Writing Equations to Model Relationships (Part 2)
5 min
Narrative
The activities in this lesson require students to observe tables of values, look for patterns, and generalize their observations into equations. This Warm-up prompts students to think about how they could go about analyzing the values in the table and to articulate their reasoning.
Launch
Arrange students in groups of 2. Display the table for all to see. Explain that the quantities in each column are related.
Ask groups to try to find a relationship, and pay attention to how they go about doing so. Emphasize that the goal is not to successfully find a relationship. It is to notice the strategies they use when attempting to figure out what the relationship might be.
Student Task
Here is a table of values. The two quantities, x and y, are related.
| x | y |
|---|---|
| 1 | 0 |
| 3 | 8 |
| 5 | 24 |
| 7 | 48 |
What are some strategies you could use to find a relationship between x and y? Brainstorm as many ways as possible.
Sample Response
Sample responses:
- See if there is an operation that could be done to x that would produce y.
- Compare how the x-values change and the y-values change, and see if there is any pattern.
- See if the numbers in one column follow a special pattern and if that pattern could be connected to the numbers in the other column.
- Plot the x- and y-values on a coordinate plane, and see if the graph looks like one of a familiar relationship.
Activity Synthesis (Teacher Notes)
Invite groups to share their strategies and record them for all to see. If not already described by students, apply each strategy using the values in the table, or ask students to give an example of how it could be applied.
Some students may notice that each time x increases by 2, y increases by 8 more than the previous time. Others may notice that the y-values are 1 less than the square numbers 1, 4, 9, 25, and 49, and that these numbers are the squares of the listed x-values, and from there concluded that the relationship is along the lines of "Square x and subtract 1 to get y." Neither observations are essential, but consider asking if they see any special patterns in either column that could help determine the relationship.
If no one mentions plotting the pairs of values as a way to understand the relationship between x and y, and if time permits, consider displaying a graph such as shown (or displaying a blank coordinate plane and plotting the points together).
Ask students to keep in mind the different strategies as they work on the activities in the lesson.
Math Community
After the Warm-up, display the Math Community Chart and a list of 2–5 revisions suggested by the class in the previous exercise for all to see. Remind students that norms are agreements that everyone in the class shares responsibility for, so everyone needs to understand and agree to work on upholding the norms. Briefly discuss any revisions and make changes to the “Norms” sections of the chart as the class agrees. Depending on the level of agreement or disagreement, it may not be possible to discuss all suggested revisions at this time. If that happens, discuss the remaining suggestions over the next few lessons.
Tell students that the class now has an initial list of norms or “hopes” for how the classroom math community will work together throughout the school year. This list is just a start, and over the year it will be revised and improved as students in the class learn more about each other and about themselves and math learners.
Standards
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- HSA-CED.A.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- HSF-LE.A.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
15 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Writing Equations to Model Relationships (Part 2)
5 min
Narrative
The activities in this lesson require students to observe tables of values, look for patterns, and generalize their observations into equations. This Warm-up prompts students to think about how they could go about analyzing the values in the table and to articulate their reasoning.
Launch
Arrange students in groups of 2. Display the table for all to see. Explain that the quantities in each column are related.
Ask groups to try to find a relationship, and pay attention to how they go about doing so. Emphasize that the goal is not to successfully find a relationship. It is to notice the strategies they use when attempting to figure out what the relationship might be.
Student Task
Here is a table of values. The two quantities, x and y, are related.
| x | y |
|---|---|
| 1 | 0 |
| 3 | 8 |
| 5 | 24 |
| 7 | 48 |
What are some strategies you could use to find a relationship between x and y? Brainstorm as many ways as possible.
Sample Response
Sample responses:
- See if there is an operation that could be done to x that would produce y.
- Compare how the x-values change and the y-values change, and see if there is any pattern.
- See if the numbers in one column follow a special pattern and if that pattern could be connected to the numbers in the other column.
- Plot the x- and y-values on a coordinate plane, and see if the graph looks like one of a familiar relationship.
Activity Synthesis (Teacher Notes)
Invite groups to share their strategies and record them for all to see. If not already described by students, apply each strategy using the values in the table, or ask students to give an example of how it could be applied.
Some students may notice that each time x increases by 2, y increases by 8 more than the previous time. Others may notice that the y-values are 1 less than the square numbers 1, 4, 9, 25, and 49, and that these numbers are the squares of the listed x-values, and from there concluded that the relationship is along the lines of "Square x and subtract 1 to get y." Neither observations are essential, but consider asking if they see any special patterns in either column that could help determine the relationship.
If no one mentions plotting the pairs of values as a way to understand the relationship between x and y, and if time permits, consider displaying a graph such as shown (or displaying a blank coordinate plane and plotting the points together).
Ask students to keep in mind the different strategies as they work on the activities in the lesson.
Math Community
After the Warm-up, display the Math Community Chart and a list of 2–5 revisions suggested by the class in the previous exercise for all to see. Remind students that norms are agreements that everyone in the class shares responsibility for, so everyone needs to understand and agree to work on upholding the norms. Briefly discuss any revisions and make changes to the “Norms” sections of the chart as the class agrees. Depending on the level of agreement or disagreement, it may not be possible to discuss all suggested revisions at this time. If that happens, discuss the remaining suggestions over the next few lessons.
Tell students that the class now has an initial list of norms or “hopes” for how the classroom math community will work together throughout the school year. This list is just a start, and over the year it will be revised and improved as students in the class learn more about each other and about themselves and math learners.
Standards
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A-CED.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- HSA-CED.A.2·Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- HSF-LE.A.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).