Transcript PowerPoint file - University of Wisconsin–Milwaukee

Fractions: Teaching with
Understanding Part 2
This material was developed for use by participants in the
Common Core Leadership in Mathematics (CCLM^2)
project through the University of Wisconsin-Milwaukee.
Use by school district personnel to support learning of its
teachers and staff is permitted provided appropriate
acknowledgement of its source. Use by others is
prohibited except by prior written permission.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Learning Intentions and Success Criteria
We are learning to:
• Understand and use unit fraction reasoning.
• Use reasoning strategies to order and compare
fractions.
• Read and interpret the cluster of CCSS standards
related to fractions.
Success Criteria:
Explain the mathematical content and language in
3.NF.1, 3.NF.2 and 3.NF.3, 4.NF.2 and provide
examples of the mathematics and language.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Fraction Strips
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Making Fraction Strips
White:
whole
Green: halves, fourths, eighths
Yellow: thirds, sixths, ninths
?:
twelfths
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Benefits of Fractions Strips
• Why is it important for students to
fold their own fraction strips?
• How does the “cognitive demand”
change when you provide prepared
fraction strips?
• How might not labeling fraction strips
with numerals support developing
fraction knowledge?
• How this tool supports 1.G.3, 2.G.3,
and 3.G.2?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
CCSSM Focus on
Unit Fractions
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Standard 3.NF.1 Unit Fractions
• Fold each fraction strip to show only
one “unit” of each strip.
• Arrange these unit fractions from
largest to smallest.
• What are some observations you can
make about unit fractions?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Fractions Composed of Unit
Fractions
• Fold your fraction strip to show ¾.
• How do you see this fraction as
‘unit fractions’?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Looking at a Whole
• Arrange the open fraction strips in front of
you.
• Look at the thirds strip. How do you see the
number 1 on this strip using unit fractions?
• In pairs, practice stating the relationship
between the whole and the number of unit
fractions in that whole (e.g., 3/3 is three parts
of size 1/3).
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Standard 3.NF.1. Non-unit Fractions
In pairs, practice using the language of the standard to
describe non-unit fractions.
3
5
7
7
9
5
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
3.NF. 1
3.NF.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.
• How do you make sense of the language in
this standard connected to the previous
activities?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Why focus on unit fractions?
• How will you explain the meaning of standard 3.NF.1
to colleagues in your schools?
• What conjectures can you make as to why the CCSSM
is promoting this unit-fraction approach?
3.NF.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.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Number Line Model
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Number Line Model
What do you know about a number
line that goes from 0 to 4?
•
0
4
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Sequential & Proportional
Strategies
Draw two number lines from 0 to 4. Use whole
numbers & fractions to show parts on the
number line.
•# line 1 show sequential reasoning
•# line 2 show proportional reasoning
Is it harder when you have to mark fractions?
Why?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
0
1
On your slate draw another
number line from 0 to 2 that shows
thirds.
Mark 5/3 on your number line.
Explain to your shoulder partner
how you marked 5/3.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
NF Progressions Document
What are the CCSSM expectations for number lines?
Read: “The Number Line and Number Line Diagrams”
on page 3.
Read: Standard 3.NF.2, parts a and b.
With a partner, explain this standard to each other while
referring to your drawing.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Standard 3.NF.2
3.NF.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.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Explain Ken’s thinking?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Explain Judy’s thinking?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
On your slate, draw a number
line from 0 to 1.
•Use proportional
thinking
to
1
1
place 4and 3 on the number
line.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Equivalency
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Equivalency
• Place the whole fraction strip that
represents 0 to 1 on a sheet of paper.
Draw a line labeling 0 and 1.
• Lay out your fraction strips, one at a time,
and make a tally mark on the line you drew.
Write the fractions below the tally mark.
• Look for patterns to help you decide if two
fractions are equivalent.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Which fractions are
equivalent? How do you
know?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
NF Progressions Document
Number off by twos: ones study Grade 3, twos study
Grade 4.
Grade 3 Equivalent Fractions Read pp. 3-4; study margin
notes and diagrams. Study standard 3.NF.3.
Grade 4: Equivalent Fractions Read p. 5; study margin
notes and diagrams.
With your shoulder partner, identify what distinguishes
student learning at each grade.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Standard 3.NF.3, Parts a, b, & c
3.NF.3. Explain equivalence of fractions in special cases, and
compare fractions by reasoning about their size.
a.
Understand two fractions as equivalent (equal) if they
are the same size, or the same point on a number line.
b.
Recognize and generate simple equivalent fractions, e.g.,
1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent,
e.g., by using a visual fraction model.
c.
Express whole numbers as fractions, and recognize
fractions that are equivalent to whole numbers.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Standard 4.NF.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.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Comparing Fractions
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Compare Fractions by
Reasoning about their Size
• 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?)
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Standards 3.NF.3d and 4.NF.2
3.NF.3d Compare two fractions with the same numerator or the
same denominator by reasoning about their size. Recognize that
comparisons are valid only when the two fractions refer to the
same whole. Record the results of comparisons with the symbols
>, =, or <, and justify the conclusions, e.g., by using a visual
fraction model.
4.NF.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.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Standard 3NF3d & 4NF2
• On your slate, provide an example of
comparing fractions as described in these
standards.
• What is the difference between the two
standards?
• Share with your partner.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Ordering Fractions #1
1.1/4, 1/2, 1/9, 1/5, 1/100
2.3/15, 3/9, 3/4, 3/5, 3/12
3. 24/25, 7/18, 8/15, 7/8
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Ordering Fractions #2
1) Write each fraction on a post it note.
2) Write 0, ½, 1, and 1 ½ on a post it note and
place them on the number line as
benchmark fractions.
3) Taking turns, each person:
Places one fraction on the number line and
explains their reasoning about the size of
the fraction.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Ordering Fractions
3/8
3/10
6/5
7/47
7/100
25/26
7/15
13/24
17/12
8/3
16/17
5/3
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Extension of Unit Fraction
Reasoning
Jason hiked 3/7 of the way around
Devil’s Lake. Jenny hiked 3/5 of the
way around the lake. Who hiked the
farthest?
 Use fraction strips and
reasoning to explain your
answer to this question.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
The Garden Problem
Jim and Sarah each have a garden. The
gardens are the same size. 5/6 of Jim’s
garden is planted with corn. 7/8 of Sarah’s
garden is planted with corn. Who has planted
more corn in their garden?
 Use fraction strips and reasoning to
explain your answer to this question.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Reflect
Summarize how you used reasoning
strategies to compare and order
fractions based on their size.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Translating the Standards to
Classroom Practice
• Discuss the progression of the
standards we did today. Is the
progression logical?
• Discuss how the standards effect
classroom practice. What will need
to change?
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012
Let’s Rethink the Day
We know we are successful when we can…
 Explain the mathematical content and language
in 3.NF.1, 3.NF.2 and 3.NF.3, 4.NF.2 and provide
examples of the mathematics and language.
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012