Dividing Unit Fractions by Whole Numbers
Lesson 11 of 13
Objective: SWBAT solve problems involving the division of fractions by whole numbers and whole numbers by fractions.
To warm students up for this lesson, I show a video clip about dividing fractions.
Then, as a class, we add to the "essential understandings" wall that we have been "building" throughout the unit.
Students agree that division with fractions needs to be added to the wall.
"When we divide with fractions, we can use the keep change flip strategy to find the answer".
After this is added to the wall, I reinforce that this procedure is only applicable to dividing fractions. The group responds to my quetsions:
• When you add with fractions can you "keep, change flip"? (NO)
• When you subtract with fractions can you "keep, change flip"? (NO)
• When you multiply with fractions can you "keep, change flip"? (NO/ some students did say yes here... as I predicted they would. It is critical that students do not over generalize this proceedure.
• When you divide with fractions can you "keep, change flip"? (YES)
• Based on what you know about fractions, what do you think you would do if you had a division problem with mixed numbers? (Make it improper fractions, like we do when we multiply mixed numbers. It is much easier to work with fractions than to try to think about the whole number and the fraction separately).
For the guided practice portion of this lesson, I write three problems on the board.
- 1/4 divided by 2
- 2 divided by 1/4
- 2 x 1/4
Without actually solving, students discuss the differences between these 3 problems. The goal is to make sense of the problem and determine what is happening in each of these scenarios. This is a more rigorous expectation than solving, because students must explore and articulate their thinking, which requires understanding rather than rote/procedural knowledge.
Sample student thinking (a reflection of think-pair- share) with prompting:
When we first looked that this, we thought they were all the same, because all three equations have 1/2 and 2. Then we noticed that 2 x 1/4 is a different operation, and it isn't a "keep change flip because that would have come from 2 divided by 4 since you flip the second fraction. If the first two equations were multiplication they would be the same, but division doesn't have a commutative property.
In the first one, you are divided 1/2 into 2 parts.
In the seconds one, you are dividing 2 wholes into 4 parts.
The third example shows multiplication, you are finding 1/4 of 2.
Then, using interactive modeling, we solve each of the problems, and draw models to match.
At this time, I revisit the questions students asked yesterday about properties of division to reinforce why 1/4 divided by 2 and 2 divided by 1/2 are different problems.
I continue to keep coming back to 2 x 1/4 to show that multiplication of fractions is different from division of fractions to avoid having students over generalize the keep change flip procedure.
Students solve problems from a fraction division sheet. This has examples of a whole number divided by a fraction and a fraction divided by a whole number.
I made this worksheet using WorksheetWorks. It was a valuable resource in this instance because I was able to choose the number of problems and also the type of division problems I wanted. I mix division of fractions by a whole number with division of whole numbers by a fraction to make sure students have to think before solving. I add more rigorous expectations by requiring that the students choose one of each type to model.
After checking their answers and revising any errors, students work on solving word problems from the text book.
The word problems I select are division problems that involve fractions and mixed numbers. When the majority of students have moved on to these problems, I regroup the students to discuss division properties once more. Together, we read one of the problems to determine the dividend (the amount/item that is being divided) and the divisor (what it is being divided into) because this is a critical step to solving division problems.
At the end of this lesson, we are running out of time. It is my goal to have a lesson wrap-up each day, even if it is short.
Today, five students share something they learned about dividing with fractions. These are posted on the board. These five lesson can be revisited at the start of the next lesson if needed.
With more time, I would facilitate a verbal assessment: Students are asked to make one statement about dividing with fractions. I allow students to repeat something that has already been said, if they need to. The purpose of this ticket out is for students to hear from their peers, and share with their peers, essential information.
The best learning happens when we learn from someone in a similar learning zone. This ticket out allows for more social interaction around the concept and increases opportunities for students to learn from their peers.