Aimathic
Login | English | Deutsch

Free math worksheets

Build your own math worksheets from 30,000+ problems for grades 3 to 12, from fractions to AP Calculus. Every problem comes with step-by-step solutions.

Understand multiplication as groups and arrays

Click problems to add them to your worksheet.

5158043
Complete the multiplication facts. a) \(8 \times 1 = \square\) b) \(1 \times 6 = \square\) c) \(9 \times 0 = \square\) d) \(0 \times 4 = \square\) What happens when you multiply a number by \(1\)? What happens when you multiply a number by \(0\)?

Hints

- Compare each product with the number that is not \(0\) or \(1\). - Look at parts a and b together. What stayed the same? - Look at parts c and d together. What do both products have in common?

Solution

1. \(8 \times 1 = 8\). 2. \(1 \times 6 = 6\). 3. \(9 \times 0 = 0\). 4. \(0 \times 4 = 0\). 5. Multiplying a number by \(1\) keeps the number the same. Multiplying a number by \(0\) gives \(0\).

Answer

a) \(8\) b) \(6\) c) \(0\) d) \(0\) Multiplying a number by \(1\) keeps the number the same. Multiplying a number by \(0\) gives \(0\).
5158053
Complete each multiplication fact. a) \(5 \times 0 = \square\) b) \(0 \times 3 = \square\) c) \(1 \times 8 = \square\) d) \(4 \times 1 = \square\) What rule do parts a and b show about multiplying by \(0\)?

Hints

- Compare the number that is not \(0\) or \(1\) with the product. - Look closely at the two facts that contain \(0\). - What do those two products have in common?

Solution

1. \(5 \times 0 = 0\). 2. \(0 \times 3 = 0\). 3. \(1 \times 8 = 8\). 4. \(4 \times 1 = 4\). 5. Multiplying any number by \(0\) gives \(0\).

Answer

a) \(0\) b) \(0\) c) \(8\) d) \(4\) Multiplying any number by \(0\) gives \(0\).
5191933
A classroom display has \(7\) equal rows with \(6\) stars in each row. a) Write a multiplication equation for the display. b) In your equation, explain what each factor represents and what the product represents.

Hints

- Find the number of equal rows and the number of objects in each row. - A multiplication equation can represent “number of groups \(\times\) amount in each group.” - The product should describe the total number of objects in the display.

Solution

1. Seven rows with six stars in each row are represented by \(7 \times 6\). 2. \(7 \times 6=42\). 3. The factor \(7\) represents the number of rows, the factor \(6\) represents the number of stars in each row, and the product \(42\) represents the total number of stars.

Answer

a) \(7 \times 6=42\) b) \(7\) is the number of rows, \(6\) is the number of stars in each row, and \(42\) is the total number of stars.
5206083
Write this repeated addition as multiplication. Then find the total. \(12\,\text{cents} + 12\,\text{cents} + 12\,\text{cents} + 12\,\text{cents} + 12\,\text{cents} + 12\,\text{cents} + 12\,\text{cents}\)

Hints

- Count how many times \(12\,\text{cents}\) appears. - Use that count as one factor. - The other factor is the amount in each equal group.

Solution

1. There are \(7\) equal groups of \(12\,\text{cents}\). 2. Write \(7 \times 12\,\text{cents}\). 3. \(7 \times 12\,\text{cents}=84\,\text{cents}\).

Answer

\(7 \times 12\,\text{cents}=84\,\text{cents}\)
5373443
The two arrays contain the same number of dots but have different shapes. Write a multiplication equation for each array. Explain how you know the products are equal without counting every dot one at a time.
Figure for problem 537344

Hints

- Find the number of rows and the dots in each row. - Use a multiplication fact for each array and compare the products.

Solution

1. Array a) has \(4\) rows of \(6\) dots, so \(4 \times 6 = 24\). 2. Array b) has \(3\) rows of \(8\) dots, so \(3 \times 8 = 24\). 3. Each array is organized into equal groups whose multiplication fact has a product of \(24\).

Answer

a) \(4\times6=24\) b) \(3\times8=24\) The products are equal because both multiplication facts equal \(24\).
5373453
Mara says, “Both arrays show the same multiplication fact because both have \(4\) rows.” Is Mara right? Explain. Then write the correct multiplication equation for each array.
Figure for problem 537345

Hints

- Compare more than the number of rows. - How many dots are in each row of each array?

Solution

1. Array a) has \(4\) rows with \(5\) dots in each row, so it shows \(4 \times 5 = 20\). 2. Array b) has \(4\) rows with \(8\) dots in each row, so it shows \(4 \times 8 = 32\). 3. Mara’s statement is incorrect. The number of rows alone does not find the multiplication equation; the number of dots in each row must also match.

Answer

Mara is incorrect. Array a) shows \(4 \times 5 = 20\), while array b) shows \(4 \times 8 = 32\). The row count and the number of dots per row must both match.
5373953
Look at the array. Use rows and equal groups instead of counting every dot. a) Write a multiplication equation for the dots in the left half. b) Write a multiplication equation for the dots in the right half. c) What does each number in your multiplication equations mean in the picture?
Figure for problem 537395

Hints

- Count the rows, not every dot. - Look at how many dots from one half appear in a single row. - Your explanation should say what both factors mean in the picture.

Solution

1. The array has \(6\) rows. Each row has \(4\) dots in the left half, so the left half has \(6 \times 4 = 24\) dots. 2. Each row also has \(4\) dots in the right half, so the right half has \(6 \times 4 = 24\) dots. 3. In each equation, \(6\) represents the number of rows and \(4\) represents the number of dots from that half in each row.

Answer

a) \(6 \times 4 = 24\) b) \(6 \times 4 = 24\) c) The factor \(6\) is the number of rows, and the factor \(4\) is the number of dots in that half of each row.
5550803
Look at pictures a), b), and c). a) Which pictures show equal groups? b) For each picture that shows equal groups, write a multiplication equation for the total. c) Why does the other picture not show equal groups?
Figure for problem 555080

Hints

- In each picture, compare the number of counters in every row. - Equal groups must all have the same number of counters. - Write a multiplication equation only when the groups are equal.

Solution

1. Picture a) has \(3\) equal groups of \(4\), so \(3\times4=12\). 2. Picture b) has \(4\) equal groups of \(3\), so \(4\times3=12\). 3. Picture c) does not show equal groups because one row has fewer counters than the others.

Answer

a) Pictures a) and b) b) a): \(3\times4=12\); b): \(4\times3=12\) c) Picture c) has groups of different sizes, so it does not show equal groups.
5165233
Look at the checkerboard array. a) How many rows are there? How many tiles are in each row? b) Write a multiplication equation for all the tiles. c) How many tiles are filled? Explain how the repeating pattern helps you know.
Figure for problem 516523

Hints

- Use the picture to count the rows and the number of tiles in one row. - For the total, think of each row as one equal group. - For part c, look for how the filled tiles repeat within every row.

Solution

1. The array has \(8\) rows with \(8\) tiles in each row. 2. The total number of tiles is \(8 \times 8 = 64\). 3. Each row has \(4\) filled tiles, and there are \(8\) rows. Thus, \(8 \times 4 = 32\) filled tiles.

Answer

a) \(8\) rows and \(8\) tiles in each row b) \(8 \times 8 = 64\) c) \(32\) filled tiles; each of the \(8\) rows contains \(4\) filled tiles.
5374343
The picture shows a rectangular array of square tiles. Do not count the tiles one at a time. a) Write a multiplication equation for the whole array. b) Write a multiplication equation for the tiles completely inside the border. c) Use those two products to find the number of border tiles.
Figure for problem 537434

Hints

- Use the row and column counts from the grid rather than counting individual tiles. - Removing the outside border removes two rows and two columns from the dimensions of the interior. - Find the whole and interior products before finding the border count.

Solution

1. The array has \(7\) rows and \(8\) columns, so the whole array has \(7 \times 8=56\) tiles. 2. Removing the top and bottom rows and the left and right columns leaves \(5\) interior rows and \(6\) interior columns, so the interior has \(5 \times 6=30\) tiles. 3. Subtract the interior from the whole array: \(56-30=26\). 4. Therefore, there are \(26\) border tiles.

Answer

a) \(7 \times 8=56\) b) \(5 \times 6=30\) c) \(56-30=26\) border tiles
5374353
An \(8 \times 9\) checkerboard uses two kinds of squares. In one row, there are \(5\) of one kind and \(4\) of the other. In the next row, the counts switch. Why are there the same number of each kind in the whole checkerboard?

Hints

- Look at two neighboring rows together. - In the second row, the \(5\) and \(4\) switch places. - How many of each kind are in one pair of rows?

Solution

1. In one row, the counts are \(5\) and \(4\). In the next row, they switch. 2. Each pair of rows has \(9\) squares of each kind. 3. There are \(4\) pairs of rows, so each kind appears \(4 \times 9=36\) times.

Answer

Each pair of rows has \(9\) squares of each kind. There are \(4\) row pairs, so each kind appears \(4 \times 9=36\) times.

All problems may be used, copied and printed free of charge for school and tutoring, including paid tutoring. Commercial adaptations as well as publication or redistribution on the internet are not permitted.

Building free math worksheets