Unit 3

Add and Subtract Fractions

 

Grade 5

Math

Unit Length and Description:

 

25 days

 

In this unit students will use prior learning to as they are adding and subtracting fractions with unlike denominators using concrete and visual models, reasoning and equations. They develop conceptual understanding of addition and subtraction of fractions and mixed numbers in order to solve real-world problems. Quantitative reasoning will be used by students to determine whether their answers are reasonable.

Students use visual models such as area models, fraction strips, or number lines as they begin to add fractions with unlike denominators. Students begin to understand the need for like denominators by using concrete models. Once students understand the need for like denominators and can identify appropriate denominators, then they begin using the algorithm. Students understand that when they are finding equivalent fractions, they are multiplying the original fraction by names for 1.

Students solve problems involving addition and subtraction of fractions with unlike denominators. Students also use benchmarks, comparisons and mental math to justify their thinking and to determine whether their answer is reasonable.

Students extend their previous work with considering a fraction as a division situation to expressing the quotient of a division problem as a fraction or mixed number. Real-life problems should provide the context in which expressing the fraction as a remainder makes sense. Students are provided with a variety of division problems to model interpreting the remainder as a fraction.

 

 

Standards:

 

NBT - Number and Operation in Base Ten

Use equivalent fractions as a strategy to add and subtract fractions.

5.NF.A.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.

For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (in general, a/b + c/d = (ab + bc)/bd.)

5.NF.A.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.

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

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 50pound 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.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 = (na)/(nb) to the effect of multiplying a/b by 1.

Foundational Standards

Represent and solve problems involving multiplication and division.

3.OA.A.1

Interpret products of whole numbers, e.g., interpret 5 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 7.

3.OA.A.2

Interpret whole-number quotients of whole numbers, e.g., interpret 56 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 8.

Understand properties of multiplication and the relationship between multiplication and division.

3.OA.B.6

 

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

3.MD.C.7b

Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

Extend understanding of fraction equivalence and ordering.

4.NF.A.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.

4.NF.A.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. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Build fractions from unit fractions.

4.NF.B.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.   Understand a fraction a/b with a > 1 as a sum of fractions 1/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. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

c.    Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 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.   Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 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.B.4

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a.   Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 (1/4), recording the conclusion by the equation 5/4 = 5 (1/4).

b.   Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 (2/5) as 6 (1/5), recognizing this product as 6/5. (In general, n (a/b) = (n a)/b.)

c.    Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

Use the four operations with whole numbers to solve problems.

4.OA.A.1

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.A.2

Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

Solve problems involving measurement and conversion of measurements.

4.MD.A.2

Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Standards for Mathematical Practices

1.   Make sense of problems and persevere in solving them.

2.   Reason abstractly and quantitatively.

3.   Construct viable arguments and critique the reasoning of others.

4.   Model with mathematics.

5.   Use appropriate tools strategically.

6.   Attend to precision.

7.   Look for and make use of structure.

8.   Look for and express regularity in repeated reasoning.

Instructional Outcomes

 

        5.NF.A.1:

o   I can find equivalent fractions.

o   I can add and subtract fractions with unlike denominators using equivalent fractions.

o   I can add and subtract mixed numbers with unlike denominators using equivalent fractions.

        5.NF.A.2:

o   I can solve word problems using addition and subtraction of fractions with like and unlike denominators referring to the same whole.

o   I can use benchmark fractions and number sense of fractions to check for reasonableness of answers.

        5.NF.B.3

o   I can interpret a fraction as division of the numerator by the denominator

o   I can solve word problems involving division of whole numbers with quotients as fractions or mixed numbers.

o   I can recognize the remainder as a fractional part of the problem.

        5.NF.B.5

o   I can explain the relationship between two multiplication problems that share a common factor ( and ).

o   I can compare the product of two factors without multiplying. Example: 2 x ? = < 1 Answer must be less than .

o   I can explain why multiplying a number by a fraction greater than one will result in a product greater than the given number.

o   I can explain why multiplying a fraction by one (which can be written as various fractions, ex. , etc.) results in an equivalent fraction.

o   I can explain why multiplying a fraction by a fraction will result in a product smaller than the given number.

 

 

 

Enduring Understandings:

 

        Landmark/benchmark numbers should be used when making decisions about other numbers.

        Models can be used to compute fractions with like and unlike denominators.

        The same fractional amount can be represented by an infinite set of equivalent fractions.

        A fraction describes the division of a whole into equal parts, and it can be interpreted in more than one way depending on the whole to be divided.

        Multiplying a whole number by a fraction involves division as well as multiplication. The product is a fraction of the whole number.

        Rounding and compatible numbers can be used to estimate the product of fractions or mixed numbers.

        The relative size of the factors can be used to determine the relative size of the product.

 

Essential Questions:

 

        Why would I need to use landmark/benchmark numbers when making decisions about other numbers?

        How can models (line plots, etc.) be used to compute fractions with like and unlike denominators?

        What strategies can be used to determine if answers are reasonable?

        How can I tell if a fraction is greater than, less than, or equal to one whole.

        How can fractions with different denominators be added together? Subtracted?

        How are fractions related to division?

        How can I multiply fractions and whole numbers?

        How does multiplying by a fraction change the other factor?