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Tell students to close their books or devices (or to keep them closed). Reveal one situation at a time and ask students to write an equation that represents the relationship. For each problem:
Give students quiet think time and ask them to give a signal when they have an answer and a strategy for how they came up with their equation.
Invite students to share their equations and strategies and record and display their responses for all to see.
Write an equation to represent each relationship.
The purpose of this discussion is for students to hear and explain strategies for writing equations to represent situations. Consider discussing:
“How are the equations for the two situations similar? How are they different?” (Both equations have two variables. Both equations include numbers used in the descriptions. Both equations describe a linear relationship. One equation has both variables on one side while the other equation has variables on both sides. The slope and vertical intercept are indicated in one of the equations but not the other.)
“What are some strategies that helped you to write your equations?” (Make a table of possible values; find an initial value and a rate of change; compare the situation to similar situations from previous lessons and activities.)
“Is the slope for each of these equations positive or negative? Why does that make sense with the situation?” (For the fruit, the slope is negative, which makes sense because if more of one fruit is bought, less can be bought of the other. For the savings account, the slope is positive, which makes sense because the more weeks go by, the more money will be in the account.)
In this activity, students are given an image with five graphs representing changes in five different savings account balances over time. By writing equations to represent each graph, and interpreting what specific points represent for each account, students contextualize their understanding of linear relationships (MP2).
By considering the meaning of non-intersecting lines or the point of intersection between two lines, this activity lays the foundation for thinking about systems of equations in following units.
Display the image from the task for all to see and explain that each line represents one person’s weekly savings account balance from the start of the year. Ask students to consider line and invite 2–3 students to describe what line shows in the context of a savings account. If not mentioned by students, explain that this line represents the situation from the Warm-up that they wrote an equation for. Instruct students not to choose line for the first question.
Arrange students in groups of 3–4. Give students 10 minutes of work time followed by a whole-class discussion.
Each line represents one person’s weekly savings account balance from the start of the year.
The purpose of this discussion is for students to share the stories they created for each graph. While students will touch on the meaning of the point of intersection between two lines in preparation for future work, the focus of this activity is on the interpretation of slope, vertical intercept, and specific points in context.
Invite students to share their stories for each line. After each story, discuss the following questions:
“What part of this story is represented by the slope?“
“Is the slope positive, negative, or zero? Why does this make sense in this situation?”
“What part of this story is represented by the vertical intercept?”
After a story for each line has been shared, discuss the answers to the last question. Emphasize that represents a point in time when both accounts have the same value, and this point is also a solution to both of the equations represented by each line.
If time allows, ask students to add a line to the graphs in the task for each of the following and then write an equation for the line:
A savings account that starts with \$10 and has \$5 added each week
A situation with a -intercept of 100 and a slope of -10
A line parallel to line
A line that intersects line at
In this activity, students write an equation to represent the combinations of fish that a market can order each day. Students plot points and interpret the meaning of points that are or are not on the graph of the line representing the equation (MP2).
Arrange students in groups of 2. Give them 3–4 minutes of quiet work time followed by a whole-class discussion.
The Fabulous Fish Market orders tilapia, which costs \$3 per pound, and salmon, which costs \$5 per pound. The graph shows how much of each type of fish can be purchased if the market budgets to spend \$210 on this order each day.
| A | B | C | D | E | F | |
|---|---|---|---|---|---|---|
| pounds of tilapia | 5 | 19 | 27 | 25 | 65 | 55 |
| pounds of salmon | 36 | 30.6 | 25 | 27 | 6 | 4 |
Which of these combinations is a possible order if the market plans to spend its entire budget of \$210? Explain your reasoning.
The purpose of this discussion is to consider what different features of the given graph mean in the context of this situation. Begin by asking students:
“Does it make sense for this graph to be only in the first quadrant?” (Yes, because purchasing a negative amount of fish is not possible, so having all positive values makes sense.)
"Does it make sense for this situation to be represented with a line? (Yes, since the fish is priced by weight, a fraction of a pound is reasonable.)
“What do the intercepts represent in this situation?” (The vertical intercept represents the pounds of salmon that can be purchased if no tilapia is ordered. The horizontal intercept represents the pounds of tilapia that can be purchased if no salmon is ordered.)