Fraction Multiplication with Clocks

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Objective

SWBAT multiply a fraction by a whole number.

Big Idea

Students will model how to multiply a fraction and whole number using a clock model, equation, and repeated addition.

Opening

20 minutes

Rationale

Whenever I teach a fraction unit, I always aim to expose students to a variety of fraction models for comparing and computing fractions. By using models such as the number line, area, and clock model, students are able to conceptualize fractions in more ways than one. Also, individual students might find that one model makes more sense than another. 

Clock Model

I printed a Clock (found at this link) for each student. I then placed each clock inside a page protector for repeated practice with a dry-erase marker. I like to give students a hand-less clock as the hands sometimes make the fraction clock model more confusing.

Fraction Identification

Prior to this lesson, I created a Powerpoint presentation, Fraction Clock Warm Up, to help demonstrate basic fractions using a clock model.

Projecting Slide2, I explained: Let's say that Tatum went to taekwondo practice for 1/2 an hour. Can you show me 1/2 an hour using your clock? This fraction was the easiest for students to identify: 1:2 a Clock. One student reasoned, "It's 30 minutes because 1/2 of 60 minutes is 30 minutes."

We then moved onto Slide3. I continued: Now, let's say that Tatum practice taekwondo for 1/4 an hour. Can you show me 1/4 an hour using your clock face? Students worked together with group mates and agreed, "One fourth an hour is equal to 15 minutes because half of 30 minutes is 15 minutes and 1/4 is half of 1/2." Here's a student board example: 1:4 a Clock

Next, we discussed Slide4. What if Tatum only practiced for 1/12 an hour? One student said, "That's not very long!" Then, they demonstrated their thinking on their clocks: 1:12 a Clock

At this point, we went on to one of the  more challenging fractions for students to demonstrate using a clock, 1/6 on Slide5. Here's a great conversation that happened between a group of students: Finding 1:6. Identifying equal parts and determining placement of lines can be challenging: 1:6 a Clock.

Finally, we went to the last slide, Slide6. What if Tatum practice for 1/3 an hour? With time, students realized that 1/3 an hour equals 20 minutes because 20 minutes x 3 = 60 minutes: 1:3 a Clock.

Factors of 12

Before moving on, I asked students to help identify the Factors of 12 on the board. I then asked students to turn and talk: Why do you think I asked you to find 1/2, 1/3, 1/4, 1/6, and 1/12 of a clock instead of 1/5 of 12? A few minutes later, a student responded, "5 isn't a factor of 12. The denominator has to be a factor of 12." 

Teacher Demonstration

30 minutes

Presentation

To build on student understanding of clock fractions and to provide fraction multiplication instruction, I created a "part 2" Powerpoint Presentation: Fraction Clock Multiplication on the same topic, Tatum going to taekwondo practice.

Goal & Lesson Introduction

Starting on Slide1, I introduced the lesson's goal: I can multiply a fraction by a whole number. I explained: Now that you know how to locate unit fractions on a clock model, we are going to take your learning a step further. Now, we are going to multiply these unit fractions by a whole number. 

Multiplying by 1/2

Going on to Slide2, I asked: What if Tatum practices for 1/2 an hour two times last week? How many hours did she practice altogether? Students demonstrated their thinking using their clock models and by writing an equation on their white boards: Student Model 1:2 x 2. After students had time to turn and talk, I demonstrated 1/2 x 2 on the board: Teacher Demonstration 1:2 x 2

Multiplying by 1/4

Following the same procedure, we solved the problem presented on Slide3. Again, students showed their thinking using the clock model and an equation: 1:3 x 4 Clock Model and 1:4 x 3 Equation

Multiplying by 1/12

Moving on to Slide4, we discussed the total practice time if Tatum went to taekwondo for 1/12 an hour six times. Students were quick to point out, "That's half an hour." Here are a couple examples of student work: 1:12 x 6 Clock Model and 1:12 x 6 Equation. After students had time to solve this problem, I demonstrated how to solve this problem on the board: Teacher Demonstration 1:12 x 6. I tried to write each multiplication equation as a repeated addition equation as well in order to build a strong connection. 

Multiplying by 1/6

On Slide5 we discussed a problem that would result in a fraction greater than a whole: 1/6 x 7. Some students had a hard time modeling an improper fraction on their boards: 1:6 x 7 Clock Model. Others drew a clock model alongside their equations: 1:6 x 7 Equation & Model. I then modeled how to use arrows to demonstrate a fraction greater than one on the clock: Teacher Demonstration 1:6 x 7.

Multiplying by 1/3

For the final slide, Slide6, students solved and discussed 1/3 x 4: 1:3 x 4 Equation

Student Practice

50 minutes

Choosing Partners

Picking math partners is always easy as I already have students placed in desk groups based upon behavior, abilities, and communication skills. Before students began working, I asked them to discuss how they would like to support each other today. I gave them many examples: Do you want to take turns talking out loud? Do you want to solve quietly and then check with each other? Or do you want to turn and talk anytime you get stuck? Students always love being able to develop a "game plan" with their partners! 

Practice Page

To provide students with further practice, I printed two fraction multiplication pages (front to back) from Math-Aids. I explained: I'd like for you to continue multiplying fractions by whole numbers! I then modeled the first five problems on the first page for the class to make sure students understood the steps. For each problem I modeled how to write each multiplication equation as an addition equation: Teacher Model.

By the time students finished the first page, most students evaluated the process of multiplying by a whole number and were ready to apply their observed shortcut: a/b x c = (a x c)b. Providing students with the opportunity to discover this algorithm on their own helps engage students in Math Practice 8: Look for and express regularity in repeated reasoning. If 1/2 x 3 = 1/2 + 1/2 + 1/2 = 3/2, then 1/2 x 3 = (1 x 3)/2. 

Monitoring Student Understanding

Once students began working, I conferenced with every group. My goal was to support students by providing them with the opportunity to explain their thinking and by asking guiding questions. I also wanted to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3). 

  1. What did you do first?
  2. What do you need to always remember?
  3. Does that feel right? 
  4. Do you see a pattern?
  5. What makes more sense to you?
  6. Does that always work?
  7. What did you do that helped you be successful?

Conferences

I loved conferencing with this student: 1:3 x 4. She did a beautiful job solving the problem using repeated addition and you can tell that she really tried hard to answer my questions! 

Another student, working on the back side, was Applying Observed Rules. He realized the easier way of multiplying fractions by whole numbers on the first side. 

Completed Work

Here are completed examples of both practice pages: Student Page 1 and Student Page 2