# 6.4 and *6.5a - Multiply and Divide Fractions

6.4   The student will demonstrate multiple representations of multiplication and division of fractions.

6.5   The student will

a)  multiply and divide fractions and mixed numbers;*  (with NO CALCULATOR)

Bloom's Level:  Apply

### BIG IDEAS

• If I have 3/4 of a pizza I will be able to evenly share it with 4 friends, I am able to increase a recipe to produce 12 dozen cookies, and if I have 16 yards of ribbon I can determine how many bows I can make if each requires 3/4 yd.
• I will better understand the concept of multiplication and division of fractions as I move into more abstract applications.

### UNDERSTANDING THE STANDARD

• Using manipulatives to build conceptual understanding and using pictures and sketches to link concrete examples to the symbolic enhance students’ understanding of operations with fractions and help students connect the meaning of whole number computation to fraction computation.
• Multiplication and division of fractions can be represented with arrays, paper folding, repeated addition, repeated subtraction, fraction strips, pattern blocks and area models.
• When multiplying a whole by a fraction such as  3 x 1/2, the meaning is the same as with multiplication of whole numbers: 3 groups the size of 1/2 of the whole.
• When multiplying a fraction by a fraction such as 2/3 ⋅ 3/4, we are asking for part of a part.
• When multiplying a fraction by a whole number such as 1/2 x 6, we are trying to find a part of the whole.
• For measurement division, the divisor is the number of groups. You want to know how many are in each of those groups. Division of fractions can be explained as how many of a given divisor are needed to equal the given dividend. In other words, for 1/4 ÷ 2/3, the question is, “How many 2/3 make 1/4?”
• For partition division the divisor is the size of the group, so the quotient answers the question, “How much is the whole?” or “How much for one?”

·  The symbol  can be used in grade six in place of “x” to indicate multiplication.

·  In exponential notation, the base is the number that is multiplied, and the exponent represents the number of times the base is used as a factor. In , 8 is the base and 3 is the exponent (e.g., ).

·  Any real number other than zero raised to the zero power is 1. Zero to the zero power () is undefined.

·  A perfect square is a whole number whose square root is an integer (e.g., ). Zero (a whole number) is a perfect square.

·  Perfect squares may be represented geometrically as the areas of squares the length of whose sides are whole numbers (e.g., , etc.). This can be modeled with grid paper, tiles, geoboards and virtual manipulatives.

·  The examination of patterns in place value of the powers of 10 in grade six leads to the development of scientific notation in grade seven.

### ESSENTIALS

• When multiplying fractions, what is the meaning of the operation?
When multiplying a whole by a fraction such as 3 x 1/2, the meaning is the same as with multiplication of whole numbers: 3 groups the size of 1/2 of the whole.
When multiplying a fraction by a fraction such as 2/3 ⋅ 3/4, we are asking for part of a part.
When multiplying a fraction by a whole number such as 1/2 x 6, we are trying to find a part of the whole.
• What does it mean to divide with fractions?
For measurement division, the divisor is the number of groups and the quotient will be the number of groups in the dividend. Division of fractions can be explained as how many of a given divisor are needed to equal the given dividend.  In other words, for  the question is, “How many for 1/4 ÷ 2/3, the question is, “How many 2/3 make 1/4?”
For partition division the divisor is the size of the group, so the quotient answers the question, “How much is the whole?” or “How much for one?”

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

6.41  Demonstrate multiplication and division of fractions using multiple representations.

6.42  Model algorithms for multiplying and dividing with fractions using appropriate representations.

·  6.5a1  Demonstrate/model multiplication and division of fractions (proper or improper) and mixed numbers using multiple representations.

### KEY VOCABULARY

multiplication, division, product, quotient, dividend, divisor, remainder, simplest form, fraction, mixed numbers, numerator, denominator, reciprocal, inverse operations

Updated: Oct 27, 2017