An Empson Approach to Teaching Fractions in a 3 rd grade Class

1. An Empson Approach to Teaching Fractions in a 3rd grade Class A 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.

2. 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.

3. 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

4. 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.

5. 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.

6. 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:

7. Students’ thinking one • One student is working on how to share • Another is thinking how to add thirds

8. Students’ thinking two • One student is working on how to share • Anther student is seeing one whole and a piece

9. 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?

10. 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:

11. 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?

12. 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.

13. Fraction Equivalency and order problems

14. 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?

15. 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.

16. 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.

17. 3 kids each get 4/3 candy bars. How many candy bars do they have all together? _____________Candy bars

18. Students’ Thinking One • One student after drawing a picture adds them together. • Another student has multiplied them

19. Students’ Two Thinking This Student Realizes that 4/3 is 1 1/3 because 3/3 equals a whole

20. Fifth graders’ explanations on equivalency ***NEW Discuss Mr. Li’s class solutions of a problem. Pages 123-125 of Empson’s Extending Mathematics

21. 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

22. 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.

23. 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.

24. 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.

25. 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.