Download Subtracting Fractions

Vacaville USD
November 4, 2014
AGENDA
• Problem Solving, Patterns, Expressions and
Equations
• Math Practice Standards and High Leverage
Instructional Practices
• Number Talks
– Computation Strategies
• Fractions
Expectations
• We are each responsible for our own
learning and for the learning of the group.
• We respect each others learning styles
and work together to make this time
successful for everyone.
• We value the opinions and
knowledge of all participants.
Cubes in a Line
How many faces (face units) are there when:
6 cubes are put together?
10 cubes are put together?
100 cubes are put together?
n cubes are put together?
Questions?
What do I mean by a “face unit”?
Do I count the faces I can’t see?
Cubes in a Line
How many faces (face units) are there when:
6 cubes are put together?
10 cubes are put together?
100 cubes are put together?
n cubes are put together?
Cubes in a Line
Cubes in a Line
Cubes in a Line
Cubes in a Line
Cubes in a Line
Cubes in a Line
We found several different number
sentences that represent this problem.
• What has to be true about all of these
number sentences?
5.OA.2. Write simple expressions that record
calculations with numbers, and interpret numerical
expressions without evaluating them. For example,
express the calculation “add 8 and 7, then multiply
by 2” as 2 × (8 + 7). Recognize that 3 × (18932 +
921) is three times as large as 18932 + 921,
without having to calculate the indicated sum or
product.
5.OA.3 Generate two numerical patterns using two
given rules. Identify apparent relationships between
corresponding terms. Form ordered pairs consisting
of corresponding terms from the two patterns, and
graph the ordered pairs on a coordinate plane. For
example, given the rule “Add 3” and the starting
number 0, and given the rule “Add 6” and the
starting number 0, generate terms in the resulting
sequences, and observe that the terms in one
sequence are twice the corresponding terms in the
other sequence. Explain informally why this is so.
Math Practice Standards
• Remember the 8 Standards for
Mathematical Practice
• Which of those standards would be
addressed by using a problem such as
this?
1. Make sense of problems and persevere in
solving them
6. Attend to precision
OVERARCHING HABITS OF MIND
CCSS Mathematical Practices
REASONING AND EXPLAINING
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the
reasoning of others
MODELING AND USING TOOLS
4. Model with mathematics
5. Use appropriate tools strategically
SEEING STRUCTURE AND
GENERALIZING
7. Look for and make use of structure
8. Look for and express regularity in repeated
reasoning
High Leverage Instructional
Practices
High-Leverage Mathematics
Instructional Practices
An instructional emphasis that
approaches mathematics learning as
problem solving.
1. Make sense of problems and persevere
in solving them.
An instructional emphasis on cognitively
demanding conceptual tasks that
encourages all students to remain
engaged in the task without watering
down the expectation level (maintaining
cognitive demand)
1. Make sense of problems and persevere
in solving them.
Instruction that places the highest value
on student understanding
1. Make sense of problems and persevere
in solving them.
2. Reason abstractly and quantitatively
Instruction that emphasizes the
discussion of alternative strategies
3. Construct viable arguments and critique
the reasoning of others
Instruction that includes extensive
mathematics discussion (math talk)
generated through effective teacher
questioning
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique
the reasoning of others
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in
repeated reasoning
Teacher and student explanations to
support strategies and conjectures
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique
the reasoning of others
The use of multiple representations
1. Make sense of problems and persevere
in solving them.
4. Model with mathematics
5. Use appropriate tools strategically
Number Talks
What is a Number Talk?
• Also called Math Talks
• A strategy for helping students develop a
deeper understanding of mathematics
– Learn to reason quantitatively
– Develop number sense
– Check for reasonableness
– Number Talks by Sherry Parrish
What is Math Talk?
• A pivotal vehicle for developing efficient,
flexible, and accurate computation
strategies that build upon key foundational
ideas of mathematics such as
– Composition and decomposition of numbers
– Our system of tens
– The application of properties
Key Components
•
•
•
•
•
Classroom environment/community
Classroom discussions
Teacher’s role
Mental math
Purposeful computation problems
Classroom Discussions
• What are the benefits of sharing and
discussing computation strategies?
• Students have the opportunity to:
– Clarify their own thinking
– Consider and test other strategies to see if
they are mathematically logical
– Investigate and apply mathematical
relationships
– Build a repertoire of efficient strategies
– Make decisions about choosing efficient
strategies for specific problems
5 Goals for Math Classrooms
•
•
•
•
•
Number sense
Place Value
Fluency
Properties
Connecting mathematical ideas
Clip 5.6 – 5th Grade
Subtraction: 1000 – 674
• Before we watch the clip, talk at your
tables
– What possible student strategies might
you see?
– How might you record them?
• What evidence is there that the students
understand place value?
• How do the students’ strategies exhibit
number sense?
• How does fluency with smaller numbers
connect to the students’ strategies?
• How are accuracy, flexibility, and efficiency
interwoven in the students’ strategies?
Clip 5.1 – 5th Grade
Multiplication: 12 x 15
• Before we watch the clip, talk at your
tables
– What possible student strategies might
you see?
– How might you record them?
• What evidence is there that students
understand place value?
• How do student strategies exhibit number
sense?
• How do the teacher and students connect
math ideas?
• What questions does the teacher use to
facilitate student thinking about big ideas?
Clip 5.5 – 5th Grade
Division String: 496 ÷ 8
• Before we watch the clip, talk at your
tables
– What possible student strategies might
you see?
– How might you record them?
• What evidence is there that students
understand place value?
• How do students build upon their
understanding of multiplication to divide?
• How does the teacher connect math ideas
throughout the number talk?
Solving Word Problems
3 Benefits of Real Life Contents
• Engages students in mathematics that is
relevant to them
• Attaches meaning to numbers
• Helps students access the mathematics.
A crane operator unloaded the following cargo:
• 5 pallets of lumber. Each pallet weighs 7.3 tons.
• 9 pallets of concrete. Each pallet weighs 4.8 tons.
a) How many pounds of cargo were unloaded?
b) Which load of cargo was heavier, the
lumber or the concrete? How many pounds
heavier?
Ava is saving for a new computer that costs
$1,218. She has already saved half of the
money. Ava earns $14.00 per hour. How
many hours must Ava work in order to save
the rest of the money?
Mrs. Onusko made 60 cookies for a bake
sale. She sold 2/3 of them and gave 3/4 of the
remaining cookies to the students working at
the sale. How many cookies did she have
left?
Equivalent Fractions
5th Grade
Use equivalent fractions as a strategy to
add and subtract fractions.
CaCCSS
• Fractions are equivalent (equal) if they are
the same size or they name the same
point on the number line.
Fraction Families
1 2 3 4 5 6
     .......
2 4 6 8 10 12
1 2 3 4 5 6
     .......
3 6 9 12 15 18
3 6 9 12 15 18
  

 .......
5 10 15 20 25 30
Equivalent Fractions
• Fraction Family Activity
• Equivalent Fraction Activity
5th Grade CCSS-M
5.F.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.)
Adding Fractions
1
10
8
1
0
• Add
3 7

8 8
2
1
8
1
1
4
2
2
5 3
 
8 4
5 3
 
8 4
5 3 5 6
   
8 4 8 8
5 3 5 6 11 3
    1
8 4 8 8 8 8
3
5 3
 
6 4
5 3
 
6 4
5 3 10 9
   
6 4 12 12
5 3 10 9 19
7
   
1
6 4 12 12 12
12
Subtracting Fractions
Subtracting Fractions
Possible sequence of instruction
• Subtracting 2 fractions less than 1
3 1

4 8
Subtracting Fractions
• Subtracting when 1 fraction is between 1
and 2 and 1 fraction is less than 1
3 5
1 
4 8
1 5
1 
4 8
Subtracting Fractions
• Subtracting mixed numbers
3 2
4 2
4 3
1 5
5 2
3 6
Subtracting Fractions
• Strategies: Change to improper fractions
1
3 2 19 8 57 32 25
4 2      2
12
4 3 4 3 12 12 12
1 5 16 17 32 17 15
3
1
5 2       2  2
3 6 3 6 6 6 6
6
2
Subtracting Fractions
• Strategies: Borrow
9
8
1
3 2
4 2  4 2 2
12 12 12
4 3
2 5
2 5
1 5
3
1
5  2  5  2  4 1  2  2  2
6 6
6 6
3 6
6
2
Subtracting Fractions
• Strategies: Shift (Compensate)
1 1 3
5  5
3 6 6
5 1
2   3
6 6
3
1
2  2
6
2
Multiplying Fractions
5.F.4 Apply and extend previous
understandings of multiplication to multiply a
fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a
partition of q into b equal parts; equivalently,
as the result of a sequence of operations a
× q ÷ b. For example, use a visual fraction
model to show (2/3) × 4 = 8/3, and create a
story context for this equation. Do the same
with (2/3) × (4/5) = 8/15. (In general, (a/b) ×
(c/d) = ac/bd.)
5.F.4 Apply and extend previous
understandings of multiplication to multiply a
fraction or whole number by a fraction.
b. Find the area of a rectangle with
fractional side lengths by tiling it with unit
squares of the appropriate unit fraction
side lengths, and show that the area is
the same as would be found by
multiplying the side