# Lesson 2

Equations and Graphs

These materials, when encountered before Algebra 1, Unit 7, Lesson 2 support success in that lesson.

## 2.1: The Word List (5 minutes)

### Warm-up

The purpose of this warm-up is to connect student interpretations of a graph and equation to a situation which will be useful when students write functions that represent situations in a later activity. While students may notice and wonder many things about these representations, the connection to the actual situation are the important discussion points.

### Launch

Display the prompt, equation, and graph for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

### Student Facing

A group is asked to memorize a list of 20 words, then recall as many as possible later. An equation that represents the position of the word on the list, $$n$$, and the number of people in the group who remembered the word, $$P$$, is modeled by the function $$P = 0.34n^2 -8.7n +97.3$$.

What do you notice? What do you wonder?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the prompt. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the connections between the equation, graph, and situation do not come up during the conversation, ask students to discuss this idea.

## 2.2: Seeing Solutions (15 minutes)

### Activity

In this activity, students work to solve linear equations and recognize the solution in graphs. This skill will help students when they examine quadratic equations and their solutions in the associated Algebra 1 lesson.

### Student Facing

1. A person is hiking from the top of a mountain into a valley. The function $$2,\!000 - 32t$$ represents their elevation in feet above sea level, $$t$$ minutes after they started their hike.
1. What does a solution to the equation $$2000-32t=0$$ mean?
2. Use technology to create a graph of $$y=2,\!000-32t$$. Where do you see the solution to that equation on the graph?
2. A new electronic device originally costs $1,000 but loses$175 worth of value every year.
1. Write a function that represents the worth of the device after $$s$$ years.
2. How many years until the device is worth $0? 3. Use technology to graph the function. Where can you see the solution to your equation on the graph? ### Student Response For access, consult one of our IM Certified Partners. ### Activity Synthesis The purpose of the discussion is to notice the connection between solutions of equations and the graph of both sides of the equation. Select students to share their solutions. Ask students: • “Why are both solutions to these questions related to the horizontal intercept?” (Since both questions asked about when the value is 0, it makes sense to look at the points where $$y = 0$$ which is along the horizontal axis.) • “How could you find the worth of the electronic device after 4.5 years?” (Substitute 4.5 for $$s$$ in the equation and solve for the value.) • “How could you find the number of minutes it takes to reach an elevation of 100 feet above sea level?” (Substitute 100 in the equation for $$y$$ and solve for $$t$$.) ## 2.3: Understanding Solutions in Situations (20 minutes) ### Activity In this activity, students practice solving equations and seeing the solution in graphical form. The questions in this activity are not restricted to where the equation is equal to zero. Students must recognize that solutions to these equations may involve either rewriting the equation to equal zero before graphing or solving a system of equations with each side of the equation representing a function to be graphed. ### Student Facing 1. The expression $$5.25+0.85x$$ represents the amount a yogurt shop charges for yogurt with $$x$$ ounces of toppings. 1. What does the equation $$5.25+0.85x=7.08$$ mean in this situation? 2. What would a solution to this equation mean? 3. Use technology to graph $$y=5.25+0.85x$$. Where can you see the solution to the equation on the graph? 2. Drinks cost$1.50, sandwiches cost $4.00, and there is a flat delivery fee of$5 for each delivery regardless of the number of orders.
1. Write an expression that represents the amount it costs to have $$x$$ meals including a drink and a sandwich delivered to an office.