Teacher Guide: Equivalent Ratios
Learn how to find and create ratios that represent the same relationship.
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10 questions on Ratios. Students join with a name, you see everyone's score.
For Teachers
- Identify equivalent ratios using multiplication and division
- Generate equivalent ratios from a given ratio
- Determine if two ratios are equivalent by simplifying
- Complete ratio tables with missing values
- Apply equivalent ratios to solve real-world problems
- • Understanding of what a ratio represents
- • Multiplication and division of whole numbers
- • Concept of equivalent fractions
- • Finding greatest common factor (GCF)
- 1. If you double both parts of a ratio, does the ratio change? Why or why not?
- 2. A store sells 3 apples for 2 dollars. What are some equivalent ratios for this? How could you use them?
- 3. Why does adding the same number to both parts NOT give an equivalent ratio?
- 4. How could you use equivalent ratios to compare prices at two different stores?
Adding the same number to both parts creates an equivalent ratio
Larger numbers mean the ratio is bigger
You can only multiply, not divide, to find equivalent ratios
For Struggling Students:
- • Use physical manipulatives (colored counters) to show ratios
- • Start with simple ratios like 1:2 and 2:4
- • Provide partially completed ratio tables
- • Use visual tape diagrams to show equivalent relationships
For On-Level Students:
- • Complete ratio tables with multiple missing values
- • Determine if given ratios are equivalent
- • Solve word problems involving equivalent ratios
- • Create their own equivalent ratio problems
For Advanced Students:
- • Work with three-part ratios (2:3:5)
- • Explore ratios with fractions or decimals
- • Solve complex scaling problems (triple the recipe, then halve it)
- • Connect equivalent ratios to proportional reasoning
- 6.RP.A.1 (CCSS.MATH.CONTENT.6.RP.A.1)
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities
- 6.RP.A.3 (CCSS.MATH.CONTENT.6.RP.A.3)
Use ratio and rate reasoning to solve real-world and mathematical problems
- 6.RP.A.3.A (CCSS.MATH.CONTENT.6.RP.A.3.A)
Make tables of equivalent ratios relating quantities with whole number measurements
- visualInteractive Ratio Table
Students fill in missing values in ratio tables
- activityRecipe Scaling Game
Adjust recipe ingredients for different serving sizes
- worksheetEquivalent Ratios Practice
Find missing values and determine equivalence
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
Worked Examples
Find three ratios equivalent to .
Multiply both parts by 2
and →
Multiply both parts by 3
and →
Multiply both parts by 4
and →
Answer: Three equivalent ratios are , , and
Common Mistakes
Adding the same number to both parts instead of multiplying
Why it's wrong: Adding changes the ratio's value. If becomes (adding 2), these are NOT equivalent.
Correct: Always multiply or divide both parts by the SAME number. (multiplying by 2).
Multiplying only one part of the ratio
Why it's wrong: To keep ratios equivalent, you must apply the same operation to BOTH parts.
Correct: , not or .
Thinking larger numbers mean larger ratios
Why it's wrong: looks bigger than , but they represent the same relationship.
Correct: The size of the numbers does not determine the ratio's value. Always simplify to compare.
Why It Matters
- Cooking: If a recipe calls for 2 cups of flour to 3 cups of sugar, you can double it to 4 cups of flour and 6 cups of sugar
- Maps: A scale of 1:100 means 1 cm on the map equals 100 cm in real life
- Mixing: If paint requires a 3:1 ratio of white to blue, you need to keep this ratio no matter how much paint you make
- Shopping: Comparing unit prices uses equivalent ratios to find the best deal
Real World Applications
Scaling Recipes
Chefs use equivalent ratios to adjust recipes for different numbers of servings.
Example:
A recipe for 4 people uses 3 eggs and 2 cups of flour. For 8 people, use 6 eggs and 4 cups of flour (multiply by 2).
A smoothie recipe uses 2 bananas for every 3 cups of milk. You want to make enough for 6 bananas.
How many cups of milk do you need?
Step 1: Write the mathematical expression
If 2 bananas need 3 cups, find the multiplier:
Map Scales
Maps use ratios to represent real distances. A scale of 1:50,000 means 1 cm on the map equals 50,000 cm (or 500 m) in reality.
Example:
On a 1:100 scale map, 5 cm represents 500 cm (5 meters) in real life.
A map has a scale of 1:25,000. Two cities are 8 cm apart on the map.
What is the real distance in kilometers?
Step 1: Write the mathematical expression
Calculate the real distance:
Mixing Paint
Artists mix colors using specific ratios to get consistent shades.
Example:
To make light green, mix blue and yellow in a 1:4 ratio. For 20 ml total, use 4 ml blue and 16 ml yellow.
A shade of orange requires red and yellow paint in a 2:5 ratio. You have 10 ml of red paint.
How much yellow paint do you need?
Step 1: Write the mathematical expression
Find the equivalent ratio:
Key Takeaways
- 1Equivalent ratios express the same relationship between quantities
- 2To create equivalent ratios, multiply or divide BOTH parts by the same number
- 3You can check if ratios are equivalent by simplifying them to lowest terms
- 4Ratio tables help organize and find multiple equivalent ratios
- 5Never add or subtract to find equivalent ratios - only multiply or divide
Frequently Asked Questions
How are equivalent ratios different from equivalent fractions?
Can ratios have decimals?
What is the simplest form of a ratio?
Glossary
- Equivalent ratios
- Two or more ratios that represent the same relationship between quantities
- Ratio table
- A table showing pairs of numbers that all form equivalent ratios
- Simplest form
- A ratio where both terms have no common factors other than 1
- Scale factor
- The number you multiply both parts of a ratio by to get an equivalent ratio