In this lesson plan, your learner will extend their understanding of ratios by creating coordinate graphs of equivalent ratios. They will begin by creating a double number line and a table of values for a word problem then progress to visualizing the data on a coordinate graph.
This lesson will explore various models of representing equivalent ratios. Here are a few concepts that are helpful to know:
Equivalent Ratios: Equivalent ratios are ratios that represent the same relationship between quantities but are expressed differently. They have the same value and can be obtained by scaling up or down the original ratio. For example, the ratios 3/2 and 6/4 are equivalent because they both represent the relationship of 3 to 2.
Double Number Line: A double number line is a visual aid that shows the relationship between two quantities with different units. It consists of two parallel number lines where equivalent ratios are connected through a pattern of repeated addition or scalar multiplication.
Table of Values: A table of values organizes equivalent ratios into a structured format, making it easier to identify patterns and relationships between quantities. Each pair of values represents an equivalent ratio.
Coordinate Graph: Equivalent ratios can also be represented on a coordinate graph, where each point corresponds to a specific ratio. The x-axis typically represents one quantity (e.g., hamburgers), and the y-axis represents another quantity (e.g., hotdogs).
Patterns in Equivalent Ratios
When exploring equivalent ratios through different representations, several patterns emerge:
Repeated Addition: In a double number line or table of values, equivalent ratios often exhibit patterns of repeated addition. For instance, moving from one point to the next may involve adding the same value repeatedly.
Scalar Multiplication: Equivalent ratios can also demonstrate patterns of scalar multiplication, where both parts of the ratio are scaled up or down by the same factor. This scaling maintains the proportional relationship between the quantities.
Linear Pattern on Graphs: On a coordinate graph, equivalent ratios show a linear pattern, forming a straight line when plotted. This linear relationship reflects the constant rate of change between the quantities, reinforcing the proportional nature of the ratio.
Teaching Plan
The following activities will help your learner become confident with creating and interpreting graphs of equivalent ratios.
Examples and visuals to support the lesson:
1. Tables and Double Number Lines
This activity will give your learner a chance to review ratio concepts by creating a table of values and double number line.
Provide your learner with a scenario that includes a ratio with different units. For example: At a baseball game, the snack bar sells 3 hotdogs for every 2 hamburgers they sell.
Have your learner create a double number line and a table of values showing the relationship between hotdogs to hamburgers.
Before progressing to the next activity, ensure that your learner can recognize and interpret the equivalent ratios on both the double number line and the table.
Skill Check
I can create tables and double number lines that show equivalent ratios.
2. Coordinate Graphs of Equivalent Ratios
In this activity, your learner will use their experience with creating a table and double number line to understand and interpret a coordinate graph.
Explain how the values from the table and double number line can also be represented on a coordinate graph. Before showing the completed graph, use a blank graph to review the parts of a coordinate graph along with the steps for plotting ordered pairs.
Next, show your learner the completed graph for the hotdog example. Guide them in making connections between the table, double number line, and graph.
One way to connect the graphical representation to the work they have already done with double number lines is to think about rotating the hotdog number line from above 90 degrees counterclockwise. You may want to cut the number lines out and physically show this rotation. This helps to solidify that each point on the graph corresponds to two quantities: the number of hamburgers and the number of hotdogs.
Help your learner see that the first column of the table (Hamburgers Sold) corresponds to the horizontal x-axis and the second column of the table (Hotdogs Sold) corresponds to the vertical y-axis.
To ensure that your learner understand these connections, ask them questions such as, "What does the ordered pair (8, 12) represent in the situation?" Or ask them to write the ordered pair that represents 30 hotdogs and 20 hamburgers being sold.
Skill Check
I understand how a coordinate graph can represent equivalent ratios.
3. Exploring Patterns
Once your learner is comfortable interpreting the coordinate graph, they can explore connections between the table, double number lines, and graph.
Provide your learner with additional scenarios to practice with. Have them complete a double number line, table of values, and graph.
Encourage them to make connections and look for patterns. Point out the pattern of repeated addition that is seen as we move from one point to the next. Also, note that the points have a linear pattern (they lie on a straight line).
Multiplication patterns can also be seen in each model. For example, the ratio of 2 to 3 can be multiplied by 4 to create an equivalent ratio of 8 to 12. These ratios can be seen as number pairs on the double number line and table of values.
Advanced learners may also notice the scalar multiplication pattern on the graph by creating slope triangles. The slope triangle for the point (2, 3) can be multiplied by a scale factor of 4 to create a slope triangle for (8, 12). Your learner may notice that the larger triangle is a dilation of the smaller triangle. These concepts will be explored in future lessons.
Skill Check
I can use tables, double number lines, and graphs that find patterns of equivalent ratios.
4. Using Graphs to Solve Problems
Use this activity to check your learner's understanding of the lesson.
Once your learner is confident with recognizing the connections between the word problem and its various representations, challenge them to complete a coordinate graph for a ratio word problem.
This time, do not provide them with a double number line or table of values, although your learner may wish to create their own before completing the graph.
Once they've had some practice creating graphs, ask your learner equations about the graphs or have them make up their own questions to answer.
Skill Check
I can create and use coordinate graphs to solve ratio word problems.
In this lesson plan, your learner has explored how to represent ratios using double number lines, tables of values, and coordinate graphs. By connecting these different representations, they will gain a deeper understanding relationships between quantities. This skill will prepare them for learning about proportional and linear relationships in future lessons.
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Skill Check