October 1, 2012
Presenter: Kathy Carpenter
[email protected]
Goals for this Session
• Examine a progression of some of the content
for Fractions.
• Discuss a variety of ways to teach standards.
• Discuss how to assess for student
understanding.
• Housekeeping:
This session is being recorded and you will be sent a link to share.
Any questions you have can be submitted in the chat box during the webinar.
How I Prepared for this Session
• First, I studied the standards as they
progressed vertically and chose ones that
built on each other.
• I consulted:
Elementary and Middle School Mathematics:
Teaching Developmentally by Van de Walle
Nimble with Numbers, available from Dale
Seymour
Shifts Happen…
Shift
Focus
From:
Laundry lists
To:
Concentrated areas that move in and
out of the curriculum and come alive
Teaching a little bit of everything using the math’al practices
every year
Coherence
Disjointed topics (vertically and Ideas that increase in sophistication
horizontally)
with each grade level, connections
within grade levels
Application
Problem solving after content
Content coming alive through
applications, purposefulness in student
learning
Conceptual
Understanding
Procedural Skill
and Fluency
Jumping right into procedures
Assessing for understanding at a
conceptual level
Drill drill drill
Fluency that goes beyond rote
memorization and facilitates the use of
multiple strategies
Today’s Third-grade Standards
• Number and Operations—Fractions: Develop an understanding of
fractions as numbers.
1. Understand a fraction 1/b as the quantity formed by 1 part when a
whole is partitioned into b equal parts; understand a fraction a/b as the
quantity formed by a parts of size 1/b.
2. Understand a fraction as a number on the number line; represent
fractions on a number line diagram.
• 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.
Today’s Fourth-grade Standards
• Number and Operations—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, with attention to how the number and size
of the parts differ even though the two fractions themselves are the same
size. Use this principle to recognize and generate equivalent fractions.
2. Compare two fractions with different numerators and different
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. Record the results of comparisons with symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
Today’s Fifth-grade Standards
• Number and Operations—Fractions: Use equivalent fractions as a
strategy to add and subtract fractions.
1. Add and subtract fractions with unlike denominators (including
mixed numbers) by replacing given fractions with equivalent
fractions in such a way as to produce an equivalent sum or
difference of fractions with like denominators. For example, 2/3 +
5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
2. Solve word problems involving addition and subtraction of
fractions referring to the same whole, including cases of unlike
denominators, e.g., by using visual fraction models or equations to
represent the problem. Use benchmark fractions and number sense
of fractions to estimate mentally and assess the reasonableness of
answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7,
by observing that 3/7<1/2.
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of
others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Big Ideas: Fractions
Big Idea #1
Denominators and Numerators
Big Idea #2
Estimation with Fractions
Big Idea #3
Equivalent Fractions
Big Idea #4
Operations with Fractions
Addition/Subtraction
Multiplication/Division
The Role of
Estimation with
Fractions
Estimation: Close to 0, ½, 1
Write a collection of 15 fractions for students to sort.
Discuss.
Scaffolding approach:
Use fractions very close to the benchmarks.
Use fractions with most of the denominators less than 20.
Use more complicated fractions, and fractions that fall exactly
halfway between the benchmarks (discuss!).
Estimation: Sorting Fractions
Name a number that could be included in each
region of the Venn diagram.
Fractions whose denominator is 10
Fractions > 1/2
Fractions < 1
Estimation: Patterning with Fractions
What comes next?
3, 3½, 4, 4½, 5, . . .
5, 4 3/5, 4 1/5, . . .
Describe the pattern.
Reinforce “counting” with fractions.
Estimation: Fitting Fractions Diagnostic
Use 1, 3, 5, and 6 to get the greatest and least
sums.
+
=
Use each digit only once. Describe your strategy.
Estimation: Fitting Fractions Diagnostic
Use 1, 3, 5, and 6 to get a difference of 3.
-
= 3
Use each digit only once. Describe your strategy.
Equivalent
Fractions
Concepts
Extending Parts and Wholes
If this rectangle is one whole,
Find one-fourth.
Find two-thirds.
Find five thirds.
Extending Parts and Wholes
If this rectangle is one-third, what could
the whole look like?
Extending Parts and Wholes
If this rectangle is three-fourths, draw a
shape that could be the whole.
Extending Parts and Wholes
If this rectangle is four-thirds, what
rectangle could be the whole?
Fractions with Pattern Blocks
Explore
Impart conditions
Create representations
Physical—actual pattern blocks
Iconic—sketch/picture of pattern blocks
Symbolic—numerical expressions
http://www.mathplayground.com/patternblocks.html
Using Sets in Understanding
Parts and Wholes
If 8 counters are in a whole set, how many are in one-fourth of a
set?
If 4 counters are one half of a set, how big is the set?
If 12 counters are three-fourths of a set, how many counters are in
the full set?
Equivalence: rearranging arrays (looking at rows, etc.)
The Importance of
Unit Fractions
List a set of unit fractions such as 1/3, 1/8, 1/5, and 1/10.
Ask students to put the fractions in order from least to
greatest.
Have students defend the way they ordered the fractions.
Equivalent Fractions Concepts v. Rules
How do you know that 4/6 is the same as 2/3?
Think of at least two different explanations.
Area Models for Equivalencies
Dot paper
Isometric grid paper
29
Other Models for Understanding
Equivalence
• Number line
• Length: fold paper
strips
• Labels: think of
denominator as
“unit” (i.e., 3 fourths)
If students have a conceptual understanding of
equivalent fractions, there is no need to teach
an algorithm for adding and subtracting
fractions!
Third Grade
• Related Standards: 3OA7, 3MD4, 3G2
• Concepts underlying these standards
Number line
Fractions don’t always “look” the same
Third-grade Task: illustrativemathematics.org
Mrs. Frances drew a picture on the board.
Then she asked her students what fraction it represents.
1. Emily said
that
the picture represents 2/6. Label the picture to show how
Third-grade
Task:
illustrativemathematics.org
Emily's answer can be correct.
2.
Raj said that the picture represents 2/3. Label the picture to show how
Raj's answer can be correct.
3.
Alejandra said that the picture represents 2. Label the picture to show
how Alejandra's answer can be correct.
Fourth Grade
• Related Standards: 4OA4, 4NBT5, 4MD4
• Concepts critical to these standards
Parts and wholes
Fluency with different fraction models
Fourth-grade Task:
illustrativemathematics.org
Fifth Grade
• Related Standards: 5NBT7, 5MD1, 5MD2
• Concepts critical to these standards:
Equivalence
Equivalence
Equivalence
Fifth-grade Task: illustrativemathematics.org
• For each of the following word problems, determine whether or
not (2/5+3/10) represents the problem. Explain your decision.
1. A farmer planted 2/5 of his forty acres in corn and another 3/10 of his land in wheat.
Taken together, what fraction of the 40 acres had been planted in corn or wheat?
2. Jim drank 2/5 of his water bottle and John drank 3/10 of his water bottle. How much
water did both boys drink?
3. Allison has a batch of eggs in the incubator. On Monday 2/5 of the eggs hatched.
By Wednesday, 3/10 more of the original batch hatched. How many eggs hatched in
all?
4. Two-fifths of the cross-country team arrived at the weight room at 7 a.m. Ten
minutes later, 3/10 of the team showed up. The rest of the team stayed home. What
fraction of the team made it to the weight room that day?
Note: even more scenarios available online
Based on today, what are 3 changes I
can make to my class this school year?
Note: Work on “chunks” at a time to
keep from getting overwhelmed and
to measure the impact of the changes.
Next webinar: December 3rd, 3:30p