# Exploring Equivalent Fractions

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## Objective

SWBAT identify equivalent fractions.

#### Big Idea

Students will investigate the meaning of equivalent fractions using models.

## Opening

30 minutes

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. 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!

Getting Started

Prior to the lesson, I placed magnetic money and fractions on the board to help students conceptualize our number talk today.  I drew a 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 1/10  to 1/4

For the first task, I asked students to compare 1/10 to 1/4. I asked: Which fraction is greater? Which is smaller? How do you know? Please show your thinking on your number line. Prove it to me!

Here, this student beautifully uses money to explain that 1/4 is greater because 1/4 = \$0.25 and 1/10 = \$0.10. To help the student make his work more precise, I ask him a few questions. Prior to sharing, I heard him say, "1/4 is  \$0.15 greater." I tried to help him model this for the rest of the class using the magnetic money as a model: Comparing 1:10 to 1:4

Here's what the student number lines looked like: Student Number Line 1:10 to 1:4.

After we had decided that 1/4 was greater because 0.25 > 0.10, I was inspired to ask students to prove this with a calculator by checking: 1 divided by 4 = 0.25 and 1 divided by 10 equals 0.1 or 0.10. Further investigating followed with the calculators. I just had to share their findings!

Task 2: Compare 1/4 to 1/2

For the final task, students compared 1/4 to 1/2: Student Number Line 1:4 & 1:2. After some time, a student explained her thinking on the board: Comparing 1:4 and 1:2

## Teacher Demonstration

40 minutes

Conjectures Poster

To begin today's lesson, I showed students a piece of chart paper and explained: I would like to keep track of your conjectures about fractions on this poster! Remember, a conjecture is a conclusion formed on the basis of incomplete information. As mathematicians, we are always developing theories about how math works and then, once we investigate further, we can go back and change or add to these theories, based on new evidence. Before we begin investigating today, does anyone have a conjecture about fractions they would like to share?

One student raised his hand and said, "The smaller the denominator, the bigger the fraction." I wrote the letter C (as his name starts with C) and his conjecture on the poster. I then asked him to provide evidence to support his thinking. He said, "For example, 1/2 is bigger than 1/4." I drew a picture of this on the poster to support student understanding. Here's what our conjecture poster will look like at the end of our math lesson today: Conjectures Poster
Next, a student said, "All fractions can be equal to a decimal. For example, 1/2 = 0.5 = 0.50." I was so impressed with this conjecture as I haven't taught this idea explicitly.
By this time, many hands were in the air! I decided to call on one more student. She explained, "Any number divided by itself equals 1... except for 0. For example, 4/4 =1, but 0/0 = 0."
Knowing we were running out of room on our poster, I explained: Today, I would like for you to continue developing conjectures about fractions, based on evidence!
I knew that including a Conjectures poster during this lesson would encourage students to engage in Math Practice 3: Construct viable arguments and critique the reasoning of others.

Goal & Problem

At this point, I explained the goal of today's lesson: I can identify equivalent fractions. To provide students with an opportunity to explore the meaning of equivalent fractions, I created the following resources: Equivalent Fraction Handouts and copied them front to back for each group. I began by passing out Resource 1 and Resource 2 (front to back).

I wanted to give students the opportunity to reason abstractly and quantitatively (Math Practice 2) by representing fractions (an abstract concept) using models (concrete concept).

Investigating 1/2

I asked students to look at Resource 1 and to begin as a team. I explained: The gray portion of each array is equal to 1/2. Please make notes on this paper as a group. Label each array. Use the bottom part of the paper to record your observations!

Students excitedly went right to work. Right away, I celebrated a group leaning in to communicate with one other. Soon, all groups were literally "putting their heads together!"

This group, Observing Halves, explained that "they're all halves of the whole." They also pointed out that one shaded part is 1/2 when there are two parts and 4 shaded parts is also half since there are 8 parts.

Another group, Shaded vs Not Shaded, explained that the shaded part is always equal to the unshaded part with fractions equivalent to 1/2.

Then, this group, Investigating Values, began applying this area model to a party where a different number of people came. We then discussed if the value of each 1/2 would be different when the 1/2 is decomposed (cut up into smaller amounts).

Here are a couple examples of the end product:

Before moving on, I returned to the Conjectures Poster and asked: Would anyone like to add a conjecture about fractions?

One student said, "No matter how you split 1/2, it will always have the same value." Another student explained, "1/2 = 2/4 = 4/8 = 8/16.... Each time, the numerator and the denominator just doubles." Then another student added, "This could go on forever!"

At this point, I challenged students to keep in mind the following while investigating today, To find equivalent fractions, you..."

Investigating 2/2

I asked students to flip over Resource 1 as Resource 2 was copied on the back side. Here's a conference with one group: Investigating 1:2 Further. I loved watching these students use another tool (calculator) to further prove their thinking!

Here's an example of student work: Group Example 2:2.

Investigating Fourths

Next, I passed out the rest of the resources (Resource 3 and Resource 4, back-to-back and Resource 5 and Resource 6 back-to-back). As exciting as this was, if I had continued to hand out one resource at a time, student interest would dwindle! By passing out more than one resource, students were more engaged in this activity!

These students explained why 1/4 = 2/8 = 4/16 = 4/64: Investigating 1:4. Another group did a great job Seeing More Equivalent Fractions within the area models, for example, 3/4 = 6/8!

Here are a couple work examples:

## Student Practice

30 minutes

Continued Practice

To provide students with more practice identifying and explaining equivalent fractions, I asked each group of students to get a fraction tile set: I explained: For continued practice today, I'd like for you to continue finding equivalent fractions using these fraction tiles.

I then asked each student to get a lined sheet of paper. I modeled how to split the paper into fourths: Teacher Model.. I continued: First, I would like you to use your fraction tiles to find equivalent fractions for 1/2... then 1/3... then 1/4... and then 1/5. Students knew exactly what to do and got right to work! It was almost as if they had been waiting all year to get their hands on these fraction tiles!

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. How do you know that these are equivalent fractions?
3. Can you show me using the fraction tiles?
4. What patterns are you noticing?
5. Do your group members agree?
6. Does this always work?
7. Do these represent the same value?

Conferences

Exploring Equivalent Fractions: I loved listening to this student develop her own explanation of equivalent fractions. She even began to work beyond the constraints of the model by identifying 60/120 as an equivalent fraction to 1/2.

Matching up Fraction Strips: These students used the tile labeled 1/3 to match up other tiles. Eventually, one student said, "I just realized something." I loved how she began to realize that fractions with denominators that are multiples of 2 (4, 6, 8, 10) can be equivalent to 1/2.

Completed Work

Here are a couple examples of completed work: