Lesson Plan: Introduction to Ratios 

Lesson Summary

In this lesson, students are introduced to the fundamental concept of ratios, which are used to compare two or more quantities. They will learn to define and identify ratios, express them in various forms (such as a:b, a to b, and a/b), and understand the relationships between the quantities involved. Through practical examples and interactive activities, students will explore part-to-part and part-to-whole ratios, as well as the concept of equivalent ratios. This foundational knowledge will prepare them for more advanced topics in proportional reasoning and problem-solving.

Lesson Objectives

• Define and identify ratios
• Express ratios in different forms (a:b, a to b, a/b)
• Understand the relationship between quantities in a ratio

Common Core Standards

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.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

Prerequisite Skills

• Basic fraction knowledge
• Understanding of division
• Plotting points on a coordinate plane

Key Vocabulary

• Ratio: A comparison between two or more quantities, indicating how many times one quantity contains or is contained within the other(s). For example, the ratio 3:2 compares 3 parts of one quantity to 2 parts of another.
  • Multimedia Resource: https://www.media4math.com/library/22157/asset-preview
• Part-to-Part Ratio: A ratio that compares different parts of a whole to each other. For instance, in a fruit basket with 4 apples and 6 oranges, the part-to-part ratio of apples to oranges is 4:6.
  • Multimedia Resource: https://www.media4math.com/library/43389/asset-preview
• Part-to-Whole Ratio: A ratio that compares one part of a whole to the entire whole. Using the same fruit basket example, the part-to-whole ratio of apples to total fruits is 4:10.
  • Multimedia Ratio: https://www.media4math.com/library/43388/asset-preview
• Equivalent Ratios: Ratios that express the same relationship between quantities, even though the numbers may be different. For example, the ratios 2:3 and 4:6 are equivalent because they represent the same proportional relationship.
  • Multimedia Ratio: https://www.media4math.com/library/43384/asset-preview
• Proportion: An equation that states that two ratios are equal. For example, 2/3 = 4/6 is a proportion showing that the two ratios are equivalent.
  • Multimedia Resource: https://www.media4math.com/library/43387/asset-preview
• Unit Rate: A ratio that compares a quantity to one unit of another quantity. For example, if a car travels 300 miles in 5 hours, the unit rate is 60 miles per hour.
  • Multimedia Resource: https://www.media4math.com/library/43395/asset-preview
• Rate: A specific type of ratio that compares two quantities with different units, such as miles per hour or price per item.
  • Multimedia Resource: https://www.media4math.com/library/22156/asset-preview

Multimedia Resources

• A collection of definitions on the topic of ratios, proportions, and percents: https://www.media4math.com/Definitions--RatiosProportionsPercents
• A student tutorial slide show on definitions on the topic of ratios, proportions, and percents: https://www.media4math.com/library/slideshow/student-tutorial-ratios-proportions-and-percents-definitions

Warm Up Activities

Choose from one or more of these activities.

Activity 1: Sorting

This discussion-based activity helps students explore how numbers can be used to describe relationships between objects, leading into the concept of ratios.

Instructions:

1. Display a collection of images or physical examples of different types of sports balls, such as:
   • 4 basketballs
   • 6 soccer balls
   • 2 footballs
   • 3 baseballs
2. Ask students: "How can we organize or categorize these balls using numbers?"
3. Guide the discussion by prompting:
   • How many basketballs are there compared to soccer balls?
   • What fraction of the total collection are footballs?
   • How might we express the number of basketballs relative to all other balls?
4. Introduce the idea that ratios allow us to compare different quantities, such as basketballs to soccer balls (4:6) or footballs to the total (2:15).
5. Conclude by explaining that ratios, like fractions, help describe relationships between numbers in a meaningful way.

Activity 2: Modeling Fractions

Before introducing ratios, students will review fractions by engaging in a hands-on activity that models fractional amounts using everyday objects.

Instructions:

1. Provide each student or group with a set of objects, such as:
   • A pizza cut into equal slices
   • A set of colored blocks (e.g., 3 red, 5 blue, and 2 yellow)
   • A collection of fruit (e.g., 6 apples and 3 oranges)
2. Ask students to determine the fractional amount of one type of object compared to the whole. For example:
   • If a pizza has 8 slices and 3 have been eaten, what fraction represents the uneaten slices?
   • If there are 3 red, 5 blue, and 2 yellow blocks, what fraction of the blocks are red?
3. Discuss how fractions compare part of a whole, while ratios compare different quantities, setting the stage for the ratio lesson.

Activity 3: Simple Ratios and Fractions

Display a group of 3 red balls and 2 yellow balls on the board or using a digital presentation. Use this companion slide show:

https://www.media4math.com/library/slideshow/simple-ratios

Ask students to:

1. Calculate the fraction of red balls (3/5)
2. Calculate the fraction of yellow balls (2/5)
3. Discuss how they can compare the number of red balls to yellow balls

Guide the discussion towards the idea of comparing quantities directly (3 to 2) rather than using fractions. This will help introduce the concept of ratios as a different way to express relationships between quantities.

Teach

Definitions

• Ratio: A comparison between two quantities.
• Part-to-part: A ratio that compares different parts of a whole to each other.
• Part-to-whole: A ratio that compares one part of a whole to the entire whole.
• Comparison: The act of evaluating two or more quantities to determine their relationship.
• Equivalent ratios: Ratios that express the same relationship between quantities, even though the numbers may be different.

This slide show of definitions also provides examples of the terms:

https://www.media4math.com/library/slideshow/basic-ratio-definitions

Overview of Ratios

Use this slide show to go over the basic concepts around ratios:

https://www.media4math.com/library/slideshow/overview-ratios

Application of Ratios: Zoo Animals

Continue with an application of ratios by looking at this example of ratios of zoo animals from the San Diego Zoo, which is also summarized in this slide show:

https://www.media4math.com/library/slideshow/application-ratios-zoo-animals

Let's say the San Diego Zoo has:

• 8 elephants
• 24 monkeys
• 12 zebras

Demonstrate different ways to express ratios using these animals:

1. Elephants to monkeys:
   • 8:24 (simplified to 1:3)
   • 8 to 24
   • 8/24 (simplified to 1/3)
2. Zebras to elephants:
   • 12:8 (simplified to 3:2)
   • 12 to 8
   • 12/8 (simplified to 3/2)
3. Monkeys to zebras:
   • 24:12 (simplified to 2:1)
   • 24 to 12
   • 24/12 (simplified to 2/1)

Explain the difference between part-to-part ratios (like elephants to monkeys) and part-to-whole ratios (like monkeys to total animals).

Application of Equivalent Ratios: Vehicle Ratios

Now, introduce equivalent ratios using a different context: a parking lot with cars, motorcycles, and SUVs. You can use this slideshow or use the information below:

https://www.media4math.com/library/slideshow/applications-equivalent-ratios-vehicle-ratios 

Let's say a parking lot has:

• 40 cars
• 10 motorcycles
• 20 SUVs

Use the ratio of cars to motorcycles to demonstrate equivalent ratios:

40:10 = 4:1 = 8:2 = 20:5

Show how to create and use ratio tables to find missing values. For example:

Explain that all these ratios are equivalent because they represent the same relationship between cars and motorcycles.

Finally, demonstrate how to plot ratio pairs on a coordinate plane using the car to motorcycle ratio (4,1), (8,2), (12,3), (16,4), (20,5).

Use this Desmos activity to demonstrate plotting ratio pairs:

https://www.desmos.com/calculator/tikwghztdz

Example 1: Counting Fruit in a Basket

Problem: A fruit basket contains 6 apples, 4 bananas, and 2 oranges. 

Find the ratio of:

1. Apples to bananas
2. Bananas to oranges
3. Apples to total fruit

Solution:

1. The ratio of apples to bananas is:

   $$6:4 \text{ or } \frac{6}{4} \text{ (can be simplified to } 3:2 \text{)}$$

2. The ratio of bananas to oranges is:

   $$4:2 \text{ or } \frac{4}{2} \text{ (simplifies to } 2:1 \text{)}$$

3. The ratio of apples to the total number of fruit:

   $$6:(6+4+2) = 6:12 \text{ or } \frac{6}{12} \text{ (simplifies to } 1:2 \text{)}$$

Example 2: Counting Vehicles in a Parking Lot

Problem: A parking lot has 10 sedans, 5 trucks, and 3 motorcycles. 

Find the ratio of:

1. Sedans to trucks
2. Motorcycles to total vehicles
3. Trucks to motorcycles

Solution:

1. The ratio of sedans to trucks:

   $$10:5 \text{ or } \frac{10}{5} = 2:1$$

2. The ratio of motorcycles to total vehicles:

   $$3:(10+5+3) = 3:18 \text{ or } \frac{3}{18} = 1:6$$

3. The ratio of trucks to motorcycles:

   $$5:3$$

Example 3: Counting School Supplies

Problem: A classroom supply box contains 12 pencils, 8 markers, and 6 erasers. 

Find the ratio of:

1. Pencils to markers
2. Markers to total items
3. Erasers to pencils

Solution:

1. The ratio of pencils to markers:

   $$12:8 \text{ or } \frac{12}{8} = 3:2$$

2. The ratio of markers to total items:

   $$8:(12+8+6) = 8:26 \text{ or } \frac{4}{13}$$

3. The ratio of erasers to pencils:

   $$6:12 \text{ or } \frac{6}{12} = 1:2$$

Example 4: Ratios in a Sports Team

Problem: A school soccer team has 15 players: 5 defenders, 6 midfielders, and 4 forwards. Find the ratio of:

1. Defenders to midfielders
2. Midfielders to total players
3. Forwards to defenders

Solution:

1. The ratio of defenders to midfielders:

   $$5:6$$

2. The ratio of midfielders to total players:

   $$6:15$$

3. The ratio of forwards to defenders:

   $$4:5$$

Example 5: Cooking Recipe

Problem: A pancake recipe calls for 2 cups of flour, 1 cup of milk, and 3 eggs. Find the ratio of:

1. Flour to milk
2. Eggs to total ingredients
3. Milk to eggs

Solution:

1. The ratio of flour to milk:

   $$2:1$$

2. The ratio of eggs to total ingredients:

   $$3:(2+1+3) = 3:6 \text{ or } 1:2$$

3. The ratio of milk to eggs:

   $$1:3$$

Example 6: Comparing Movie Genres

Problem: A survey of students' favorite movie genres finds that 18 students prefer action, 12 prefer comedy, and 10 prefer drama. Find the ratio of:

1. Action to comedy
2. Drama to total students
3. Comedy to action

Solution:

1. The ratio of action to comedy:

   $$18:12 \text{ or } 3:2$$

2. The ratio of drama to total students:

   $$10:(18+12+10) = 10:40 \text{ or } 1:4$$

3. The ratio of comedy to action:

   $$12:18 \text{ or } 2:3$$

Review

Summary of Concepts Covered

In this lesson, students explored the concept of ratios as a way to compare two or more quantities. They learned how to express ratios in different forms, simplify ratios, and apply them to real-world scenarios. Key takeaways include:

• Ratios can compare part-to-part or part-to-whole relationships.
• Ratios can be written in three formats: $a:b$, $a \text{ to } b$, and $\frac{a}{b}$.
• Equivalent ratios represent the same relationship even if the numbers differ.
• Ratios appear in various real-world contexts such as cooking, sports, business, and science.

Key Vocabulary

• Ratio: A mathematical comparison between two or more quantities.
• Part-to-Part Ratio: A ratio comparing two different groups within a whole.
• Part-to-Whole Ratio: A ratio comparing one part to the total amount.
• Equivalent Ratios: Two or more ratios that express the same relationship.
• Proportion: An equation that states two ratios are equal.
• Unit Rate: A ratio where the second term is 1, used to compare quantities per unit.

Multimedia Resources

Use this video to review the basics of ratios:

https://www.media4math.com/library/1789/asset-preview

Additional Worked-Out Examples

Example 1: Book Categories in a Library

Problem: A school library organizes its books into three categories: 24 adventure books, 18 science books, and 12 history books. Find the ratio of:

1. Adventure books to history books
2. History books to total books
3. Science books to adventure books

Solution:

1. The ratio of adventure books to history books:

   $$24:12 \text{ or } 2:1$$

2. The ratio of history books to total books:

   $$12:(24+18+12) = 12:54 \text{ or } 2:9$$

3. The ratio of science books to adventure books:

   $$18:24 \text{ or } 3:4$$

Example 2: Student Attendance at an Event

Problem: A school event had 120 students attend. Among them, 48 were in 6th grade, 36 in 7th grade, and 36 in 8th grade. Find the ratio of:

1. 6th graders to 8th graders
2. 7th graders to total students
3. 8th graders to 6th graders

Solution:

1. The ratio of 6th graders to 8th graders:

   $$48:36 \text{ or } 4:3$$

2. The ratio of 7th graders to total students:

   $$36:120 \text{ or } 3:10$$

3. The ratio of 8th graders to 6th graders:

   $$36:48 \text{ or } 3:4$$

Example 3: Ingredients in a Recipe

Problem: A cookie recipe requires 3 cups of flour, 2 cups of sugar, and 1 cup of butter. Find the ratio of:

1. Flour to sugar
2. Butter to total ingredients
3. Sugar to flour

Solution:

1. The ratio of flour to sugar:

   $$3:2$$

2. The ratio of butter to total ingredients:

   $$1:(3+2+1) = 1:6$$

3. The ratio of sugar to flour:

   $$2:3$$

Example 4: Comparing Sales

Problem: A sports store sells different types of jerseys: 30 basketball, 45 soccer, and 15 baseball jerseys. Find the ratio of:

1. Basketball jerseys to baseball jerseys
2. Soccer jerseys to total jerseys
3. Baseball jerseys to basketball jerseys

Solution:

1. The ratio of basketball to baseball jerseys:

   $$30:15 \text{ or } 2:1$$

2. The ratio of soccer to total jerseys:

   $$45:(30+45+15) = 45:90 \text{ or } 1:2$$

3. The ratio of baseball to basketball jerseys:

   $$15:30 \text{ or } 1:2$$

Quiz

Answer the following questions.

1. Express the ratio of cars to SUVs in the parking lot in three different forms.
   40 cars
   10 motorcycles
   20 SUVs
   
    
2. What is the part-to-whole ratio of motorcycles to all vehicles in the parking lot?
   
    
3. If the ratio of cars to motorcycles is 4:1, how many cars are there if there are 15 motorcycles?
   
    
4. Complete the equivalent ratio table for zebras to elephants: 3:2, 6:4, __:6, 12:__
   
    
5. Suppose you were plotting the ratio pair (12,3). Which is the x-coordinate? Which is the y-coordinate?
   
    
6. What is the simplified form of the ratio 20:70 (SUVs to total vehicles)?
   
    
7. If the ratio of cars to SUVs is 2:1, and there are 30 cars, how many SUVs are there?
   
    
8. Express the ratio of motorcycles to total vehicles as a simplified fraction.
   
    
9. If the ratio of cars to parking attendants is 20:1, how many attendants are there if there are 40 cars?
   
    
10. Create a ratio table for the ratio of cars to motorcycles (4:1), with 4 equivalent ratios.
    
     

Answer Key

1. 40:20, 40 to 20, 2/1
2. 10:70 or 1:7
3. 60 cars
4. 9:6, 12:8
5. x: 12, y: 3
6. 2:7
7. 15 SUVs
8. 10/70 or 1/7
9. 2 attendants
10. 4:1, 8:2, 12:3, 16:4 (or any other correct equivalent ratios)

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