Teacher Guide: Comparing Fractions by Cross Multiplication
Learn the cross multiplication method to quickly compare any two fractions, even with different denominators.
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Class quiz
10 questions on Comparing & Ordering. Students join with a name, you see everyone's score.
For Teachers
- Apply cross multiplication to compare two fractions
- Determine which of two fractions is greater, lesser, or if they are equal
- Explain why cross multiplication works mathematically
- Use cross multiplication in real-world comparison scenarios
- • Understanding of fractions as parts of a whole
- • Basic multiplication of whole numbers
- • Familiarity with inequality symbols (<, >, =)
- 1. Why might cross multiplication be faster than finding a common denominator?
- 2. Can you think of a time when you needed to compare two quantities that were not in the same units?
- 3. If two fractions cross multiply to give equal products, what does that tell you?
- 4. How could you use cross multiplication when shopping?
The cross products are the values of the fractions
Cross multiplication only works with proper fractions
For Struggling Students:
- • Use visual arrow diagrams showing the cross pattern
- • Start with fractions where one denominator divides evenly into the other
- • Provide multiplication tables for reference
For On-Level Students:
- • Compare fractions with two-digit numbers in numerators or denominators
- • Apply to word problems involving rates or unit prices
- • Verify answers by converting to decimals
For Advanced Students:
- • Compare three or more fractions by pairwise cross multiplication
- • Explore algebraic proofs of why cross multiplication works
- • Create real-world problems for classmates to solve
- 5.NF.A.1 (CCSS.MATH.CONTENT.5.NF.A.1)
Add and subtract fractions with unlike denominators by replacing given fractions with equivalent fractions
- 5.NF.A.2 (CCSS.MATH.CONTENT.5.NF.A.2)
Solve word problems involving addition and subtraction of fractions referring to the same whole
- 6.NS.C.7 (CCSS.MATH.CONTENT.6.NS.C.7)
Understand ordering and absolute value of rational numbers
- visualCross Multiplication Diagram
Interactive arrows showing the cross pattern
- activityFraction Face-Off
Students compete to quickly compare fractions using cross multiplication
- worksheetCross Multiply Practice
20 fraction comparison problems with increasing difficulty
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Definition
- - If , then
- - If , then
- - If , then
Worked Examples
Which is greater: or ?
Set up the cross multiplication
vs → Ready to cross multiply
Multiply first numerator by second denominator
→ Left product:
Multiply second numerator by first denominator
→ Right product:
Compare the products
, so → is greater
Answer:
Common Mistakes
Multiplying numerator by numerator instead of cross multiplying
Why it's wrong: Students sometimes confuse cross multiplication with regular multiplication. Cross multiplication means diagonally: numerator of first with denominator of second.
Correct: Always multiply across: numerator of one fraction times denominator of the other.
Mixing up which product corresponds to which fraction
Why it's wrong: After calculating both products, students forget which product belongs to which fraction.
Correct: The product using the first fraction's numerator tells you about the first fraction. Write the products directly below or beside each fraction.
Thinking cross multiplication gives the actual fraction value
Why it's wrong: Students may think the products are the fractions' values. They are not; they are only useful for comparison.
Correct: Cross multiplication tells you which fraction is larger, not what the fractions equal as decimals or in lowest terms.
Why It Matters
- Speed: No need to find the least common denominator
- Accuracy: Works for any two fractions, even with large or unfriendly denominators
- Foundation: The same technique is used later to solve proportions and equations
- Comparing prices per unit (which deal is better?)
- Determining which recipe uses more of an ingredient proportionally
- Checking if two ratios are equivalent
Real World Applications
Shopping Comparisons
Stores often display different package sizes at different prices. Cross multiplication helps find the better deal.
Example:
If a 3-pack costs 8 euros and a 5-pack costs 12 euros, compare and (items per euro). Cross multiply: vs . Since , you get more items per euro with the 5-pack.
A bakery sells 4 muffins for 6 euros or 7 muffins for 10 euros.
Which deal gives more muffins per euro?
Step 1: Write the mathematical expression
Compare and :
Recipe Scaling
When comparing recipes, cross multiplication helps determine which uses more of an ingredient proportionally.
Example:
Recipe A uses cup of sugar for 12 cookies. Recipe B uses cup for 12 cookies. Which is sweeter? Cross multiply: vs . Since , Recipe A uses more sugar.
Recipe A uses cup of flour. Recipe B uses cup of flour.
Which recipe uses more flour?
Step 1: Write the mathematical expression
Cross multiply to compare:
Key Takeaways
- 1Cross multiplication compares fractions by multiplying each numerator by the opposite denominator
- 2For vs : compare with
- 3The fraction whose numerator gives the larger product is the greater fraction
- 4If both products are equal, the fractions are equivalent
- 5This method works for any fractions, regardless of their denominators
Frequently Asked Questions
Why does cross multiplication work?
Can I use cross multiplication with more than two fractions?
Does the order matter?
Glossary
- Cross multiplication
- A method to compare fractions by multiplying each numerator by the opposite fraction's denominator
- Numerator
- The top number in a fraction, representing parts taken
- Denominator
- The bottom number in a fraction, representing total equal parts
- Equivalent fractions
- Fractions that represent the same value, such as and