Common Denominator Comparison
Lesson 11 of 16
Objective: SWBAT compare fractions using common denominators.
Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using a number line model, Student Number Line, and a hundreds grid, Hundred Grids. For each task today, students shared their strategies with peers (sometimes within their group, sometimes with someone across the room). It was great to see students inspiring others to try new methods and it was equally as great to see students examining each other work for possible mistakes!
Prior to the lesson, I placed magnetic money and fractions on the board to help students conceptualize our number talk today.
I invited students to join me on the front carpet with their number lines. I then drew a number line on the board, Number Line on the Board, and marked 0, 1, and 2 on the line. I asked students to do the same on their own number lines.
Task 1: Compare (3 x 30/100) to (1/4 x 2)
For the first task, I asked students to compare 3 x 30/100 to 1/4 x 2. I asked: Which expression is greater? Which is smaller? How do you know? Please show your thinking on your number line. Prove it to me!
After students had time to compare these fractions on their number lines, a couple students volunteered to share their thinking in front of the class. I encouraged students to think about money conversions to help them make further sense of fractions.
Here's an example of student work during this time: Number Line Example 3:100 x 3 < 1:4 x 2. After students turned and talked about their findings, two students modeled their thinking in front of the class: Student Explaining using a Hundreds Grid and Student Number Line 13 x 1:10 < 3 x 1:2.
Task 2: Compare (1/10 x 13) to (1/2 x 3)
For the next task, students compared 1/10 x 13 to 1/2 x 3. Just as before, students compared the expressions using their number lines: Student Number Line 13 x 1:10 < 3 x 1:2 and hundreds grids: Hundreds Grid 3 x 1:2 vs 1:10 x 13.
Then, a student modeled her thinking on the board: Student Modeling 13 x 1:10 < 1:2 x 3.
To help teach today's lesson and provide students with guided practice, I created a Powerpoint Presentation: Finding Common Denominators.
Goal & Introduction
For today's lesson, I invited students to the front carpet with their whiteboards. I began by introducing today's goal using the first slide of the presentation: I can compare fractions using common denominators. I explained: Today, we are going to focus on finding common denominators with and without an area model. When we compare two fractions with uncommon denominators, it's kind of like comparing apples to oranges. I continued on to the next slide featuring a cartoon on apples and oranges: Comparing Apples to Oranges. We then discussed the meaning behind the phrase, comparing apples to oranges.
Next, we discussed the difference between common denominators and uncommon denominators using the next slide of the presentation: Uncommon vs Common Denominators.
Equivalent Fractions Poster
Over the past two days, students have been comparing fractions using a variety of strategies. Before moving on the the next slide, I introduced the final strategy poster for comparing fractions: Equivalent Fractions Poster. I explained: When we are comparing two fractions that have uncommon denominators, we can find equivalent fractions with common denominators to make the comparing process easier. For example, 2/6 is equivalent to 4/12 and 3/4 is equivalent to 9/12. Comparing 4/12 to 9/12 is much easier because they have common denominators. Notice that the denominator is 12 with each fraction.
I then asked students to apply the strategy by comparing 1/3 to 3/5. Without wasting a minute, most students immediately converted 1/3 to 5/15 and 3/5 to 9/15: 1:3 < 3:5. After providing students with time to turn and talk, we continued on to the next slide of the presentation.
Finding Common Denominators With an Area Model
Once the next slide, 1:2 vs 1:4, was projected, I explained: I know many of you can find equivalent fractions with common denominators, but I want to make sure you understand how to model this process! Let's take a look at 1/2 and 1/4.
First, we made a list of common multiples of 2 and 4: Multiples of 2 and 4.. Then, we determined the least common multiple. Even though this isn't a fourth grade concept, exposure to future math concepts can help front load students for what is to come.
Next, I modeled how to find equivalent fractions in order to compare fractions in common denominators: Teacher Model 1:2 > 1:4. We discussed and modeled how 1/2 is equal to 2/4 and how 1/4 is equal to 1/4 (by multiplying by 1/1). Next, I colored in 2/4 of one array and 1/4 of the other.
I then asked: How many squares are in each grid? Students responded, "24 squares!" How many squares out of 24 is equal to 1/2? "Twelve!" I then modeled how 1/2 = 2/4 = 12/24. We then discussed how 1/4 = 6/24 as well. I said, Do you see how 1/2 really is equal to 2/4 and 12/24 and how all three fractions name the same part?
We then discussed the next five slides in the same fashion: Comparing 1:6 to 2:4. During this time, students completed similar work on their own boards: Student Board 1:6 < 2:4. Students excitedly volunteered to model each task!
I knew that this presentation would engage students in Math Practice 7: Look for and make use of structure. Students would look for patterns between slides and they would make use of this structure to solve more complex tasks. Also, by asking students to model how to find common denominators using an area model, students were engaged in Math Practice 2: Reason abstractly and quantitatively. I wanted students to develop an understanding of this abstract process beyond memorization of steps.
Finding Common Denominators Without an Area Model
Once we arrived at slide 10, 1:2 vs 1:10, we transitioned to finding common denominators without an area model. I modeled how to find the multiples of each denominator in order to identify the least common denominator: Teacher Model 1:2 > 1:10. Students continued to complete similar work on their own white boards: Student Board 1:2 > 1:10.
We then completed the rest of the slides together, using the same process: Teacher Model 1:8 < 1:5 and Student Board 1:8 < 1:5. Again, at this point, students were excited to take over the modeling process while I took on more of a coaching role.
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!
To provide students with further practice, I printed a fraction comparison page from Math-Aids. I explained: For continued practice today, I'd like for you to continue comparing fractions by finding common denominators! I then modeled the first five problems for the class to make sure students understood the steps.
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).
- What did you do first?
- What do you need to always remember?
- Does that feel right?
- What makes more sense to you?
- Does that always work?
- Why is important to find common denominators?
- What did you do that helped you be successful?
Here is a perfect example of how students sometimes need a little direction from the teacher to remember the steps of abstract processes, such as finding common denominators: Tracking Your Thinking.
Most students did a great job with this activity, especially when they caught on to the steps of this strategy. Here's an example of a student's Completed Work.