Solved Exercises on Ordering Fractions on Number Lines in Grade 4

Master ordering fractions on number lines: rules for placing and arranging fractions with visual representations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Order fractions with like denominators
Exercise 1
Order: \(\frac{2}{8}\), \(\frac{5}{8}\), \(\frac{1}{8}\), \(\frac{7}{8}\) on a number line
Definition:

Ordering fractions: Arranging fractions from least to greatest (or greatest to least) based on their value

Method for like denominators:
  1. Compare the numerators directly
  2. Arrange fractions based on numerator values
  3. Plot on number line according to their relative positions
Fractions
\(\frac{2}{8}, \frac{5}{8}, \frac{1}{8}, \frac{7}{8}\)
Numerators
\(1, 2, 5, 7\)
Order
\(\frac{1}{8} < \frac{2}{8} < \frac{5}{8} < \frac{7}{8}\)
Step 1: Identify denominators

All fractions have denominator 8, so we compare numerators directly

Step 2: Order numerators

From least to greatest: \(1 < 2 < 5 < 7\)

Step 3: Arrange fractions

Since denominators are equal, \(\frac{1}{8} < \frac{2}{8} < \frac{5}{8} < \frac{7}{8}\)

0
\(\frac{1}{8}\)
\(\frac{2}{8}\)
1/2
\(\frac{5}{8}\)
\(\frac{7}{8}\)
1
\(\frac{1}{8}\)
<
\(\frac{2}{8}\)
<
\(\frac{5}{8}\)
<
\(\frac{7}{8}\)
\(\frac{1}{8} < \frac{2}{8} < \frac{5}{8} < \frac{7}{8}\)
Final answer:

\(\frac{1}{8} < \frac{2}{8} < \frac{5}{8} < \frac{7}{8}\)

Applied rules:

• Like denominators rule: When denominators are equal, order by numerators

• Number line positioning: Greater fractions appear further to the right

• Ascending order: From least to greatest value

2 Order fractions with like numerators
Exercise 2
Order: \(\frac{3}{4}\), \(\frac{3}{8}\), \(\frac{3}{5}\), \(\frac{3}{10}\) on a number line
Definition:

Like numerators ordering: When fractions have the same numerator, the one with the smaller denominator is greater

Fractions
\(\frac{3}{4}, \frac{3}{8}, \frac{3}{5}, \frac{3}{10}\)
Denominators
\(4, 8, 5, 10\)
Order
\(\frac{3}{10} < \frac{3}{8} < \frac{3}{5} < \frac{3}{4}\)
Step 1: Identify numerators

All fractions have numerator 3, so we compare denominators inversely

Step 2: Order denominators (ascending)

From least to greatest: \(4 < 5 < 8 < 10\)

Step 3: Arrange fractions (inverse)

Since smaller denominators mean larger fractions: \(\frac{3}{10} < \frac{3}{8} < \frac{3}{5} < \frac{3}{4}\)

0
\(\frac{3}{10}\)
\(\frac{3}{8}\)
1/2
\(\frac{3}{5}\)
\(\frac{3}{4}\)
1
\(\frac{3}{10}\)
<
\(\frac{3}{8}\)
<
\(\frac{3}{5}\)
<
\(\frac{3}{4}\)
\(\frac{3}{10} < \frac{3}{8} < \frac{3}{5} < \frac{3}{4}\)
Final answer:

\(\frac{3}{10} < \frac{3}{8} < \frac{3}{5} < \frac{3}{4}\)

Applied rules:

• Like numerators rule: When numerators are equal, order by denominators (inverse)

• Denominator effect: Smaller denominator means larger fraction

• Number line positioning: Greater fractions appear further to the right

3 Order mixed fractions
Exercise 3
Order: \(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{1}{4}\), \(\frac{3}{4}\) on a number line
Definition:

Mixed fraction ordering: Ordering fractions with different numerators and denominators requires finding common denominators or converting to decimals

Fractions
\(\frac{1}{2}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}\)
Common Denominator
\(12\)
Order
\(\frac{1}{4} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}\)
Step 1: Find common denominator

LCD of 2, 3, 4 is 12

Step 2: Convert fractions

\(\frac{1}{2} = \frac{6}{12}\), \(\frac{2}{3} = \frac{8}{12}\), \(\frac{1}{4} = \frac{3}{12}\), \(\frac{3}{4} = \frac{9}{12}\)

Step 3: Order converted fractions

\(\frac{3}{12} < \frac{6}{12} < \frac{8}{12} < \frac{9}{12}\), so \(\frac{1}{4} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}\)

0
\(\frac{1}{4}\)
1/2
\(\frac{2}{3}\)
\(\frac{3}{4}\)
1
\(\frac{1}{4}\)
<
\(\frac{1}{2}\)
<
\(\frac{2}{3}\)
<
\(\frac{3}{4}\)
\(\frac{1}{4} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}\)
Final answer:

\(\frac{1}{4} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4}\)

Applied rules:

• Common denominator method: Convert all fractions to same denominator for comparison

• Conversion technique: \(\frac{a}{b} = \frac{a \times n}{b \times n}\) where n is the multiplier

• Number line positioning: Greater fractions appear further to the right

Rules and methods, laws,...
\(\frac{a}{c} < \frac{b}{c} \text{ when } a < b\)
Order Fractions with Like Denominators
Rule 1
\(\frac{a}{c} < \frac{b}{c} \text{ if } a < b\)
When denominators are equal, compare numerators
Rule 2
\(\frac{c}{a} > \frac{c}{b} \text{ if } a < b\)
When numerators are equal, smaller denominator is larger
Rule 3
\(\frac{a}{b} < \frac{c}{d} \text{ if } ad < bc\)
Cross multiplication method for unlike denominators
Key definitions:

Number line: A visual representation of numbers arranged in order from left to right

Ascending order: Arranging numbers from least to greatest

Descending order: Arranging numbers from greatest to least

Like denominators: Fractions with the same denominator

Like numerators: Fractions with the same numerator

Common denominator: A shared denominator among fractions to facilitate comparison

Complete methodology:
  1. Identify fraction types: Determine if fractions have like denominators, like numerators, or are mixed
  2. Select appropriate method: Use direct comparison for like denominators, inverse comparison for like numerators, or conversion for mixed fractions
  3. Perform conversions: Find common denominators or convert to decimals if needed
  4. Order the fractions: Arrange from least to greatest based on converted values
  5. Plot on number line: Place fractions at appropriate positions on the number line
Tip 1: For like denominators, focus only on numerators.
Tip 2: For like numerators, focus only on denominators (inverse relationship).
Tip 3: Use benchmarks (0, 1/2, 1) to estimate fraction positions.
Tip 4: Cross multiplication can be used: \(\frac{a}{b} < \frac{c}{d}\) if \(ad < bc\).
Common errors: Confusing numerator and denominator roles, forgetting that larger denominators mean smaller fractions for like numerators, miscalculating common denominators.
Memory aid: "More parts, more value" for like denominators; "Fewer divisions, bigger pieces" for like numerators.
Rules to know by heart:

• Like denominators: \(\frac{a}{c} < \frac{b}{c}\) when \(a < b\)

• Like numerators: \(\frac{c}{a} < \frac{c}{b}\) when \(a > b\)

• Cross multiplication: \(\frac{a}{b} < \frac{c}{d}\) when \(ad < bc\)

• Number line positioning: Greater values appear to the right

Solution: Exercises 4 to 5
4 Order fractions using cross multiplication
Exercise 4
Order: \(\frac{2}{5}\), \(\frac{3}{7}\), \(\frac{1}{3}\), \(\frac{4}{9}\) on a number line
Definition:

Cross multiplication method: Comparing fractions by multiplying the numerator of one by the denominator of the other

Fractions
\(\frac{2}{5}, \frac{3}{7}, \frac{1}{3}, \frac{4}{9}\)
Cross Products
Compare pairs
Order
\(\frac{1}{3} < \frac{2}{5} < \frac{3}{7} < \frac{4}{9}\)
Step 1: Compare \(\frac{1}{3}\) and \(\frac{2}{5}\)

\(1 \times 5 = 5\) and \(2 \times 3 = 6\), so \(5 < 6\), therefore \(\frac{1}{3} < \frac{2}{5}\)

Step 2: Compare \(\frac{2}{5}\) and \(\frac{3}{7}\)

\(2 \times 7 = 14\) and \(3 \times 5 = 15\), so \(14 < 15\), therefore \(\frac{2}{5} < \frac{3}{7}\)

Step 3: Compare \(\frac{3}{7}\) and \(\frac{4}{9}\)

\(3 \times 9 = 27\) and \(4 \times 7 = 28\), so \(27 < 28\), therefore \(\frac{3}{7} < \frac{4}{9}\)

0
\(\frac{1}{3}\)
\(\frac{2}{5}\)
1/2
\(\frac{3}{7}\)
\(\frac{4}{9}\)
1
\(\frac{1}{3}\)
<
\(\frac{2}{5}\)
<
\(\frac{3}{7}\)
<
\(\frac{4}{9}\)
\(\frac{1}{3} < \frac{2}{5} < \frac{3}{7} < \frac{4}{9}\)
Final answer:

\(\frac{1}{3} < \frac{2}{5} < \frac{3}{7} < \frac{4}{9}\)

Applied rules:

• Cross multiplication: \(\frac{a}{b} < \frac{c}{d}\) when \(ad < bc\)

• Sequential comparison: Compare fractions pairwise to establish order

• Number line positioning: Greater fractions appear further to the right

5 Descending order exercise
Exercise 5
Order: \(\frac{5}{6}\), \(\frac{3}{4}\), \(\frac{7}{8}\), \(\frac{2}{3}\) from greatest to least on a number line
Definition:

Descending order: Arranging fractions from greatest to least value

Fractions
\(\frac{5}{6}, \frac{3}{4}, \frac{7}{8}, \frac{2}{3}\)
Common Denominator
\(24\)
Order
\(\frac{7}{8} > \frac{5}{6} > \frac{3}{4} > \frac{2}{3}\)
Step 1: Find common denominator

LCD of 6, 4, 8, 3 is 24

Step 2: Convert fractions

\(\frac{5}{6} = \frac{20}{24}\), \(\frac{3}{4} = \frac{18}{24}\), \(\frac{7}{8} = \frac{21}{24}\), \(\frac{2}{3} = \frac{16}{24}\)

Step 3: Order converted fractions (descending)

\(\frac{21}{24} > \frac{20}{24} > \frac{18}{24} > \frac{16}{24}\), so \(\frac{7}{8} > \frac{5}{6} > \frac{3}{4} > \frac{2}{3}\)

0
\(\frac{2}{3}\)
\(\frac{3}{4}\)
5/6
\(\frac{7}{8}\)
1
\(\frac{7}{8}\)
>
\(\frac{5}{6}\)
>
\(\frac{3}{4}\)
>
\(\frac{2}{3}\)
\(\frac{7}{8} > \frac{5}{6} > \frac{3}{4} > \frac{2}{3}\)
Final answer:

\(\frac{7}{8} > \frac{5}{6} > \frac{3}{4} > \frac{2}{3}\)

Applied rules:

• Common denominator method: Convert all fractions to same denominator for comparison

• Descending order: Arrange from greatest to least value

• Number line positioning: Greater fractions appear further to the right

Complete Summary: Order Fractions on Number Lines
\(\frac{a}{c} < \frac{b}{c} \text{ when } a < b\)
Like Denominators Rule
\(\frac{c}{a} < \frac{c}{b} \text{ when } a > b\)
Like Numerators Rule
\(\frac{a}{b} < \frac{c}{d} \text{ when } ad < bc\)
Cross Multiplication Rule
Key definitions:

Number line: A straight line where numbers are placed in order from left to right, with equal distances representing equal differences in value.

Ordering fractions: The process of arranging fractions from least to greatest (ascending) or greatest to least (descending) based on their value.

Ascending order: Arranging numbers from smallest to largest: \(\frac{1}{4} < \frac{1}{2} < \frac{3}{4} < 1\)

Descending order: Arranging numbers from largest to smallest: \(1 > \frac{3}{4} > \frac{1}{2} > \frac{1}{4}\)

Like denominators: Fractions that have the same denominator, allowing direct comparison of numerators.

Like numerators: Fractions that have the same numerator, where the fraction with the smaller denominator is greater.

Common denominator: A shared denominator that allows fractions with different denominators to be compared directly.

Complete methodology:
  1. Identify fraction characteristics: Determine if fractions have like denominators, like numerators, or are mixed
  2. Select appropriate ordering method: Choose from direct comparison, inverse comparison, common denominator, or cross multiplication
  3. Convert fractions if needed: Find common denominators or convert to equivalent forms for comparison
  4. Establish the order: Arrange fractions based on their comparative values
  5. Plot on number line: Mark the positions of each fraction on the number line in their correct order
  6. Verify accuracy: Check that the number line positions reflect the established order
Tip 1: For like denominators, simply order the numerators.
Tip 2: For like numerators, order denominators in reverse (smaller denominator = larger fraction).
Tip 3: Use benchmarks like 0, 1/2, and 1 to estimate positions before precise ordering.
Tip 4: Cross multiplication: \(\frac{a}{b} < \frac{c}{d}\) if \(ad < bc\).
Tip 5: Draw the number line first with key benchmarks before placing fractions.
Tip 6: Practice with visual models like pie charts or fraction bars to understand relative sizes.
Common errors: Confusing numerator and denominator roles, forgetting that larger denominators mean smaller fractions when numerators are equal, miscalculating common denominators, misplacing fractions on the number line.
Memory aids: "More parts, more value" for like denominators; "Fewer divisions, bigger pieces" for like numerators; "Cross multiply to compare" for unlike fractions.
Visualization: Draw circle models or bar models to see fraction sizes before ordering.
Real-world connection: Think of sharing different amounts of the same size pizza to understand fraction relationships.
Rules to know by heart:

• Like denominators: \(\frac{a}{c} < \frac{b}{c}\) when \(a < b\)

• Like numerators: \(\frac{c}{a} < \frac{c}{b}\) when \(a > b\)

• Cross multiplication: \(\frac{a}{b} < \frac{c}{d}\) when \(ad < bc\)

• Common denominator: Convert fractions to have the same denominator, then compare numerators

• Number line positioning: Greater values appear to the right of lesser values

• Transitive property: If \(\frac{a}{b} < \frac{c}{d}\) and \(\frac{c}{d} < \frac{e}{f}\), then \(\frac{a}{b} < \frac{e}{f}\)

Fraction Ordering Framework

📊
Like Denominators Method

\(\frac{2}{7} < \frac{5}{7}\) because \(2 < 5\)

Compare numerators directly when denominators are equal

2
5
Like Numerators Method

\(\frac{3}{4} > \frac{3}{7}\) because \(4 < 7\)

Smaller denominator means larger fraction when numerators are equal

3/4
3/7
Cross Multiplication Method

\(\frac{2}{3} < \frac{3}{4}\) because \(2 \times 4 < 3 \times 3\)

\(8 < 9\), so \(\frac{2}{3} < \frac{3}{4}\)

2×4
3×3
Fraction Ordering Workflow
1
Identify Type
2
Select Method
3
Convert
4
Order
5
Plot
Number Line Positioning

Fractions positioned according to their relative values on the number line

Master These Ordering Techniques!

Questions & Answers

Question: I'm confused about why \(\frac{3}{4}\) is greater than \(\frac{3}{7}\) when 4 is smaller than 7. Shouldn't the fraction with the bigger denominator be larger?

Answer: This is a very common source of confusion! The key is understanding what the denominator represents:

  • The denominator tells us how many equal pieces the whole is divided into
  • \(\frac{3}{4}\) means 3 pieces from a whole divided into 4 equal parts
  • \(\frac{3}{7}\) means 3 pieces from a whole divided into 7 equal parts

When you divide a pizza into 4 pieces, each piece is much bigger than when you divide the same pizza into 7 pieces. So 3 big pieces (fourths) are more than 3 small pieces (sevenths).

Think of it this way: If you had to share a cake with 3 friends (making 4 pieces total) versus sharing with 6 friends (making 7 pieces total), you'd get a bigger slice when there are fewer people to share with!

The rule for like numerators is: when numerators are the same, the fraction with the smaller denominator is larger.

Question: How do I find the common denominator when ordering fractions like \(\frac{2}{3}\) and \(\frac{5}{8}\)?

Answer: Finding a common denominator involves finding a number that both denominators can divide into evenly. Here's how:

  • List multiples: Write out multiples of each denominator until you find a common one
  • For 3: 3, 6, 9, 12, 15, 18, 21, 24, 27...
  • For 8: 8, 16, 24, 32, 40...
  • First common multiple is 24, so LCD = 24

Now convert each fraction:

  • \(\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}\)
  • \(\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}\)

Since \(\frac{16}{24} > \frac{15}{24}\), we have \(\frac{2}{3} > \frac{5}{8}\).

Alternative method: Multiply the denominators together (3 × 8 = 24), though this may not give you the least common denominator.

Question: What is cross multiplication and how does it help order fractions?

Answer: Cross multiplication is a method to compare two fractions without finding a common denominator. Here's how it works:

To compare \(\frac{a}{b}\) and \(\frac{c}{d}\):

  • Multiply the numerator of the first fraction by the denominator of the second: \(a \times d\)
  • Multiply the numerator of the second fraction by the denominator of the first: \(c \times b\)
  • Compare the results: if \(a \times d < c \times b\), then \(\frac{a}{b} < \frac{c}{d}\)

Example: Compare \(\frac{2}{5}\) and \(\frac{3}{7}\):

  • \(2 \times 7 = 14\)
  • \(3 \times 5 = 15\)
  • Since \(14 < 15\), we have \(\frac{2}{5} < \frac{3}{7}\)

This method is especially useful when dealing with large denominators where finding a common denominator would be difficult.