Transcript Laying the Groundwork for Fractions *Sharing Cookies*

Welcome Highland Ranch and
Midland!
December 10, 2014
Facilitator: Andrea Barraugh
Introducing Division while Laying the
Groundwork for Fractions
“Sharing Cookies”
Table Talk
How are fractions and division related?
Focus Practices
3. Construct viable arguments and critique the
reasoning of others.
5. Model with mathematics
6. Attend to precision.
7. Look for and make use of structure.
LAUNCH
Sharing Cookies
What if there is one more person at the door?
(12 cookies with 13 people)
How many cookies will each person get?
More or less than 1 whole cookie?
More or less than ½ cookie?
Sequencing the Social Interactions
1.
2.
3.
4.
Independent
Partner Exploration
Table Sharing
Whole Group Sharing
Exploration Choices
• Choose a sequence of sharing situations to explore.
• Model each situation using the cookies provided or with
another visual model.
• What observations can you make about equal shares?
Share paper cookies with 4
children.
6 cookies with 4 children
5 cookies with 4 children
4 cookies with 4 children
3 cookies with 4 children
2 cookies with 4 children
1 cookies with 4 children
Share paper cookies using
these scenarios:
4 cookies with 4 children
5 cookies with 2 children
2 cookies with 4 children
7 cookies with 6 children
4 cookies with 5 children
13 cookies with 10
children
12 cookies with 20
children
Math Summary
• What did you learn about the relationship
between division and fractions?
Professional Reflections
In which practices were you engaged during this
“sharing” experience?
How might you use this experience with your
students?
3rd Grade Highlighted
CC “Division as Sharing” Standard
• Interpret quotient of 56 ÷ 8:
– 56 cookies shared by 8 people – how many cookies
each? (number of objects)
– 56 cookies, each person gets 8 cookies – how many
people? (number of shares)
3rd Grade CC Fraction Standards
Understand Fractions:
3/4
0
1/4
0/4
1
1/4
1/4
2/4
3/4
Simple Equivalent Fractions: 1/2 = 2/4
2/3 = 4/6
Whole Numbers as Fractions:
4/1 = 4
3/3 = 1
4/4
Compare Two Fractions with Same Numerator or Same Denominator:
2/3 > 1/3
2/3 > 2/4
Highlighted 4th Grade Fraction Standard
CCSS.MATH.CONTENT.4.NF.B.3.A
Understand addition and subtraction of fractions as
joining and separating parts referring to the same whole.
Highlighted 5th Grade CC Fraction Standard
Apply and extend previous understandings of multiplication and
division.
CCSS.MATH.CONTENT.5.NF.B.3
• Interpret a fraction as division of the numerator by the denominator
(a/b = a ÷ b).
• Solve word problems involving division of whole numbers leading
to answers in the form of fractions or mixed numbers, e.g., by using
visual fraction models or equations to represent the problem.
– For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied
by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person
has a share of size 3/4.
– If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds
of rice should each person get? Between what two whole numbers does your answer
lie?
Making Sense of Fractions
• Changing the Whole
• Modeling Operations
Fraction Operations Standards
• What are the fraction operations standards for
your grade level?
• How do they progress across 3-5?
Pattern Blocks
Investigate the pattern
blocks by looking for
mathematical
relationships.
Fraction Exploration
Changing the Whole
=1
=1
= 1/2
= 1/6
= 1/3
= ___
= ___
= ___
32
Challenges
=1
=1
=
=
=
=
=
=
=
33
Modeling Operations with
Fractions
Fraction Problems
(Solve each problem with pencil and paper.)
1/3 + 1/3 =
2/3 + 1/2
2/3 – 1/2
1/2 x 1/3
1/2 ÷ 1/3
35
How can you model the same problems with pattern blocks?
1/3 + 1/3 =
2/3 + 1/2
2/3 – 1/2
1/2 x 1/3
1/2 ÷ 1/3
36
Multiplication as Grouping
Connecting to Whole Numbers
3x4=
3 groups of 4 stars =
4
+
4
+
4
= 12
Using Pattern Blocks and Language
to Connect Whole Number and Fraction Multiplication
3 x 1/6 =
5x½=
1/3 x 6 =
3 x 2/3 =
Connecting to Grouping Whole Numbers:
Multiplying a Fraction by a Fraction
1/2 x 2/3
1/2 groups of 2/3
1/2 of 2/3
Try These . . .
½x½=
2/3 x 1/2 =
½ x 1/3 =
3/2 x 2/3 =
½ x 2/3 =
1/3 x ½ =
Division with Fractions
How can interpretation, language, and modeling
support our understanding?
20 ÷ 4 =




Twenty divided into 4 groups
How many in each group?
How many groups of 4 are in 20?
How many 4s are in 20?
1/2 ÷ 1/3 =
How many one thirds are in one half ?
How many
are in
2÷½=
Two divided into groups of ½, how many ½s?
How many 1/2s are in two?
?
More Opportunities to Grapple
Practice
1/6 + 1/2 =
5/6 + 1/3 =
Challenges
5/6 ÷ 2/6
5/6 – 1/2 =
5/6 – 2/3 =
2/3 ÷ 1/6
1/2 x 5/6 =
5/6 x 1/2 =
1/6 ÷ 3/6
1/3 ÷ 2/6 =
1 ÷ 1/6 =
1 2/3 ÷ 1/2
42
Processing the Experience
• What practices were you engaged with as you
explored these mathematical ideas?
• What new insights do you have into operations
with fractions?
• What new insights do you have into teaching
fractions?