Today's lesson is multiplying a whole number by a decimal. We have learned to multiply 2-digit numbers by 2-digit numbers using the expanded algorithm. For this lesson, we use the standard algorithm to multiply a whole number by a decimal. We use the hundreds grid to validate our answer and give us a more concrete understanding of the skill.
To review, I ask the students, "What does multiplication mean?" Student response: repeated addition. Therefore, you multiply when a number is being repeated.
My mom is baking 3 cakes for dinner. She needs 2.5 cups of flour for each cake. How much flour does she need in all?
I ask the students to tell me the clue word(s) to help them solve this problem. Student response: in all. I remind the students that "in all" tells you that you are trying to find the total. I like to question my students to get their input as we work whole class. This gives me an idea of how they are thinking before we get into the lesson. I ask, "Which number is repeating?" Student response: 2.5 cups. "How many times is it repeating?" Student response: 3. As I write the multiplication problem on the board, the students write the problem on their paper. I write 2.5 x 3 on the Smart board. (I tell the students to put the number with the most digits at the top because it is easier to multiply with the one digit at the bottom.)
I let the students know that when you multiply with decimals, you should multiply the problem as if there are no decimals there. First, we multiply 3 x 5 = 15. I write the 5 at the bottom, then regroup the 1 to the ones place. Then, we multiply 3 x 2 = 6 + 1 = 7. I tell the students that now we have multiplied the problem, we can now look at the decimal. To determine where to put the decimal, you count the total numbers behind the decimal. In 2.5, there is one number behind the decimal. In our answer, we need one number behind the decimal. In our answer, we need to put the decimal between the 7 and the 5. The answer is 7.5.
To get more practice, we work another problem.
.28 ounces of salt is needed for each of the 3 cakes. How much salt is needed in all?
The students determine that our multiplication problem is .28 x 3. Together we use the standard algorithm to multiply 3 x 8 = 24. We put the 4 at the bottom, then regroup the 2 to the tenths place. "What is the value of the 2?" Student response: 2 tenths. Next, we multiply 3 x 2= 6 + 2 = 8. Last, we put the decimal in the number. "How many numbers are behind the decimal in .28?" Student response: 2. I let them know that you must have two numbers behind the decimal in your answer. Our answer is .84.
To give the students a conceptual understanding of multiplying a whole number by a decimal, I use the hundreds grid to show them. I tell the students that if I have .28, then I know that it is not a whole. That is like having 28 cents out of a dollar. I know that it has been cut into 100 pieces, because the number is 28 hundredths. How do I write 28 hundredths as a fraction? Student response: 25/100. This means that I have 28 out of 100. Therefore, I can show you a model of the answer using the hundreds grid. Lets see if the model will match our answer. In the problem, .28 is being repeated 3 times. On the Smart board, there are 3 hundred grids. I ask the students, "How many should I shade in the first grid?" Student response: 28. If I shade 28 in the first grid, how many will I shade in each of the other two grids?" Student response: 28. I let the students know that they are correct because 28 repeated 3 times.
To validate our answer of .84, we count all of the pieces shaded in the three grids. There is a total of 84 boxes shaded. Therefore, 84 hundredths are shaded. Our answer is validated, and the students get a visual of the multiplication problem.
For this activity, I let the students work independently first, then share with a partner after they have had time to work on the concept. (By doing this, it allows me to see what each student is doing on their own. Also, the students have a chance to hear their classmates thinking on the skill.)
I give each student an activity sheet and hundreds grids. The students must multiply a whole number by a decimal. The students use the hundreds grids to validate their answers. By using the hundreds grids, the students get a conceptual understanding of the skill because they get a visual of the product.
As they work, I monitor and assess their progression of understanding through questioning.
1. How many numbers are behind the decimal?
2. What number is being repeated?
3. How many do you need to shade in each grid?
4. Does your model validate your answer?
As I walk around the classroom, I am questioning the students and looking for common misconceptions among the students. Any misconceptions are addressed at the point, as well as whole class at the end of the activity.
Any student that finishes the assignment early, can go to the computer to practice fractions at the following site until we are ready for the whole group sharing: http://www.coolmath.com/prealgebra/02-decimals/decimals-cruncher-multiplication.htm
To close the lesson, I bring the students back together as a whole class. I feel that it is very important to let the students share their answers as a whole class. This gives those students who still do not understand another opportunity to learn it. I like to use my document camera to show the students' work during this time. Some students do not understand what is being said, but understand clearly when the work is put up for them to see.
I feel that by closing each of my lessons by having students share their work is very important to the success of the lesson. Students need to see good work samples (Student Work) (Student Work - Multiplying a whole number by a decimal.jpg), as well as work that may have incorrect information. More than one student may have had the same misconception. During the closing of the lesson, all misconceptions that were spotted during the activity will be addressed whole class.
I collect all papers from the students. All struggling students identified as I monitored during their independent activity will receive further instruction in small group.
I noticed that a few students were making mistakes on placing the decimal in the answer. They wanted to place the decimal in the answer aligned with the decimal in the problem. This is because when we add and subtract decimals, we line the decimals in the problem and answer. I worked with the students to place the decimal in the answer last by counting the number of decimal places.