Understanding Ratios, Proportional Relationships, and Rational Numbers

Lesson Summary

In this lesson, students will deepen their understanding of ratios and proportional relationships by exploring more advanced applications of proportional reasoning. This lesson builds on prior knowledge from Grade 7 and introduces new problem-solving strategies involving proportional equations, graphing relationships, and identifying constant rates of change.

Students will practice:

• Identifying and analyzing proportional relationships in tables, graphs, and equations.
• Determining the constant of proportionality ($k$) and understanding its role in proportional relationships.
• Comparing proportional and non-proportional relationships using multiple representations.
• Solving real-world problems involving proportional reasoning, including scale models, percent applications, and unit rates.

This lesson will help students make connections between proportional relationships and linear equations, laying the foundation for algebraic concepts covered later in the curriculum.

Lesson Objectives

• Recognize and represent proportional relationships between quantities
• Identify the constant of proportionality (unit rate) in various representations
• Understand rational numbers as ratios of integers
• Convert repeating decimals to rational numbers

Common Core Standards

• CCSS.MATH.CONTENT.7.RP.A.2 Recognize and represent proportional relationships between quantities.
• CCSS.MATH.CONTENT.7.RP.A.2.B Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
• CCSS.MATH.CONTENT.8.NS.A.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Prerequisite Skills

• Basic understanding of ratios
• Graphing on a coordinate plane
• Knowledge of fractions and decimals

Key Vocabulary

• Ratio: A comparison of two numbers, often written as $\frac{a}{b}$ or using a colon (e.g., 3:5).
  • Multimedia Resource: https://www.media4math.com/library/22157/asset-preview
• Proportional Relationship: A relationship between two variables where the ratio remains constant.
  • Multimedia Resource: https://www.media4math.com/library/43387/asset-preview
• Constant of Proportionality: The constant $k$ in the equation $y = kx$, representing the unit rate.
  • Multimedia Resource: https://www.media4math.com/library/43411/asset-preview
• Unit Rate: A rate with a denominator of 1, such as $\frac{60}{1}$ miles per hour.
  • Multimedia Resource: https://www.media4math.com/library/43395/asset-preview
• Linear Relationship: A relationship that forms a straight line when graphed, often written in the form $y = mx + b$.
• Graph of a Proportional Relationship: A straight line that passes through the origin (0,0) with a slope equal to the constant of proportionality.
• Scale Factor: A multiplier used to enlarge or reduce a figure while maintaining proportionality.
  • Multimedia Resource: https://www.media4math.com/library/43390/asset-preview
• Rational Number: A number that can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. Rational numbers include integers, terminating decimals, and repeating decimals.

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 activities.

Activity 1: Review of Ratios

Start the lesson by reviewing basic ratios and how they are used to compare two quantities.

Example: There are 15 boys and 10 girls in a classroom. Write the ratio of boys to girls in three different ways.

Solution:

• Fraction form: $\frac{15}{10}$ (which simplifies to $\frac{3}{2}$)
• Standard form: 15:10 (which simplifies to 3:2)
• Word form: "3 to 2"

Activity 2: Review of Rates

Introduce the concept of rates by showing how they compare different quantities with different units.

Example: A car travels 180 miles in 3 hours. What is the unit rate?

Solution:

1. Divide the total miles by the total hours: $$\frac{180}{3} = 60$$
2. The unit rate is 60 miles per hour.

Discuss how unit rates are used in real life, such as miles per hour, price per item, and heartbeats per minute.

Activity 3: Review of Proportions

Reinforce the concept of proportions by solving for an unknown value in a proportion equation.

Example: Solve for $x$ in the proportion:

Solution:

1. Use cross-multiplication: $$4 \times 20 = 5 \times x$$
2. Simplify: $$80 = 5x$$
3. Divide both sides by 5: $$x = \frac{80}{5} = 16$$

Encourage students to think about proportional relationships in different contexts, such as maps, scale models, and recipe adjustments.

Teach

Definitions

• Ratio: A comparison of two quantities expressed as a fraction or using the word "to"
• Proportion: An equation stating that two ratios are equal
• Constant of proportionality: The constant ratio between two proportional quantities
• Unit rate: A rate expressed as a quantity per one unit of another quantity
• Rational number: A number that can be expressed as the ratio of two integers
• Irrational number: A number that cannot be expressed as the ratio of two integers
• Repeating decimal: A decimal in which a digit or group of digits repeats indefinitely

This slide show provides additional support for these terms by providing examples of the term:

https://www.media4math.com/library/slideshow/definitions-ratios-proportions-and-rational-numbers

Instruction

Rational Numbers. Introduce rational numbers as ratios of integers, emphasizing that they can be positive or negative. You can use this slideshow to introduce rational numbers:

https://www.media4math.com/library/slideshow/ratios-and-rational-numbers

Summary:

• Rational numbers are expressed as a/b where a and b are integers and b ≠ 0
• Examples: 3/4, -2/5, 7/-3, -8/-9
• Discuss how the signs of the numerator and denominator affect the overall sign of the rational number

Remind students that rational numbers are based on ratios. Define proportional relationships and demonstrate how to identify them in tables, graphs, and equations, including negative proportions. 

A number that consists of a repeating decimal can be written as a rational number:

For a decimal with a repeating pattern of two digits, multiply by 100 and go through the same process. 

Proportional Relationships. Use the following examples of ratios and proportions to see rational numbers in context. 

Example 1: Real World Connection - Water Pressure vs Depth

As a diver descends underwater, the water pressure increases proportionally to the depth. The pressure increases by 1 atmosphere for every 10 meters of depth.

This History channel video provides additional context:

This table summarizes the change in pressure versus change in depth. Note the negative numbers in column 1.

Graph: Plot points (0,0), (-10,1), (-20,2), (-30,3) on a coordinate plane.

Equation: y = -0.1x, where x is the depth in meters and y is the pressure change in atmospheres.

Solution: This is a proportional relationship because the ratio of y to x is constant (-0.1:1) and the graph passes through the origin. The negative constant of proportionality indicates that as depth decreases negatively, pressure increases positively.

Use this Desmos activity to explore this activity:

https://www.desmos.com/calculator/jg3dnokjjx

 Have students use the slider to confirm the constant of proportionality and the equation of the line shown.

Example 2: Real World Connection - Currency Exchange

Use this slide show to walk students through an application of proportional reasoning in the context of currency exchange rates:

https://www.media4math.com/library/slideshow/applications-proportions-exchange-rates

Summary:

• The exchange rate between US dollars and euros is 0.85 Euros per dollar. Write this as an equation and find the constant of proportionality
• Find the dollars-to-Euros exchange rate. Find the corresponding equation.
• An item sells for 45 Euros. Find the price in dollars.
• You want to exchange $500 into Euros. How many Euros will you get?

Solution:

• Unit rate = 0.85 euros per dollar.
• y = 0.85 x. The constant of proportionality is 0.85, representing the exchange rate.
• Dollars-to-Euros: y = 1.18x. The constant of proportionality is 1.18.
• Exchanges:
  • 45 Euros = $53.10
  • $500 = 425 Euros

Example 3: Real World Connection - Freezer Temperature Drop

The temperature in a freezer is dropping at a rate of -3°C every 5 minutes.

Equation: y = -0.6x, where x is the time in minutes and y is the temperature change in °C.

Solution: This is a proportional relationship with a negative constant of proportionality (-0.6). The graph is a straight line passing through the origin, with a negative slope indicating the temperature decrease over time.

Use this Desmos activity, if time allows:

https://www.desmos.com/calculator/dmlvt6tpl7 

Review

Lesson Summary

In this lesson, students explored ratios and proportional relationships through real-world applications and mathematical reasoning. They learned how to identify proportional relationships in tables, graphs, and equations and how to determine the constant of proportionality. The lesson also introduced the connection between proportional relationships and linear equations, emphasizing how proportional graphs always pass through the origin.

Additionally, students examined rational numbers and their role in proportional reasoning. They learned that rational numbers include fractions, terminating decimals, and repeating decimals—concepts that are critical when working with proportional equations and unit rates.

Key takeaways from this lesson include:

• Ratios compare two quantities and can be written in fraction, colon, or word form.
• Rates express a ratio where the two quantities have different units.
• Proportional relationships maintain a constant ratio and can be represented by the equation $y = kx$, where $k$ is the constant of proportionality.
• Unit rates and scale factors are useful in solving real-world problems involving speed, pricing, and scaling.
• Proportional graphs are straight lines that pass through the origin.
• Rational numbers, including fractions and decimals, play an important role in solving ratio and proportion problems.

By reviewing these concepts, students will reinforce their understanding of proportional reasoning and be prepared for more advanced algebraic topics.

Key Vocabulary

• Ratio: A comparison of two numbers, often written as $\frac{a}{b}$ or using a colon (e.g., 3:5).
• Rate: A ratio that compares two quantities with different units, such as miles per hour or cost per item.
• Proportional Relationship: A relationship between two variables where the ratio remains constant.
• Constant of Proportionality: The fixed ratio ($k$) in the equation $y = kx$, representing the unit rate.
• Unit Rate: A rate where the denominator is 1, such as $\frac{60}{1}$ miles per hour.
• Rational Number: A number that can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. Rational numbers include integers, terminating decimals, and repeating decimals.
• Linear Relationship: A relationship that forms a straight line when graphed, often written in the form $y = mx + b$.
• Graph of a Proportional Relationship: A straight line that passes through the origin (0,0) with a slope equal to the constant of proportionality.
• Scale Factor: A multiplier used to enlarge or reduce a figure while maintaining proportionality.

By reviewing these concepts, students will reinforce their understanding of proportional reasoning and be prepared for more advanced algebraic topics.

Group activity: Students create posters showing proportional relationships in different representations and examples of rational numbers expressed as ratios.

Example 1: Positive Constant of Proportionality - Cycling and Grade

Introduce this video, which is an application of proportional reasing in the context of cycling. Grade is the slope of a cycling trail expressed as a percent.

https://www.media4math.com/library/1814/asset-preview

What is the slope of a hill with a grade of 12%? Write the equation that represents this.

Solution:

• The slope is 0.12.
• y = 0.12x 

Example 2: Negative Constant of Proportionality - Underwater Temperature Change

As a submarine descends, the water temperature decreases by 1°C for every 100 meters of depth. Create a table and graph showing the temperature change as the depth increases.

You can also use this Desmos activity:

https://www.desmos.com/calculator/23lw6ctq3r 

Quiz

Answer the following questions.

1. Is the relationship between x and y proportional? x: 2, 4, 6; y: 6, 12, 18
   
    
2. What is the constant of proportionality in the equation y = 3x?
   
    
3. If a car travels 240 miles in 4 hours at a constant speed, what is the unit rate?
   
    
4. Does the graph of y = 2x + 1 represent a proportional relationship? Why or why not?
   
    
5. In a recipe, 2 cups of flour are used for every 3 cups of milk. What is the ratio of flour to milk?
   
    
6. What is the unit rate if 15 items cost $45?
   
    
7. Convert the repeating decimal 0.363636... to a fraction.
   
    
8. Express 5/8 as a ratio in proportion to 1.
   
    
9. Convert 0.25 to a fraction in its simplest form.
   
    
10. Is 0.333333... a rational or irrational number? Explain using ratio terminology.
    
     
11. What type of decimal expansion do all rational numbers have?
    
     
12. Convert 2.7777... to a fraction and express it as a ratio.
    
     

Answer Key

1. Yes
2. 3
3. 60 miles per hour
4. No, because it doesn't pass through (0,0)
5. 2:3
6. $3 per item
7. 4/11
8. 0.625:1 or 5:8
9. 1/4
10. Rational, because it can be expressed as the ratio 1:3.
11. Terminating or repeating. 
12. 25/9 or 25:9

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