Writing Equations Using the Distributive Property MAFS.3.OA.2.5

Lesson Opening If you combined the arrays below, what multiplication equation could represent it? If the two equations below were combined, what multiplication equation could represent it? (Use square tiles to model if needed) 2 × 5 = 10 2 × 3 2 × 3

2 × 3 2 × 3 2 × 6 = 12 2 × 10 = 20 or 4 × 5 = 20 Lesson Opening If you combined the arrays below, what multiplication equation could represent it? If the two equations below were combined, what multiplication equation could represent it? (Use square tiles to model if needed) 2 × 5 = 10 2 × 3 2 × 3 2 × 6 = 12 The second problem is challenging. If students need square tiles, hand them out. 2 × 10 = 20 or 4 × 5 = 20

Yesterday, we learned… 3 × 2 3 × 2 3 × 4 can be broken apart

Here is how we would write it as an equation: Yesterday, we learned… 3 × 2 3 × 2 3 × 4 can be broken apart Here is how we would write it as an equation: + 3 x 4 = (3 × 2) (3 × 2)

3 x 4 = (3 x 2) + (3 x 2) 3 x 2 3 x 2 Put parentheses around each multiplication array to show that it NAMES the array. Also, if we solve the equation we have to do the operation inside the parentheses first!

3 x 4 = (3 x 2) + (3 x 2) = 12 3 x 2 3 x 2 Then, put an addition sign in between the arrays to show that you ADD the products of the array to find the TOTAL product.

Write the equation for this array: 4 x 3 4 x 2 4 x 5 = (4 x 2) + (4 x 3)

Write the equation for this array: 4 x 3 4 x 2 4 x 5 = (4 x 2) + (4 x 3) 2 3 The 5 was decomposed into 2 and 3. Then each part was multiplied by 4.

Write the equation for this array: 3 x 5 6 x 5 = (3 x 5) + (3 x 5) = 30 3 3 3 x 5 The 6 was decomposed into 3 and 3. Then each part was multiplied by 5.

Here’s another way to think about it… 6 tens 6 6 x 10 4 tens 4 2 2 tens 4 x 10 2 x 10 This is a number bond that shows 6 decomposed into 4 and 2. This is a number bond that shows 6 tens decomposed into 4 tens and 2 tens. 6 x 10 = (4 x 10) + (2 x 10)

7 Let’s try another one. 7 x 4 7 x 4 = (5 x 4) + (2 x 4) 5 x 4 2 x 4 fours This is a number bond that shows 7 fours decomposed into 5 fours and 2 fours. 7 x 4 7 x 4 = (5 x 4) + (2 x 4) 5 fours 2 fours 5 x 4 2 x 4

On your dry-erase board, write an equation for the following: 8 threes 8 x 3 = (5 x 3) + (3 x 3) = 24 8 x 3 or 8 x 3 = (4 x 3) + (4 x 3) = 24 or 5 threes 3 threes 8 x 3 = (6 x 3) + (2 x 3) = 24 5 x 3 3 x 3 or 8 x 3 = (7 x 3) + (1 x 3) = 24

Use the Distributive Property to write an equation (think of a number bond or an array). 6 × 6 (3 × 6) + (3 × 6) = 36 (4 × 6) + (2 × 6) = 36 (5 × 6) + (1 × 6) = 36 (6 × 3) + (6 × 3) = 36 (6 × 4) + (6 × 2) = 36 (6 × 5) + (6 × 1) = 36

Use the Distributive Property to write an equation (think of a number bond or an array). 7 × 8 (5 × 8) + (2 × 8) = 56 (4 × 8) + (3 × 8) = 56 (6 × 8) + (1 × 8) = 56 (7 × 4) + (7 × 4) = 56 (7 × 5) + (7 × 3) = 56 (7 × 6) + (7 × 2) = 56

Your Turn Draw the array that represents the multiplication expression. Use two different colors to represent how you can break apart the array. Write the equation that represents how you broke apart the array. Don’t forget to write the product.

3 × 12 (5 × 3) + (5 × 3) + (5 × 2) = 40 Bonus Questions The following equation represents an array broken apart into 3 smaller arrays. What is the equation for the original array? How do you know? Can you break apart this array into 3 smaller arrays? What would the equation be? (5 × 3) + (5 × 3) + (5 × 2) = 40 3 × 12

3 × 12 (5 × 3) + (5 × 3) + (5 × 2) = 40 Bonus Questions The following equation represents an array broken apart into 3 smaller arrays. What is the equation for the original array? How do you know? Can you break apart this array into 3 smaller arrays? What would the equation be? rows columns rows columns rows columns (5 × 3) + (5 × 3) + (5 × 2) = 40 Since there are 5 rows in each of the arrays, I know the array has 5 rows. So then I add up all the columns. 3 + 3 + 2 = 8. So the original array is 5 × 8. 3 × 12

Bonus Question #2 Answer Are there other ways? Bonus Question #2 Answer Choose to decompose either the 3 or the 12. 3 × 12 12 threes 12 threes 3 twelves 10 threes 1 three 1 three 4 threes 1 twelve 4 threes 1 twelve 4 threes 1 twelve (10 x 3) + (1 x 3) + (1 x 3) = 36 (4 x 3) + (4 x 3) + (4 x 3) = 36 (1 x 12) + (1 x 12) + (1 x 12) = 36

Exit Ticket: Which expressions could represent the array? 6 × 4 4 × 6 (6 × 4) + (6 × 4) (3 × 4) + (3 × 4) (6 × 2) + (6 × 2) (6 × 4) × (6 × 2)