Teaching and Learning Fractions
with Conceptual Understanding
Algebra Forum IV
San Jose, CA
May 22, 2012
Compiled and Presented by
April Cherrington
Region 5
Joan Easterday
Region 1
Susie W. Hakansson, Ph.D.
California Mathematics Project
Description
Fractions from a number line approach
represents a shift in thinking about fractions,
moving beyond part-whole representations to
thinking of a fraction as a point on the number
line. Included in this session will be the
following: rationale, comparing and ordering,
and justification.
2
Outline for Today



(20 minutes) Introduction
(90 minutes) Breakout session
(20 minutes) Reflection
3
Introduction


CaCCSS-M Task Force
Conceptual
understanding

Why number line?

Fraction progressions


Order problems

Cognitive level


Language issues

Standards for
Mathematical Practice
Challenges students face
Overview of break out
session
4
Fractions Task Force

Greisy Winicki-Landman, Chair

Nadine Bezuk

Pam Hutchison

April Cherrington

Natalie Mejia

Pat Duckhorn

Gregorio Ponce

Joan Easterday

Debbie Stetson

Doreen Heath Lance

Kathlan Latimer
5
Demands of CaCCSS-M
“… almost all teachers are placing a lower priority on
student understanding in recent years, ….”
“… the sort of high quality PD that an really affect
teachers in their ability to produce students who
understand is very, very difficult to do, and very few
people have much clue about how to do it.”
Scott Farrand
6
Fraction Sense: Comparing
Solve these problems mentally without using algorithms.
Justify your thinking.
 8 15
> 1/2 (?)
 6 11
 7 22
> 1/3 (?)
 7 8
/
/
/
> 7/15 (?)
/ > 8/9 (?)
7
Cognitive Demand Spectrum
Memorization Procedures
Without
Connections
to understanding,
meaning, or concepts
Tasks that require
memorized procedures in
routine ways
Procedures
With
Connections
Doing
Mathematics
to understanding,
meaning, or concepts
Tasks that require
engagement with concepts,
and stimulate students to
make connections to
meaning, representation,
and other mathematical
ideas
Why Is English So Hard?


The soldier decided to desert his dessert in the
desert.
Upon seeing the tear in the painting, I shed a
tear.

After a number of injections, my jaw got number.

A minute is a minute part of a day.
9
Why Is English So Hard?


There is no egg in eggplant and no ham in
hamburger.
How can a slim chance and a fat chance be the
same, while a wise man and a wise guy are
opposites?

Did you say thirty or thirteen?

Did you say two hundred or two hundredths?

Did you say fifty or sixty?
10
Dual Meaning Words
Math Word
Meaning 1 (Math)
Meaning 2
Solution
The answer
Two or more substances
mixed together
Exercise
Math problems to solve
Physical movement to stay fit
Product
The answer when you
multiply two or more
numbers
Something made by humans
or machines
Expression
Math statement with
numbers and/or variables
To show emotion
The Guinevere Effect
9th and 10th graders’ responses

Tom had 5 apples. He ate 2 of them. How many
apples were left?


A. 10
B. 7
C. 5
D. 3 (100%)
Guinevere had 5 pomegranates. She ate 2 of them.
How many pomegranates were left?

A. 10 (22%)
B. 7 (24%)
C. 5 (23%)
D. 3 (31%)
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Key Strategies for English Learners

Access prior knowledge

Frontload language

Build on background knowledge

Extend language

Be aware of multiple meanings of words

Have students Think, Ink, Pair, Share (TIPS)
13
Teachers learn to amplify and enrich--rather
than simplify--the language of the classroom,
giving students more opportunities to learn the
concepts involved.
Aída Walqui, Teacher Quality Initiative
14
Why Number Line?
“Hung-Hsi Wu attempts to bring coherence to the teaching
and learning of fractions by beginning with the definition of a
fraction as the length on the number line (1998). This
approach eliminates the ‘conceptual discontinuity’ (2002)
encountered moving from work with whole numbers to
fractions; it also brings coherence to the various meanings of
fractions and allows for both conceptual work to operations on
fractions (2008). Wu asserted that ‘The number line is to
fractions what one’s fingers are to whole numbers ...”
15
Basic Assumptions about the
Number Line and Its Use




Numbers go on infinitely in both directions.
On a conventional horizontal number line, the numbers
increase from left to right.
The numbers to the right of zero are the positive
numbers, and those to the left are negative numbers.
0 is not positive nor negative.
16
Basic Assumptions about the
Number Line and Its Use

Using the number line, there are basically two types of
tasks:



Given a point on the number line, assign a number to it (its
coordinate)
Given a number, place it as a point on the number line.
The length of the interval from 0 to 1 is called the unit
and it determines the distance between every pair of
consecutive integers on the line.
17
Basic Assumptions about the
Number Line and Its Use


Fractions can be placed on the number line by
partitioning the length from 0 to 1 into d equal parts. One
of these parts has the length 1/d; n of those parts has the
length n/d. The fraction n/d is the point at the end of the
segment of length n/d.
Given the unit interval, each point on the number line can
be associated to infinitely many fractions: the name will
depend on the partition chosen. All of these fractions are
equivalent.
18
WHY THE Number Line?


It serves as a visual/physical model to represent the
counting numbers and constitutes an effective tool to
develop estimation techniques, as well as a helping
instrument when solving word problems.
It constitutes a unifying and coherent representation for
the different sets of numbers (N, Z, Q, R), which the
other models cannot do.
19
WHY THE Number Line?


It is an appropriate model to make sense of each set of
numbers as an expansion of other and to build the
operations in a coherent mathematical way.
It enables to present the fractions as numbers and to
explore the notion of equivalent fractions in a
meaningful way.
20
WHY THE Number Line?



The number line, in some way, looks like a ruler,
fostering the use of the metric system and the decimal
numbers.
It fosters the discovery of the density property of
rational numbers.
It provides an opportunity to consider numbers that are
not fractions.
21
Common Core Standards Mathematics
Grades 3, 4 and 5
Number and Operations - Fractions
Grade 3
• Develop understanding of
fractions as numbers
Grade 4
•Extend understanding of fraction
equivalence and ordering
•Build fractions from unit fractions
by applying and extending previous
understandings of operations on
whole numbers.
•Understand decimal notation for
fractions, and compare decimal
fractions.
Grade 5
• Use equivalent fractions as a
strategy to add and subtract
fractions.
• Apply and extend previous
understandings of multiplication
and division to multiply and
divide fractions.
Common Core Standards Mathematics
Grades 6 and 7
Number and Operations - Fractions
Grade 6
•Apply and extend previous
understandings of multiplication
and division to divide fractions
by fractions.
Grade 7
•Apply and extend previous
understandings of operations
with fractions to add, subtract,
multiply, and divide rational
numbers.
CaCCSS-M: Mathematical Practice

We will focus two of the Standards for
Mathematical Practice:


Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning
of others
24
25
Reason Abstractly and
Quantitatively
DO STUDENTS:
 Explain a problem to themselves, determine what it
26
Construct Viable Arguments and
Critique the Reasoning of Others

Use multiple representations (verbal descriptions,?
Standards for Mathematical Practice
27
Question
What are some of the challenges that students
have with fractions?
28
Overview of Breakout Sessions

Appropriate grade level problem

Twelve (12) cards

Videos of students working with 12 cards

Human Number Line activity

Reflection
29
Slides for Breakout
30
Breakout Reflection



What mathematics did you use in the activities?
How did your reasoning and explaining support
and expand your understanding of the
mathematics?
How did these activities provide access to all
students? Give specific examples.
31
Return to main room
32
33
Part IB: Equivalent Fractions,
Comparing and Ordering Fractions


Use the structure of the number line and
benchmarks to determine value of the “?” of each
number line strip.
With a partner agree on the value of the “?” and
share your strategies.
34
Ordering 12 Cards
Hardest
Easiest
Order Number Lines
Explain Reasoning
1.
2.
.
.
.
.
.
.
.
12.
CCSS-M Task Force: CAMTE, CDE, CISC, CMC, CMP
M2 A3
Human Number Line: Ordering

Divide into groups of 10.

Each of you will be given a card.



One person will place himself/herself on a line to
establish a point.
Each subsequent person is to place himself/herself to
the left, right, or in between the existing numbers to
maintain proper order.
This task focuses on order and not proportionality.
36
Reflection: Think, Ink, Pair, Share

How has this experience with these
activities expanded your concept of
fractions?

How will this inform your instruction in
the classroom?
How will you amplify and enrich the
language to support English learners?

37
SOLVE THIS WITHOUT USING
ALGEBRA OR a/b = c/d
What is the ratio of men to women in a town
where two-thirds (2/3) of the men are married
to three-fourths (3/4) of the women?
38
44
Goals for Institute




Support English learners in mathematics with high
cognitive demand tasks.
Become familiar with the CaCCSS-M, particularly the
Standards for Mathematics Practice.
Gain a conceptual understanding of the number line as
a big idea in the CaCCSS-M.
Use the number line in working with fractions.
45
Reflection




How will we support English learners in mathematics
with high cognitive demand tasks?
What did we learn about the CaCCSS-M, particularly
the Standards for Mathematics Practice?
What understanding of the number line as a big idea in
the CaCCSS-M did we gain?
How will we use the number line in working with
fractions?
46
Delivery of Instruction


Engage students in high cognitive demand tasks.
Assess by walking around (ABWA). Provide
access to the language of mathematics.

Allow for discourse (Think, Ink, Pair, Share--TIPS)

Set high expectations and increase expectations

Allow students to explore why—metacognition
47
Acquiring the Knowledge


Become familiar with the content and academic language of
your lesson and possible misinterpretations.
Frontload the academic/mathematics/English language of the
mathematics content.

Amplify and enrich the language.

English language learners are trying to catch a moving target.

Be aware of how your students interpret the academic and
mathematics language.
48
Putting This Together
When you design instruction, you start with the
cognitively demanding mathematics you want
students to learn. You gain an in-depth understanding
of the content. Then you incorporate the most
effective instructional practices to meet the needs of
your students. You access prior knowledge, build
background knowledge, and extend language.
49
Putting This Together
You instinctively know to frontload the language, to
ask questions of students, to have students think,
ink, pair, share, to increase discourse, to increase
student engagement, to assess, etc. This process
becomes second nature to you. You are addressing
the needs of ALL students, particularly English
learners.
50
Equity and Quality
How do we provide access to ALL students? We
want students to make sense of rigorous, high
quality, and cognitively demanding mathematics. We
want them to approach the zone of proximal
development, not the zone of minimal effort (e.g.,
assigning only lower level problems, not requiring
homework, expecting less).
51
Equity and Quality
We want equity and quality. Equity without quality is
meaningless. Quality without equity is unjust. We
must always ask ourselves, what can we do to
incorporate both?
52
Final Thoughts
It is up to us to provide greater access and opportunity to high
cognitive level mathematics by enriching the language and by
using instruction that supports the learning needs of ALL
students. It is up to us to acquire the understanding of
fractions from a number line approach. What will you do to
acquire the knowledge so that the understanding of the
content and the integration/infusion of mathematics and
language becomes second nature to you?
53
Assessment




What did I learn?
What do I plan to implement in my
classroom?
Questions I have remaining.
I suggest . . .
54
Contact Information
April Cherrington: acherrington@smcoe.k12.ca.us
Patricia Duckhorn: pduckhorn@att.net
Susie W. Hakansson: shakans@ucla.edu
Greisy Winicki-Landman: greisyw@csupomona.edu
55
Links
CaCCSS-M Task Force Materials: http://caccssm.cmpso.org
TODOS: Mathematics for ALL: http://www.todos-math.org
California Mathematics Project: http://www.cmpso.org
56