Transcript Operations with Fractions - elementary-math

Operations with Fractions
Learning goals:
1. Know the key teaching strategies for
elementary mathematics
2. Understand the depth of the content
regarding fractions in 4th and 5th grades
3. Consider additional teaching strategies
for engagement and scaffolding higher
order thinking
In which of the following
are the three fractions
arranged from least to
greatest?
NAEP 8th Grade, 49%
correct
Why so few?
How did it go this past month?
Key Points
1. Use manipulatives and drawings.
2. Build knowledge of fraction operations on
the underlying structures of word
problems.
3. Help students reason with fractions.
4. Focus on the reasons behind operations,
to develop the procedures.
Problems that represent key content
Procedures
1. Adding or subtracting by finding
equivalent fractions
- How to find equivalent fractions
- Why add numerators when the denominators
are the same
Procedures
2. Multiplying fractions by multiplying the
numerators and multiplying the
denominators.
- Where does this come from?
• Marty made two types of cookies. He used
1/5 cup of flour for one recipe and 2/3 cup
of flour for the other recipe. How much
flour did he use in all? Is it greater than 1/2
cup or less than 1/2 cup? Is the amount
greater than 1 cup or less than 1 cup?
• Explain your reasoning in writing.
When first learning…
• Allow students to use manipulatives or
drawings to figure this out.
• Use fraction circles or fraction bars to
determine whether 1/5 + 2/3 is less than or
greater than ½ or 1.
• Eventually the image of the manipulatives
or drawings will become second nature so
students can “see” in their heads the
fraction relationships.
Basic concepts
• A fraction is a part of a whole…
• The numerator means… the denominator
means…
• Unit fractions get smaller as their denominators
get larger.
• Fractions are numbers on the number line (1/2 is
half the way from 0 to 1…)
• Fractions that are the same size are called
equivalent fractions.
Reasoning questions
• Which is larger, 2/8 or 5/8? Why?
• Which is larger, 2/4 or 2/6? Why? How can you
prove this?
• Which is larger, 2/3 or 3/4? 2/5 or 5/10?
3.NF.3 d. Compare two fractions with the same
numerator or the same denominator by reasoning
about their sizes.
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.
Reasoning questions
• Where would you place 5/6 on the number line?
• Can you use other fraction pieces with different
denominators to show 1/2? 1/4? 3/4?
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.
C-R-A for equivalence
Concrete
Representational
Abstract
Illuminations fraction game (Fraction Tracks)
http://illuminations.nctm.org/ActivityDetail.aspx?ID=18
• How can playing a game like Fraction Tracks help a
student build understanding about the relative sizes of
fractions?
• How can playing a game like Fraction Tracks help a
student build understanding about the equivalence of
fractions?
• What characteristics of the classroom environment
would support students as they use a game like Fraction
Tracks to help them deepen their understanding of
fractions?
• Smarter Balanced Assessment items
• Other virtual manipulatives on our Elementary
Math Resources web pages
Which approach?
Is it A… or B…
A: “You can find equivalent fractions by
multiplying the numerator and denominator
by the same number.” (Teacher explains
procedure, shows worked out examples, students
practice with new problems)
4.NF.1 Explain why a fraction a/b is equivalent to a fraction
(n × a)/(n × b) by using visual fraction models
Why does this work
mathematically? Why don’t they mention this?
Which approach?
Is it A… or B…
B: See pages 12-13 in Operations with
Fractions packet
Fraction Addition and Subtraction
Understanding the reasons behind a
procedure is just as important as being
able to do the procedure.
Adding or subtracting fractions by finding
equivalent fractions… What’s the procedure and
why does it work?
1 5
+
4 8
Write examples for each
Learning Progression
4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and
separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator
in more than one way, recording each decomposition by an equation.
Justify decompositions, e.g., by using a visual fraction model.
c. Add and subtract mixed numbers with like denominators, e.g., by
replacing each mixed number with an equivalent fraction, and/or by using
properties of operations and the relationship between addition and
subtraction.
d. Solve word problems involving addition and subtraction of fractions
referring to the same whole and having like denominators, e.g., by using
visual fraction models and equations to represent the problem.
4.NF.5 Express a fraction with denominator 10 as an
equivalent fraction with denominator 100, and use this
technique to add two fractions with respective
denominators 10 and 100. For example, express 3/10 as
30/100 and add 3/10 + 4/100 = 34/100. (Addition and
subtraction with unlike denominators in general is not a
requirement at this grade.)
Estimation and visualization are important.
These abilities will help students monitor
their work when finding exact answers.
Using reasoning about size
• For each of the following problems, explain if you think
the answer is a reasonable estimate or not.
• Estimation and visualization are important.
These abilities will help students monitor their
work when finding exact answers.
• Students need to experience acting out addition
and subtraction concretely with an appropriate
model before operating with symbols.
Learning Progression
• Step 1: Learn what it means to add
fractions with the same denominator.
• Pictures, analogies, methods, etc.
• How does this generalize into adding
fractions with different denominators?
– Step 2: one is a multiple of the other
– Step 3: both scale up to a common multiple
5.NF.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.
5.NF.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.
C-R-A for adding/subtracting
Concrete
Use manipulatives to
model this.
Representational
Abstract
NLVM Fractions – Adding
You try it
Concrete
Story problem?
1. Adding to or putting
together
2. Taking from or taking
apart
3. Comparing
4. Part-whole
Representational
Abstract
2
1
3 2 
3
2
… by using visual fraction models or equations to represent the problem
• Estimation and visualization are important.
These abilities will help students monitor their
work when finding exact answers.
• Students need to experience acting out addition
and subtraction concretely with an appropriate
model before operating with symbols.
• Making connections between concrete actions
and symbols is an important part of
understanding. Students should be encouraged
to find their own way of recording with symbols.
Using circle fractions
• Page 494, last paragraph 1st
column, through end.
• Mark the text.
• For the problems in Figure 6,
see the packet with rulers
(like number lines).
For students who struggle …
• Manipulatives and drawings
• Partner work
• Explicit teaching:
– Teacher verbalizes thought processes
– Works together with student
– Allows for practice with guided feedback
Pair up and try this with
2 3

5 10
Start with “Can you show me how to make 2/5 from the fraction circle pieces?”
Collaborative cards
• What do you think of this game as a
teaching tool?
What about decimals?
• What does the common core say?
• How would you sequence this in a learning
progression?
• What manipulatives and visual
representations are helpful?
• How are decimals related to fractions?
NLVM Place Value Number Line (3-5 Number and Operations)
4.NF.5 Express a fraction with denominator 10 as an
equivalent fraction with denominator 100, and use this
technique to add two fractions with respective
denominators 10 and 100. For example, express 3/10 as
30/100 and add 3/10 + 4/100 = 34/100. (Addition and
subtraction with unlike denominators in general is not a
requirement at this grade.)
4.NF.6 Use decimal notation for fractions with
denominators 10 or 100. For example, rewrite 0.62 as
62/100; describe a length as 0.62 meters; locate 0.62 on a
number line diagram.
4.NF.7 Compare two decimals to hundredths by reasoning
about their size. Recognize that comparisons are valid only
when two decimals refer to the same whole. Record the
results of comparisons with the symbols >, =, or <, and
justify the conclusions, e.g., by using a visual model.
5.NBT.3 Read, write, and compare decimals to
thousandths.
5.NBT.4 Use place value understanding to round decimals
to any place.
5.NBT.7 Add, subtract, multiply, and divide decimals to
hundredths, using concrete models or drawings and
strategies based on place value, properties of operations,
and/or the relationship between addition and subtraction;
relate the strategy to a written method and explain the
reasoning used.
Decimal place value
• Why do we line up the decimal point when
adding or subtracting?
• How is this like adding or subtracting
three-digit whole numbers?
• See the NLVM simulation Base Blocks
Decimals
Multiplying with fractions
• Multiplication of a fraction by a whole
number. 4.NF.4
• 10 people at a party eat a half sandwich each. How many
whole sandwiches is that? 10 x ½ (Count on the number
line. This is repeated addition.)
• Multiplication of a fraction or whole number
by a fraction. 5.NF.4
• A bag has 20 apples in it. You want to give ¼ of the bag to
a friend. How much is ¼ of 20? ¼ x 20 (How else can
you solve this? How much is ¾ of 20?)
• It is important to emphasize the underlying
structure of these word problems.
• 10 people at a party eat a half sandwich each. How many
whole sandwiches is that? 10 x ½ (How could you solve
this with a number line? What kind of problem is it?)
• A bag has 20 apples in it. You want to give ¼ of the bag
to a friend. How much is ¼ of 20? ¼ x 20
Multiplication of a fraction or whole number
by a fraction.
• You have 2/3 of a pumpkin pie left over
from Thanksgiving. You want to give 1/2 of
it to your sister. How much of the whole
pumpkin pie will this be? (Use fraction pieces
or drawings to figure this out, or mental
reasoning.)
• 3/4 of a pan of brownies needs to be divided equally
among three classes. How much does each class get?
3
3
4
Fair shares division
• You have 3/4 of a pan of brownies left over from a party.
You want to give 1/3 of that to your neighbor. How much
does your neighbor get?
1 3
of
3 4
1 3

3 4
• You have 3/4 of a pan of brownies left over from a party.
You want to give 2/3 of that to your neighbor. How much
does your neighbor get?
3 2

4 3
or
2 3
of
3 4
• There is 1/2 of a pan of brownies left. You want
to give 2/3 of it to your sister. What fraction of
the whole pan of brownies will she get?
• Try this with fraction circles or fraction bars.
What did you have to do to get the answer?
• What drawing might you make that could help?
• We see 2/6 as the answer from the drawings?
Can you also see 1/3 in the drawings? Is it
necessary to force the answer to be 1/3?
Look at the previous problem. Do we need to force the reduced fraction as the
answer?
• What operation can you do with 2/3 and
3/4 to get your answer? View the C-R-A to
develop the answer.
• Then create a C-R-A for 2/3 x 3/4.
• Read and do pp. 23-25 in Operations with
Fractions.
Try these (drawings and symbols)
• You have 7/8 of a cup of sugar. You need
2/3 of this for a recipe. How much is that?
• A piece of wood is 2 1/8 feet long. How
much is 3/4 of that?
Solve real world problems
• 5.NF.6 Solve real world
problems involving
multiplication of fractions
and mixed numbers, e.g.,
by using visual fraction
models or equations to
represent the problem.
• Write a problem for
this multiplication.
1
3
5
3 8
Here’s one for you
• It takes ¾ of a yard 16 and 1/2 of a yard
of fabric to make
16 and 2/3 of a
one pillow case.
pillowcase
How many can be
made from 12 ½
yards?
• How much is left
over?
Division involving fractions
• Thirteen big cookies need to be divided
equally among 4 people. How much does
person get?
5.NF.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?
• Divide a fraction by a
whole number to
solve problems.
• Divide a whole
number by a fraction
to solve problems.
• Think up problems to
go with each of these.
1
4
2
1
10 
4
5.NF.7 Apply and extend previous understandings of division to divide unit
fractions by whole numbers and whole numbers by unit fractions.
• 4 runners on a relay
team will run equal
portions of a race that
is 1/2 mile long. How
far does each race?
• How many 1/4 cup
servings are in 10
ounces of cereal?
1
4
2
1
10 
4
Partitive
division –
equal shares.
Is this the
same as 1/4 of
1/2?
Measurement
division – how
many times
does the
fraction go into
the whole
number?
Homework
• Is there a Number
Talks in this?
• Bring back stories
to share about
work with whole
numbers or
fractions. Samples
of student work are
nice, too!