8.1: Notice and Wonder: Crimes (10 minutes)
The purpose of this warm-up is to elicit the idea that information about situations can be learned from graphs, which will be useful when students draw graphs to represent situations in a later activity. While students may notice and wonder many things about this graph, descriptions of individual points and rates of change are the important discussion points.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
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.
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 graph. 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 individual points such as \((2015, 1.199)\) and rates (even relative) such as the idea that the number of crimes tends to be decreasing does not come up during the conversation, ask students to discuss this idea.
8.2: Dining Out (15 minutes)
In this activity, students interpret graphs representing the number of customers at a restaurant as a function of time. Students are given scenarios and select the best restaurant for each scenario. In the associated Algebra 1 lesson, students describe graphs on their own. This activity supports students by giving them some options for interpreting the graphs.
These graphs show how busy restaurants are at different times of the day.
For each situation, select the best restaurant. Be prepared to explain your reasoning.
- Which restaurant is busy in the morning, then has fewer customers in the evening?
- If Lin’s mom wants to go to a popular dinner restaurant, which restaurant should Lin take her mom to eat?
- Noah’s dad prefers breakfast places with few customers so that he can start on work while eating. Which restaurant should Noah’s dad go to for breakfast?
- Which restaurant would you visit during a 30 minute lunch break? 1 hour lunch break?
The purpose of the discussion is to interpret the graphs in the situations. Select students to share their solutions and reasoning, then ask for additional solutions or interpretations, especially about where to go during lunch. As students explain their reasoning, listen for precise mathematical language (MP6).
- “What type of restaurant do you think is represented by the line with decreasing slope?” (Maybe it is a place that specializes in breakfast, or a donut shop.)
- “Which graph is most likely a fast food restaurant?” (The constant one is probably closest to what I expect for a fast food restaurant since people would come in throughout the day.)
8.3: Draw the Graphs (20 minutes)
In this activity, students practice drawing graphs that could represent situations described. Student solutions do not need to be exact, but attention should be paid to the descriptions.
For each situation, draw a graph that could represent it.
Diego starts at home and walks away from home at a steady rate of 3 miles per hour.
Mai starts 5 miles from home and walks at a steady rate of 3 miles per hour toward her home until she gets there and stays.
A soccer player kicks a ball that’s on the ground so that it goes up to a height of about 10 feet and then comes back down to hit the ground 1.55 seconds later.
The amount of charge left in a phone battery as a percentage is a function of time. Clare runs her phone until it is completely dead, then charges it all the way back up at a steady rate.
The purpose of the discussion is to recognize considerations students have when drawing their graphs. Select students to share their work. If possible, display selected graphs for all to see while the students describe how they drew the graph. Ask students:
- “How did you use the given rate for Diego’s walk?” (I used it as the slope. I knew that after 1 hour he would be 3 miles from home, so I connected the points \((0,0)\) and \((1,3)\) with a line.)
- “How did you decide how long it would probably take to run down the phone battery to 0? How long until it was charged to full?” (I thought about how long my phone usually lasts and I think it can go almost a full day. Then it doesn’t take very long to charge, so I drew the graph so that it took 1 hour to charge.)