# Lesson 6

Finding Interesting Points on a Graph

These materials, when encountered before Algebra 1, Unit 4, Lesson 6 support success in that lesson.

## 6.1: Notice and Wonder: Unemployment Percentage (10 minutes)

### Warm-up

The purpose of this warm-up is to introduce the idea of unemployment percentage which will be useful when students examine the graph in more detail in a later activity. While students may notice and wonder many things about the graph, an understanding of what the graph is showing is the important discussion point.

### Launch

Display the 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.

If necessary, explain to students that the unemployment percentage means the percentage of people who are seeking jobs that are unable to gain employment.

### Student Facing

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

## 6.2: Examining Unemployment Percentages (10 minutes)

### Activity

In this activity, students examine graphs for unemployment percentages in the United States and in Michigan. Students are asked to interpret the highest and lowest points on the graph in terms of the situation. In the associated Algebra 1 lesson, students look at features of graphs including the maximum and minimum.

### Student Facing

This graph shows the percentage of the workforce that is unemployed in the United States and Michigan for several years.

1. For the United States, what are the highest and lowest points on the graph? What do the points mean in the situation?
2. For Michigan, what are the highest and lowest points on the graph? What do the points mean in the situation?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to examine graphs to relate important points such as the maximum and minimum to the situation described by the graphs. Select students to share their solutions. Ask students:

• "Are there other features that you think are important besides the highest and lowest points on the graph?” (In 2005, there is another low point for Michigan’s unemployment percentage even if it is not the lowest on the entire graph. There is also something significant happening during 2008 since there is such a great jump during that year.)
• “Why do you think it might be important to pick out the highest and lowest values on a graph like this?” (It gives an idea of the range of values that are possible and can lead to an interesting discussion about the circumstances surrounding those points. For example, why is the maximum for these graphs around 2009 and 2010? What was happening in the US and Michigan during those years?)

## 6.3: The Wire (20 minutes)

### Activity

In this activity, students examine graphs to select interesting features. At this stage, it is not important for students to describe a set list of features. Anything that is interesting to the students should be honored. In the associated Algebra 1 lesson, students will develop a more concrete list of features to look for in a graph.

### Student Facing

1. Use technology to graph the function $$f(x) = x^4 - 16x^3 + 86x^2 - 176x + 105$$.
2. What are some points on the graph that you think are interesting? Explain your reasoning.
3. Examine the graph representing electrical voltage in a wire as a function of time. What interesting points do you see? Explain your reasoning.

4. Use the points you found to describe what is happening to the voltage within the wire.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to collect information about what students may find interesting about the graphs. Select students to share the features they found interesting. For each question, press students to share any additional answers they have. There may be points that some people found interesting and others did not, and there may be different ways to describe the voltage within the wire (for example, by referencing the duration, referencing the point, referencing the rate at which the voltage changes, or other methods).