Welcome to today's lesson on ratios and double number lines! In this lesson plan, we will begin by exploring the relationships between quantities by using a table of values and identifying equivalent ratios. We will then introduce double number lines as a visual representation of ratios with different units. Finally, your learner will interpret and analyze the patterns on double number lines to solve problems.
Here are a few concepts that are helpful to know for the lesson:
Ratio: A ratio is a comparison of two quantities expressed as a quotient or fraction. It represents the relationship between two different units of measurement. For 6 miles in 1 hour, the ratio of distance to time is 6 miles to 1 hour or 6 miles/1 hour.
Equivalent Ratios: Equivalent ratios are ratios that have the same value but may be expressed in different forms. They maintain the same relationship between quantities. For instance, the ratios 6 miles/1 hour, 12 miles/2 hours, and 18 miles/3 hours are equivalent ratios because they represent the same relationship between distance and time.
Double Number Lines: Double number lines are visual representations used to illustrate equivalent ratios. They consist of two parallel number lines, with values on the top line representing one quantity (e.g., distance) and corresponding values on the bottom line representing another quantity (e.g., time).
Patterns in Equivalent Ratios
Double number lines show connections between equivalent ratios and patterns of repeated addition or scalar multiplication. These patterns are useful for solving problems that involve equivalent ratios.
Repeated Addition: Equivalent ratios often exhibit patterns of repeated addition. For example, the equivalent ratios 6/1 , 12/2 , and 18/3 , demonstrate a pattern of adding 6 to the numerator every time 1 is added to the denominator.
Scalar Multiplication: Equivalent ratios can also demonstrate patterns of scalar multiplication, where both parts of the ratio are multiplied by the same factor. For instance, multiplying the numerator and denominator of 6/1 by two results in the equivalent ratio of 12/2 , maintaining the same relationship between the values but scaled up by a factor of 2.
Teaching Plan
The following activities will help your learner become confident with ratios and double number lines.
Examples and visuals to support the lesson:
Ratios with Different Units
Use this activity to review the concept of a ratio and discuss ratios with different units.
Provide your learner with a scenario that includes a ratio with different units. For example: Harmony runs 6 miles per hour.
Point out that the ratio shows the relationship between the distance (in miles) Harmony runs and the time (in hours) it takes her to run the distance.
Have your learner complete a table that shows Harmony's distance at various times.
To make sure that your learner understands the connection between the values, have them list use the table to list equivalent ratios showing the relationship between distance and time. Remind your learner to include the units of measurement (e.g. miles, hours) with the values.
Using the example of running 6 miles per hour, a list of ratios might include 6 miles/1 hour, 12 miles/2 hours, 18 miles/3 hours, etc.
Skill Check
I can create a table and list of equivalent ratios to represent a ratio word problem.
Introducing Double Number Lines
This activity will introduce your learner to representing ratios with double number lines.
Explain that double number lines are often used to represent ratios where the units of measure are different.
Point out that, just as with a table, you can see that pairs of values are connected. The values on top are connected to the values directly below them.
Your learner may also notice that the double number line shows patterns of repeated addition and scalar multiplication.
Ask your learner various questions about the values in the number line. For example, "How far does Harmony run in 2 hours?" Or "How long does it take Harmony to run 30 miles?"
Have your learner explain their answers using the information on the double number line and their understanding of ratios.
Skill Check
I can use a double number line to find equivalent ratios and solve word problems.
Missing Values on Double Number Lines
In this activity, your learner will use patterns on double number lines to find missing values.
Once your learner has shown that they can interpret the values and patterns on the double number line, provide them with double number lines that have missing values.
To find the missing values, your learner will likely apply the same strategies as with the previous activities. They may look at each pair of values and notice the pattern of repeated addition. Or they may use multiplication/division to find the missing values.
Whatever method they choose, encourage your learner to explain their thought process.
Skill Check
I can use ratio patterns to fill in missing values on a double number line.
Double Number Lines and Word Problems
Now that your learner has practiced finding missing values, you can have them create their own double number lines to solve word problems.
Provide your learner with a word problem that has ratios with different units. Guide them in creating a double number line to represent the word problem.
Have your learner use their double number line to answer questions about the word problem. For each question, encourage them to explain their reasoning.
Here are a few scenarios to consider:
Dave paid $24 for 8 pounds of fruit salad for a company picnic. Dave realizes that he needs more fruit salad so he goes back to the store and buys 2 more pounds of fruit salad. How much more money will he spend? (Solution: He will spend $6 more.)
A broken faucet is leaking 2 ounces of water every 5 minutes. How much water will leak out of the faucet in 1/2 an hour? (Solution: 12 ounces of water.)
Skill Check
I can create a double number line that represents a word problem and use it to answer questions.
Summary
The activities in this lesson plan will provide your learner with a strong foundation in ratios and double number lines. Through interpreting values and solving for missing values on double number lines, your learner will hone their problem-solving skills and deepen their understanding of ratios. Applying this knowledge to real-life word problems with self-created double number lines fosters a hands-on approach that will prepare your learner for working with math concepts such as proportional and linear relationships.
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Skill Check