
45
6
0
6
12
18
24
30
36
42
48
54
7
0
7
14
21
28
35
42
49
56
63
8
0
8
16
24
32
40
48
56
64
72
9
0
9
18
27
36
45
54
63
72
81
A
Every product is 0.
B
Every product is 1 more than the other factor.
C
Every product is 1.
D
Every product equals the other factor.
Show the answer and how to solve it
The correct answer is "Every product equals the other factor." When you multiply $1$ by any number, the answer is that same number.
1. 1Look at the highlighted row in the multiplication table.
2. 2The highlighted row shows multiplying $1$ by each column header. The products are:
3. 3Look closely: $1 \times 0 = 0$ (column header $0$ ), $1 \times 6 = 6$ (column header $6$ ), $1 \times 9 = 9$ (column header $9$ ). Each product equals the column header — the other factor.
4. 4So the always-true statement is: "Every product equals the other factor."
Common mistake. A common mistake is thinking the row labeled $1$ means every product is $1$ . But only $1 \times 1 = 1$ ; the other products in the row are different ( $0, 2, 3, 4, \ldots, 9$ ), and each one equals the column header above it.
The idea underneath. When you multiply by $1$ , you get the number back. The number $1$ is special in multiplication — it does not change the value of the other factor.
3On level
Look at the two rows of numbers in the addition table. Which statement describes the +1 pattern in the rows as shown?
$\displaystyle \begin{array}{c|ccccc} + & 1 & 2 & 3 & 4 & 5 \\ \hline 2 & 3 & 4 & 6 & 6 & 7 \\ 6 & 7 & 8 & 9 & 10 & 11 \end{array}$
A
Only the top row follows the +1 step pattern.
B
Only the bottom row follows the +1 step pattern.
C
Each row follows the +1 step pattern.
D
Neither row follows the +1 step pattern.
Show the answer and how to solve it
The top row's third sum is 6 instead of 5, so the steps go 3, 4, 6 (a jump of +2) and then 6, 6 (a step of +0) — that breaks the pattern. The bottom row's sums are 7, 8, 9, 10, 11 — every step is +1. So only the bottom row follows the +1 step pattern.
1. 1In a correct addition table, each entry equals the row number plus the column number. Here, the top row uses 2, and the bottom row uses 6. Check the entries as shown.
2. 2Check the top row. Each expected sum is 2 plus the column number:
3. 3The displayed top row is 3, 4, 6, 6, 7. The third cell shows 6, but $2 + 3$ should be 5. The step from 4 to 6 is +2, and the step from 6 to 6 is +0. So the top row does NOT follow the +1 step pattern.
4. 4Check the bottom row. Each expected sum is 6 plus the column number:
5. 5The displayed bottom row is 7, 8, 9, 10, 11. Every cell matches the expected sum, and every step adds 1. So the bottom row DOES follow the +1 step pattern. Only the bottom row follows the +1 step pattern.
Common mistake. A common mistake is checking only one row before answering. Both rows must be inspected — even when one row is clearly correct, the other might still break the pattern, and vice versa.
The idea underneath. In a correct addition table, each entry equals the row number plus the column number. As the column number increases by 1, the sum increases by 1. Check the numbers shown in each row to see whether every step adds 1.
4Stretch
Maria wrote this claim about multiplication: "I think $6 \times 8 = 48$ , so $8 \times 6$ must be more because the bigger number comes first."
Select the statement that best describes Maria's reasoning.
A
Maria's reasoning is incorrect because you can multiply two numbers in any order and get the same product.
B
Maria's reasoning is correct because the bigger factor first gives a bigger product.
C
Maria's reasoning is incorrect because she should have added the two numbers instead of multiplying.
D
Maria's reasoning is incorrect because the larger factor should come first when multiplying.
Show the answer and how to solve it
Maria's reasoning is incorrect because when you multiply, the order of the factors does not change the product. Both $6 \times 8$ and $8 \times 6$ equal $48$ .
1. 1Read Maria's claim carefully. She thinks that $6 \times 8 = 48$ , but she thinks $8 \times 6$ must give a bigger answer because the bigger number comes first.
2. 2The dots below make $6$ rows of $8$ , or $6 \times 8$ . Count down instead: there are $8$ columns of $6$ , or $8 \times 6$ . Both ways count the same dots. No dots are added or taken away.
3. 3Both products equal $48$ , so Maria's reasoning is incorrect. The order of the factors does not change the product.
Common mistake. A common mistake is thinking that because the bigger number comes first in $8 \times 6$ , it must give a bigger product than $6 \times 8$ . Both arrangements always have the same total.
The idea underneath. You can count the same rectangle of dots by rows or by columns. Switching the factors changes how you count the dots, not how many dots there are.