**Module 2 Lesson 6** Objective: Connect area diagrams and the distributive property to partial products of the standard algorithm without renaming.

**Fluency – Multiply Mentally**

**Fluency – Multiply by Multiples of 100**

**Fluency – Multiply Using the Area Model** • 43 x 12 • 243 x 12 • 312 x 23 10 + 2 243 2430 486 2430 + 486 = 2,916 10 + 2 43 20 + 3 430 86 312 6240 936 430 + 86 = 516 6240 + 936 = 7,176

**Application Problem** • Scientists are creating a material that may replace damaged cartilage in human joints. This hydrogel can stretch to 21 times its original length. If a strip of hydrogel measures 3.2 cm, what would its length be when stretched to capacity? Give the final answer in a statement of solution. Check to make sure your answer is reasonable by rounding 3.2 to the nearest whole number. 32 X 21 32 +640 672 tenths tenths = 67.2

**Concept Development – Problem 1** • 64 x 73 • Method 1 Area Model • Please draw a rectangle in you math notebook. • Write 64 x 73 above rectangle. Let’s represent units of 73 • How many seventy-threes are we counting? • 64 • How can we decompose (break apart)(distributive property) to make out multiplication easier? Show this on your area model? • 60 + 4 (easiest way) • Can we do 73 x 4 and 73 x 60 easy mentally? • Yes for some people, but not for all students. • Can we decompose 73? • Yes, 70 + 3 • When splitting 73 and 64 how many total boxes do we need in our rectangle and why? • 4 because we both are decomposed we have 4 numbers to place.

**Concept Development – Problem 1** • Sample area model. • Now let’s fill in the numbers 60 + 4 and 70 +3. • Now let’s add all the numbers. 70 + 3 4 280 12 + 4200 180 60 4200 + 280 + 180 + 12 = 4,672

**Concept Development – Problem 1** • Now let’s look at a different method (Method 2 – Standard algorithm.) • How would we write the problem for the standard algorithm? • Up and down similar to addition and subtraction problem. • What is the first step? • Times 73 by 4. • Where will our answer go? • Right under the problem. 73 X 64 1 73 X 64 292

**Concept Development – Problem 1** • What is the next step? • The value of 60 x 73. • Why 60 x 73 and not 64 x 73? • Because we have already done 4 x 73. • Where will our answer start and why? • Decompose 60 • 6 tens • What is 6 tens times 3? • 18 tens • What is 18 tens in standard form? • 180 • How many hundred(s) do I have and how many tens and how many ones do I have? • 1 hundred, 8 tens, and 0 ones 1 73 X 64 292

**Concept Development – Problem 1** • 1 hundred, 8 tens, and 0 ones • Where will I write the 0 ones? • Under the 2 in the ones place value. • Where will I write the 8 tens? • Under the 9 • Where will I write the 1 and why? • Above the 7 because we have to carry or just under or above the 2 in the hundreds place value (small to show our carry.) • Now we will do 6 tens (60) times 7 tens (70) • What is 6 tens times 7 tens? • 42 hundred • Where will our answer go? • The 2 will go in the hundreds and then we have to place the 4 in the thousands place because 42 hundred is 4200. 1 1 1 73 X 64 292 73 X 64 292 +4280 4672

**Concept Development – Problem 1** • Another way to look at the problem is a cross between the area model and standard algorithm is to write each answer on a separate line and then add them all up. • For example: • First do 3 x 4 and write down the answer • Next do 7 tens (70) x 4 ones (Remember to line up place values for adding later. 73 X 64 12 73 X 64 12 280

**Concept Development – Problem 1** • Next move to the second line. • We will start in the tens place value because we have already done the ones. • 6 tens (60) x 3 • Next do 6 tens (60) x 7 tens (70). • (Remember to line up place values for adding later. 73 X 64 12 280 180 73 X 64 12 280 180 +4200 4,672

**Concept Development – Problem 2-3** • Complete the problems using the standard algorithm and using the area model. • 814 x 39 = • 624 x 82 = 24,000 + 7,200 + 300 + 120 + 90 + 36 = 31,746 48,000 + 1,600 + 1,200 + 320 + 40 + 8= 51,168 9 2 30 80 + + 4 120 36 4 320 8 + + 10 20 300 1600 90 40 + + 800 600 24,000 48,000 7,200 1200

**Concept Development – Problem 2-3** • Complete the problems using the standard algorithm and using the area model. • 814 x 39 = • 624 x 82 = Step 1 multiply 624 by 2 624 X 82 1248 1 3 Step 1 multiply 814 by 9 814 X 39 7326 3 1 Step 2 multiply 624 by 80 624 X 82 1248 + 49920 51,168 1 Step 2 multiply 814 by 30 814 X 39 7326 + 24420 31,746

**Concept Development – Problem 4-5** • Complete the problems using the standard algorithm or using the area model. • 391 x 59 • 874 x 63 15,000 + 2,700 + 4,500 + 810 + 50 + 9 = 23,069 9 50 + 1 2 874 X 63 2622 1 50 9 + 4 2 874 X 63 2622 +52440 55,062 90 4500 810 + 300 15,000 2,700

**Problem Set** • Draw an area model, and then solve using the standard algorithm. • 48 x 35 648 x 35 • Solve using the standard algorithm. • 758 x 92 958 x 94 • Carpet costs $16 a square foot. A rectangle is 14 feet long by 16 fee wide. How much would it cost to carpet the floor? • General admission to The American Museum of Natural History is $19. • If a group of 125 students visits the museum, how much will the group’s tickets cost? • If the group also purchases IMAX movie tickets for an additional $4 per student, what is the new total cost of all the tickets? Write an expression that shows how you calculated the new price.

**Exit Ticket** • Draw an area model, and then solve using the standard algorithm. • 78 x 42 • 783 x 42

**Homework** • Draw an area model and solve using the standard algorithm. • 27 x 36 527 x 36 • Solve using the standard algorithm. • 649 x 53 758 x 46 • Solve using an area model. • 496 x 53 529 x 48 • Each of the 25 students in Mr. McDonald’s class sold 16 raffle tickets. If each ticket cost $15 how much money did Mr. McDonald's students raise? • Jayson buys a car and pays by installments. Each installment is %567 per month. After 48 months, Jayson owes$1,250. What was the total price of the vehicle?