Transcript Developing Fraction Concepts Math Alliance July 13, 2010 Beth Schefelker, DeAnn Huinker, Chris Guthrie & Melissa Hedges.

Developing Fraction Concepts
Math Alliance
July 13, 2010
Beth Schefelker, DeAnn Huinker,
Chris Guthrie & Melissa Hedges
Learning Intentions
and Success Criteria
We are learning to…
•
deepen our knowledge of fractions and
rational numbers
We will be successful when…
•
we can apply knowledge of fraction
benchmarks and conceptual thought patterns
to reason and compare fractions.
3
4
•
How do students see this fraction?
Students often see fractions as two
whole numbers (Behr et al., 1983).
•
What are ways we want students to
“see” and “think about” fractions?
Different models offer different
opportunities to learn.
Area model – visualize part of the
whole
Length or linear model –
emphasizes that a fraction is
a number as well as its
relative size to other numbers
Use the grey triangles to cover ¾ of
the octagon.
Set Model – the whole is set of
objects and subsets of the whole
make up fractional parts.
1
½
2
Where would ¾ fall on this
number line? Why?
3/4 of the smiley faces are blue
What is a fraction?
What is a rational number?
Are they the same?
Rational Number vs Fraction
•
Rational Number = How much?
Refers to a quantity or relative amount,
expressed with varied written symbols.
•
Fraction = Notation
Refers to a symbol or numeral used to
represent a rational number.
(Lamon, 1999)
Reasoning About Fractions
Name the fraction shown in the shaded
region of the figure below:
What do I need
to consider as I
decide on an
answer?
Share your responses.
What do you notice?
Who’s right?
Go to a poster that has a different answer
than yours.
Defend why that response could be
correct.
Now come up with a second argument to
defend your answer.
“The study of fractions offers many
delightful and challenging opportunities to
practice mathematical reasoning.”
p. 65 Beckmann
2¼
¾
9/12
9/4 9
Consider each response above as you respond to each
question:
• What’s the whole?
• What are the parts?
Big Ideas
• A fraction tells us the relationship between the part
and the whole.
• A fraction is always a fraction of some whole. The
whole needs to be understood while working with the
fraction even if it is not made explicit.
• Models help clarify ideas and visualize the
relationships between numerator and denominator.
What does the research say about
how students use fractions?
•
A majority of U.S. students have
learned rules but understand very little
about what quantities the symbols
represent and consequently make
frequent and nonsensical errors.
(NRC, 2001)
Share one error you’ve seen your
students make.
Reason with “Rational Numbers”
and Use Benchmarks
Is it a small part of the whole unit?
Is it a big part?
More than, less than, or equivalent:
to one whole?to one half?
Close to zero?
Finish these fractions so they are close
to but greater than one-half.
9
12
15
21
Finish these fractions so they are close
to but less than 1 whole.
11
16
85
24
Comparison of Fractions
Consider ways to reason with benchmarks
when comparing these fractions.
•
•
•
•
•
5/7
3/8
5/4
15/16
1 1/3
or
or
or
or
or
3/7
3/4
8/9
9/10
6/3
Conceptual Thought Patterns
for Comparing Fractions
More of the same-size parts.
Same number of parts but different sizes.
More or less than one-half or one whole.
Distance from one-half or one whole
(residual piece).
Ordering Fractions
on the Number Line
Deal out fraction cards (1-2 per person).
Allow quiet time to think about placements.
Taking turns, each person:
Places one fraction on the number line, and
Explains his/her reasoning using benchmarks
and conceptual thought patterns.
Warning: No conversions to decimals! No
common denominators! No cross multiplying!
Fraction Cards
3/8
3/10
6/5
7/47
7/100
25/26
7/15
13/24
14/30
16/17
11/9
5/3
8/3
17/12
Using Representations to
Conceptualize Fractions
How did you think about the fraction 8/3?
What does 8/3 mean?
Develop a real-life context for 8/3?
 Make a representation for your story that
helps develop an understanding of 8/3.

Reflect
As you placed the fractions
on the number line, summarize
some new reasoning or
strengthened understandings.
Walk Away
Fractions as quantities.
Benchmarks: 0, 1/2, 1, 2
Conceptual thought patterns.
Homework
Beckmann
Read pp. 65-70
Class Activities: p. 33
1 and 4
Also recommended though not required “Practice
Problems for Section 3.1” p. 70 #7
Using reasoning other than finding common
denominators, cross-multiplying, or converting to
decimal numbers to compare the sizes (greater than,
equal to, or less than) of the following fractions:
•1/49
1/39
•7/37
7/35
•13/25
5/8
•17/18
19/20