Multiplying with Decimal Quantities

1. 473 • 7 =     (Answer: 3,311) 
   
• With this problem, I am looking to ensure that students are able to regroup properly, know their multiplication tables, and I am looking to see if they know the proper algorithm for multiplying a multi-digit quantity by a single digit quantity.
2. 3,249 • 21 =   (Answer: 68,229) 
   
• With this problem, I am looking to to see if my students are able to regroup, know their multiplication tables, and I am looking to see if they know the proper algorithm for multiplying a mult-digit quantity by a double digit quantity... Specifically, I am looking to see if they know that they have to annex a zero when they multiply the multiplicand by the tens place digit in the multiplier to denote that they are actually multiplying by 20 NOT 2.
3. 9,514 • 355 =   (Answer: 3,377,470) 
   
• With this problem, I am looking to to see if my students are able to regroup, know their multiplication tables, and I am looking to see if they know the proper algorithm for multiplying a mult-digit quantity by a triple digit quantity... Specifically, I am looking to see if they know that they have to annex two zeros when they multiply the multiplicand by the hundreds place digit in the multiplier to denote that they are actually multiplying by 300 NOT 3.

To begin this discussion, I will first ask the following question:

• Do I have to line up the place values to multiply decimal quantities? (Students should answer "No") Should I receive a "yes" response then, I will show students their misconception using the example in the opening. I will multiply 0.3 by 5 and then, I will multiply 0.3 by 5.0 and ask, "Is there a difference?" The students should be able to see that either way, you will arrive at the same solution. That being the case, they should understand that lining up the place values is unnecessary.

Once arriving to the desire answer through strategic questioning, I will then ask:

• Why don’t I have to line up the decimals when multiplying?

I am looking for my students to conclude that the reason is due to its multiplicative nature. Because, it is essentially repeated addition, the place values are, in essence, already lined up. To prove this thinking, I will demonstrate this by writing the same decimal quantity over and over directly beneath the previously written quantity to show how the place values line up.

Then, I will write the problem 2.54 • 1.7 =  and ask…

• What if we are not multiplying by a whole number but by another decimal quantity? Teacher will instruct students to multiply the two quantities as if they had no decimal.

Then, I will ask…

• What must we do with the decimal? And Why?

I am looking for the students to know that the decimal must be moved according to the number of place values behind the decimals in the quantities being multiplied. I want my students to make the connection of place value to the placement of the decimal in the product.

I will have my students complete five problems then we will go over those five problems as a transition into independent practice.

Guided Skill Practice: Where Does the Decimal Go?

• For this guided practice, I will have my students take out the sheet of paper that they completed the Warm Up exercise on. Using that same piece of paper, I will have the students copy down the three problems indicated below. The students should recognize that these are the same problems except for the fact that the quantities contain decimals. For this reason, students should be able to understand that they can simply copy their answers form the Warm Up then, simply properly place the decimal to provide the correct solution to the problems.
• Should the students not come to these conclusions on their own, I will ask some questions to prompt them to come to these conclusions. I would ask them, "So, do you see any correlation between these problems and the problems you worked on for the warm up?" Someone should answer yes. After a students answers "yes" I will then ask them what correlation does he/she sees? And, what does that mean for you in completing the problems for guided practice?

1. 4.73 • 7 =
2. 32.49 • 2.1 =
3. 9.514 • 3.55 =

Guided Practice:

1. Andrew has 8 wooden boards that are 2.25 feet each. How many feet of wooden board does Andrew have altogether?
• The students should be able to see that multiplication is indicated in the fact that the problem presents 8 wooden boards that are all the same length. That being the case, we have the same length 8 times letting us know that we should multiply in this problem.
2. Julie bought 3 outfits and a pair of shoes at a department store. The subtotal for all of the items she purchased was $127.96. If she paid $0.07 in tax for every dollar spent, how much did Julie pay in tax? What was the grand total of the items Julie bought?
• This problem is purposely worded so that the students can see that the problem is asking how much is $0.07 for every dollar Julie spent. For this reason, students should, at least be able to see that they are figuring out $0.07 for every $127 which, would indicate that you must multiply $127 times $0.07.
• Be aware that many students will get confused as to what to do with the cents portion... the 96 cents. It would be at this point that I would go ahead and explain that the $0.07 for every dollar in tax is the same as saying 7% of every dollar in tax. I would explain that what this is telling us is that Julie is paying 7 pennies in tax for every 100 pennies she spends on merchandise. I would then show them the fraction 7/100. The reason I would show them the fraction is because many students will remember that a percentage can be written as a fraction by simply placing that percentage over a denominator of 100.
• After ensuring that the students understand the concept of tax and percent, I would then ask, "Then, is it possible to have 7% of 96 cents?" The students should answer "yes." From that point, we should be able to figure out how many cents Julie has to pay in tax for 96 cents using multiplication.

To explore the concepts that were taught in today's lesson, I will have students pretend to be waiters/waitresses as well as customers at a restaurant. The students will work with a partner. Each student will place an “order” using a menu provided by the teacher. They will then give their “order” to their partner to figure out the subtotal and grand total using a gratuity of 15% and a tax of 7%.

The scenario should be presented as follows. You are a waiter/waitress at Bistro 305. You have one customer left at a table. The customer has just finished their meal and is ready to go. But, when you go to ring up the cost of their meal the register shorts out and will no longer work! You must calculate your customer’s meal taking into account a 15% gratuity and a 7% tax.

While the students are completing this task, I will travel the classroom looking for misconceptions to clear up. I will specifically be looking to ensure that students are understanding and demonstrating the following:

• When adding the decimal quantities, they understand that they must line up the place values and properly regrouping.
• When multiplying the decimal quantities, I will make sure that the students are properly following the steps of the multiplication algorithm for multiplying multi-digit decimal quantities (i.e. regrouping properly, annexing zeros when necessary...etc).
• Ensure that the students don't confuse when they should line up the decimal in their solution (adding) and when they should count the digits behind the decimals to determine where the decimal belongs in their solution (multiplication).

After calculating the Grand Total of the meal, my students will then switch papers so that they have been given their “bill” for what they have ordered. After switching papers, they will then be given three minutes to check their partner’s work. If a student determines that their partner has made a mistake or more than one mistake, the student must identify the mistake(s) write them on a sheet of paper and tell what their partner should have done instead.

Then, I will give my students approximately 4 minutes to confer with their partner concerning the problem, what they encountered while solving the problem and their solutions. I will provide each set of partners with a large piece of construction paper upon which each partner will display their “bill” and together they will write things that they learned during this process, difficulties that they came across, and anything else that they deem significant during the completion of this task.

Then, I will choose sets of partners to come to the front of the class to present their work. The class will be given time to ask questions and discuss with the presenting partners.

TOTD: You bought a new video game controller for $19.97. How much will you pay in total for the controller if you pay a tax of 6%? (Tax would be $1.19 therefore, you would pay a total of $21.16 for the controller)

I have chosen this TOTD to allow the students the opportunity to show that they truly understand the concepts taught during this lesson. To show me that they understood this lesson, the students will need to calculate the correct solution to this problem. Their ability to do so shows me that they...

• Can recognize when they need to multiply and when they need to add decimal quantities.
• Have the ability to correctly add and multiply decimal quantities.