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Understand multiplication as groups and arrays

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5191933
In \(14 \times 6 = 84\), name the mathematical terms for \(14\), \(6\), and \(84\).

Hints

- Identify the operation shown by \(\times\). - The numbers being multiplied share one term. - The result of multiplication has a different term.

Solution

1. The expression is a multiplication equation. 2. The numbers \(14\) and \(6\) are factors. 3. The result \(84\) is the product.

Answer

\(14\) and \(6\) are factors. \(84\) is the product.
5158043
Imagine each dot array and write a matching multiplication equation. a) How many dots are in an array with \(1\) row and \(10\) columns? b) How many dots are in an array with \(6\) rows and \(1\) column? c) An array has \(5\) columns but \(0\) rows. How many dots are shown? What do you notice about the results in parts a and b compared with the other factor?

Hints

- Rows run horizontally, and columns run vertically. - Picture an array with only one row. - Can an array contain dots if it has no rows?

Solution

1. One row of \(10\) dots gives \(1 \times 10 = 10\). 2. Six rows of \(1\) dot give \(6 \times 1 = 6\). 3. Zero rows contain no dots, so \(0 \times 5 = 0\). 4. Multiplying by \(1\) leaves the other factor unchanged.

Answer

a) \(1 \times 10 = 10\) b) \(6 \times 1 = 6\) c) \(0 \times 5 = 0\) Observation: A number multiplied by \(1\) stays the same.
5158053
Complete each statement and equation. - If you take \(5\) groups of \(0\) counters, you have ... counters in all. \(\left(5 \times 0 = \square\right)\) - If you take \(0\) groups of \(3\) counters, you have ... counters in all. \(\left(0 \times 3 = \square\right)\) - If you take \(1\) group of \(8\) counters, you have ... counters in all. \(\left(1 \times 8 = \square\right)\) - If you take \(4\) groups of \(1\) counter, you have ... counters in all. \(\left(4 \times 1 = \square\right)\) What rule can you write about multiplication by \(0\)?

Hints

- Picture the counters in equal groups. - How many counters are there when every group is empty? - What do the equations with a factor of \(0\) have in common?

Solution

1. The completed equations are \(5 \times 0 = 0\), \(0 \times 3 = 0\), \(1 \times 8 = 8\), and \(4 \times 1 = 4\). 2. Multiplying any number by \(0\) gives a product of \(0\), no matter which factor is \(0\).

Answer

- \(5\) groups of \(0\) counters make \(0\) counters. \(\left(5 \times 0 = 0\right)\) - \(0\) groups of \(3\) counters make \(0\) counters. \(\left(0 \times 3 = 0\right)\) - \(1\) group of \(8\) counters makes \(8\) counters. \(\left(1 \times 8 = 8\right)\) - \(4\) groups of \(1\) counter make \(4\) counters. \(\left(4 \times 1 = 4\right)\) Rule: The product of any number and \(0\) is \(0\).
5206083
Write the repeated addition as a multiplication expression, then find the total: \(12\text{ cents} + 12\text{ cents} + 12\text{ cents} + 12\text{ cents} + 12\text{ cents} + \$0.12 + \$0.12\).

Hints

- Express every amount in cents. - Count how many equal addends there are. - Replace the repeated addition with one multiplication expression.

Solution

1. Convert each dollar amount to cents: \(\$0.12 = 12\text{ cents}\). 2. There are \(5 + 2 = 7\) equal addends of \(12\) cents. 3. Write the multiplication expression: \(7 \times 12\text{ cents}\). 4. Calculate: \(7 \times 12\text{ cents} = 84\text{ cents}\), which is \(\$0.84\).

Answer

\(7 \times 12\text{ cents} = 84\text{ cents} = \$0.84\)
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

- Identify 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) \(4 \times 6 = 24\) b) \(3 \times 8 = 24\) Both arrays contain \(24\) dots.
5373453
Mara says, “Both arrays show the same multiplication equation because both have four rows.” Check her statement and write a precise correction.
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 determine 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
The array has a blue left half and an orange right half. How many blue dots and how many orange dots are there? Explain without counting every dot of each color.
Figure for problem 537395

Hints

- Examine just one row. - The same left-right split repeats in all \(6\) rows.

Solution

1. Each of the \(6\) rows contains \(4\) blue dots on the left and \(4\) orange dots on the right. 2. Therefore, each color appears \(6 \times 4 = 24\) times.

Answer

There are \(24\) blue dots and \(24\) orange dots. Each row contains \(4\) dots of each color, and the same left-right split repeats across \(6\) rows.
5165233
Anna is making a square checkerboard design with alternating blue and yellow square tiles. The design has \(8\) rows with \(8\) tiles in each row. a) How many tiles are in the design altogether? b) How many tiles are blue? c) Each tile has a side length of \(10\,\text{cm}\). What is the side length of the entire square design?

Hints

- Think of the design as an array with equal numbers of rows and columns. - In an \(8 \times 8\) alternating pattern, the two colors occur equally often. - To find the full side length, count how many tile side lengths fit along one edge.

Solution

1. The \(8 \times 8\) array contains \(8 \times 8 = 64\) tiles. 2. In an even-sized checkerboard pattern, half the tiles are blue. Thus, \(64 \div 2 = 32\) tiles are blue. 3. One side contains \(8\) tiles, each \(10\,\text{cm}\) long. The side length is \(8 \times 10\,\text{cm} = 80\,\text{cm}\).

Answer

a) \(64\) tiles b) \(32\) blue tiles c) \(80\,\text{cm}\)
5374343
A rectangular array has \(7\) rows and \(8\) columns of square tiles. The tiles on the outside border are red, and the tiles completely inside are blue. Find the number of border tiles and interior tiles without counting them one at a time.
Figure for problem 537434

Hints

- Remove the top and bottom rows and the left and right columns to describe the interior array. - Find the total number of tiles, then subtract the number of interior tiles.

Solution

1. The interior has \(7-2=5\) rows and \(8-2=6\) columns, so it contains \(5\times6=30\) tiles. 2. The whole array contains \(7\times8=56\) tiles. 3. Therefore, the border contains \(56-30=26\) tiles.

Answer

There are \(26\) border tiles and \(30\) interior tiles.
5374353
An \(8 \times 9\) array alternates between two categories in a checkerboard pattern. In one row, the first category appears \(5\) times and the second appears \(4\) times; in the next row, the counts switch. Why do the two categories appear equally often in the entire array?

Hints

- Group neighboring rows in pairs. - In each pair, the \(5\)-and-\(4\) counts switch and balance.

Solution

1. In one row, the two category counts are \(5\) and \(4\). In the next row, the counts switch. 2. Each pair of neighboring rows therefore contains \(9\) items from each category. 3. There are \(4\) pairs of rows, so each category appears \(4 \times 9 = 36\) times.

Answer

Each pair of neighboring rows contains \(9\) items from each category. Four row pairs give \(4 \times 9 = 36\) items in each category.

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