Gail Burrill's presentation at Green Lake 2012 of H. Wu's work with understanding fractions

1. Rethinking Fractions:
Implications for Teaching and
Learning Algebra
Gail Burrill
Michigan State University
Burrill@msu.edu

2. Fractions
1. On the portion of the number line below, a dot shows
where 1/2 is. Use another dot to show where 3/4 is.
(NAEP 2003)
QuickTimeª and a
decompressor
are needed to see this picture.
37% correct, grade 4
64% correct grade 8
2. Jose ate 1/2 of a pizza.Ella ate 1/2 of another pizza.
Jose said that he ate more pizza than Ella, but Ella
said they both ate the same amount. Use words and
pictures to show that Jose could be right. (NAEP, 1992)
Grade 4, 57% omitted; 8%
satisfactory

3. An Overview
Why a different approach?
What does this approach look like?
The role of interactive dynamic technology
What might we gain for algebra?

4. Procedural & Conceptual
Knowledge
Understanding requires both procedural and
conceptual knowledge
Conceptual supports the development of
procedural but not the other way around
NRC, 2001

5. Conceptual Knowledge:
– Makes connections more visible,
– enables reasoning about the mathematics,
– less susceptible to common errors,
– less prone to forgetting.
Procedural Knowledge:
– strengthens and develops understanding
– allows students to concentrate on relationships
rather than working out results
NRC, 1999; 2001

6. Research on learning algebra:
Making links to the classroom
1988 NCTM Yearbook on Algebra:
Common Mistakes in Algebra (Marquis, 1988)
a2. b5 = (ab)7 (x+4)2 = x2+ 16
x+y y x r x+r
= + =
x+z z y s y+s
3a-1= 1
10 of 22 were related to
3a
fractions

7. A fraction
Is typically thought of as three things:
Quotient,
Part to Whole, or
Ratio
Rethinking Fractions:
Based on Chapter 2, Fractions by H. Wu
Department of Mathematics #3840
University of California, Berkeley
Berkeley, CA 94720-3840
http://www.math.berkeley.edu/_wu/

8. Grade 3: Understand
A fraction 1/b is the quantity formed by 1 part when a whole is
partitioned into b equal parts; a fraction a/b is the quantity
formed by a parts of size 1/b.
a fraction is a number on the number line; represent fractions on
a number line diagram.
2. a. Represent a fraction 1/b on a number line diagram by
defining the interval from 0 to 1 as the whole and partitioning it
into b equal parts. Recognize that each part has size 1/b and that
the endpoint of the part based at 0 locates the number 1/b on the
number line.
b. Represent a fraction a/b on a number line diagram by marking
off a lengths 1/b from 0. Recognize that the resulting interval has
size a/b and that its endpoint locates the number a/b on the
number line.

9. Grade 3: Understand
A fraction 1/b is the quantity formed by 1 part when a whole is
partitioned into b equal parts; a fraction a/b is the quantity
formed by a parts of size 1/b.
a fraction is a number on the number line; represent fractions on
a number line diagram.
2. a. Represent a fraction 1/b on a number line diagram by
defining the interval from 0 to 1 as the whole and partitioning it
into b equal parts. Recognize that each part has size 1/b and that
the endpoint of the part based at 0 locates the number 1/b on the
number line.
b. Represent a fraction a/b on a number line diagram by marking
off a lengths 1/b from 0. Recognize that the resulting interval has
size a/b and that its endpoint locates the number a/b on the
number line.

10. Grade 3: Understand
3. equivalence of fractions in special cases, and compare fractions
by reasoning about their size.
a. two fractions as equivalent (equal) if they are the same size, or
the same point on a number line.
b. Recognize and generate simple equivalent fractions. Explain
why the fractions are equivalent,
c. Express whole numbers as fractions, and recognize fractions
that are equivalent to whole numbers. Express whole numbers as
fractions, and recognize fractions that are equivalent to whole
numbers.
d. Compare two fractions with the same numerator or
denominator by reasoning about their size. Recognize that
comparisons are valid only when the two fractions refer to the
same whole.

11. Grade 3: Understand
3. equivalence of fractions in special cases, and compare fractions
by reasoning about their size.
a. two fractions are equivalent (equal) if they are the same size,
or the same point on a number line.
b. Recognize and generate simple equivalent fractions. Explain
why the fractions are equivalent,
c. Express whole numbers as fractions, and recognize fractions
that are equivalent to whole numbers. Express whole numbers as
fractions, and recognize fractions that are equivalent to whole
numbers.
d. Compare two fractions with the same numerator or
denominator by reasoning about their size. Recognize that
comparisons are valid only when the two fractions refer to the
same whole.

12. Grade 3: Geometry
Partition shapes into parts with equal areas.
Express the area of each part as a unit fraction of
the whole

13. Grade 4: Fractions
Extend understanding of fraction equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction
(n × a)/(n × b) by using visual fraction models. Use this to
recognize and generate equivalent fractions.
2. Compare two fractions with different numerators and
denominators, e.g., by creating common denominators or
numerators, or by comparing to a benchmark fraction such as
1/2. Recognize that comparisons are valid only when the two
fractions refer to the same whole.
Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.

14. Grade 4: Fractions
Extend understanding of fraction equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction
(n × a)/(n × b) by using visual fraction models. Use this to
recognize and generate equivalent fractions.
2. Compare two fractions with different numerators and
denominators, e.g., by creating common denominators or
numerators, or by comparing to a benchmark fraction such as
1/2. Recognize that comparisons are valid only when the two
fractions refer to the same whole.
Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.

15. Grade 4: Fractions
3. A fraction a/b with a > 1 is a sum of fractions 1/b.
a. addition and subtraction of fractions is joining and separating
parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same
denominator in more than one way, recording each
decomposition by an equation.
c. Add and subtract mixed numbers with like denominators,
4. Apply and extend previous understandings of multiplication to
multiply a fraction by a whole number.
a. A fraction a/b is a multiple of 1/b. 5/4 = 5 × (1/4).
b. A multiple of a/b is a multiple of 1/b, and use this
understanding to multiply a fraction by a whole number.

16. Grade 4: Fractions
3. A fraction a/b with a > 1 is a sum of fractions 1/b.
a. addition and subtraction of fractions is joining and separating
parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same
denominator in more than one way, recording each
decomposition by an equation.
c. Add and subtract mixed numbers with like denominators,
4. Apply and extend previous understandings of multiplication to
multiply a fraction by a whole number.
a. A fraction a/b is a multiple of 1/b. 5/4 = 5 × (1/4).
b. A multiple of a/b is a multiple of 1/b, and use this
understanding to multiply a fraction by a whole number.

17. Fraction as a point on a number line
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decompressor
are needed to see this picture.
N/3 is a multiple of
1/3 k/p is the length of the
concatenation of k segments
A whole number is each of which has length 1 p .
also a fraction
No difference k/p is k copies of 1/p .
between proper and
improper fractions

18. QuickTimeª and a
decompressor
are needed to see this picture.
Describe in words where each fraction is on the number
line and sketch a picture to show its location.
(a) 7/9 (b) 6/11 (c) 9/4 (d) 13/5
(e) 10/ 3
(f) k/5 , where k is a whole number between 11 and14.
(g) k/6 , where k is a whole number between 25 and 29.

19. Equality
The equality a/b = k/l means that the two
points a/b and k/l are precisely the same
point on the number line.
Locate 16/3; 84/17. Which one is larger and
why?

20. A unit other than 1
If the weight of an object X is our unit, then
5/7 would mean 5 parts of X after it has
been partitioned into7 parts of equal weight.
Unit is the weight of a piece of ham that
weighs three pounds. 1/3 of the weight
would be one pound.
Unit is a pack of 15 pencils; 2/3 of a unit
would be to partition into three equal parts
of 5 pencils or 10 pencils.

21. Unit squares
The area of the unit square is by definition
1.
If two regions are congruent, then their areas
are equal. (If two shapes are congruent, then
one figure can be rotated, translated or
reflecting to fit on exactly on top of the
other.)
Find four representations of 1/4.
Find two representations of 5/6, 7/4, 9/4.

22. Area of unit square as unit 1
Prove 15/6 = 5/2

23. Area of unit square as unit 1
Prove 15/6 = 5/2
5/2 is 2 and one half so it is
2 and a half unit squares
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24. Find the fraction if
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a b c

25. Find the fraction if
The shaded area is one unit
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a b c

26. Exercises
1. Indicate the approximate position of each of the following
on the number line, and also write it as a mixed number. (a)
67/4 . (b) 205/11 (c) 459/23 (d) 1502/24 .
2. (a) After driving 352 miles, we have done only two-thirds of
the driving for the day. How many miles did we plan to drive
for the day?
(b) After reading 168 pages, I am exactly four-fifths of the way
through. How many pages are in the book?
(c) Helena was three quarters of the way to school after having
walked 22 5 miles from home. How far is her home from
school?
3. (a) I have a friend who earns two dollars for every three
times she walks her parents’ dog. She knows that this week she
she will walk the dog twelve times. How much will she earn?

27. Exercises
Indicate the approximate position of each of the following on
the number line, and also write it as a mixed number. (a) 67/4 .
(b) 205/11 (c) 459/23 (d) 1502/24 .
2. (a) After driving 352 miles, we have done only two-thirds of
the driving for the day. How many miles did we plan to drive
for the day?
(b) After reading 168 pages, I am exactly four-fifths of the way
through. How many pages are in the book?
(c) Helena was three quarters of the way to school after having
walked 22 5 miles from home. How far is her home from
school?
3. (a) I have a friend who earns two dollars for every three
times she walks her parents’ dog. She knows that this week she
she will walk the dog twelve times. How much will she earn?

28. Equivalence
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5/2=15/3
km/kl = m/l and m/l = km/kl
Use the representation of 1 as the area of a unit square
to give a proof of why 6/14 = 3/7 and 30/12 = 5/2 .

29. Addition
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are needed to see this of 1/L is (m + k) copies of 1/L
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30. Multiplication
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31. Peopl e learn by
Engaging in a concrete experience
Observing reflectively
Developing an abstract
conceptualization based upon the
reflection
Actively experimenting/testing
based upon the abstraction
Zull, 2002

32. Research on learning algebra:
Making links to the classroom
1988 NCTM Yearbook on Algebra:
Common Mistakes in Algebra (Marquis, 1988)
a2. b5 = (ab)7 (x+4)2 = x2+ 16
x+y y x r x+r
= + =
x+z z y s y+s
3a-1= 1
10 of 22 were related to
3a
fractions

33. References Mathematics. (2010). Council of Chief State
Common Core State Standards
School Officers & National Governors Association.
www.corestandards.org/
Marquis, J. Common mistakes in algebra.The Ideas of Algebra K-12. 1988
Yearbook. Coxford, A. (Ed). Reston, VA. National Council of Teachers of
Mathematics.
National Assessment for Educational Progress (2003, 1992). Released
Items. National Center for Educational Statistics.
National Research Council (2001). Adding It Up. Kilpatrick, J., Swafford,
J., & Findell, B. (Eds.) Washington DC: National Academy Press. Also
available on the web at www.nap.edu.
National Research Council. (1999). How People Learn: Brain, mind,
experience,and school. Bransford, J. D., Brown, A. L., & Cocking, R. R.
(Eds.). Washington, DC: National Academy Press.
Wu, H. (2006). Fractions, Chapter 2. In Understanding Numbers in
Elementary School Mathematics, Amer. Math. Soc., 2011.]
//www.math.berkeley.edu/_wu/
James Zull, ( 2002). The Art of Changing the Brain: Enriching the Practice
of Teaching by Exploring the Biology of Learning. Association for
Supervision and Curriculum Development, Alexandria, Virginia.