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Level F Lesson 13 Multiply Decimals In lesson 13 the objective is the student will multiply using decimals to the thousandths place. We will have three essential questions that will be guiding the lesson. Number 1, what happens to the decimal point when decimals are multiplied? Number 2, how is multiplying a decimal by a whole number like repeated addition? Number 3, how can you describe the product of two decimals less than 1? The SOLVE problem for this lesson is, Tadarius and Rocko are purchasing a new basketball. They plan to play on the school team and need it to practice. The cost of the basketball is 15 dollars and 50 cents, but it is on sale for 75 hundredths of he cost. How much is the basketball, rounded to the nearest penny? We’re going to start with the S step and the first thing we’re going to do is we’re going back to our word problem and we’re going to underline the question. How much is the basketball, rounded to the nearest penny? The next thing we’re going to do is to complete the thought, this problem is asking me to find in our own words. This problem is asking me to find the sale price of the basketball, rounded to the nearest penny. We’re going to start by modeling multiplication of decimals. When we look at our model here we first want to ask what does the completely shaded model represent? In each of these the completely shaded model represents 1 whole. The partially shaded model represents 3 tenths, because we’ve shaded 3 of the 10 pieces in our model. The next question we want to look at is, what addition problem does the model represent? If this is 1 and this is 3 tenths then each group of models is 1 and 3 tenths. If we’re going to represent that as an addition problem it would be 1 and 3 tenths plus 1 and 3 tenths plus 1 and 3 tenths. Because we know that multiplication can be expressed as repeated addition, we can also write that as a multiplication problem. Since we have three groups of 1 and 3 tenths then we can write our problem as 3 times 1 and 3 tenths is equivalent to 3 and 9 tenths. 3 groups of 1 and 3 tenths is 3 and 9 tenths. When we multiply a decimal with one place value by a whole number then in our product there is going to be one decimal place. What does this number sentence mean? Three groups of 1 and 3 tenths. Based on the model and the solution to the problem, what can you say about the position of the decimal point when multiplying a whole by a tenth? As we said in our model when you’re multiplying a whole number by a decimal with a tenth, there should be one decimal place to the right of the decimal. We have two hundredths boards, we shaded in 20 of the sections of each of those hundredths boards. What addition problem does the model represent? We have 20 hundredths plus 20 hundredths. We can also represent this with a multiplication problem. What multiplication problem does this model represent? Two groups of 20 hundredths or 2 times 20 hundredths and we have a total of 40 hundredths. What does this number sentence mean? Two groups of 20 hundredths. Based on the models and the solutions to this problem, what can you say about the decimal point position when multiplying whole numbers by hundredths? There should be two decimal places to the right of the decimal to show hundredths. We’re going to model multiplication of decimals using an array. An array is going to describe our arrangements of objects in equal rows and equal columns. The problem we’re going to look at is 2 tenths times 6 tenths. Our grid is going to represent 1 whole unit. Our first fraction 2 tenths we’re going to represent with rows. Our second fraction 6 tenths we’re going to represent with columns. If you look at our model we have marked our rows with diagonal lines starting from the right corner and going to the left. Our rows we’ve marked with diagonal lines starting from the left and going to the right. The intersection of those two lines or the boxes that have an X because they’re marked with both lines is our product of these two decimals. The hundredths that are in our intersected area are 12. So the product of 2 tenths times 6 tenths is 12 hundredths. When we’re multiplying a tenth by a tenth there are two decimal places in our product. What does this expression mean? We have 2 tenths of a group of 6 tenths. In example 2 we have 5 tenths times 3 tenths. Remember our model here is representative of 1 whole. Our first number 5 tenths we’re going to represent with rows. So we’re going to need 5 out of the 10 rows marked. Our 3 tenths is our column so we are going to mark 3 columns the other direction diagonally. The intersection of those marks is going to be our product. There are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 5 tenths of 3 tenths is 15 hundredths. Remember when we’re multiplying a tenth by an tenth, one decimal place by one decimal place we’re going to have two decimal places in the answer. What does this expression mean? 5 tenths of a group of 3 tenths. Example 3 we have our hundredths board again, which is going to represent 1 whole. The example we’re going to look at this time is 4 tenths times 7 tenths. Remember our 4 tenths is how many rows we want to mark. So we’re going to come up to our model and we’re going to mark 4 tenths with a diagonal line. 7 tenths is going to be our columns so we’re going to come back to our model and we’re going to model with diagonal lines marking 7 of the columns. The squares that have an intersection of the two lines are our product. We’re going to count those, 1, 2, 3, 4, 5, 6, 7, 8 ,9 ,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28. 28 hundredths are shaded. When we multiply a decimal in tenths by another decimal with tenths our answer is going to be hundredths. What does this expression mean? 4 tenths of a group of 7 tenths Our last model is going to be 9 tenths times 6 tenths. Remember our first number represents the number of rows. So we’re going shade with diagonals 9 tenths of our whole unit. After we shade 9 of our rows, we’re going to shade 6 columns, because we have 6 tenths with our diagonals the other direction. If we look at the intersection of the two diagonals we have 6 columns and 9 rows where they intersect. If we count the squares where the lines intersect there are 54 out of the 100 that are crossed by both diagonal lines. 9 tenths times 6 tenths is 54 hundredths. What does the expression mean? 9 tenths of a group of 6 tenths. We’re going to be moving to multiplying decimals by using power of 10. For this activity your students may use a calculator to find the product and fill in the remainder of the chart. We’re going to go down and start at the fourth row where the multiplicand or the first number in our multiplication is 15 and 35 hundredths. We’re going to multiply that by 1 on our calculator by entering 15.35 times 1 and our product is 15.35 or 15 and 35 hundredths. How many places did the decimal point move in our answer? If you look at the multiplicand and then you look at the product the decimal point did not move at all, so our answer for how many places the decimal point moved is zero. The next column says, which direction did the decimal point move, we’re going to write did not move, because we did not have any movement in the decimal point. Why? Your answers here from your class may vary but when you multiply any number times 1 it’s not going to change the value. Let’s move up the chart to row three. We’re going to multiply 15 and 35 hundredths by 10. On the calculator I’m going to enter 15.35 times 10. The product is going to be 153 and 5 tenths or 153.5. How many places did the decimal point move? It moved one place. If you look in our original multiplicand the decimal point was after the first 5. It’s now after the 3. Which direction did the decimal point move? It moved to the right. Why did that happen? Well 10, one possible answer your students might offer is 10 has one zero. So from the 1 to the 10 our decimal point moved 1 place to the right. So in our product the decimal point also moved one place to the right. Let’s go up to row two. We’re going to multiply 15.35 or 15 and 35 hundredths by 100. In my calculator I’m going to enter 15.35 times 100. My product is going to be 1,535. My decimal point moved 2 places to the right from my original answer. I’m going to write the direction of right. Back again to the decision of why? A possible answer here with the student discussion is going to be from the 1 to the hundredths we’ve moved two places by adding 2 zero’s. So that when we multiply that by 100 we’re going to move 2 decimal places to the right. At the top row we’re going to enter 15.35 into our calculator and multiply that by 1000. Our product is 15, 350. We’ve now moved our decimal point 3 places to the right. The reasoning because we’ve added 3 zeros, the decimal point has moved 3 places to the right. Let’s look down at row 5 in our chart. We’re going to multiply 15.35 hundredths times 0.1. So I’m going to enter in my calculator 15.35 times 0.1. My product is now 1.535 or 1.535. My decimal point this time has moved one place value but the direction is now to the left. Your decision on why that’s happening could include the fact that you have multiplied by a tenth so the decimal point has moved 1 place to the left. So in the product it also has to move 1 place to the left. In the next row we have 15 and 35 hundredths or 15.35. We’re going to enter that into our calculator 15 point 35 times 0 point 0 1. Our product is going to be 0.1535, Our decimal point has now moved 2 places to the left. Why? Because when we multiply by hundredths our decimal point is going to move 2 places to the left. The last row we have 15 and 35 hundredths times 1 thousandth. We’re going to enter in our calculator 15.35 times 0.001. Our answer is going to be 1.01535. We’ve moved our decimal point this time 3 places to the left. When you multiply by 1 thousandth the decimal point will move 3 places to the left. We have the example 2 tenths times 6 tenths which is equivalent to 12 tenths. This expression means 2 tenths of a group of 6 tenths. The decimal point is one place to the left in the 2 tenths, one place to the left in the 6 tenths, and in our product it’s two places to the left in our 12 hundredths. Tenths times tenths equals hundredths or 0.1 times 0.1 equals 0.01. Tenth times tenth equals hundredths. On this side of the equals sign we have one arrow and a second arrow, so we have two arrows on this side. On this side of the equal sign we also have two arrows. What place value is indicated to the right of the decimal in the numbers? Tenths. What happened to the place value to the right of the decimal in the product? The place value now is hundredths. Is the decimal in the product larger or smaller than the two factors? It is smaller. Hundredths is smaller than tenths. Why is that? Because you are multiplying parts of wholes. Our next example is 2 hundredths times 6 tenths which is going to equal a product of 12 thousandths. Hundredths times tenths equals thousandths or 0.01 times .1 equals 0.001. Hundredths times tenths equals thousandths. You’ll notice that on the left side of our equation or our equal sign we have 2 arrows to indicate hundredths, 1 arrow indicates tenths. So we have 3 place value moves on the left side of the decimal. On the right side of the decimal we have the same number of place values. 1, 2, 3, thousandths is a movement of 3 decimal places. What place value is indicated to the right of the decimal in our numbers? Hundredths and tenths. What happened to the place value to the right of the decimal in the product? The place value in the product is thousandths. Compare the decimal places in the product to the decimal places in the number sentence. As we counted before there are a total of 3 places to the right of the decimal in the problem and the product has 3 decimal values to the right of the decimal. What is the rule for multiplying decimals based on the problems above? Count the number of decimal places to the right of the decimal point in the factors and match that number in the product. We’re going to move to multiplying decimals without models. The first thing I’m going to do is multiply my 2 factors and ignore the decimal points. 2 times 2 is 4, 2 times 3 is 6, 2 times 0 is 0. Now I’m going to go back to my first factor and I’m going to count the number of decimal places. I have 1, 2 decimal places. In my second factor I have 1 decimal place. That’s a total of 3 decimal places in my problem. We know that our problem and our product have to have the same number of decimal places. So when we’re placing our decimal point in our product we’re going to move 3 places to the right. 32 hundredths times 2 tenths is 64 thousandths. Our second problem is 3 and 8 tenths times 7 tenths. The first thing we’re going to do is write our problem vertically and then we’re going to multiply ignoring our decimals. 7 times 8 is 56, we put our 6 down and we carry our 5, 7 times 3 is 21 plus our 5 is 26. Now we’re going to place our decimal in the product. We have 1 decimal place in our first factor and 1 in our second, that gives us a total of 2 decimal places. We know that we have to have the same amount of decimal places in our product. The decimal point is placed 2 decimal places to the left. We have 2 decimal places in our product and 2 in our factors. 3 and 8 tenths times 7 tenths is 2 and 66 hundredths. In your decimal foldable you should include the following information in the section called multiplication of decimals. We have a sample problem here of 10 and 51 hundredths times 1 and 2 tenths. Factor and factor and product are the terms that identify the 2 numbers we multiplying and the answer. Step 1, we going to multiply. Notice there’s no decimal in the answer, we’re just multiplying the 2 factors. Step 2, we’re going to count the number of places after the decimal point. In the first factor we have 2 places. In the second factor we have 1 for a total of 3 places after the decimal point. In step 3, we’re going to place the decimal point into the answer the same number of places. 1, 2, 3. Let’s go back to the SOLVE problem from the beginning of the lesson. Tadarius and Rocko are purchasing a new basketball. They plan to play on the school team and need to practice. The cost of the basketball is 15 dollars and 50 cents, but it is on sale for 75 hundredths of the cost. How much is the basketball, rounded to the nearest penny? We’ve already completed the S step. We’ve underlined the question and we’ve completed the idea of this problem is asking me to find, the sale price of the basketball, rounded to the nearest penny. In our O step we’re going to start by identifying our facts. Tadarius and Rocko are purchasing a new basketball, fact. They plan to play on the school team and need to practice, fact. The cost of the basketball is 15.50, fact, but it is on sale for 75 hundredths of the cost, fact. The next thing we’re going to do is go back to our word problem and eliminate the unnecessary facts. We can identify a fact that was unnecessary by looking back at our question and see if it is going to help us find how much is the basketball, rounded to the nearest penny. Tadarius and Rocko are purchasing a new basketball. We do not need to know that fact in order to complete our word problem. They plan to play on the school team and need to practice. Another unnecessary fact. The cost of the basketball is 15 dollars and 50 cents, that is a necessary fact, but it is on sale for 75 hundredths of the cost. Those are both necessary facts and we’re going to list those in our O step. In our L step we’re going to start by choosing an operation or operations. We know we have the original cost but it’s on sale for 75 hundredths of the cost. Since we’re going to be finding a decimal part of our original number, we’re going to multiply. Then we’re going to write in words our plan of action. We’re going to multiply the cost of the basketball by the decimal number which represents the discount. We’re moving to the V step we’re going to verify our plan of action. We’re going to first begin by estimating our answer. We know our original cost is 15.50 and we’re going to want 75 hundredths of that, which is three-fourths of that. So our estimate is about 12 dollars. When we carry out our plan we’re going to multiply 15 dollars and 50 cents times 75 hundredths. Remember in the lesson we talked about when you multiply hundredths times hundredths then we should have 4 place values in our answer. We have 2 place values in this factor, 2 place values in this factor. We have 4 place values in our answer. We also needed to in our plan round that number to the nearest penny because we’re dealing with money. When we move to the E step we first ask our self if our answer make sense? We needed to have a money amount, and our answer is 11 dollars and 63 cents. So the answer is yes, because we’re looking for the cost of the basketball. Is my answer reasonable? I’m going to look at my answer and compare it to the estimate. My estimate was about 12 dollars. So the answer is yes, because it is close to our estimate of the about 12 dollars. Is my answer accurate? I’m going to go back and double check my work of multiplication. Again, this is a good place to use a calculator so that you can double check your work. Write your answer in a complete sentence. The sale price of the basketball is 11 dollars and 63 cents, rounded to the nearest penny. Now we’re going to go back and answer the essential questions from the beginning of our lesson. What happens to the decimal point when decimals are multiplied? The number of decimal places in the factors is equal to the number of decimal places in the product. How is multiplying a decimal by a whole number like repeated addition? It is a shorter way to add decimals. How can you describe the product of two decimals less than 1? The decimal value in the product is smaller than either of the two decimal factors.