Fractions: Add and Subtract, Multiply and Divide, Line Plots with Fractions
Unit description: Students will extend their knowledge of adding and subtracting fractions with like denominators from fourth grade to fractions with unlike denominators. Students will extend their knowledge of multiplying fractions by whole numbers to multiply fractions by other fractions and mixed numbers. Students will consider the contexts for dividing unit fractions by whole numbers and dividing whole numbers by unit fractions and the relationship of the quotient to the dividend and divisor. Students will also use their knowledge of fractions to create and interpret line plots with fractional measures on the scale.
Essential Outcomes of the Unit
Number and Operations—Fractions- 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 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. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers
Number and Operations—Fractions- Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
- 5.NF.3 Interpret a fraction as division of the numerator by the denominator (𝑎𝑎 𝑏𝑏 = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
- 5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number or a fraction.
- 5.NF.4a Interpret the product 𝑎𝑎 𝑏𝑏 × 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.
- 5.NF.4b Find the area of a rectangle with fractional side lengths by tiling it with rectangles 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.6 Solve real world problems involving multiplication of fractions and mixed numbers.
Measurement and Data- Convert like measurement units within a given measurement system.
- 5.MD.2 Convert among different-sized standard measurement units within a given measurement system when the conversion factor is given. Use these conversions in solving multi-step, real world problems.
Other Standards Addressed in the Unit
Operations and Algebraic Thinking- Operations and Algebraic Thinking
- 5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
Number and Operations—Fractions- Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
- 5.NF.5 Interpret multiplication as scaling (resizing).
- 5.NF.5a Compare 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.5b Explain 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). Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number. Relate the principle of fraction equivalence 𝑎𝑎 𝑏𝑏 = 𝑎𝑎 𝑏𝑏 × 𝑛𝑛 𝑛𝑛 to the effect of multiplying 𝑎𝑎 𝑏𝑏 by 1.
- 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 until grade 6 (NY-6.NS.1).
- 5.NF.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
- 5.NF.7b Interpret division of a whole number by a unit fraction, and compute such quotients.
- 5.NF.7c Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.
Essential Questions and Big Ideas
How do operations with fractions relate to operations with whole numbers?
- Addition, subtraction, multiplication, and division can be completed with fractions.
- Addition and subtraction still represent putting together or taking apart and part and whole relationships.
- Multiplication can represent finding a total made from equal groups or scaling.
- Division represents splitting an amount into equal groups.
What do equivalent fractions represent and why are they useful when solving equations with fractions?
- Equivalent fractions are fractions that are equal or take up the same amount of space.
- To add and subtract fractions, they must have the same denominator or unit size.
How can I find the area of a rectangle with fractional side lengths?
- Length times width equals area.
- Side lengths can be broken up and multiplied using the distributive property.
What are the effects of multiplying by quantities greater than 1 compared to the effects of multiplying by quantities less than 1?
- Multiplying by an amount greater than one creates a product that is greater than the other factor.
- Multiplying by an amount less than one creates a product that is less than the other factor.
What does it mean to divide by a fraction? Or to divide a fraction by a whole number?
- Dividing by a fraction represents splitting a fraction into equal parts.
- Dividing a whole number by a fraction represents splitting a whole into fractional parts.
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