Transcript Reasoning with Rational Numbers (Fractions) Math Alliance Project July 20, 2010 DeAnn Huinker, Chris Guthrie, Melissa Hedges,& Beth Schefelker,

Reasoning with
Rational Numbers (Fractions)
Math Alliance Project
July 20, 2010
DeAnn Huinker, Chris Guthrie,
Melissa Hedges,& Beth Schefelker,
Learning Intentions & Success Criteria
Learning Intentions: We are learning to…
Develop “operation sense” related to adding and
subtracting common fractions through contextual
situations.
Understand how estimation should be an integral
part of fraction computation development.
Success Criteria: You will be able to…
Justify your thinking when adding and subtracting
fractions using concrete models and estimation
strategies.
What’s in common?
2
6
0.333333...
1
3
1
33 %
3
50
150
Big Idea: Equivalence
Any number or quantity can be
represented in different ways.
Example, these all represent the same quantity.
1
3
2
6
0.333333…
33 1 %
3
Different representations of the same quantity
are called “equivalent.”
Benchmarks
for “Rational Numbers”
7
13
Is it a small or big part of the whole unit?
How far away is it from a whole unit?
More than, less than, or equivalent to:
one whole? two wholes?
one half?
zero?
Conceptual Thought Patterns
for Reasoning with Fractions
8/15 or 11/15
More of the same-size
parts.
7/20 or 7/9
Same number of parts
but different sizes.
6/10 or 9/5
More or less than onehalf or one whole.
11/12 or 7/8
Distance from one whole
or one-half (residual strategy–
What’s missing?)
Estimate
12
7
+
=
13
8
NAEP 13 yr
MPS 6-7-8
• 1
7%
13%
• 2
24%
9%
• 19
28%
23%
• 21
27%
41%
• Don’t Know
14%
9%
National Assessment of Education Progress (NAEP); MPS n=72)
Task:
Estimation with Benchmarks
Facilitator reveals one problem at a time.
Each individual silently estimates.
On the facilitator’s cue:
Thumbs up = greater than benchmark
Thumbs down = less than benchmark
Wavering “waffling” = unsure
Justify reasoning.
Research Findings:
Operations with Fractions
Students do not apply their understanding
of the magnitude (or meaning) of fractions
when they operate with them.
(Carpenter, Corbitt, Linquist, & Reys, 1981)
Estimation is useful and important when
operating with fractions and these students
are more successful (Bezuk & Bieck, 1993)
Students who can use and move between
models for fraction operations are more
likely to reason with fractions as quantities.
(Towsley, 1989)
Fraction Kit
Fold paper strips. Only mark the folds,
no words or symbols on the strips.
Yellow: Whole strip
Purple: Halves, Fourths, Eighths
Green: Thirds, Sixths, Ninths, Twelfths
Representing Operations
Envelope #1
Pairs
Each pair gets one word problem.
Estimate solution with benchmarks.
Use the paper strips to represent
and solve the problem.
Table Group
Take turns presenting your reasoning.
Repeat
Representing Operations
Envelope #2
As you work through the problems in this
envelope, identify ways the problems and
your reasoning differ from envelope #1.
Pairs: Estimate. Solve with paper strips.
Table Group: Take turns presenting.
Repeat
Representing Operations
Envelope #3
Individual
Each person gets a reflection prompt.
Consider your thoughts on it.
Table Group
Take turns to facilitate a table group
discussion of your prompt.
Representing Your Reasoning
Pose a word problem for each equation (context).
On paper, clearly represent your reasoning steps
with diagrams, words, and symbols.
11
12
–
1
4
=
3
4
+
5
6
=
Big Idea: Algorithms
Algorithms for operations with rational
numbers use notions of equivalence to
transform calculations into simpler ones.
Adding or subtracting fractions may involve
renaming or trading fractions until the parts
are the same size. Sometimes one fraction
needs renaming, sometimes both, and
sometimes neither.
Walk Away
Estimation with benchmarks.
Word problems for addition and
subtraction with rational numbers.
Representing situations.
Turn to a person near you and
summarize one idea that you are
hanging on to from today’s session.
Homework
If you would like more practice on placing fractions on the
number line, Class Activity 3F (p. 49) is recommended.
Required homework Due July 27:
Class Activity 3G: Equivalent Fractions (p. 41)
Justify using conceptual thought patterns and your fraction strips
if the sum or difference for the expressions below are Greater
than or Less than the indicated benchmark:
1/8 + 4/5
5/6 + 7/8
11/12 – 1/3
6/4 – 5/3
Greater than or Less than 1
Greater than or Less than 1 ½
Greater than or Less than ½
Greater than or Less than 0
Pose a word problem for each problem below. Next use your
fraction strips to solve it, then clearly represent your reasoning
with diagrams, pictures, words, and symbols:
11/12 - ¼ = _____
¾ + 5/6 = ______
Estimation Task
Greater than or Less than
Benchmark: 1
•
4/7 + 5/8
•
1 2/9 – 1/3
Benchmark: 1
•
1 4/7 + 1 5/8
Benchmark: 3
•
6/7 + 4/5
Benchmark: 2
•
6/7 – 4/5
Benchmark: 0
•
5/9 – 5/7
Benchmark: 0
•
4/10 + 1/17
Benchmark: 1/2
•
7/12 – 1/25
Benchmark: 1/2
•
6/13 + 1/5
Benchmark: 1/2
Word Problems: Envelope #1
Alicia ran 3/4 of a marathon and Maurice ran 1/2 of the
same marathon. Who ran farther and by how much?
Sean worked on the computer for 3 1/4 hours. Later,
Sean talked to Sonya on the phone for 1 5/12 hours.
How many hours did Sean use the computer and talk on
the phone all together?
Katie had 11/12 yards of string. One-fourth of a yard of
string was used to tie newspapers. How much of a yard
of string is remaining?
Khadijah bought a roll of border to use for decorating her
walls. She used 2/6 of the roll for one wall and 6/12 of the
roll for another wall. How much of the roll did she use?
Word Problems: Envelope #2
Elizabeth practices the piano for 3/4 of an hour on Monday
and 5/6 of an hour on Wednesday. How many hours per
week does Elizabeth practice the piano?
On Saturday Chris and DuShawn went to a strawberry
farm to pick berries. Chris picked 2 3/4 pails and DuShawn
picked 1 1/3 pails. Which boy picked more and by how
much?
One-fourth of your grade is based on the final. Two-thirds
of your grade is based on homework. If the rest of your
grade is based on participation, how much is participation
worth?
Dontae lives 1 5/6 miles from the mall. Corves lives 3/4 of
a mile from the mall. How much closer is Corves to the
mall?
Envelope #3. Reflection Prompts
Describe adjustments in your reasoning to solve
problems in envelope #2 as compared to
envelope #1.
Summarize your general strategy in using the
paper strips (e.g., how did you begin, proceed,
and conclude).
Describe ways to transform the problems in
envelope #2 to be more like the problems in
envelope #1.
Compare and contrast your approach in using
the paper strips to the standard algorithm.