To warm up for this lesson, students share their solutions to the Marching Band Open Response problem from the independent practice of the previous lesson. In this problem, students use subtraction of fractions, and multiplication of a fraction by a whole number to solve.
Many students recognize that when multiplying a fraction by a whole number, it was similar to dividing the whole number by the denominator of the fraction. I choose to start with this problem to emphasize the connection between multiplying by a fraction (finding a part of a whole number) and dividing by a fraction (finding how many of a given fraction fit equally into a whole number).
The video clips show students collectively sharing their thinking process for solving the marching band open-response problem.
During the warm-up, it was determined that there are 40 students in the marching band. To transition student thinking, I move the students to carpet where there are 40 cubes. These 40 cubes represent the marching band members. We are going to do some more thinking with these marching band musicians. Today, we use modeling to help us answer the questions.
• Can someone come to the middle and show us 1/2 of the 40 students in the band?
• How can we represent this with an equation? (1/2 x 40)
• Is there another equation that also represents the exact same thing? (40 divided by 2)
I continue this modeling process with a few more problems. (It is important to note that the models are not always a 1:1 correspondence. In different problems the models can switch from one stick of 10 representing a whole, to representing a 10. This keeps the thinking flexible from problem to problem and allows students to use models in ways that meet the immediate need.)
• Find 1/5 of the 40 band members. Write this using 2 different equations.
• Find 1/3 of 3. Write this using 2 equations.
These first few examples review multiplication of a fraction and a whole number. I do this to transition students thinking. It is important to recap what students know before teaching them something new. Doing this brings multiplication of fractions to the front of their working memory, so they can connect division of whole numbers to this prior learning in the right way.
• Next, I ask students to model 3 divided into 1/2s to determine "how many 1/2s fit into 3".
I write the division equation on the board while a student models for the group. 3 divided by 1/2 = 6.
• How can this be represented using another equation? (3x2=6)? To make the reciprocal more clear, I write (3 x 2/1 = 6).
At this point I introduce the term reciprocal and as students to find the reciprocal of a variety of fractions that I write on the white board.
To help students process this new information, I show a rap video. This video introduces the "keep, change, flip" idea, but it also explains why it works. Even though this explanation is short, it is an important component of the video. I will revisit the explaining through the instruction too.
Following the "Keep, Change, Flip Rap", I lead the students in guided practice to solve a few problems involving division of whole numbers by fractions.
During each problem, I revisit the WHY the procedure "keep change flip works". I connect this with fact families for multiplication and division and the idea of when you divide you ask "how many ___ fit into ___". For each division problem I model on the board, I also model a similar multiplication problem. I have found in the past that students over generalize "keep change flip" and use this procedure to multiply fractions as well. I believe that working with division and multiplication at the same time allows students to better see and understand the differences in these operations.
3 divided by 1/3 - in this problem, we are trying to find out how many 1/3 size pieces make up 3 wholes. I draw 3 wholes (cut into thirds) and then together we count the thirds to get 9 pieces. This problem can also be expressed as 3 x 3/1
On the same screen, I write 3 x 1/3. Is this problem the same as the other 2? (No, because it is a multiplication problem, not a division problem. It is multiplied by 1/3 not 3/1. This problem is different.) When multiplying a whole number by a fraction, we find a part of the whole. In this problem we are finding out what 1/3 of 3 wholes is. If you take 1/3 from each that is 1/3 + 1/3 +1/3 that makes 1 whole. Or you can divide the set of 3 into 3 groups and there is 1 whole in each.
This guided practice provides students with support for the independent practice where they have to solve problems involving dividing a fraction by a whole number and also multiplying a fraction by a whole number.
Before practicing with a partner, I focus students attention to the essential understandings for multiplying and dividing fractions.
When you divide a whole number by a fraction, you are determining how many of that fraction fit into the whole number. You can multiply by the reciprocal (keep change flip) to find the answer.
When you multiply fractions by a whole number, you are finding a part of the whole number. You have to split the whole number into equal parts and find the value of one of those parts.
Students put these ideas (the italics) into their own ideas, but they do this before working on problems so the essential understandings are fresh in their minds.
After recording the essential understandings in their own words (on the bottom of the handout with practice problems) students work in pairs to complete the six problems (3 dividing whole numbers by fractions and 3 multiplying whole numbers by fractions).
Students use models to show their thinking on each of these problems. I make my reasoning for having the students model their thinking transparent to the students. "The easy part of these problems is the math. The challenging part is showing HOW and WHY the math makes sense. I am asking you to show models because we deserve this challenge." I also keep the numbers manageable so that students can really spend time thinking about the math and the models rather than rushing to solve problems.
The focus for the group share is to take a closer look at problems #1 and #2 on the hand out.
4 ÷ 1/2
1/2 x 4
I ask the students to think about these two problems and determine if they are the same or different and explain why.
I hear from 5 students. They all share that these problems are different because one is division and the other is multiplication. If you solve the division equation by multiplying by the reciprocal you will have a new problem 4 x 2 not 4 x 1/2 so these problems are not the same.
Even though the share reveals everyone has the same answer, I call on one student from each group to restate this. I do this because it allows all students to hear this important idea multiple times.