1. What are Fractions? Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
Monday July 20, 2015

2. Learning Intention
We are learning to … Explain the mathematical content and language in 1.G.3, 2.G.3, 3.G.2 and provide examples of the mathematics and language.

3. A commentary on the status of fraction knowledge in the U.S.....

4. 3 4 Please jot down your responses:
What is the meaning of this fraction?
What is the meaning of the top number?
What is the meaning of the bottom number?
Draw a visual model to represent it.
3 fourths of something

5. A teacher asked her students to read this fraction
A teacher asked her students to read this fraction. For each response, consider whether it is a mathematically accurate or appropriate way to refer to the fraction. 3 4
3 fourths
3 out of 4
3 over 4
3 shares of 1/4
3 parts shaded and 4 parts in the whole
3 one-fourths
3 parts of size 1/4
Some of these promote more understanding than others…. We will think about these ways of saying what ¾ means over the next few days.

6. More importantly, do your explanations support these visual models?
Visual models for ?
More importantly, do your explanations support these visual models?
Which are correct? 3 4 1

7. 7 4 Do your explanations work for ?
What is the meaning of this fraction?
What is the meaning of the top number?
What is the meaning of the bottom number?

8. Student Thinking
How should students “know” fractions?
How do students “really think” about and reason with fractions?

9. 8 9 5 4 or
Student 1
T: Which of these fractions is larger? S: Student circled 8/9 T: Why did you circle that fraction? S: Because it has a bigger number on the bottom.

10. 8 9 5 4 or
Student 2
T: Which of these fractions is larger? S: Student circled 8/9 S: Because 8 ninths makes a little more sense than 5 fourths. T: Why does it make more sense? S: Because how can you get 5 out of 4?

11. CCSSM Fractions Progression

12. CCSSM Fraction Progression
Grades 1-3
Geometry
Grades 3-5
Fractions
Number Systems
Grade 6
Pass out the CCSSM Fraction progression handout.

13. Fraction Progression: Grades 1-6
As you scan the fraction standards,
Highlight the word Understand.
Highlight the words visual model(s).
As a table group, discuss common themes that surfaced regarding:
Expectations for students?
Implications for teaching? .

14. Grades 1,2,3 Domain: Geometry
The CCSSM gives us so much more than just a list of topics.
It offers suggestions on the “ways of knowing” the mathematics that we should be developing with students.
Conceptual topic: : halves - students should be developing an understanding of halves
How is that students should understand halves?
What are the specific understandings we are trying to develop in students?

15. CCSS 1.G.3, 2.G.3, 3.G.2 What does 1.G.3 mean?
Individually do steps 1, 2 and 3.
Read the standard.
Highlight key words and key phrases from the standard and restate it in your own words.
Provide examples for the highlighted ideas.
Table share.
Repeat this 4 step process for 2.G.3 and 3.G.2
Equal shares is probably the biggest piece to focus on here – we want kids to be able to describe the shares. # of shares and the size of the shares
This is the first time that the participants will begin to look at a Progression. How did the ideas grow from 1st grade to 2nd grade to 3rd grade?

16. CCSS 1.G.3, 2.G.3, 3.G.2
Count off by 3’s at your table. #1: Read pp. 7-9 in Uncomplicating Fractions. #2: Read pp in Uncomplicating Fractions. #3: Read pp in Uncomplicating Fractions. Explain to your group how your reading further informed your thinking about the standards.
discuss how ideas of partitioning are developing across Grades 1– 3
After participants read, they will share ideas that surfaced within their triad. As they listen and share participants should be recording new ideas on their standards worksheet. This will help them add to their examples of the standards.
Bring the group together….Chart out ideas that were shared and added to the standards worksheet. Chart the standards/grade level in a new color and the new idea in another.

17. Geometry: Reason with shapes and their attributes.
Grade 1 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths… and use the phrases half of, fourth of... Describe the whole as two of, or four of the shares. Understand that decomposing into more equal shares creates smaller shares. (1.G.3) Grade 2 Partition circles and rectangles into two, three, or four equal shares, describe the shares using words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. (2.G.3) Grade 3 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. (3.G.2)

18. Geometry: Reason with shapes and their attributes.
Grade 1 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths… and use the phrases half of, fourth of... Describe the whole as two of, or four of the shares. Understand that decomposing into more equal shares creates smaller shares. (1.G.3) Grade 2 Partition circles and rectangles into two, three, or four equal shares, describe the shares using words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. (2.G.3) Grade 3 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. (3.G.2)

19. Pulling Ideas Together
What‘big ideas’ about these standards would you share with your 1st through 3rd grade teachers?
Whole group to add to chart – 5 min.
Summary statement: Table group 5 min.
Summary statement should be a concise and written on a post-it note. Put the note on the standards form.
Practice the language of shares.

20. PRR: Focus on Fractions
Read Petit, A Focus on Fractions, Chapter 4
Pages 59 – 61 (end at the top)
In your notebook (mark the page with a tab).
Write your own definition of partitioning.
Why is partitioning a key idea in Standards 1.G.3, 2.G.3, and 3.G.2?

21. Reflections

22. Learning Intention
We are learning to … Explain the mathematical content and language in 1.G.3, 2.G.3, 3.G.2 and provide examples of the mathematics and language.

23. Summary of Daily Reflections
Revisiting our learning intentions: 1.G.1, 2.G.1, 3.G2 Summarize two key points that you are taking away from the initial study of these standards that will strengthen your fraction instruction.

24. Fraction Instruction An International View Comparing the Common Core to the Japanese Curriculum Part 1

25. We are learning…
How fraction instruction in Japan compares to the Common Core State Standards for Mathematics.
How we might strengthen our fraction instruction through international comparisons of expectations for fraction teaching, learning, and curriculum.

26. Study the Grade 2 Japanese lesson.
What visual models?
What contexts?
What language?
What symbols?
What else do you notice?
Study 3 minutes
individually!!
Then wait for new directions.

27. Table group facilitators/recorders, 5 whiteboards.
Grade 2 Japanese lesson
Table group facilitators/recorders, 5 whiteboards.
What visual models?
What contexts?
What language?
What symbols?
What else do you notice?

28. Grade 2 As you study Japanese lesson, make note of . . .
important conceptual ideas to be developing with students and how it is similar or different than the CCSSM in Grades 1-2.
(1) Partitioning, folding and cutting.
(2) Strong emphasis on equal parts and matching up the pieces to see they are equal.
(3) Emphasis on describing the size of a part with words, one half, one fourth.
(4) Emphasis on “half of” and “fourth of” something, of the whole, which is the original size piece of paper.

29. Grade 2 Same and Different (1) Partitioning, folding and cutting.
(2) Strong emphasis on equal parts and matching up the pieces to see they are equal.
(3) Emphasis on describing the size of a part with words, one half, one fourth.
(4) Emphasis on “half of” and “fourth of” something, of the whole, which is the original size piece of paper.
Also (5) The number of parts or shares in one whole.
Different
(1) Focus on “equal parts” not “equal shares.”
(2) Brings in idea of length.

31. The Grade 2 Japanese lesson
– Students partitioning various sized wholes.
– Emphasis on relating one part (unit fraction) to the whole.
– Emphasis on the number of parts in one whole.
– Emphasis on oral language.
– Quick movement to using a tape diagram model.
– Quick movement to length as a context.

32. “The Teaching and Learning of Fractions: A Japanese Perspective”
Tad Watanabe Article
“The Teaching and Learning of Fractions: A Japanese Perspective”
Read pp
Section on “Unique Features”
Jot down 3 unique features that might be important for you to ponder further for your own instruction.

33. Grades 1-2: Comparing CCSSM & Japan
Similar
Different
Partitioning (folding, cutting).
Describe with words: half, third, fourth.
Describe as half of, third of, fourth of one whole ... always relate the size of a fraction to the size of a whole.
Emphasis on number of equal parts or shares in one whole: two halves, three thirds, four fourths.
Japan
Begins in Grade 2.
Focus on “equal parts” not “equal shares.”
Introduces idea of length.
Use of fraction notation.
CCSSM
Begins in Grade 1 with partitioning.

34. CCSSM “Common Core Tools” Blog
Question on Notation for Fractions March 22, 2013 #1829
Good Morning,
A team of 2nd grade teachers were working on 2.G.3 yesterday. The group was wondering to what extent fraction notation for halves, thirds, and fourths was required. We checked a couple of progressions and could not see the answer. Thank you in advance for any guidance you can provide.
Bill McCallum response: March 25, 2013 #1841
The standards don’t require fraction notation at all in Grade 2. This [standard] is not really about fractions as numbers, although it is clearly a precursor to that concept. Fractions as numbers and fraction notation are introduced in Grade 3.

35. "Are Fractions Numbers and Why Does It Matter?"
Webinar
Tue, Jul 21, :00 PM - 7:00 PM CDT

36. PRR: Focus on Fractions
Read Petit, A Focus on Fractions, Chapter 4
Pages 71 – 74 (Stages of Partitioning)
In your notebook (mark the page with a tab).
Summarize the Stages of Partitioning.
Why is partitioning a key idea in Standards 1.G.3, 2.G.3, and 3.G.2?

37. Reflections

38. We are learning…
How fraction instruction in Japan compares to the Common Core State Standards for Mathematics.
How we might strengthen our fraction instruction through international comparisons of expectations for fraction teaching, learning, and curriculum.

39. Summary of Daily Reflections
Revisiting our learning intentions: 1.G.1, 2.G.1, 3.G2 Summarize two key points that you are taking away from the initial study of these standards and the Japanese lessons that will strengthen your fraction instruction.

40. Disclaimer Core Mathematics Partnership Project
University of Wisconsin-Milwaukee,
This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.