Equivalent Fractions

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Objectives:
-Students will define what it means for something to be equivalent.
-Students will express shaded in shapes and regions as fractions of a whole.
-Students will explain how to tell whether or not two things are equivalent.
-Students will identify equivalent fractions and shapes.

Introduction:
This lesson was adapted and created by experimentation on the idea that fractions can be represented as shapes.
The lesson is heavily focused on classroom dialogue, and the sharing of ideas.

Setup:
Separate your classroom into groups of three with the instructions that the students may read the assignment but are not allowed to pick up their pens or pencils. In other words, the students are not allowed to work on the assignment.

Make sure that you have a copy of the assignment as a transparency for an overhead projector, it is important for the students to be able to use the actual shapes to explain their reasoning.

Activity:
Once all the students are in groups, explain again that you do not want to see anyone working on the assignment until you give the word. Instruct your students to read in their groups the first question and discuss what they think is the answer and why, for three minutes. Again, no writing, only talking in the groups. If they run out of things to say, encourage them to find as many ways to support their conclusions as possible.

After three minutes gather the students’ attention and choose someone to give their reasoning. If they say “I don’t know.” give the class a little bit of additional time and come back to them. If they have an answer, have them explain it and make sure that you ask questions of their explanation; play ignorant and hammer the concepts you want to emerge until you receive acceptable explanations. Take explanations from other students as well, and allow them to show their thinking on the overhead.

Some questions you may want to ask are:
1)How did you get fractions, all I see are shapes.
2)What do you mean by multiply the shapes?
3)Are their any other ways to show that these shapes are equivalent.

After the students have discussed the first question for about 5-7 minutes make sure there are no questions and have the students move onto number 2 without picking up their pencils.

Give the students only 1.5 minutes this time and then again draw the class together and have students explain their work. Ask questions if the explanations are not clear or if you want to emphasis a particular point.

After explanation have the students again discuss as a group question 3 and question 4 and then have yet again another class discussion; and still no working on the assignment.

After you discuss as a class questions three and four go onto the last question. Have the students explain what they think they are suppose to do on this section. Ask them if you think it will be a problem that their are more fractions than shapes, and what their answers might look like. After they understand what they are to do have the students pick up their pencils and get to work on the last section of the assignment.

When you see that most groups are about halfway done with the questions, gather the classes attention yet again and have the students put down their pencils and explain briefly how they are solving the problem.

After discussion give the students the remainder of the period to work on the assignment. Total it will take about 45 - 60 minutes, depending on the length of your discussion.

Expected Student Behavior:
You will see a few students fight not being able to work at their own pace, but if you emphasis the importance of group discussion and have them share their ideas in front of the class they work well.

During class discussion students seem to want to give the briefest explanation possible when first asked, but after you drill them for detail and reasons why many students become eager to present the whys and hows behind their reasoning and answers.

Some students will discuss the answer in their group in 10 seconds and think that they have adequately met your expectations of them. If this is the case ask them, while the other groups are working, to explain their answers to you. This will let them realize that they still have a long way to go and their answers may have room for improvement.

After the first 5 minutes of class discussion students really do get into the grove of the assignment and activity and start producing richer more substantive dialogue with one another.

Extensions:
-Have the students draw equivalent shapes to a given fraction.
-Have students draw three shapes with only two being equivalent and have a partner -try and identify the un-equivalent shape.
-Have students explain how equivalent fractions may be useful.

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