4.1 Visualize Fractions

Fractions are like slices of pizza – they represent parts of a whole. We'll learn how to divide things into equal parts, compare different-sized slices, and even deal with more than one whole pizza at a time.

We'll also explore how fractions line up on a number line and how to find different ways to represent the same amount. This helps us compare and order fractions, which is super useful in everyday life, from cooking to sharing things fairly.

Understanding Fractions

Fractions as parts of wholes

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  Using Models to Represent Fractions and Mixed Numbers | Prealgebra View original

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  Using Models to Represent Fractions and Mixed Numbers | Prealgebra View original

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• Represents a portion or part of a whole object or group of objects (pizza slice, piece of pie)
• Denominator specifies the total number of equal parts the whole is divided into (cutting a cake into 8 equal slices)
• Numerator indicates the number of parts being considered or taken from the whole (taking 3 slices out of the 8)
• A unit fraction has a numerator of 1 and represents one part of a whole divided into equal parts (1/4, 1/8)

Visual models for improper fractions

• Numerator is greater than or equal to the denominator represents a value greater than or equal to 1 (5/4, 3/3)
• Visualized as multiple wholes and/or parts of a whole combined (2 whole pizzas and 1/4 of another pizza)
• Mixed numbers represent a value greater than 1 combination of whole objects and parts of a whole (1 1/2 cakes, 2 3/4 pies)

Conversion of improper fractions

1. Divide the numerator by the denominator (17 ÷ 5 = 3 with remainder 2)
2. Quotient becomes the whole number part (3)
3. Remainder becomes the numerator of the fractional part (2)
4. Denominator remains the same (5)
5. Result: 3 2/5

To convert a mixed number to an improper fraction:

1. Multiply the whole number by the denominator and add the numerator (3 × 5 + 2 = 17)
2. Write the result as the numerator of the improper fraction (17)
3. Keep the original denominator (5)
4. Result: 17/5

Equivalent Fractions and Number Lines

Diagrams for equivalent fractions

• Represent the same value or proportion of a whole (1/2 of a pizza = 2/4 of a pizza)
• Different numerators and denominators but represent the same part of a whole (3/6 of a cake = 1/2 of a cake)
• Diagrams or drawings show equivalent fractions (1/2, 2/4, and 3/6 as equal parts of a rectangle or circle)

Generation of equivalent fractions

• Multiply or divide both the numerator and denominator by the same non-zero number ($\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$)
• Proportion of the whole remains the same when both parts are multiplied or divided by the same value ($\frac{3}{4} = \frac{3 \div 3}{4 \div 3} = \frac{1}{4/3}$)

Number line representation of fractions

• Visual representation of numbers in order
• Divide the space between 0 and 1 into equal parts based on the denominator (split into 4 parts for quarters)
• Locate the position of the fraction based on the numerator (3/4 is at the third mark out of four)
• For mixed numbers:
  1. Locate the whole number part on the number line (2 for 2 3/8)
  2. Divide the space between the whole number and the next whole number into equal parts based on the denominator (split into 8 parts)
  3. Locate the position by moving the appropriate number of parts to the right based on the numerator (move 3 parts to the right of 2)
• Benchmark fractions (such as 1/2, 1/4, 3/4) serve as reference points on the number line for comparing and estimating other fractions

Comparison and sequencing of fractions

• Compare fractions with the same denominator by comparing their numerators larger numerator is greater (5/8 > 3/8)
• Compare fractions with different denominators by finding a common denominator using equivalent fractions (1/3 < 3/8 because 8/24 < 9/24)
• Compare mixed numbers by first comparing the whole number parts if equal, then compare the fractional parts (2 3/4 > 2 1/2)
• Sequence fractions and mixed numbers by arranging them from the smallest to the largest value (1/4, 1/2, 3/4, 1 1/8, 1 1/2)

Fractions and Ratios

• Fractions can represent ratios, comparing quantities or parts of a whole
• A ratio compares two quantities and can be written as a fraction (3:4 can be written as 3/4)
• Ratios can be simplified like fractions to find equivalent ratios (6:8 simplifies to 3:4)
• Common denominators are used when comparing ratios, similar to comparing fractions