An Empson Approach to Teaching Fractions in a 3rd grade ClassA fraction project based on the research of Empson and Levi conducted by Mr. Joe Bysiek, 3rd grade teacher, and Dr. Gayle Millsaps and Diana Underwood****Slides marked new were added since the original presentation, for a Professional Development Project for Crown Point Community Schools.

Empson and Levi, 2011• Susan Empson and Linda Levi have developed a method

for teaching children fractions across grades K-6.• Their method is based on research observing children’s

learning of fractions in K-6 classrooms.• It builds on the work of the Cognitively Guided Instruction

working group that documented children’s learning of whole number sense and operations.

• Their approach encourages students to build meaning for fractions through solving and discussing word problems.

• They articulate a learning progression of students’ strategies for solving fraction problems that are indicative of students’ growth in understanding fractions.

A TRUE UNDERSTANDING ***NEW

Third Grade Problem:You have 4 sticks of butter. It takes ¾ sticks of butter to make a cake. How many cakes can you make?

YOU TRY

One third graders solution ***NEW

Work Explanation

5 1/3…..You can make 5 cakes and then you have one of the fourths of the three fourths needed so that’s a third of another.

Mr. Bysiek’s Learning Goals• Students understand that unit fractions are defined as equal parts of

a unit whole—i.e., 1/3 is one part of a unit whole broken into 3 equal parts.

• Students understand how fractions relate to a unit whole—what is ¾? 5/4?

• Students understand that a numerator and denominator that are the same equal one whole—3/3=4/4=5/5=1

• Students understand that the bigger the denominator the smaller the pieces—1/4 > 1/5.

• Students begin to develop an understanding of equivalent fractions—1/2=2/4=4/8 or 1/3=2/6.

• Students know how to add fractions with like denominators, and higher students can add unlike denominators—1/4+1/4+1/4=3/4 or ¼+1/2=3/4.

• Students begin to understand multiplying of unit fractions by whole numbers—3x1/4=3/4.

Fair Share Example problem• A teacher gave 4 sandwiches to 3 children to

share. If the students shared the sandwiches equally, how many sandwiches would each child get?

Answer: ________________sandwiches Work:

Students’ thinking one

• One student is working on how to share • Another is thinking how to add thirds

Students’ thinking two

• One student is working on how to share• Anther student is seeing one whole and a piece

Class Discussion

Here students begin discussing which is correct. Questions are asked if both are fair. As they agree, a discussion is started about how much each person would get. Is it the same? How do you name it? Can they both be right?

Multiple Groups Example Problem• There are 6 children and I want to give them

each ¼ of a twizzler. How many twizzlers do I need? _____twizzlers

• Show Work:

Students’ thinking one• ¼+ ¼ + ¼ + ¼ + ¼ + ¼ = 6/4 • ¼+ ¼ + ¼ + ¼= 1 and ¼ + ¼ = 2/4 • ¼ + ¼ + ¼ + ¼ =1 and ¼ + ¼ = ½

• Here a powerful discussion occurs with who is really correct? Is it fair each way? Does everyone get the same amount?

Students’ thinking two• This student says 6. Even though it is not the

correct answer, the student is still working on sharing fairly. A powerful discussion can be used and discuss what to do with the extra ¾ of each twizzler.

Fraction Equivalency and order problems

Ordering problem one• Jen and Robert got into an “argument” of who ate

more. Jen ate 1/4 and Robert ate 1/5. Who ate more? Why do you think that?

Students’ thinking one• One child states ¼ gets more because you are

sharing with less pieces.• Another child says states the smaller the

denominator the bigger the pieces, because you are sharing a candy bar with less people.

Students two thinking• 1/5 because five is a bigger number. Here a

powerful discussion arises. The student is asked to show what each person would get and gets an “ah ha” moment.

• 3 kids each get 4/3 candy bars. How many candy bars do they have all together? _____________Candy bars

Students’ Thinking One• One student after drawing a

picture adds them together. • Another student has multiplied

them

Students’ Two Thinking

This Student Realizes that 4/3 is 1 1/3 because 3/3 equals a whole

Fifth graders’ explanations on equivalency ***NEW

Discuss Mr. Li’s class solutions of a problem.

Pages 123-125 of Empson’s Extending Mathematics

Can you figure out which one is greater WITHOUT a common denominator? ***NEW

2/7 or 2/11

3/5 or 9/11

9/10 or 6/7

95/100 or 70/75

Taken from Empson’s Extending Children’s Mathematics p138

Key Findings• Empson’s approach allows more students to make

sense of fractions and how they relate to whole numbers.

• More students are able to achieve the fraction learning goals for 3rd grade than with traditional approaches.

Day 2 ***NEW• Review what a fair share problem is• Break into groups and create grade level

problems• Solve select problems together as a group

and discuss, how it meets Marzano’s scales.

Day 3 ***NEW• Review what a multiple group problem is• Break into groups and create grade level

problems• Solve select problems together as a group

and discuss, how it meets Marzano’s scales.

Day 4 ***NEW• Review what a “Problems for Fraction

Equivalence and Order” are• Break into groups and create grade level

problems• Solve select problems together as a group

and discuss, how it meets Marzano’s scales.