Engage NY Math Module 2 Lesson 11: Multiply decimal fractions by multi-digit whole numbers through conversion to a whole number problem and reasoning about the placement of the decimal.

Multiply Decimals 3 x 3 = 4 x 3 = 0.06 x 2 = 0.3 x 3 = 5 x 5 =

Multiply then Divide by the Same Number 3 x 4.1 = ? 12.3 12.3 x 10 ÷ 10 is? 3 x 4.1 x 1 is?

Multiply then Divide by the Same Number 3 x 2.4 = ? 7.2 7.2 x 10 ÷ 10 is? 3 x 2.4 x 1 is? 3 x 4 x 17.6 ÷17.6 is? 12 How do you know this is true? Discuss it with your tablemates.

Application Problem Use pictures, numbers, and words to complete this problem. Mr. Mohr wants to build a rectangular patio using concrete tiles that are 12 inches square. The patio measures 12.5 feet by 43 feet. What is the area of the patio? How many concrete tiles will he need to complete the patio? 1 2 5 X 4 3 3 7 5 +5 0 0 0 5 3 7 5 (tenths) The patio’s area is 537.5 ft² so Mr. Mohr will need to buy 538 tiles because each tile is 1 ft². (Tenths) = 537.5

Concept Development – Problem 1: 7.38 x 41 Compare this problem to our application problem. It’s still multiplication of a decimal by a whole number. The decimal in the Application Problem was tenths and this is hundredths. Estimate the product. 7 x 40 = 280 Predict whether our estimate is greater than or less than the actual product. Less than because both factors were rounded to numbers less than the actual factors. Our actual answer will be more than 280, but it will still be in the hundreds. We have 41 units of 7.38. I’d like to rename 7.38 as only hundredths. How many hundredths would that be? How do you know? 738 hundredths because 7 is 700 hundredths plus another 38 hundredths equals 738 hundredths. 7 and 38 hundredths times 100 equals 738 hundredths.

Concept Development – Problem 1: Let’s use an area model to find the actual product of this expression. Decompose those 738 hundredths into expanded for along the length of our rectangle. Write hundredths out to the right to remind us that we’ve named 7.38 hundredths. Our rectangles width is 41 whole units. Decompose 41 into expanded form along the width. What two partial products do these rows represent? 1 x 738 hundredths and 40 x 738 hundredths. Find the partial products and the final product. 700 30 8 hundredths hundredths 700 30 8 28,000 1,200 320 738 +29,520 30,258 1 40 hundredths

Concept Development – Problem 1: We found that we have 30,258 of what unit? Hundredths We need to write this in standard form. How can our estimate help us convert our product back to wholes and hundredths? The estimate told us that our answer was in the hundreds, not the ten-thousands or the thousands. 30,258 is about 100 times as large as our estimate said the real answer should be, so we need to divide by 100 to make the answer make sense. What is 30,258 hundredths written in standard form? 302.58 Let’s solve this same problem using the algorithm. Yesterday we rewrote our first factor as a whole number with the unit name to the right. Today let’s think about the units without removing the decimal from our first factor. We see 7.38, but we think 738 hundredths. Multiply 738 x 41 and find the product. Look back at our area model to confirm the partial products in your algorithm. 738 x 41 +29,520 30,258

Concept Development – Problem 1: This product is 100 times as large as the product of our original problem. What should we do to adjust this product so that it answers our original problem of 7.38 x 41? We should divide by 100. Let me record what I hear you saying. So is our adjusted product of 302.58 reasonable given our estimate? Yes.

Concept Development – Problems 2-3: In your journal, estimate the product first, then solve using the standard algorithm. 8.26 x 128 8.26 x 128 ≈ _____ x _____ = ______ 8 2 6 x 1 2 8 82.51 x 63 82.51 x 63 ≈ _____ x _____ = ______ 8 2 5 1 x 6 3

Exit Ticket 407.61 Explanation: 6.47 is the same as 647 hundredths so just divide by 100.

Problem Set Display Problem Set on the board. Allow time for the students to complete the problems with tablemates.

Homework Task