# Lesson 3

Associations and Relative Frequency Tables

These materials, when encountered before Algebra 1, Unit 3, Lesson 3 support success in that lesson.

## 3.1: Estimation (5 minutes)

### Warm-up

The purpose of an Estimation warm-up is to practice the skill of estimating a reasonable answer based on experience and known information. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking yourself “Does this make sense?” is a component of making sense of problems (MP1), and making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).

### Launch

Display the image for all to see. Ask students to silently think of a number they are sure is too low, a number they are sure is too high, and a number that is about right, and write these down. Then, write a short explanation for the reasoning behind their estimate.

### Student Facing

What percentage of the graph is labeled C?

1. Record an estimate that is:
too low  about right   too high

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask a few students to share their estimate and their reasoning. If a student is reluctant to commit to an estimate, ask for a range of values. Display these for all to see in an ordered list or on a number line. Add the least and greatest estimate to the display by asking, “Is anyone’s estimate less than $$\underline{\hspace{.5in}}$$? Is anyone’s estimate greater than $$\underline{\hspace{.5in}}$$?” If time allows, ask students, “Based on this discussion does anyone want to revise their estimate?” Then, reveal the actual value and add it to the display. Ask students how accurate their estimates were, as a class. Was the actual value inside their range of values? Was it towards the middle? How variable were their estimates? What were the sources of the error? Consider developing a method to record a snapshot of the estimates and actual value so that students can track their progress as estimators over time.

## 3.2: Relative Frequency Tables (20 minutes)

### Activity

The purpose of this activity is for students to recall how to read a relative frequency table and to determine if variables have an association. This prepares students for work in the associated Algebra 1 lesson when they create and interpret relative frequency tables.

### Launch

Students should have an understanding of what STEM careers are. STEM careers include careers in the fields of science, technology, engineering, math, and medicine. Ask students, “What are some examples of careers in STEM?” Responses include: doctors, computer scientists, civil engineers, computer scientists, and architects.

### Student Facing

The relative frequency tables display data collected from 230 students.

1. participates in afterschool activity  no afterschool activity  total
arrives home within 2 hours of school dismissal 3% 40% 43%
arrives home 2 or more hours after school dismissal 42% 15% 57%
total 45% 55% 100%
1. What percentage of students participate in after-school activities? How many students participate in after-school activities?
2. What percentage of students arrive home 2 or more hours after dismissal? How many students arrive home 2 or more hours after school dismissal?
2.  aspiring professional athlete  aspiring STEM career  total
prefer physical education 77% 23% 100%
prefer math  18% 82% 100%
1. What percentage of students who prefer math aspire to have a career in STEM?
2. What percentage of students who prefer physical education aspire to have a career in STEM?
3. Are these two percentages close?
4. Is there evidence of an association between students’ career aspirations and subject preference? Explain your reasoning.
curfew 95% 90%
no curfew 5% 10%
total 100% 100%
1. Of the students in 12th grade, what percentage have a curfew?
2. Of the students in 9th grade, what percentage have a curfew?
3. Is there evidence of an association between students’ grade level and whether they have a curfew? Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Discuss how students interpreted the tables and how they decided if there is an association. Remind students that a decimal for the number of students is unrealistic, since there cannot be a fraction of a person. It is approporiate for students to round down to the nearest whole number for this reason. Here is a question for class discussion:

• "How do you determine if there is an association between the two variables?"(Find the percentage of each category in a row or in a column. Compare the percentages for the categories to see if they are similar. If the percentages are similar, there is not likely an association. If they are very different, there is a possible association)

## 3.3: Associate Your Variables (15 minutes)

### Activity

The purpose of this activity is for students to practice creating pairs of variables that they think are associated. It is not important for students to focus on the type of association they think the variables may have. This preprares students for work in the associated Algebra 1 lesson when they create pairs of variables that are and are not associated, and create data to represent each.

### Launch

Display for all to see examples of variables that have an association and ones that do not. Allow a whole-class discussion about how each pair of variables might be associated, and why it makes sense for some variables to have no association.

Variables with an association:

• season and whether half or more of the people outside are wearing jackets or not
• growth in plant length and whether someone waters their plants

Variable with no association:

• mobile device preference and eye color
• taking a visual art elective or not and mode of transportation to school

### Student Facing

1. Invent a pair of variables that you think will have an association. Explain your reasoning.
2. Invent a pair of variables that you think will not have an association. Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Discuss students' understandings of associations. Here are sample questions for class discussion:

• "How did you determine if two variables are associated?"(I consider if the two things have any type of relationship. I also think about what would happen to one variable if the other changes at all.)
• “Can you think of two variables that might seem unrelated at first, but might actually be associated? Explain your reasoning.” (Shark attacks and whether a person has eaten ice cream recently. Although they seem unrelated, both things tend to increase in the summer and decrease in the winter.)