Multiplying Fractions Notes

Units 19-22: Multiplying Fractions

5th Grade Math

Math Vocabulary

  • Unit Fraction: A fraction with a numerator of 1. Other fractions are constructed from unit fractions.

    • Examples include:

    • 12\frac{1}{2}

    • 14\frac{1}{4}

    • 13\frac{1}{3}

    • 18\frac{1}{8}

  • Scaling: The process of changing the size (either enlarging or shrinking) of a quantity, shape, or object by multiplying its dimensions by a constant value referred to as a scale factor.

Back to the Basics

Multiplying a Fraction by a Whole Number

  • Before multiplying fractions by another fraction, review how to multiply a fraction by a whole number.

  • Example Problem: 5×135 \times \frac{1}{3}

Steps to Multiply a Fraction by a Whole Number

  1. Rewriting the Whole Number as a Fraction:

    • Begin by expressing the whole number as a fraction over 1.

    1. Example: 5=515 = \frac{5}{1}

  2. Multiply Straight Across:

    • Multiply the numerators and the denominators together.

    • From the equation 51×13\frac{5}{1} \times \frac{1}{3}, multiply:

      • Numerators: 5×1=55 \times 1 = 5

      • Denominators: 1×3=31 \times 3 = 3

    • The result is 53\frac{5}{3}.

  1. Simplify the Fraction:

    • If the result is an improper fraction, convert it to a mixed number to simplify.

    • For example: 53=123\frac{5}{3} = 1 \frac{2}{3}

Examples of Multiplying Fractions by Whole Numbers

  • 6×346 \times \frac{3}{4}

  • 3×453 \times \frac{4}{5}

  • 2×572 \times \frac{5}{7}

  • 4×264 \times \frac{2}{6}

Modeling It

  • Visual representation can also aid in understanding multiplication:

    • Example: For 3×123 \times \frac{1}{2}, draw three rectangles and label each as 12\frac{1}{2}. This visual method can be helpful for those who prefer visuals.

Simplifying Fractions

  • Simplifying fractions helps in finding the smallest equivalent fraction.

    • Step 1: Find the Greatest Common Factor (GCF) for the numerator and the denominator.

    • Example: For 612\frac{6}{12}, the GCF is 6.

    • Step 2: Divide the numerator and the denominator by the GCF.

    • 612→6÷612÷6=12\frac{6}{12} \rightarrow \frac{6 \div 6}{12 \div 6} = \frac{1}{2}

  • When fractions are simplified, they reflect equivalent fractions, such as 612\frac{6}{12} and 12\frac{1}{2}. Both represent equal quantities, with 6 being half of 12 and 1 being half of 2.

More Examples of Simplifying Fractions

  • Simplify:

    • 520\frac{5}{20}

    • 1218\frac{12}{18}

    • 220\frac{2}{20}

    • Final Example: Simplify 1228\frac{12}{28}.

  • Note: Not all fractions require simplification.

    • Examples of prime numbers that do not yield further simplification include:

    • 7, 17, 19, 13, 23, 29

Multiplying Fractions by Fractions

  • The procedure to multiply fractions by fractions mimics that of multiplying a fraction by a whole number.

  • Example Problem: 24×57\frac{2}{4} \times \frac{5}{7}

Steps for Multiplying Fractions

  1. Multiply Straight Across:

    • For 24×57\frac{2}{4} \times \frac{5}{7}:

      • Numerators: 2×5=102 \times 5 = 10

      • Denominators: 4×7=284 \times 7 = 28

    • Result: 1028\frac{10}{28}

  2. Simplification Check:

    • Check if simplification is possible (if prime numbers are involved, simplification isn’t necessary).

  • Practice Examples:

    • 12×45\frac{1}{2} \times \frac{4}{5}

    • 24×78\frac{2}{4} \times \frac{7}{8}

    • 25×12\frac{2}{5} \times \frac{1}{2}

Applications to Area

  • Understanding fractional multiplication aids in computing area. The formula for area of rectangles/squares is used often.

  • Example Word Problem:

    • Mr. Morales designs a square park with a side length of 1 mile. He includes a dog play space with sides measuring 510\frac{5}{10} miles. What is the area designated for the dog play area?

  • Further Examples:

    • Tamera has a rectangular block measuring 12\frac{1}{2} inch long and 14\frac{1}{4} inch wide.

    • Area Calculation: 12×14=18\frac{1}{2} \times \frac{1}{4} = \frac{1}{8} square inches.

    • Calculate the area for a postage stamp measuring 32\frac{3}{2} inches long and 34\frac{3}{4} inches wide.

    • Alex's rug dimensions are 56\frac{5}{6} yard long and 23\frac{2}{3} yard wide.

    • Rashon's poster is 74\frac{7}{4} yards by 23\frac{2}{3} yard.

    • A square's area is calculated with side lengths measuring 68\frac{6}{8} inch.

Multiplication as Scaling

Concept of Scaling

  • Scaling: Can be viewed as either stretching or shrinking.

    • Stretching: When multiplying by a whole number, the object or number increases in size.

    • Shrinking: When multiplying by a fraction, the object or number decreases in size.

Stretching Examples

  1. Take 3 squares and use a scale factor of 2:

    • 3×2=63 \times 2 = 6 (the shape doubles in size).

  2. For stretching 4 circles using a scale factor of 3:

    • 4×3=124 \times 3 = 12 (the quantity increases).

Numbers in Scaling

  • Scaling is not limited to shapes; it can also apply to numbers.

    • Example: Scaling the number 6 with a scale factor of 4 results in:

    • 6×4=246 \times 4 = 24

  • Exercises for Stretching Numbers:

    • Scale 5 by 7: 5×7=?5 \times 7 = ?

    • Scale 12 by 5: 12×5=?12 \times 5 = ?

    • Scale 26 by 8: 26×8=?26 \times 8 = ?

Stretching with Fractions

  • Yes, fractions can also be stretched!

    • Key Consideration:

    • Whole number scale factor = stretching/getting bigger.

    • Fraction scale factor = shrinking/getting smaller.

  • Collaborative Exercises:

    • Scale 14\frac{1}{4} by 5: 14×5=?\frac{1}{4} \times 5 = ?

    • Scale 25\frac{2}{5} by 3: 25×3=?\frac{2}{5} \times 3 = ?

    • Scale 37\frac{3}{7} by 2: 37×2=?\frac{3}{7} \times 2 = ?

Finding Missing Scale Factors

  • To determine an unknown scale factor:

    • If given a starting number (e.g., 4) and an end number (e.g., 24), divide to find the scale factor:

    • 244=6\frac{24}{4} = 6

  • Exercise Examples for Missing Variables:

    • Starting number: 3, end number: 21; Scale factor: 213=?\frac{21}{3} = ?

    • Starting number: 5, end number: 55; Scale factor: 555=?\frac{55}{5} = ?

    • Starting number: 6, end number: 36; Scale factor: 366=?\frac{36}{6} = ?

Shrinking with Fractions

  • Shrinking a number involves using a fractional scale factor to yield a smaller number.

    • Example: For 3 circles and a scale factor of 13\frac{1}{3}:

    • Calculate: 3×13=?3 \times \frac{1}{3} = ?

  • More Shrinking Examples:

    • 4 with a scale factor of 58\frac{5}{8}: Calculate 4 x 58\frac{5}{8}.

    • 12 with a scale factor of 23\frac{2}{3}: Calculate 12 x 23\frac{2}{3}.

    • 15 with a scale factor of 45\frac{4}{5}: Calculate 15 x 45\frac{4}{5}.

Word Problems

Example Problems

  • Jabilo's Walk: Jabilo lives 45\frac{4}{5} mile from the park. He walks 34\frac{3}{4} of the way to the park. To find the distance he walked: 45×34\frac{4}{5} \times \frac{3}{4}.

  • Lorenzo's Frittata: Lorenzo’s dad leaves 34\frac{3}{4} of a frittata. If he eats 23\frac{2}{3} of it, how much of the whole does he consume? 34×23\frac{3}{4} \times \frac{2}{3}.

  • Luis and Tiana's Walk: Luis walks 810\frac{8}{10} mile. Tiana walks 34\frac{3}{4} of Luis’s distance. How far did Tiana walk?

  • Mei's Garden: Mei has a rectangular garden with dimensions 2342 \frac{3}{4} yards long and 1 yard wide. Calculate how many square yards contain flowers: 12\frac{1}{2} of the garden.

  • Lisa's Audiobook: Lisa's audiobook is 4124 \frac{1}{2} hours long. If she aims to finish 13\frac{1}{3} this week, the hours needed: 4.53\frac{4.5}{3}.

  • Cameron's Fabric: Cameron has 1591 \frac{5}{9} yards of fabric and plans to use 34\frac{3}{4} for a pillow. Calculate the fabric needed: 159×341 \frac{5}{9} \times \frac{3}{4}.