2.1: The Word List (5 minutes)
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.
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.
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?
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)
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.
- 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.
- What does a solution to the equation \(2000-32t=0\) mean?
- Use technology to create a graph of \(y=2,\!000-32t\). Where do you see the solution to that equation on the graph?
- A new electronic device originally costs $1,000 but loses $175 worth of value every year.
- Write a function that represents the worth of the device after \(s\) years.
- How many years until the device is worth $0?
- Use technology to graph the function. Where can you see the solution to your equation on the graph?
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)
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.
- The expression \(5.25+0.85x\) represents the amount a yogurt shop charges for yogurt with \(x\) ounces of toppings.
- What does the equation \(5.25+0.85x=7.08\) mean in this situation?
- What would a solution to this equation mean?
- Use technology to graph \(y=5.25+0.85x\). Where can you see the solution to the equation on the graph?
- 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.
- Write an expression that represents the amount it costs to have \(x\) meals including a drink and a sandwich delivered to an office.
- Write an equation that has a solution representing the number of drink and sandwich orders it would take to cost $80.
- Graph \(y=1.5x+4x+5\). Where can you see the solution to the equation on the graph?
- The temperature in a deep freezer in a laboratory is -40 degrees Celsius. The freezer breaks, so the temperature starts to rise by 2.5 degrees per hour.
- Use technology to graph \(y=\text-40+2.5x\).
- Explain how to use this graph to find the time (after breaking) when the freezer temperature reaches 0 degrees Celsius.
- The expression \(400 - 10x^2\) represents the height in meters of an object above the ground \(x\) seconds after falling off a 400 meter building.
- Write an equation that has a solution that would give the time in seconds when the object hit the ground.
- Use technology to graph \(y=400-10x^2\) and explain where you can see the solution to your equation on the graph.
The purpose of the discussion is for students to see how solutions to equations can be seen in a graph. Select students to share their solutions. Ask students,
- “How do the equations for the first 2 questions differ from the others in this lesson?” (The expression is equal to a non-zero constant.)
- “What does the horizontal intercept mean in the sandwich situation? Is it realistic?” (It means the number of meals to order so that it costs $0. The solution is a negative number of meals since there is a delivery fee, so this does not make sense in reality.)