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Divide and Conquer: Equivalent Fractions

1996-1997 National Teacher Training Institute
Master Teacher: Susan Deese-Laurent, Educational Consultant, Certified Elementary Teacher, Lee, NH 
Subject: Science, Mathematics
Grades: E, M 
Series: The Eddie Files Episode 4: Any Way You Slice It 
It Figures Episode 19: Finding Equivalent Fractions

Pre-viewing Activities

Focus for Viewing

Viewing Activities

Post-viewing Activities

Action Plan






"On your mark....get set....GO!" marks the beginning of the great pizza swap, where students will learn a true understanding of equivalent fractions. This lesson begins by having students free explore with pattern blocks to gain a hands-on, conceptual understanding of fractions being parts of a whole. They continue on to discover that a whole can be represented in a number of ways as they create number equations to go with the pattern block designs. 

Next, by viewing an appealing video about people working at different careers and using fractions in a number of ways, students learn how important it is to have a knowledge of fractions in the working world. 

In a second video, students watch as two boys work out the problem of how to evenly share two pizzas, which have been cut up in different ways. They then watch a cartoon version of Robin Hood and see the King make the mistake of thinking that 3 out of 6 piles of gold is better than 1 out of 2 piles. 

The final activity allows students to make their own paper pizzas, including appropriate toppings, and take part in a swap of pieces to see who can end up with a whole and unique-looking pizza. 


Learning Objectives

Students will be able to:
  • demonstrate an understanding of fractions being parts of a whole by using pattern blocks . 
  • construct a paper model to depict different fractions. 
  • list, in writing, known facts about fractions. 
  • demonstrate an understanding of equivalent fractions. 
  • list several examples of how fractions are used in real life situations. 



  • pattern blocks, 1 set per 8 students 
  • 1 or 2 sets of overhead pattern blocks (not absolutely necessary if none are available)
  • chart paper

Per student: 

  • math journal (or just a blank piece of paper)
  • one round piece of oak tag (in different colors) or construction paper, 12 inches in diameter 
  • one round, brown piece of oak tag or construction paper, slightly larger than 12 inches in diameter (this will be the pizza pan)
  • small cutouts of mushrooms, pepperoni, peppers, anchovies and other toppings
  • glue sticks
  • scissors
  • one sign for each table that has a fraction written on it with numbers, i.e. fourths, sixths, eighths, twelfths, sixteenths, (or a model of a figure divided into that many piece)


Previewing Activities

In order to insure that students have an appropriate level of understanding of equivalent fractions for the post-viewing activity of this lesson, do the following activities with your students. Distribute a small container full of pattern blocks (approximately one quarter to one half of a set) to each group of four students. Allow time for free exploration (10-15 minutes, or more) if students have not worked with these manipulatives previously. Then elicit from students what they have learned about the pattern blocks by asking questions such as: "What do you notice about these pattern blocks?" (This step will not be necessary if your students have used pattern blocks before and are familiar with these manipulatives). Establish that there are different colors and sizes of pattern blocks and that they fit onto each other in various ways, i.e., 1 yellow hexagon = 2 red trapezoids. Using an overhead projector (or the blackboard), record a list of equivalences that students offer: 
  • 1 yellow hexagon = 2 red trapezoids 
  • 1 yellow hexagon = 3 blue rhombi 
  • 1 yellow hexagon = 6 green triangles 
If students suggest other equivalences, i.e., for using the orange squares, allow them to try to show these using the overhead pattern blocks. 

Tell students that for now they are to assume that a yellow hexagon = 1 whole. Ask: "If that is so, then what would 1 red trapezoid equal?" (one half) Record this information on a class chart. Ask: "What would 1 blue rhombus equal?" (one third) Record on the chart. Ask, "What would 1 green triangle equal?" (one sixth) Record on the chart. Having them keep the above information in mind, begin asking other equivalence questions such as, "What is another way of making one half of a yellow hexagon (other than with 1 red trapezoid)?" (3 greens) Then ask, "How would we write this with fractions?" (The red trapezoid, one half = 3 green triangles, three sixths) Have students experiment with a number of these discoveries and record on the class chart the equivalent fractions they come up with.

Distribute worksheet #1, and have students complete it while using the pattern blocks to figure out the equivalent fractions. (There is also a worksheet #2, depending on the age and level of your students.)


Focus for Viewing

To give students a specific responsibility while viewing, tell them that they will now see parts from a video entitled, The Eddie Files: Any Way You Slice It. Tell them that they will not be seeing the beginning of the video because it would give away the surprise activity they will be participating in later on. The part of the video they will be viewing shows different people and how they use concepts of fractions in their jobs. Tell students that whenever they see a person using their knowledge of fractions at their job, they should give a "thumbs up" signal so you can pause the video. When the video is paused, students will be asked to tell how they think a knowledge of fractions is being used by that individual at his/her job. This information will be recorded on a class chart. 


Viewing Activities

Prior to beginning the lesson with students, fast forward the video to where Sal, the owner of the camera shop, is talking about pizza and fractions. Start video at that point. 

Pause video after Eddie says, " There they were again...fractions. Everyone uses them." Remind students to give a "thumbs up" signal when they see people using their knowledge of fractions on their jobs. 

Resume video with no sound. Pause video when you see a "thumbs up" signal from a student, or after the slow motion picture of the two entrees being placed up on the shelf for pick-up. Ask: "How were fractions used in that setting?" Record student ideas on a class chart. 

Resume video to show Sal in his pizza shop telling his secret about how he cuts his pizzas into nine pieces instead of eight so people get more pieces. 

Pause video when you see a "thumbs up" signal or when the picture leaves the camera shop and shows the Jets raceway. Again ask how fractions were used. Record on chart. 

Resume video. Pause video after a "thumbs up" signal or when the picture leaves the football game and shows the silhouettes of buildings in a city. Ask the same fraction question and record. 

Resume video. Pause video after a "thumbs up" signal or when the picture goes back to the buildings and a bridge. Ask the same fraction question and record. 

Resume video for the section back in the photo shop and the heated discussion between the two men of how many pieces Sal had cut the pizza into. 

Pause video when two men are arguing back and forth and ask students, "Is it possible to cut a pizza into nine pieces? Why or why not?" This would be an appropriate time to have students respond to this question in a journal. 

Stop the video when it shows a picture of the moon in the sky. 

Remove The Eddie Files video and insert It Figures, Episode 19.
Tell students that they are now going to watch a short section from another video on fractions that will give them useful information for the activity they will work on later. (Have video ready at the part right before the man at the pizza store says, "Hey guys...your pizza's ready!" ) Explain to the students that what they will be watching is where two boys are in a pizza shop and have ordered two small pizzas. Start video.

Pause video after boy says, "Let's split it fifty/fifty." Ask students if they know what that means. Discuss briefly. Have students predict how many pieces they think each boy will get in order for the pizza to be divided evenly. Resume video. 

Pause video after the boy has evenly distributed the eight pieces of pizza and ask again why eight pieces evenly distributes the pizza. Resume video.

Pause video after boy passes out pieces of second pizza and blonde boy gives him a "look." Ask students why they think the blonde boy is not happy. Resume video. 

Pause video after boy with brown hair says, "I took four pieces, the same as I took with the last one." Ask students: "What is wrong with this thinking?" (You could also stop the video at this point and ask students to write in a journal explaining what is going on with the pizzas at this point). Resume video.

Pause video after blonde boy says, "Right...I mean wrong." Ask students, "What would YOU do to explain to your friend what is going on, i.e., why he only gets three pieces of pizza this time instead of four like last time?" Discuss ideas; this could be used for a time of journal writing. Then say, "Let's see what the boy on the video said to help his friend understand." Resume video. 

Pause video after the brown-haired boy says, "Are you going to eat your three sixths (of the pizza)...because if you're not....." Tell students that the last part of what they'll watch on this video is a cartoon version of Robin Hood. Have them watch and be ready to discuss how Robin Hood outfoxes the King and his men. Resume video. 

Stop video at the end of the cartoon when the narrator says, "Robin Hood and his merry men were as merry as ever." Discuss the concept of equivalent fractions and where the King's thinking went wrong.


Post-viewing Activities

For student motivation, dress up as a pizza maker would with chef's hat, moustache, apron. Either have a real ball of pizza dough or make believe to have one and throw it up in the air and catch it. Tell students that the first pizza in the United States was made in 1905. Ask them if they know any facts about pizza. Add that today they are going to make their own pizzas out of paper and put on the toppings they like. 

Distribute precut paper circles and assorted "toppings." Have students decorate their own pizzas. 

When students have finished making their pizzas, put one fraction sign (see materials list) on each table. Explain that they will be cutting up their pizzas into that many pieces. Do one pizza together as an example. For instance, if the sign on their table reads one sixths, ask students, "How would we go about cutting up this pizza into six equal pieces?" Listen to suggestions and discuss. Instruct students to watch as you turn over a paper pizza to the backside that does not have the toppings on it. Then, using a pencil and ruler, mark out lines that show how to cut it into six equal pieces. (You may wish to do this part of the lesson on scratch paper at first to allow students to try different methods that may not work before cutting up the pizzas they have made. You may also choose to draw lines on the paper circles prior to the lesson for younger students so they can just cut on the lines. There should be one group of students for each of the following fractions: fourths, sixths, eighths, twelfths, and sixteenths.) 

Once the groups of students have cut up their pizzas into the fractional pieces indicated at their table, tell them it is time for "THE GREAT PIZZA SWAP." (This activity will be noisy so be prepared for that.) Tell students that during the swap they are to trade equal amounts of pizza, even though the pizzas are cut out differently, so that when the bell rings to indicate that the swap is over, they each end up with a full pizza, no more or less than will fit onto their pizza pan (brown circles). Answer any questions that may arise. Say, "On your mark....get set... GO!" Allow 2 or 3 minutes (adjust to your grade and comfort level) and then ring a bell or yell loudly, "STOP!" Students must sit down with what they have on their pizza pan. 

Have students share whether or not they ended up with a whole pizza. Ask questions such as, "How did you make sure you did end up with a whole pizza?" "Give an example of a trade you made." "If you did not end up with a whole pizza, what happened?" "What would you do next time?" (You may wish to do the swap more than once to allow students to try out different strategies.) 

After discussion of the swap, have students take out their math journals (or just sheets of paper) and make an entry about the activity. Things they can address include: 

  • a description of what they did. 
  • what strategies did and did not work. 
  • what they would do differently next time. 
  • how their knowledge of fractions helped them during the swap. 
    an equation in writing showing what fractional pieces they now have on their pizza, i.e., two sixths + 2 twelfths + 2 eighths + 4 sixteenths = 1 whole pizza


Action Plan

Contact a local pizzeria and take students on a field trip to see real pizza being made. At some places, students are even allowed to get into the process themselves. 

Plan field trips to observe people working at jobs where a knowledge of fractions is utilized. This could include visiting a photographer, a musician, a chef, a seamstress, and many others. Have these career people point out to students how the concept of fractions relates to their work. 



Give students the opportunity, either before or after doing the above lesson, to make real pizzas at school. Students could each have a small piece of dough to make their own tiny pizza and add preferred toppings. 

Students may benefit from having several "pizza swaps" to enable them to feel successful in ending up with a whole and unique pizza. They may also enjoy teaching another classroom how to make paper pizzas and have their own swap. Students could share the different equations they came up with to describe their pizzas. 

Social Studies:
Have students choose a career (ideas from The Eddie Files video could be suggested) and write a report on how and why a knowledge of fractions is necessary to pursue that job. Allow students time in class to share these projects. 

Replay the section on The Eddie Files, #4 video of the drummer and his explanation of how you need to know the different sizes of notes in order to read and play music. Talk with the music teacher in your school to find out what activities your students have done with reading music in his/her classes. Have students do some of their own recording of notes. Use their examples and drum sticks to have students beat out the rhythm. 

Have students conduct science experiments in which they use fractions to measure out substances. More specifically, have students consider real objects such as bridges and skyscrapers that are found in a city. Have pictures available with statistics regarding sizes of these objects, i.e., heights, widths, and lengths. Teach students how to create drawings and models that are made to scale. This will necessarily involve the use of fractions as students make a table containing the real measurements and the ones they will use to make their models and drawings. 

There are many times when students use the concept of equivalent fractions when dividing up their paper for an art lesson. Remind them of this when you ask them to fold their papers into fourths or eighths to divide up a paper for an art project or any kind of recording. 



Print :
Chapman, Steve. How Much? How Many? A Funny Numbers Book. Chicago: Follett, 1972.

Froman, Robert. The Greatest Guessing Game. New York: Crowell, 1978.

Lauber, Patricia. The Story of Numbers. New York: Random House, 1961.

Paysan, Klaus. Birds of the World in Field and Garden . Minneapolis: Lerner Publication Co., 1970.

Peppe, Rodney. Humphrey the Number Horse. New York: Viking Press, 1979.

Pierce, Georgia. Junior Science Book of Bird Life . Illinois: Gerard Publishing Co., 1967.

Sitormer, Mindel. How did Numbers Begin. New York: Crowell, 1976.

Scrivastava, Jane. Number Families. New York: Crowell, 1979.

Trivett, John. Building Tables on Tables. New York: Crowell, 1975.

Whitney, David. The Early Book of Multiplication. New York: F. Watts, 1969.


Cool Math for Kids: Fractions

Fun Brain: Fresh Baked Fractions Game

AAA Math: Identifying Equivalent Fractions

Virtual Manipulatives: Equivalent Fractions

Harcourt Brace: Find the Equivalent Fraction Match Game

Interactive Stuff: Equivalent Fractions


NH Framework Correlations

6c. Curriculum Standard: Students will understand the meaning of models, their appropriate use and limitations, and how models can help them in understanding the natural world.

1a. Broad Goal: Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content.
1b. Broad Goal: Students will use mathematical reasoning.
2a. Broad Goal: Students will communicate their understanding of mathematics.
2b. Broad Goal: Students will recognize, develop, and explore mathematical conditions.
3a. Broad Goal: Students will develop number sense and an understanding of our numeration system.
3b. Broad Goal: Students will understand the concepts of number operations.
4b. Broad Goal: Students will develop spatial sense.



Worksheet 1

Worksheet 2

Worksheet 3

Note: Worksheets are in Adobe PDF format. You will need Adobe Acrobat Reader to access them.