# Exploring Equivalence

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## Objective

SWBAT simplify fractions and find equivalent fractions using representations such as models, pictures, or symbols.

#### Big Idea

Students have opportunities to connect the different representations, including models, to make sense of equivalent fractions.

## Launch

10 minutes

Write the words, "equivalent fractions" on the board.  Ask students to take a few minutes to discuss what these words mean with their group.

Call on students (at least one from each group) to share.

I write a fraction on the board, 6/12, and ask students to make at least 5 fractions that are equivalent to 6/12 and represent their thinking in each case using a model, words, pictures, symbols. They do this work independently, on paper.

This work time time is an important opportunity for me, as I circulate I check in on student thinking, silently if I can (I want them to do the work, not me) and using open-ended questioning if students are struggling.

Can you show me how to use......(shape, model, words) to represent 6/12? What can you tell me about the parts of the model?

How do you know this is true? Can you prove this is 6/12? Explain / show your thinking. (Creating a visual helps students to explain, because they are now describing a concrete rather than abstract model. Make sure students use mathematical vocabulary when describing.)

OR

Why did you draw the model this way?

* At this time, I find that my students, despite their clear description of equivalent fractions, had many misconceptions about how to find equivalent fractions.  I altered my lesson to address the two I saw the most frequently.

1. Finding equivalent fractions changes the numerator and denominator, but doesn't change the value of the fraction, because it is the same as multiplying and dividing by one.

2. Equivalent fractions, when modeling, require the same size parts.

In addition, I think these two understandings are foundational and critical.

## Guided Practice

15 minutes

Use 6/12 and the equivalent fractions students created in the launch as an anchor.

Make two columns on the white board, one with simplified as the heading and one as equivalent.

Throughout the guided practice, I use reduce and simplify interchangeably.

To reach all of my learners, I project fractions tiles that can be manipulated to represent the fractions we are discussing. I have specialized program for this, but you could use a document camera or an overhead projector and fraction tiles. I flip back and forth from the image of equivalent fraction tiles to the notes pages. This helps students visualize the equivalence of fractions, and enriches their discussion because they have a concrete referent.

Which of these fractions show a simplified version of 6/12?  As I call on students, I begin to list the student generated fractions in the "simplified" column.  When all are listed, students are asked to think about how the simplified fractions compare to 6/12.  Students will notice that there are fewer tiles, but the tiles are larger. As I call on students, my strategy is to insure equity in making sure different voices are heard but because I am working toward a specific understanding I may call on specific students as I end the discussion. One way to know who's on track is to circulate and listen during student discussion.

Because the model has only 12 pieces, I repeat the process with equivalent fractions but don't limit students to to the fraction range of one whole to 12ths (e.g., 24ths).  Help students notice that these tiles are (or would be if we had them that small) smaller, and there are more of them. You may have to try and draw it, or have students share how they would divide the 12ths to make the 24ths, and also explain why 24ths.

Call on students to "explain, mathematically, how we could simplify fractions".  * See reflection.

Many students say "divide by 2" This requires coaching to help students learn to be clear in their speaking.  They are not dividing the fraction by 2, rather the numerator by 2 and the denominator by 2.  I ask students to explain their strategy for reducing 6/12 to emphasize that dividing by 2 isn't really what simplify means.  My end goal is to show the students that dividing by 2/2, 3/3, or 6/6 are all means for simplifying 6/12.  In all instances, they are dividing the fraction by 1 whole.

## Manipulatives Intro

10 minutes

Students are given various manipulatives that can be used to represent fractions (fraction tiles, number lines, fraction circles, cubes, Cuisenaire Rods).

This activity provides students time to thoughtfully work with various manipulatives and reestablishes the concept of equivalence.  It allows the students to have some fun with fractions and fraction tools before jumping right into the topic.

## Group Share

5 minutes

Students are given an opportunity to share discoveries made about simplifying fractions and/or finding equivalent fractions while using math tools.

Throughout the school year, I work to build a community of learners would value learning from their mistakes and others.  I always ask, "who made a mistake?" "Were you able to see where you went wrong?" "What did you learn from it?"

This classroom routine helps students feel comfortable, confident, and willing to share even when they were not completely accurate.  Because of this, the group share portion of the lesson is very beneficial.