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5th Grade Mathematics
Unit #3: Interpreting Fractional Values and 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 Vocabulary Mathematical Practices 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
Fraction Numerator Denominator Simplify Equivalent Convert Mixed Number Improper Fraction Size
Benchmark Fraction Position Sum Location Difference Attributes Estimate Sides Reasonable Common Denominator Part:Whole
MP.1: Make sense of problems and persevere in solving them
MP.2: Reason abstractly and quantitatively MP.3: Construct viable arguments and critique the reasoning of others
MP.4: Model with mathematics 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
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Common Core State Standards Progression of Skills According to the PARCC Model Content Framework, Standard 3.NF.2 should serve as an opportunity for in- depth focus:
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.”
4th 5th 6th
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.
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.
6.NS.1: Interpret and compute quotients of fractions, and solve 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?.
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.
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.
5.NF.3: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b)
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
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Big Ideas Students Will… • 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
• 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
Know/Understand Be Skilled At… • 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).
• 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.
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Unit Sequence Student Friendly Objective
SWBAT… Key Points/
Teaching Tips Exit Ticket Instructional
Resources 1 Model part to whole
relationships.
• 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)
1. How many eighths are in one whole? Explain and draw a visual to justify your thinking: 2) How many fourths are in 3/1? Draw a visual to justify your thinking 3) How many thirds are in 3 2/3? 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.
My Math Chapter 8 “Am I Ready?” “Fraction Kits” (Appendix C) “Pattern Block Fractions” (Appendix C)
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2 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
• 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. (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.
For each of the problems below, draw a visual model to represent the problem and then write your answer in fraction form: 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? 2) If three chicken pies are shared equally among 5 people, what fraction of a pie will each person have? 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 Lessons 2-3 (Appendix C) My Math Chapter 8, Lesson 1
3
4 Model fractions as division using tape diagrams
Engage NY Module 4 Lesson 4 (Appendix C)
5 Evaluate the context of a real world situation in order to interpret the fractional quotient. Represent and solve using visual fraction models.
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:
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
“Sharing Candy Bars” “Sharing Candy Bars Differently” (Appendix C) Engage NY Module 4 Lesson 5 (Appendix C)
Student 1 Student 2 Student 3 Student 4 1 2 3 4 1 2 3 4 1 2 3 4
Pack 1 pack 2 pack 3 pack 4 pack 5 pack 6 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.
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6 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.
• 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.
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
“Red Rectangles” “Pattern Blocks” (Appendix C) Engage NY Module 3 Lesson 1 (Appendix C)
7 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.
• 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) 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 Show your work and explain how you got your answer:
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
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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
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)
• 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 3 Then shade in 1/5 + 1/5 • 0 1 2 3
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: 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:
This mixed number is between the whole numbers _____ and 3 _____
and is closest to _______.
“Closest to 0, ½, 1” (Appendix C) My Math Chapter 9 Lesson 1
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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/lessons/98-compare-fractions-to-the-benchmark-of-12
http://learnzillion.com/lessons/99-compare-fractions-to-the-benchmark-of-14 http://learnzillion.com/lessons/100-compare-fractions-to-the-benchmark-of-34
10 Compare fractions with different denominators by generating equivalent fractions with common denominators.
• 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
11 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)
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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
3/12 + 5/12 + ? = 12/12 Represent and solve this problem using the number line below:
Engage NY Module 3 Lesson 2 (Appendix C) Additional Practice: My Math Chapter 9 Lesson 2
13 Explore adding fractions with unlike denominators 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
Use fraction tiles to find the sum of:
3/4 + 1/5
Draw a picture that shows your solution.
Engage NY Module 3 Lesson 3 (Appendix C) My Math Chapter 9 Lesson 4 http://learnzillion.com/lessons/973-add-fractions-with-different-denominators-using-fraction-bars
14 Add fractions resulting in sums greater than one using visual models
• 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-add-mixed-number-fractions-with-different-denominators-using-area-models
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
Engage NY Lesson 3.4 (Appendix C) Resource for remediation and/or additional practice: My Math Chapter 9 Lesson 10
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15 Explore subtracting fractions with unlike denominators 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
Use fraction models to find the difference of:
4/5 – 2/3 Draw a picture that shows your solution.
Engage NY Lesson 3.5 (Appendix C) My Math Chapter 9, Lesson 6
http://learnzillion.com/lessons/974-subtract-fractions-with-different-denominators-using-fraction-bars
16 Subtract fractions from numbers between 1 and 2 using visual fraction models
• 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.
This diagram models a way to show how 3 1/6 and 1 ¾ can be expressed with a denominator of 12.:
Engage NY Lesson 3.6 (Appendix C)
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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 beehives. From the first hive she collected !
! gallon of honey. The last two
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.
Engage NY Lesson 3.7 (Appendix C)
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18 Add fractions to and subtract fractions from whole numbers using fraction models and the number line
Engage NY Lesson 3.8 (Appendix C)
19 Add fractions with unlike denominators by creating equivalent fractions with common denominators. Estimate sums (closest to/less than/greater than 0, ½, 1 whole) to judge the reasonableness of your sum.
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 combined your candy, what fraction of the bag would you have? Estimate your answer and then calculate. How reasonable was your estimate? 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.
1) Which equation below gives the correct value of the following sum?
3/8 + 14/12
A. 3/8 + 7/6 = 10/14 B. 9/24 + 28/24 = 37/24 C. 3/12 + 14/12 = 17/12 D. 3/8 + 14/12 = 17/20 2) Find the sum of 1/5 and 2/3 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.
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20 Add fractions with sums greater than 2
• Pacing: up to 2 days
1) Find the sum: 3 !!+ 1 !
!=
4 5/7 + 3 ¾ =
Use estimation to judge the reasonableness of each sum:
2) Erin jogged 2 !
! miles on Monday.
Wednesday, she jogged 3 !! miles, and on
Friday, she jogged 2 !! miles. How far
did Erin jog altogether?
3) Clayton says that 2 !!+ 3 !
! will be
more than 5, but less than 6 since 2 + 3 is 5. Is Clayton’s reasoning correct? Prove him right or wrong.
Engage NY Lesson 3.10 (Appendix C) My Math Chapter 9 Lesson 11
21
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
1. Find the difference: 5/8 – 1/12 (b) Use estimation to judge the reasonableness of your difference
My Math Chapter 9, Lesson 7 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.
