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Unit 3, Lesson 3

Associations and Relative Frequency Tables

Extra Support Materials for Algebra 1

3.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Aspects of Mathematical Modeling Estimation

Materials

None

Activity Narrative

The purpose of this Warm-up is to practice the skill of estimating a reasonable answer based on experience and known information, and also to help students develop a deeper understanding of the meaning of standard units of measurement. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking, “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. 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. Invite students to record these numbers and to write a short explanation for the reasoning behind their estimate.

Activity

None

What percentage of the graph is labeled C?

<p>Circle divided into sections. A and B are same size, each about 33% of circle. C, D, and E are same size and together take up rest of circle </p>

Record an estimate that is:

too low about right too high
Explain your reasoning.

Student Response

to access Student Response.

Activity Synthesis

The goal of this discussion is for students to analyze the reasonableness and accuracy of their estimates.

Ask a few students to share their estimates and their reasoning. If a student is reluctant to commit to an estimate, ask for a range of values. Display these responses 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 ? Is anyone’s estimate greater than ?” If time allows, ask students, “After this discussion does anyone want to revise their estimate?”

Then reveal the actual value, and add it to the display.

Encourage students to think about and discuss the accuracy of their estimates and the estimates that are displayed by asking questions such as:

"Is the actual value inside the range of values?"
"Is it toward the middle?"
"How variable are the estimates?"
"What are the sources of error?"

Consider developing a method to record a "snapshot" of the estimates and actual value so that students can, over time, track their progress as estimators.

3.2

Activity

Instructional Routines

None

Materials

None

Activity Narrative

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. Students attend to precision when making decisions about rounding their calculations (MP6).

Launch

In this activity, students answer questions about three relative frequency tables. For the second table, 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.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color-code the column or row being used.
Supports accessibility for: Visual-Spatial Processing

Activity

None

3.3

Activity

Instructional Routines

MLR3: Critique, Correct, Clarify

Materials

None

Activity Narrative

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 prepares students for work in the associated Algebra 1 lesson when they create pairs of variables that are and are not associated, and when they create data to represent each. As students explain why pairs of variables do or do not have an association, they construct viable arguments and critique the reasoning of others (MP3).

This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.

Launch

Display the following examples of variables that have an association and ones that do not.

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 the plants

Variable with no association:

Mobile device preference and eye color
Taking a visual art elective or not and mode of transportation to school

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response explaining how each pair of variables does or does not have an association, by correcting errors, clarifying meaning, and adding details.

Display this first draft:
The season and whether half or more of the people outside are wearing jackets or not does have an association because I wear a jacket when it is cold.
Mobile device preference and eye color does not have an association because they are different things.
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

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.

The relative frequency tables display data collected from 230 students.

	participates in after school activity	no after school 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%

What percentage of students participate in after school activities? How many students participate in after school activities?
What percentage of students arrive home 2 or more hours after dismissal? How many students arrive home 2 or more hours after school dismissal?

	aspiring professional athlete	aspiring STEM career	total
prefer physical education	77%	23%	100%
prefer math	18%	82%	100%

What percentage of students who prefer math aspire to have a career in STEM?
What percentage of students who prefer physical education aspire to have a career in STEM?
Is there evidence of an association between students’ career aspirations and subject preference? Explain your reasoning.

9th grade 12th grade

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.

	9th grade	12th grade
curfew	95%	90%
no curfew	5%	10%
total	100%	100%

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Lesson 3

Associations and Relative Frequency Tables

Warm-up

Standards Alignment

Building On

6.RP.A.3.c

Addressing

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Building on Student Thinking

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

6.RP.A.3.c

Addressing

HSS-ID.B.5

Building Toward

Standards Alignment

Building On

8.SP.A.1

Addressing

HSS-ID.B.6

Building Toward