Download Extending Children`s Mathematics: Fractions

Grade 5 UbD Math Unit Planning 2014 to 2015
PS 105
Grade/Unit#/Book(s)/Topic
Grade 5 / Unit 4/ Book 4 /Fractions
Approximate Days or Dates
35
Stage 1 - Identify Desired Results
Learning Outcomes
What relevant goals will this unit address?
(must come from curriculum; include specific Common Core standards)
Use equivalent fractions as a strategy to add and subtract fractions
5.NF.1: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing 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.
Apply and extend previous understandings of multiplication and division
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?
5.NF.B.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.4.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.NF.B.4.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 lengths. Multiply fractional side lengths to find areas of rectangles, and
represent fraction products as rectangular areas.
5.NF.B.5: Interpret multiplication as scaling (resizing), by:
5.NF.B.5.A: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated
multiplication.
5.NF.B.5.B: Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing
multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a
product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.
5.NF.B.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.
5.NF.B.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1
5.NF.B.7.A: Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for
(1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 =
1/12 because (1/12) × 4 = 1/3.
5.NF.B.7.B: Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5),
and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because
20 × (1/5) = 4.
5.NF.B.7.C: Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions,
e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share
1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Enduring Understandings
What understandings about the big ideas are desired?
What misunderstandings are predictable?
Essential Questions
What are the Essential Questions?
Are there any potential cross-curricular connections during this chapter?
Students will understand that...
Essential Question:

Fractions are numbers that have a place on a number line.

Fractions can only be added or subtracted when the parts are
alike. Just like when we add and subtract whole numbers by
adding tens with tens and ones with ones, we need to add like
fractional parts.


Fraction multiplication can be understood similarly to whole
number multiplication as equal groups. Thus 1/3 x 9 can be
thought of as 9 groups of 1/3 or 1/3 of a group of 9.

Dividing a whole number by a fraction (e.g., 3 / ½) is best
understood as how many ½’s make up 3 wholes?

Dividing a fraction by a whole number (e.g., ½ / 3 is best
understood as ½ shared in 3 equal groups.
Related misconceptions…

Students have a hard time with the idea that dividing a number
by a fraction makes the number bigger, whereas multiplying by a
fraction (less than 1) makes the number smaller.
How is computing with fractions similar to and different from
computing with whole numbers?
Cross-curricular connections…
Knowledge:
What knowledge will student acquire as a result of this unit? This content
knowledge may come from the chapter’s goals, or might also address prerequisite knowledge that students will need for this unit.
Skills:
What skills will students acquire as a result of this unit? List the skills and/or
behaviors that students will be able to exhibit as a result of their work in this
unit.
Students will know...
Students will be able to…
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


There are an infinite number of ways to express any fraction, but it
is customary to use the simplest form when writing final answers.
Similarly, we generally change improper fractions to mixed numbers
when writing a final answer.
Multiplying a number by a fraction less than 1 decreases the
number.
Multiplication of fractions is commutative. Therefore, taking 2/3 of 9
is equivalent to taking 9 groups of 2/3.










Compare and order fractions and mixed numbers.
Add fractions and mixed numbers with different denominators.
Subtract fractions and mixed numbers with different denominators.
Estimate sums and differences of fractions and mixed numbers.
Solve complex story problems involving adding and subtracting fractions
or mixed numbers.
Create area (array) models for multiplying fractions, especially for
fractions of fractions.
Find a fraction of a whole number or a whole number times a fraction.
Multiply fractions and mixed numbers using models, the distributive
property or the basic algorithm for multiplying proper fractions.
Divide fractions by whole numbers without algorithms.
Divide whole numbers by fractions without algorithms.
Stage 2 – Assessment Evidence
Evidence
Through what evidence (work samples, observations, quizzes, tests,
journals or other means) will students demonstrate achievement of the
desired results? Formative and summative assessments used throughout
the unit to arrive at the outcomes.
Student Self-Assessment
How will students reflect upon or self-assess their learning?
Stage 3 – Learning Plan
#
Content Goal
Lesson Notes/Planned Differentiation
Unit Note: We are going to use the CGI approach to understanding
fractions to supplement this unit. The nine suggested CGI lessons are
numbered from CGI-1 to CGI-9.
Additional Resources or Math Centers
Extending Children's Mathematics: Fractions
& Decimals: Innovations In Cognitively
Guided Instruction
By Empson and Levi
1.1
Everyday Uses of
Rational Numbers
2.2
Comparing Fractions
2.3
Ordering Fractions
2.4
Solving Problems with
Fractions and Percents
3.4
Fractions on Number
Lines
Comparing and
Ordering Fractions
Equal Sharing
Problems and
equivalent fractions
3.5
CGI-1
CGI-2
CGI-3
Equal Sharing
Problems and Adding
Fractions
Equal Sharing
Problems and Adding
Fractions
This lesson serves as an introduction and pre-assessment of what
students currently understand about fractions, decimals, and percents.
We won’t be doing most of the percent lessons in this book, but it is
fine to do a basic introduction. (SKIP the rest of the first investigation.)
The emphasis throughout this unit should be on ways to use number
sense to compare and order fractions, NOT on getting common
denominators.
“In Between” is an excellent game. Even though we skipped the
percent lessons from the first investigation, they should be fine playing
it. Just let them know that they can think of 10% as 1/10 and 90% as
9/10.
Everyone can play “In Between” and do the “Which is Greater”
problems. Advanced students can do the “Fraction and Percent
Problems” as enrichment.
This is a very valuable set of lessons as the Common Core is big on the
Number Line model for fractions.
This lesson is fine as written.
Spend three days (CGI-1, CGI-2, and CGI-3) working on equal sharing
problems from the CGI book. After the first day, focus on problems that
may lead students to have to add fractions with different denominators.
CGI book, chapters 1 and 2 (see pages 3235 for advice on problem selection)
Start today with two problems: 8 students share 6 brownies equally
and 6 students share 8 brownies equally
Try these problems today:
For homework and/or last ten minutes of
classwork, give a sheet reviewing equivalent
fractions (a fourth grade skill).
Ditto
__ children in art class have to share __ packages of clay so that
everyone gets the same amount. How much clay can each child have?
(12, 9) (12, 8) (8, 3) (20, 16)
Note: the first one may lead students to give everyone half and then
everyone ¼. The second one may lead to everyone getting ½ + 1/6.
Encourage students to use pictures to try to add these fractions.
Try these problems today:
Ditto
__children want to share __ sub sandwiches so that each one gets the
same amount. How much should each person get?
(2, 1 ¾) (3, 2 ¼) (2, 1 7/8) (3, 3 ¾) (3, 3 ¼)
Note: these should definitely lead students to have to add fractions.
CGI-4
Adding and
Subtracting fractions
and mixed numbers
with friendly
denominators
CGI-5
Adding and
Subtracting fractions
and mixed numbers
with friendly
denominators
CGI-6
Adding and
Subtracting fractions
and mixed numbers
with friendly
denominators
3.6
Fraction Track Game
3.7
3.8
3.9
Fraction Story
Problems
CGI-7
Multiple Groups
Problems
Try these problems today:
CGI book, chapter 8 (see pages 209-211 and
218-221 for advice on problem selection)
Sara likes to eat __ sandwich for snack and __ sandwich for lunch. If
Sara is going to eat her snack and lunch at school, how many
sandwiches does she need to bring to school?
(1/2, 3/4) (3/4, 1 ½) (1/2, 3/8) (1/2, 7/8) (3/8, ¾) (3/4, 1 1/8)
Yvette had __ jars of jam. After she made sandwiches for her friends,
she had __ jars of jam left. How much jam did she use?
(3, 2 ½) (3 ¼, 2 ½) (4, 2 ½) (4, 2 ¾) (3, 2 7/8)
Note: Encourage students to draw pictures to solve these. If students
are struggling allow them to use fraction strips, but try to move them
from strips to drawings and then from drawings to numbers.
Today and tomorrow move from story problems to equations. Use all of
the Addition and Subtraction Equations on pages 210 to 211. See if
students can do these without manipulatives or drawings, but allow
them to use these models if necessary. The goal is to move students to
using equivalent fractions to create common denominators.
Ditto
Continue the work from yesterday. Today students should focus on the
last set of problems, where both denominators need to be changed. If
these prove difficult, let students go back to drawings or fraction strips
and encourage them to discover the algorithm, rather than telling them
today. They should begin to discover that when one denominator is not
a multiple of the other denominator, they can use the product of the
denominators as their common one. Don’t worry about LCD. It is not
required for fifth grade.
The idea is to play this game WITHOUT the use of the standard
algorithm for addition of fractions, but instead to use what they have
learned about Equivalent Fractions.
This is where students get to combine all of their skills for adding and
subtracting fractions. Don’t do Roll Around the Clock. Supplement this
section with the problems referenced to the right.-->
Ditto
Spend three days working on multiple groups problems from the CGI
book. Use equations to summarize student results so that they
understand they are multiplying fractions by whole numbers. And
highlight students’ strategies that use multiplicative relationships to
solve these problems (For example, listen for students who say things
like, “I know 4 groups of ¾ is 3 so 20 groups of ¾ would be 5 times 3
or 15.”
CGI book, chapter 3 (see pages 69-71 for
advice on problem selection)
Start today with two problems: It takes ¾ pound of clay to make a
bowl. How much clay is needed to make 12 bowls? It takes 3/8 cup of
sugar to make a loaf of bread. How much sugar would you need for 16
loaves?
http://www.k5mathteachingresources.com/5th-gradenumber-activities.html
For homework and/or last ten minutes of
classwork, give a sheet reviewing the skills of
addition and subtraction of fractions.
CGI-8
Multiple Groups
Problems
Start off today with a number talk using the open number sentences on
page 67 of the CGI book. Then do this problem:
CGI-9
Multiple Groups
Problems
Eve’s Gecko eats 2/7 jar of baby food a day. How many jars should she
buy to last for two weeks? If she buys a case of 24 jars, how many
days would that last?
Today skip the contexts and solve a variety of open number sentences
such as these:
¾ x 14 = ?
¾ x ? = 12 ¾
4A.1
Multiplying a Whole
Number by a Fraction
4A.2
Multiplying Whole
Numbers by Fractions
and Mixed Numbers
Multiply fractions or
Mixed Numbers
Multiplying Fractions
by Fractions
A Rule for Multiplying
Fractions
4A.3
4A.4
4A.5
4A.6
2 days
4A.7
4A.8
4A.9
4A.10
Test
Using Arrays for
Multiplying Fractions
Multiplying Fractions
and Multiplying Mixed
Numbers
Dividing a Whole
Number by a Fraction
Dividing a Fraction by
a Whole Number
Dividing with Fractions
Unit Review
Unit Assessment
NOTE: The Multiple Groups problems of the past few days involved
finding a whole number of groups of a fraction. Today it is the
opposite, finding a fraction of a group. For these types of problems, the
bar model should be used.
For 4/5 of 20, it would look like this:
4
4
4
4
4
This lesson is fine as written.
This lesson is fine as written.
This lesson is fine as written.
It is OK to use the algorithm for fraction times a fraction, but continue
to have students practice creating models, because they are difficult
and absolutely necessary for the state test. Students who already knew
the rule should be focusing on understanding and explaining WHY the
rule works.
This is a tricky model that appears often on the state test, so take an
extra day to practice this.
This lesson is fine as written, but be sure that students include array
models with their work.
These types of problems are essentially CGI Multiple Groups problems
with the number of groups unknown. That is how they should be
solved. Do NOT teach or allow the use of the “Invert and Multiply”
algorithm. It is completely unnecessary for grade 5 and harmful to
students’ thinking and understanding of the meaning of division of
fractions.
Again, emphasize the use of models to solve these problems and
promote understanding.
This lesson is fine as written.
Give students a day to practice solving story problems with all four
operations mixed up. Note that students often have trouble identifying
the correct operation to use when solving fractions problems.
Post-Unit Reflection
Considerations
Comments
Required Areas of Study:
Was there alignment between outcomes, performance
assessment and learning experiences?
Adaptive Dimension:
For struggling students:
Did I make purposeful adjustments to the curriculum
content (not outcomes), instructional practices, and/or
the learning environment to meet the learning needs and
diversities of all my students?
For students who need a challenge:
Suggested Changes:
How would I do the unit differently next time?