Madison County Schools

MADISON COUNTY PUBLIC SCHOOLS

District Curriculum Map for Mathematics: Grade 5

Unit 6

Unit Description / Unit 6
Operations with Fractions
Suggested Length: 6 weeks
Big Idea(s)
What enduring understandings are essential for application to new situations within or beyond this content? / Enduring Understanding(s)

Develop fluency with fraction operations of addition & subtraction
Develop understanding of fraction multiplication & division

Enduring Skills Rubric measures competency of the following skills:

Solves word problems involving addition and subtraction of fractions, using equivalent fractions in cases of unlike denominators by using visual fractions or equations
Recognizes reasonableness of answer using benchmark fractions
Uses equivalent fractions to add/subtract fractions and/or mixed numbers with unlike denominators
Constructs and communicates a well-organized and complete written response using a logical progression of steps.
Makes a line plot to display a data set of measurements in fractions of a unit with denominators limited to 2, 4, and 8, and uses operations on fractions to solve problems involving information on line plots
Solves word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers AND interprets the fraction as division of the numerator by the denominator AND identifies a simple model representing the situation.
Solves real world problems by multiplying a mixed number by a fraction, a fraction by a fraction, and a whole number by a fraction and interprets the product by using given context
Solves real world problems by dividing a fraction by a whole number and a whole number by a fraction using visual fraction models and interpreting the quotient by using given context
Multiplies a mixed number by a fraction, a fraction by a fraction, and a whole number by a fraction and interprets the product
Divides a fraction by a whole number and a whole number by a fraction using visual fraction models and interpreting the quotient
Assesses reasonableness of a product of two fraction factors without performing the multiplication with either factor being greater or less than one

Essential Question(s)
What questions will provoke and sustain student engagement while focusing learning? /

How can I solve real-world problems that involve addition and subtraction of fractions?
How does multiplying fractions relate to real world problems?
How do I show multiplication of fractions using a visual model?
How does dividing fractions relate to real world problems?
How do I show division of fractions using a visual model?

Standards / Standards for Mathematical Practice

Make sense of problems and persevere in solving them. Students make sense of the meaning of addition, subtraction, multiplication and division of fractions with whole- number multiplication and division.
Reason abstractly and quantitatively. Students demonstrate abstract reasoning to create and display area models of multiplication and both sharing and measuring models for division. They extend this understanding from whole numbers to their work with fractions.
Construct viable arguments and critique the reasoning of others. Students construct and critique arguments regarding their understanding of fractions greater than, equal to, and less than one whole and one half. Students recognize reasonable answers when performing operations with fractions.
Model with mathematics. Students draw representations of their mathematical thinking as well as use words and numbers to explain their thinking
Use appropriate tools strategically. Students select and use tools such as visual models, measuring sticks/line plots, and manipulatives of different fraction sizes to represent situations involving the relationship between fractions.
Attend to precision. Students attend to the precision when performing operations with fractions. Students use appropriate terminology when referring to fractions.
Look for and make use of structure. Students develop the concept of all operations with fractions with fractions including drawing visual representations and working with like and unlike denominators through the use of various manipulatives
Look for and express regularity in repeated reasoning. Studentsrelate new experiences to experiences with similar contexts when allowing students to develop relationships for fluency and understanding of fractional computation. Students explore operations with fractions with visual models and begin to formulate generalizations.

Standards for Mathematical Content
5.NF.1Add 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. (Ingeneral, a/b + c/d = (ad + bc)/bd.)
5.NF.2Solve 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. Forexample, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that3/7 < 1/2.
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, notingthat 3/4 multiplied by 4 equals 3, and that when 3 wholes are sharedequally among 4 people each person has a share of size 3/4. If 9 peoplewant to share a 50-pound sack of rice equally by weight, how manypounds of rice should each person get? Between what two whole numbersdoes your answer lie?
5.NF.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 ofoperations 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.)
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.5 Interpret multiplication as scaling (resizing), by:
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.
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.6Solve 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.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Note: Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.)
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.
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.
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?
Supporting Standard(s)
Which related standards will be incorporated to support and enhance the enduring standards? / 5.MD.1 Convert among different sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use the conversion to solve multi-step real world problems.
5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, ¼, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.
Instructional Outcomes
What must students learn and be able to by the end of the unit to demonstrate mastery? / I am learning to…

Calculate equivalent fractions in order to add or subtract fractions and mixed numbers
Interpret the product of a multiplication problem based on the size of the factors
Explain the inverse relationship between multiplication and division
I can explain how fractions are related to division by using models and/or equations
I can solve word problems that include division of whole numbers and explain the quotient as it relates to whole numbers, mixed numbers, or fractions
Model the multiplication of fractions
Explain why when multiplying fractions
Model the division of whole numbers and fractions
Solve real world problems that require operations of fractions

Use benchmark fractions to determine reasonability of an answer

Vocabulary
What vocabulary must students know to understand and communicate effectively about this content? / Essential Vocabulary
Area model
Array
Denominator
Dividend
Divisor
Equal parts
Equivalent fraction
Factor
Fraction
Improper fraction
Like/common denominator
Mixed number
Numerator
Of
Product
Quotient
Skip count
Unit fraction
Supporting Vocabulary
Associative property
Distributive property
half
Inverse relationship
Remainder
Whole
Resources & Activities
What resources could we use to best teach this unit? / Stepping Stones
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Module 2: Lessons 1-5
Module 4: Lessons 1-7
Module 6: Lessons 2-7

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Module 4: Lesson 8
Module 6: Lesson 8
Module 9: Lesson 9

5.NF.3

Module 11: Lessons 1-2

5.NF.4

Module 9: Lessons 1-8

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Module 9: Lesson 4,6,7

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Module 9: Lessons 8-9

5.NF.7

Module 11: Lessons 3-8

Engage NY
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5.NF.1

Module 3: Topics A, B, C, D

5.NF.2

Module 3: Topics B, C, D

5.NF.3

Module 4: Topic B

5.NF.4

Module 4:Topics C, D, E
Module 5: Topic C

5.NF.5

Module 4: Topic F

5.NF.6

Module 4: Topics D, E, F
Module 5: Topic C

5.NF.7

Module 4: Topic G

Howard County Website
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K-5 Math Teaching Resources
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Created by Beth Richardson ():

Fraction of the Day

Making Math Magic

Teachers Pay Teachers
Dividing Fractions Task Cards (TPT $2.75):
Multiplying Fractions Task Cards (TPT $2.50):
Adding and Subtracting Fractions Task Cards (TPT $2.50):
Common Core Assessment Pack-Common Core Grade 5 (TPT $12.50):
Illustrative Mathematics
Illuminations
Remember there are other sources in your school that may not be listed on this common resources list due to variation in each individual school. Examples of other great resources your school may have access to include: Everyday Math Games, Investigations, Everyday Partner Games, AVMR file folders, Ongoing Assessment Project, etc.