Download LEVEL F LESSON 13 – MULTIPLY WITH DECIMALS

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.