Fractions: Teaching with Understanding Part 2

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Fractions Teaching with Understanding Part 2
This material was developed for use by
participants in the Common Core Leadership in
Mathematics (CCLM2) 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.
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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.

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• Fraction Strips

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Making Fraction Strips

• White whole
• Green halves, fourths, eighths
• Yellow thirds, sixths, ninths
• ? twelfths

Note relationships among the fractions as you
fold. Remember no labels.
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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?

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• CCSSM Focus on
• Unit Fractions

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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?

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Fractions Composed of Unit Fractions

• Fold your fraction strip to show ¾.
• How do you see this fraction as unit fractions?

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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).

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Standard 3.NF.1. Non-unit Fractions

• In pairs, practice using the language of the
  standard to describe non-unit fractions.

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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?

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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.

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• Number Line Model

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Number Line Model

• What do you know about a number line that goes
  from 0 to 4?

0
4
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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?

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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.
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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.

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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.
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Explain Kens thinking?
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Explain Judys thinking?
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• On your slate, draw a number line from 0 to 1.
• Use proportional thinking to place and on
  the number line.

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• Equivalency

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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.

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Which fractions are equivalent? How do you know?
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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.
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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.

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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.

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• Comparing Fractions

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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
  strategyWhats missing?)

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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 gt, , or lt, 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 gt, , or lt, and justify the
conclusions, e.g., by using a visual fraction
model.
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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.

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Ordering Fractions 1

• 1/4, 1/2, 1/9, 1/5, 1/100
• 3/15, 3/9, 3/4, 3/5, 3/12
• 24/25, 7/18, 8/15, 7/8

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Ordering Fractions 2

• Write each fraction on a post it note.
• Write 0, ½, 1, and 1 ½ on a post it note and
  place them on the number line as benchmark
  fractions.
• Taking turns, each person
• Places one fraction on the number line and
  explains their reasoning about the size of the
  fraction.

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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

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Extension of Unit Fraction Reasoning

• Jason hiked 3/7 of the way around Devils 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.

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The Garden Problem

• Jim and Sarah each have a garden. The gardens are
  the same size. 5/6 of Jims garden is planted
  with corn. 7/8 of Sarahs 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.

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Reflect

• Summarize how you used reasoning strategies to
  compare and order fractions based on their size.

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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?

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Lets 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.