Transcript Letting Fraction Algorithms Emerge through Problem Solving

Reasoning with
Rational Numbers (Fractions)
DeAnn Huinker, Kevin McLeod, Bernard Rahming,
Melissa Hedges, & Sharonda Harris,
University of Wisconsin-Milwaukee
This material is based upon
work supported by the
National Science Foundation
Grant No. EHR-0314898.
Mathematics Teacher Leader (MTL) Seminar
Milwaukee Public Schools
March 2005
www.mmp.uwm.edu
Reasoning with
Rational Numbers (Fractions)
Session Goals
To deepen knowledge of rational number
operations for addition and subtraction.
To reason with fraction benchmarks.
To examine “big mathematical ideas”
of equivalence and algorithms.
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.
For example,
1 , 2 , 0.333333..., 33 %1
3all represent
6
the same quantity.3
Different representations of the same
quantity are called “equivalent.”
Big Idea: Algorithms
What is an algorithm?
Describe what comes to mind when
you think of the term “algorithm.”
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
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 strategy–What’s missing?)
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.
Rational Number vs Fraction
Rational Number = How much?
Refers to a quantity,
expressed with varied written symbols.
Fraction = Notation
Refers to a particular type of symbol or
numeral used to represent a rational number.
Characteristics of
Problem Solving Tasks
1: Task focuses attention on the
“mathematics” of the problem.
2: Task is accessible to students.
3: Task requires justification and
explanation for answers or methods.
Characteristics of
Problem Solving Tasks
Individually
Read pp. 67-70, highlight key points.
Table Group
Designate a recorder.
Discuss characteristics & connect to task.
Whole Group
Report key points and task connections.
Discuss
Identify benefits of using
problem solving tasks:
for the teacher?
for the students?
1
5
1
–
=
2
8
Task
Write a word problem for this equation.
In other words, situate this computation
in a real life context.
1
3
+
=
2
4
1
5
1
–
=
2
8
Task
Write a word problem for each equation.
Draw a diagram to represent each word
problem and that shows the solution.
Explain your reasoning for how you
figured out each solution.
Which is
accurate? Why?
1
1
1 –
=
3
5
1
5
Alexis has 1 yards of felt. She used
1
of a yard of felt to make a costume.
3
How much is remaining?
Alexis has 1 1 yards of felt. She used
5
1
of it for making a costume. How
3
much felt is remaining?
Notes for comparing the two fraction situations.
Whole = 1 yard of felt
1 1/5 yards of felt. Use 1/3 of a yard of felt to make a costume.
1 1/5 yards – 1/3 yards = 2/3 yards + 1/5 yards = 13/15 yards
Whole = 1 1/5 yards of felt
1 1/5 yards of felt. Use 1/3 of the whole piece of felt to make a costume.
6/5 yards – (1/3 x 6/5) = 6/5 yards – 2/5 yards = 4/5 yards
Examining Student Work
Establish two small groups per table.
Designate a recorder for each group.
Comment on accuracy and reasoning:
Word Problem
Representation (Diagram)
Solution
Summarize
Strengths and limitations
in students’ knowledge.
Implications for instruction.
1
3
+
=
2
4
NAEP Results: Percent Correct
Age 13
35%
Age 17
67%
National Assessment of Education Progress (NAEP)
1
3
+
=
2
4
MPS Results
Grade 5
Grade 6
51
86
53
60
250
Correct
Solution
51%
25%
30%
30%
33%
Correct Word
Problem
39%
24%
23%
28%
28%
Accurate
Diagram
37%
15%
21%
20%
22%
Clear
Reasoning
24%
15%
34%
15%
21%
n
Grade 7 Grade 8 Overall
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).
Source: Vermont Mathematics Partnership (funded by NSF (EHR-0227057) and US DOE (S366A020002))
Fraction Kit
Fold paper strips
Purple: Whole strip
Green: Halves, Fourths, Eighths
Gold: 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.
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.
Representing Your Reasoning
Using plain paper and markers,
clearly represent your reasoning with
diagrams, words, and/or symbols for:
1
11
–
=
4
12
5
3
+
=
6
4
Representing Operations
Envelope #3
Pairs
Each pair gets one reflection prompt.
Discuss and respond.
Table Group
Take turns, pairs facilitate a table
group discussion of their prompt.
Big Idea: Algorithms
Algorithms for operations with
rational numbers use notions of
equivalence to transform
calculations into simpler ones.
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.
Estimation Task
4/7 + 5/8
Greater than or Less than
Benchmark: 1
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 the yard
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. Twothirds 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.