NCTM Standards

Understand numbers, ways of representing numbers, relationships among numbers and number systems.

In grades 6-8

• All students should work flexibly with fractions, decimals, and percents to solve problems;

Understand the meaning of operations and how they relate to one another.

In grades 6-8

• All students should understand the meaning and effects of arithmetic operations with fractions, decimals, and integers;

Compute fluently and make reasonable estimates

In grades 6-8

• All student should select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods;
• All students should develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use;

NCTM Focal Points

Grade 6: Students use the meanings of fractions, multiplication and division, and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions and explain why they work. Students use common procedures to multiply and divide fractions and decimals efficiently and accurately. They multiply and divide fractions and decimals to solve problems, including multistep problems and problems involving measurement.

Grade 4 - Number and Operations – Fractions

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

4.NF.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?

Grade 5 – Number and Operations - Fractions:

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

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 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.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.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.

Vocabulary

product, multiplier, starting value, paraphrase, unit, denominator, numerator, area, convert, simplify, parts of, groups of

IEP Goals (sample)

• Given a multiplication story problem containing at least one proper fraction, the student will identify the starting value and multiplier and then use models to represent and solving the problem accurately for 5 out of 6 examples for 5 consecutive sessions.

• Given a multiplication story problem and a paraphrasing strategy, the student will paraphrase the problem, write an equation that illustrates the problem, find the product and simplify the answer for 5 out 6 examples by the end of the first marking period.

• Given a story problem using numbers with common factors the student will represent the problem as both a paraphrase and an equation, find the product, simplify the answer and identify the units without the use of models with 90% accuracy by the completion of this IEP.

• Given a multiplication equation with a variety of number combinations, (i.e. proper fractions, mixed numbers, whole numbers), the student will apply previous learned strategies to solve the problem and explain his thinking with drawings or models for 9 out of 10 examples in 3 consecutive sessions.