Unit 5, Lesson 1
Proportional Relationships and Equations
Accelerated 6
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This is the first Which Three Go Together? routine in the course. In this routine, students are presented with four items or representations and asked: “Which three go together? Why do they go together?”
Students are given time to identify a set of three items, explain their rationale, and refine their explanation to be more precise or find additional sets. The reasoning here prompts students to notice common mathematical attributes, look for structure (MP7), and attend to precision (MP6), which deepens their awareness of connections across representations.
This Warm-up prompts students to compare four expressions. In making comparisons, students have a reason to use language precisely. The activity also enables the teacher to hear the terminologies students know and how they talk about characteristics of algebraic expressions.
Before students begin, consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. Signals might include a thumbs-up or a certain number of fingers that tells the number of responses they have. Using such subtle signals is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.
Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three expressions that go together and can explain why. Next, tell students to share their response with their group, and then together find as many sets of three as they can.
Which three go together? Why do they go together?
A
B
C
D
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “sum,” “product,” “factor,” “term,” “expression,” “equation,” “variable,” or “unknown,” and to clarify their reasoning as needed. Consider asking:
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In this activity, students write an equation to represent a proportional relationship between distance and time. The context of airplane flight is similar to that of a previous activity, but not exactly the same. This activity prompts students to move back and forth between the abstract representation and the context (MP2) as they create an equation and then use it to find other values that aren’t in the table. This task increases the level of difficulty by having so much missing information and by using decimals in the table.
There are various ways students may approach the last question. Monitor for students who:
Plan to have students present in this order to support moving them from arithmetic methods towards algebraic methods.
Tell students that this activity revisits the context of flying in an airplane but it is not exactly the same situation as the airplane activity in the earlier lesson.
Select students with different strategies, such as those described in the Activity Narrative, to share later. Aim to elicit both key mathematical ideas and a variety of student voices, especially from students who haven't shared recently.
A plane flew at a constant speed between Denver and Chicago. It took the plane 1.5 hours to fly 915 miles.
| time (hours) | distance (miles) |
|---|---|
| 1 | |
| 1.5 | 915 |
| 2 | |
| 2.5 | |
The purpose of this discussion is to show the value of finding an equation that represents a proportional relationship.
Ask previously selected students to share their solutions to the last question (the distances the plane would travel in 3 and 3.5 hours). Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see.
Connect the different responses to the learning goals by asking questions such as:
If not mentioned by students, display the equation
The key takeaways are:
Help us improve by sharing suggestions or reporting issues.
If students have trouble representing each relationship with an expression, encourage them to draw diagrams or to describe the relationship in words.
Students who are having trouble understanding the task can draw a segment between Denver and Chicago on the map and label it with the distance and the time. Consider prompting them to find the speed in miles per hour (the distance the plane travels in 1 hour at this speed). Then ask students how this number can help them complete the table and answer the questions.