Multiplying with Decimals

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Objective

SWBAT: • Make estimates of multiplication problems involving decimals • Multiply multi-digit decimals together

Big Idea

How do you multiply decimals? Where does the decimal point go? Students work on answering these questions and developing an algorithm for multiplying decimals.

Do Now

5 minutes

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Here, I want students to connect their knowledge of fractions to decimals.  I also want them to apply what they did in the previous lesson (Adding and Subtracting Decimals) to answer number 3.

 I ask students if they have an efficient way of counting the shaded boxes.  For number three I ask for a volunteer to explain how we can use estimation to answer this question.  I want students to recognize that if you are starting with the same amount (1) then 1 – 0.001 has a larger answer because you are subtracting a smaller piece.  If students struggle with this, I ask them first to compare 0.01 and 0.001.  0.01 is one hundredth or 1/100.  0.001 is one thousandth or 1/1000.  Which decimal is smaller?  How do you know?  Then I give an example using whole numbers. Which answer is larger 10 – 2 or 10 – 3?

I also have a volunteer show and explain his/her work showing the procedure of subtracting the decimals.  Some students will struggle with regrouping.  Other students will struggle with figuring out how to line up the numbers.  I have even seen some students stack up 0.01 – 1 (with the 1 under the hundredths place) because they do not realize which number is larger.  I look for these mistakes and review them with the class.

Estimation

5 minutes

Here I want students to quickly review the vocabulary word product and then work on estimating.  I give students a couple minutes to make and write explanations for their estimates.  I stress to students that estimates need to be relatively quick. 

I walk around and monitor student work.  I’m looking to see what strategies students are using.  Some students may round numbers to whole numbers and multiply.  Other students may make connections to fractions or percent.  For example, for number 2 a student may recognize that 0.5 is equal to ½ and find ½ of 10. 

I call one student to explain one of their estimates.  These students are students I observed using a particular strategy as I was walking around.  I am not giving exact answers at this time.  We will work on finding exact products later in the lesson.  It is important that students are able to use their number sense to make reasonable estimates.

Problems

10 minutes

I have a student read problem 1.  I ask students to make an estimate.  Do you think it is more or less than 12 liters?  Why?  Then I ask them to work on the two problems independently.  I stress that there are many ways they can find their answer, it is important that they show their work.  Some students may struggle and this is okay. I want them to do most of the heavy lifting during this lesson so that they understand what is happening when we multiply with decimals.  Students are engaging with MP1: Make sense of problems and persevere in solving them.

As students work, I walk around and monitor their progress.  I am taking note of different strategies students are using.  If a student finishes problems before other students, I prompt them to find other ways of finding the same answer.  If students are struggling with working with decimals I will give them 10x10 grids to represent the decimals.

For problem one, some students may add 3.7 four times.  Other students may double 3.7 and then double their answer.  Other students may multiply 3 x 4 and then add up 0.7 four times.  Other students may make connections with their knowledge of adding or multiplying fractions.  Other students may be familiar with the algorithm.

I ask 3 students to share and explain their work for #1.  If I don’t see one of the strategies I have mentioned, I will present it and ask students if it works and why or why not.  Then I ask students to compare our answer with our estimate.    How did we do?  Does our answer make sense?

What about the decimal point?

15 minutes

A common struggle with students is that they do not know where the decimal point should go in their product.  The first step is that they can estimate to see if their answer makes sense.  I Create Homogeneous Groups using data from the ticket to go from Adding and Subtracting Decimals. 

I have a volunteer read the task and pass out the calculators.  I give students 3-5 minutes to make their estimates and calculate their answers.  I walk around to ensure that students are on tasks and write neatly. 

We come back together as a class.  I ask students to think about what patterns they notice in their answers and what questions they have.  I read the directions for “Comparing Sets” and explain that they will participate in a Think Write Pair ShareStudents are engaging in MP8: Look for and express regularity in repeated reasoning and MP3: Construct viable arguments and critique the reasoning of others.

Then I ask for volunteers to share out with the class.  What patterns did you and your partner notice?  I want students to see the relationship between the number of decimal places in the factors and the number of decimal places in the product.  We test their ideas using the problems in Set A and Set B.

I have a volunteer collect the calculators.

Practice

15 minutes

We do number 1 and 2 together.  I ask students how they want to find the exact product.  I also show them the algorithm.  I stress that students need to use their number sense and the patterns they noticed earlier to decide if their final answer makes sense. 

Students work independently on the rest of the problems.  They can check in with their partner if they are stuck.  I Post A Key so that students can check their work when they finish a page.  I am looking that students are making reasonable estimates and that they are successfully multiplying the decimals using a strategy of their choice. 

If students successfully complete the practice problems, they can work on the college practice problems.

If students are struggling, I may intervene in one of the following ways:

  • Ask them what their estimate is and how they got it.
  • Give them a word problem situation that represents the problem.
  • Pull a small group of students who are struggling to work together.

Closure and Ticket to Go

10 minutes

I begin the Closure by asking students to look at questions 2 and 3.  What do you notice?  How did you find the exact product.  I want students to notice that the answers were the same.  Why is it?  I want students to recognize that 0.6 is equal to 0.60 and that .24 (or 24/100) is equivalent to .240 (240/1000).  Then I ask students to share out how they found their estimates for number 6.  How did you find the exact product? I have a student show and explain his/her work.

With about five minutes left I pass out the Ticket to Go for students to complete independently.  Then I pass out the HW Multiplying with Decimals.