Download 5th Grade Mathematics

 5th Grade Mathematics Unit #3: Interpreting Fractional Values a nd Calculating Sums and Differences Pacing: 32 Days Unit Overview
In this unit students use what they’ve learned in Grades 3 and 4 about equivalency in terms of visual models and benchmarks to extend their
understanding of fraction concepts to include interpreting fractions as quotients. Students will then build on and apply these foundational concepts to
add and subtract fractions and mixed numbers with unlike denominators. They reason about size of fractions to make sense of their answers- e.g.
they understand that the sum of ½ and 2/3 will be greater than 1. Please note, that the most important part of this unit is NOT learning the algorithm
for adding and subtracting fractions, rather it is building a reasonable understanding of how to add and subtract using benchmark fractions,
equivalent fractions, number sense, and visual fraction models. In many cases it may not be necessary to find least common denominator to add
fractions with unlike denominators. Students should be encouraged to use their conceptual understanding of fractions rather than just using the
algorithm for adding fractions.
This unit will build students’ understanding of fractions as numbers that lie between whole numbers on a number line. Number sense in fractions also
includes moving between decimals and fractions to find equivalents, and being able to use reasoning such as 7/8 is greater than ¾ because 7/8 is only
1
/8 less than a whole and ¾ is ¼ less than one whole. Students should use benchmark fractions to estimate and examine the reasonableness of their
answers.
Prerequisite Skills
1) Add, subtract, and multiply fluently 2) Use number lines to show comparisons of numbers 3) Represent fractions with visual models 4) Recognize and create basic equivalent fractions 5) Create arrays to model whole numbers using pairs of factors 6) Add and subtractions fractions and mixed numbers
with like denominators
Vocabulary
Fraction
Numerator
Denominator
Simplify
Equivalent
Convert
Mixed Number
Improper Fraction
Benchmark Fraction
Sum
Difference
Estimate
Reasonable
Common
Denominator
Size
Part:Whole
Mathematical Practices
MP.1: Make
sense of problems and persevere in solving them
Position
MP.2: Reason
abstractly and quantitatively
Location
MP.3: Construct viable arguments and critique the reasoning
Attributes
of others
Sideswith mathematics
MP.4: Model
MP.5: Use appropriate tools strategically
MP.6: Attend to precision
MP.7: Look for and make use of structure
MP.8: Look for and express regularity in repeated reasoning Common Core State Standards
Progression of Skills
Major Standards (70%) 5.NF.1: Add and Subtract Fractions with Unlike Denominators 5.NF.2: Solve Word Problems Requiring Addition and Subtraction of Fractions 5.NF.3: Interpret Fractions as Quotients 4th
5th
6th
According
to
the
PARCC
Model
Content
Framework,
4.NF.1: Explain
5.NF.1: Add and subtract
6.NS.1: Interpret and
3.NF.2
should
why a fraction a/b is fractions with unlike Standard
compute
quotients
of serve as a
depth
focus:
equivalent to a
denominators (including
fractions, and solve
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.
According to the PARCC Model Content Framework,
A Key Advance in Fraction Concepts Between Grades 4 and 5 is:
“Students use their understanding of fraction equivalence and their skill in
generating equivalent fractions as a strategy to add and subtract fractions,
including fractions with unlike denominators.”
An Opportunity for In-Depth Focus is:
“When students meet this standard, they bring together the threads of fraction
equivalence (grades 3–5) and addition and subtraction (grades K–4) to fully
extend addition and subtraction to fractions.”
4.NF.3: Understand
a fraction a/b with a
> 1 as a sum of
fractions 1/b. Add
and subtract
fractions and mixed
numbers with like
denominators.
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.
5.NF.3: Interpret a fraction
as division of the numerator
by the denominator (a/b = a
÷ b)
word problems
involving division of
fractions by fractions,
e.g., by using visual
fraction models and
equations to represent
the problem. For
example, create a
story context for (2/3)
÷ (3/4) and use a
visual fraction model
to show the quotient;
use the relationship
between multiplication
and division to explain
that (2/3) ÷ (3/4) = 8/9
because 3/4 of 8/9 is
2/3. (In general, (a/b)
÷ (c/d) = ad/bc.) How
much chocolate will
each person get if 3
people share 1/2 lb of
chocolate equally?
How many 3/4-cup
servings are in 2/3 of a
cup of yogurt? How
wide is a rectangular
strip of land with
length 3/4 mi and area
1/2 square mi?.
2 | P a g e Big Ideas
•
•
•
Fractions are numbers with
special names that tell how
many parts of that size are
needed to make the whole,
written in the form a/b (when b
is not zero).
Every fraction is equivalent to
an infinite number of other
fractions
When comparing, adding and
subtracting fractions with
unlike denominators, use
equivalent fractions with
common denominators
Students Will…
Know/Understand
•
•
•
•
•
•
•
•
I can reason about fraction
values by comparing them to
common benchmarks in order
to judge the reasonableness of
my work (e.g. recognize an
incorrect result 2/5 + 1/2 = 3/7,
by observing that 3/7 < 1/2.)
•
You can only compare, add
and subtract fractions when
they refer to the same whole.
It would be irrelevant to add
and subtract fractions of a
different whole because a
fraction has different values
based on the size of its whole
The product of denominators to unlike fractions will
always yield a common denominator (In general, a/b
+ c/d = (ad + bc)/bd.)
Fractions are numbers that lie between whole numbers
on a number line.
Equivalent fractions can be generated by determining
a similar relationship between a set of numbers.
Fractions with unlike denominators can be added or
subtracted by creating equivalent fractions with like
denominators.
To add or subtract fractions, they must refer to the
same whole.
Fraction bars representing fractions with different
denominators can be added or subtracted by further
dividing one or both bars into the same number of
pieces.
Fractions being added or subtracted on a number line
can be further divided into the same number of
sections as the other denominator to create equal
pieces. (e.g. ¾ + 1/3 represented on a number line
would need to be further divided in order to add, so
for ¾ each fourth would be divided into thirds because
3 is in the other denominator. Similarly, for 1/3, each
third would be divided into fourths because 4 is the
other denominator. This produces a number line
divided into 12ths for both fractions).
Be Skilled At…
•
•
•
•
•
•
Estimating reasonableness of answers to problems involving addition and subtraction of fractions by using benchmark fractions and “fraction sense” Using equivalent fractions as a strategy to add and subtract fractions with unlike denominators (including mixed numbers) Drawing visual fraction models (area models, number lines, etc…) to find a common denominator between fractions. Fluently adding and subtracting fractions with unlike denominators (including mixed numbers) using the algorithm (finding the common denominator). Discussing how to add and subtract fractions using manipulatives and mathematical representations. Solving 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.
3 | P a g e Unit Sequence
1
Student Friendly Objective
SWBAT…
Model part to whole
relationships.
•
•
•
Key Points/
Teaching Tips
Assess students’ prerequisite skills
using the “Am I Ready” resource and
address misconceptions as needed
Explain that a proper fraction is a
number between 0 and 1 and that an
improper fraction is a number that
exists between two whole numbers.
Students should observe that the larger
the numerator, the smaller the size of
each part and vice versa (the smaller
the denominator the larger the size of
each part)
Exit Ticket
1. How many eighths are in one whole?
Explain and draw a visual to justify your
thinking:
Instructional
Resources
My Math
Chapter 8
“Am I Ready?”
2) How many fourths are in 3/1? Draw a
visual to justify your thinking
“Fraction Kits”
(Appendix C)
3) How many thirds are in 3 2/3?
“Pattern Block
Fractions”
(Appendix C)
4) Draw the following ribbons.
a. 1 ribbon. The piece shown below is
!
only ! of the whole. Complete the
drawing to show the whole piece of
ribbon.
b. 1 ribbon. The piece shown below is
!
of the whole. Complete the
!
drawing to show the whole piece of
ribbon.
c. 2 ribbons, A and B. One third of A is
equal to all of B. Draw a picture of
the ribbons.
d. 3 ribbons, C, D, and E. C is half the
length of D. E is twice as long as D.
Draw a picture of the ribbons.
4 | P a g e 2
3
Interpret and represent
fractions as division.
Use manipulatives, visual
fraction models, or drawings
to model fractions as the
division of a numerator by
the denominator
4
Model fractions as division
using tape diagrams
5
Evaluate the context of a real
world situation in order to
interpret the fractional
quotient. Represent and solve
using visual fraction models.
Pacing: 2 days
Note: skip the application problem
from engage ny lesson 2
Example Problem for Mini-Lesson:
After a class potluck, Emily has three
equally sized apple pies left and she wants
to divide them into eight equal portions to
give to eight students.
For each of the problems below, draw a
visual model to represent the problem
and then write your answer in fraction
form:
(a) Draw a picture showing how Emily
might divide the pies into eight equal
portions.
(b) What fraction of a pie will each of the
eight students get?
(c) Explain how your answer to (b) is
related to the division problem 3 ÷ 8.
2) If three chicken pies are shared
equally among 5 people, what fraction of
a pie will each person have?
•
•
Example Problem for Mini-Lesson:
Your teacher gives 7 packs of paper to
your group of 4 students. If you share the
paper equally, how much paper does each
student get?
See visual representation below:
Student 1
Pack 1
Student 2
pack 2
Student 3
pack 3
Student 4
pack 4
Engage NY
Module 4 Lessons 2-3
(Appendix C)
My Math
Chapter 8, Lesson 1
1) If a piece of wood with a length of 5
feet is cut into 6 equal pieces, what is the
length of each piece?
3) If 8 pounds of grass seed are divided
equally into 5 piles, what is the weight of
one of these piles?
Engage NY
Module 4 Lesson 4
(Appendix C)
“Sharing Candy Bars”
“Sharing Candy Bars
Differently”
(Appendix C)
1) Write a division word problem for
31 ÷ 4 where the answer is a mixed
number. Show how to solve your
problem.
2) A carpenter used exactly 25 pieces of
wood to make 9 shelves of equal length.
Each shelf measured between —
A. 1 and 2 feet C. 3 and 4 feet
B. 2 and 3 feet D. 4 and 5 feet
1
2
3
pack 5
4
1
2
3
4
pack 6
Engage NY
Module 4 Lesson 5
(Appendix C)
1
2
3
4
pack 7
Each student receives 1 whole pack of paper and ¼ of the each of the 3 packs of paper. So each student gets 1 ¾ packs of paper.
5 | P a g e 6
7
Explore the concept of
equivalent fractions using
tiles and models. Observe
patterns in numerators and
denominators to deduce the
mathematical process for
generating equivalent
fractions using
multiplication.
•
•
Use division to generate the
•
simplest equivalent fraction
(i.e. the simplest form of a
fraction). Observe patterns
when simplifying fractions to •
deduce efficient processes
for simplifying.
Explain that when you can
no longer divide the
numerator and denominator
by any other factors besides
1, the fraction is now in
simplest form.
•
•
After students have had the
opportunity to observe patterns in
numerators and denominators, provide
time for them to apply the process of
multiplying numerators and
denominators by the same whole
number to create equivalent fractions
(Note: this is a review from 4th grade)
Explain that when you multiply the
numerator and denominator by the
same number, it does not change the
value of the fraction but only changes
the number and size of its parts.
Suggestion for the “I can apply what I
learned yesterday” box of the do now:
Use multiplication to generate three
equivalent fractions for 2/3
Suggestion for the “I’m Ready for
Today’s Lesson” box of the do now:
List all of the factors for 18
List all of the factors for 24
Circle the factors they have in
common
Segway into the lesson by connecting
to the first box of the do now
(reviewing yesterday’s concept) and
having students list the next three
equivalent fractions for 2/3 by
multiplying the numerator and
denominator by 2). Push students to
consider how they would use the
inverse operation to find equivalent
fractions if they had been given 18/24
to start (divide each by 2)
Then ask students to try dividing by
1) James is bowling. He knocked down 4
out of 10 bowling pins.
What fraction of the bowling pins were
not knocked down? Use tiles or fraction
models to solve:
A. 1/3
B. 2/3
C. 2/5
D. 3/5
1) Show two different ways you could
divide to simplify 24/30:
2) Alicia opened her piggy bank and
counted the coins inside. Here is what
she found:
22 pennies
5 nickels
5 dimes
8 quarters
What fraction of the coins in the piggy
bank are dimes?
A. 1/10
B. 1/8
C. 1/5
D. 11/20
“Red Rectangles”
“Pattern Blocks”
(Appendix C)
Engage NY
Module 3 Lesson 1
(Appendix C) My Math
Chapter 8, Lesson 3
*Modify resource by
not requiring students
to use prime
factorization in this
lesson
Resource for
Remediation:
My Math
Chapter 8 Lesson 2
(GCF – which is a
review from 4th grade)
*Note: do not teach
prime factorization
Show your work and explain how you
got your answer:
6 | P a g e •
8
Attend to precision when
plotting fractions and mixed
numbers on a number line.
Round fractions to their
nearest benchmark (0, ¼, ½,
¾ and 1 whole)
•
•
each of common factors they found in
box #2 of their do now à they should
observe that dividing by the greatest
common factor allows them to
generate the simplest equivalent
fraction in one step
In addition to using common factors,
encourage students to consider basic
divisibility rules to determine right
away if they can divide by 2, 5 or 10
This is a review; may be treated as a
flex day based on your students’
prerequisite understandings/skills
involving fractions
Mixed Number Example: 2 2/5 = 2
wholes + 1/5 + 1/5
On the number line below, I will shade in
two wholes, and then decompose the
whole between 2 and three into fifths
0
1
2
1) Plot the fraction 3/8 on the number
line below:
b.) Describe the value of the fraction by
explaining its location on the number line
between benchmark fractions:
“Closest to 0, ½, 1”
(Appendix C)
My Math
Chapter 9 Lesson 1
This fraction can be found on the number
line between ___________ and
__________, and is closest to
____________.
2) Show the mixed number 2 ¾ on the
number line below:
3
Then shade in 1/5 + 1/5
•
0
1
2
3
This mixed number is between the
whole numbers _____ and 3 _____
and is closest to _______.
7 | P a g e 9
Use number lines and
fraction models to compare
fractions with different
denominators.
•
•
Students should observe that the larger
the numerator, the smaller the size of
each part and vice versa (the smaller
the denominator the larger the size of
each part)
Encourage students to move from
concrete (i.e. using models and
visuals) to abstract (reasoning) when
comparing fractions
Byron says that 3/5 is greater than ½
because the denominator 5 is greater than
the denominator 2.
a. Use a number line to determine if he is
correct than 3/5 > ½:
b. Is his thinking correct? Explain:
http://learnzillion.com/l
essons/98-comparefractions-to-thebenchmark-of-12
http://learnzillion.com/l
essons/99-comparefractions-to-thebenchmark-of-14
http://learnzillion.com/l
essons/100-comparefractions-to-thebenchmark-of-34
10 Compare fractions with
different denominators by
generating equivalent
fractions with common
denominators.
11
•
As with simplifying, allow students an
opportunity to explore different
strategies and observe which ones are
most efficient (i.e. multiplying across
the denominators, determining the
LCD, etc.)
Leah and Jamal were swimming laps in
an Olympic size pool. They timed each
other to see who could swim the farthest
in just thirty seconds. Leah swam 11/12
of a lap and Jamal swam 7/8 of a lap.
Who swam the farthest in thirty seconds?
My Math
Chapter 8 Lesson 6
Flex Day (Instruction Based on Data)
Recommended Resources:
“Relating Fractions to Division” (Appendix C)
“Fraction Compare” (Appendix C)
“Snack Time” (Appendix C)
“Decomposing Fractions” (Appendix C)
My Math Chapter 8 Review (Pages 601 – 604)
8 | P a g e 12 Find the sum of fractions
with like denominators using
number lines.
•
Express a given fraction as
the sum of up to three
fractional parts and/or a
combination of whole
numbers and fractional parts.
Students added fractions with like
denominators using visual models in
4th grade – you may incorporate visual
models for students who are struggling
with the number line
Noah, Daniel, and Ava collected a total
of 12 pounds of aluminum cans. Noah
collected 3 pounds, Daniel collected 5
pounds, and Ava collected an unknown
number of pounds of aluminum cans.
Noah wrote this number sentence to
show how many pounds they collected
altogether
Engage NY
Module 3 Lesson 2
(Appendix C)
Additional Practice:
My Math
Chapter 9
Lesson 2
3/12 + 5/12 + ? = 12/12
13 Explore adding fractions
with unlike denominators
using visual fraction models
14 Add fractions resulting in
sums greater than one using
visual models
•
The Engage NY resource should be
the primary resource for this lesson –
use fraction tiles as needed for
struggling students and the My Math
resource for additional practice
Encourage students to reason about
fractional values using benchmarks to
estimate and/or predict if their sum
will be greater than one whole (i.e. “I
notice that both fractions are greater
than ½, therefore I know my sum will
be greater than 1”)
http://learnzillion.com/lessons/975-addmixed-number-fractions-with-differentdenominators-using-area-models
•
Represent and solve this problem using
the number line below:
Use fraction tiles to find the sum of:
Engage NY
Module 3 Lesson 3
3/4 + 1/5
(Appendix C)
Draw a picture that shows your
solution.
1) Elijah went to dinner and a school
concert for 3 7/12 hours.
The concert lasted 1 2/3 hours. Create an
area model to determine how many hours
dinner lasted:
A. 1 11/12
B. 2 1/12
C. 2 5/12
D. 2 5/9
My Math
Chapter 9 Lesson 4
http://learnzillion.com/l
essons/973-addfractions-withdifferent-denominatorsusing-fraction-bars
Engage NY
Lesson 3.4
(Appendix C)
Resource for
remediation and/or
additional practice:
My Math
Chapter 9 Lesson 10
9 | P a g e 15 Explore subtracting fractions
with unlike denominators
using visual fraction models.
16 Subtract fractions from
numbers between 1 and 2
using visual fraction models
•
•
•
The Engage NY resource should be
the primary resource for this lesson –
use fraction tiles as needed for
struggling students and the My Math
resource for additional practice
Example from unpacked standards
guide on how to use models to
subtract mixed numbers:
This model shows 1 ¾ subtracted from
3 1/6 leaving 1 + ¼ = 1/6 which a
student can then change to 1 + 3/12 +
2/12 = 1 5/12. 3 1/6 can be
expressed with a denominator of 12.
Once this is done a student can
complete the problem, 2 14/12 – 1
9/12 = 1 5/12.
Use fraction models to find the difference Engage NY
of:
Lesson 3.5
(Appendix C)
4
/5 – 2/3
My Math
Draw a picture that shows your solution. Chapter 9, Lesson 6
http://learnzillion.com/l
essons/974-subtractfractions-withdifferent-denominatorsusing-fraction-bars
Engage NY
Lesson 3.6
(Appendix C)
This diagram models a way to show how
3 1/6 and 1 ¾ can be expressed with a
denominator of 12.:
10 | P a g e 17 Make sense of real world
problems involving fractions
by representing and solving
them using visual models
•
Lila collected the honey from 3 of her Engage NY
beehives.
From the first hive she Lesson 3.7
!
collected ! gallon of honey. The last two (Appendix C)
!
hives yielded ! gallon each.
a. How many gallons of honey did Lila
collect in all? Draw a diagram to
support your answer.
b. After using some of the honey she
collected for baking, Lila found that
!
she only had ! gallon of honey left.
How much honey did she use for
baking?
!
c. With the remaining ! gallon of honey,
Lila decided to bake some loaves of
bread and several batches of cookies
for her school bake sale. The bread
!
needed ! gallon of honey and the
!
cookies needed ! gallon. How much
honey was left over?
d. Lila decided to make more baked
!
goods for the bake sale. She used !
lb less flour to make bread than to
!
make cookies. She used ! lb more
flour to make cookies than to make
!
brownies. If she used ! lb of flour to
make the bread, how much flour did
she use to make the brownies?
Explain your answer using a diagram,
numbers, and words.
11 | P a g e 18 Add fractions to and subtract
fractions from whole
numbers using fraction
models and the number line
19 Add fractions with unlike
denominators by creating
equivalent fractions with
common denominators.
Engage NY
Lesson 3.8
(Appendix C)
Example of estimating to judge the
reasonableness of a sum:
Your teacher gave you 1/7 of the bag of
candy. She also gave your friend 1/3 of
the bag of candy. If you and your friend
Estimate sums (closest
combined your candy, what fraction of the
to/less than/greater than 0, ½, bag would you have? Estimate your
1 whole) to judge the
answer and then calculate. How
reasonableness of your sum. reasonable was your estimate?
1) Which equation below gives the
correct value of the following sum?
3/8 + 14/12
A.
B.
C.
D.
3/8 + 7/6 = 10/14
9/24 + 28/24 = 37/24
3/12 + 14/12 = 17/12
3/8 + 14/12 = 17/20
2) Find the sum of 1/5 and 2/3
Student 1
1/7 is really close to 0. 1/3 is larger than
1/7, but still less than 1/2. If we put them
together we might get close to 1/2.
1/7 + 1/3= 3/21 + 7/21 = 10/21. The
fraction does not simplify. I know that 10
is half of 20, so 10/21 is a little less than
½.
Another example: 1/7 is close to 1/6 but
less than 1/6, and 1/3 is equivalent to 2/6,
so I have a little less than 3/6 or ½.
a. Use estimation to judge the
reasonableness of your sum
My Math
Chapter 9, Lesson 5
Engage NY
Lesson 3.9
(Appendix C)
*Modify resources to
require
estimation/reasoning
about fraction values
in order for students to
judge the
reasonableness of their
work.
12 | P a g e 20 Add fractions with sums
greater than 2
•
Pacing: up to 2 days
1) Find the sum:
!
!
3! + 1! =
Engage NY
Lesson 3.10
(Appendix C)
4 5/7 + 3 ¾ =
Use estimation to judge the
reasonableness of each sum:
My Math
Chapter 9 Lesson 11
!
2) Erin jogged 2 ! miles on Monday.
!
Wednesday, she jogged 3 ! miles, and on
21
!
Friday, she jogged 2 ! miles. How far
did Erin jog altogether?
!
!
3) Clayton says that 2 ! + 3 ! will be
22 Subtract fractions with
unlike denominators by
creating equivalent fractions
with common denominators.
Estimate differences (closest
to/less than/greater than 0, ½,
1 whole) to judge the
reasonableness of your sum.
•
Require students to use estimation to
evaluate the reasonableness of their
differences, as well as to critique the
work of their peers
more than 5, but less than 6 since 2 + 3
is 5. Is Clayton’s reasoning correct?
Prove him right or wrong.
1. Find the difference:
My Math
Chapter 9, Lesson 7
5/8 – 1/12
(b) Use estimation to judge the
reasonableness of your difference
Engage NY
Lesson 3.11
(Appendix C)
*Modify resources to
require
estimation/reasoning
about fraction values
in order for students to
judge the
reasonableness of their
work.
13 | P a g e