Unit 4, Lesson 2
Writing Equations to Model Relationships (Part 1) (Optional)
Integrated Math 1
2.1
Warm-up
Standards Alignment
Building On
6.RP.A.3.c
Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
Addressing
Building Toward
Materials
NoneActivity Narrative
This Math Talk focuses on finding a percentage of a value. It encourages students to think about fractions, decimals, and the meaning of percent to mentally solve problems and rely on the structure of finding different percentages of the same value to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students calculate prices that involve a percent increase and write an equation to generalize the calculation.
Students are likely to approach the problems in different ways. They may:
- Convert each percentage into a fraction and multiply the fraction by 200. For example, they may think of 25% as , 12% as or , and 8% as or .
- Convert each percentage into a decimal and multiply it by 200.
- Notice that 1% of 200 is 2, and that any percentage of 200 can be found by multiplying the percentage by 2. For example, 25% of 200 is , and % of 200 is , or .
- Use that % of is the same as % of to find that % of 200 is equivalent to doubling .
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneEvaluate mentally.
- 25% of 200
- 12% of 200
- 8% of 200
- % of 200
Student Response
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?"
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Advances: Speaking, Representing
2.2
Activity
Instructional Routines
MLR2: Collect and DisplayMaterials
NoneActivity Narrative
This activity gives students an opportunity to examine multiple quantities and relationships in a geometric context, and to use letters to represent quantities.
As students analyze the number of faces, vertices, and edges in several Platonic solids and try to identify relationships, they practice looking for structure (MP7). Some of the relationships could be represented by inequalities. One particular relationship can be represented by equations, setting the stage for upcoming work on equivalent equations.
If work time is coming to an end and no students are able to find an equation that relates the parts of the Platonic solids, suggest that students try adding the vertices and faces in each row.
Launch
Ask students to keep their books or devices closed.
Display the images of the three Platonic solids. If physical polyhedra are available, consider displaying them as well. Ask students: "In what ways are the three figures alike? In what ways are they different?"
Students may say that the figures are alike in that:
- They are all three-dimensional figures.
- Each Platonic solid has only one kind of polygon for its faces (triangle for the tetrahedron, square for the cube, and pentagon for the dodecahedron).
- The faces of each figure seem to be regular polygons (or polygons whose sides are equal in length).
- They all have an even number of faces, vertices, and edges.
They may say that the figures are different in that:
- Each figure has a different polygon for its faces (a triangle for the tetrahedron, a square for the cube, and a pentagon for the dodecahedron).
- They all have a different number of faces, vertices, and edges.
If students refer to edges and vertices as “lines” and “points," ask if they remember the “math names” for these things. Review the terms "vertices," "edges," and "faces" as needed.
Tell students that they will now investigate the relationships between the faces, vertices, and edges in each polyhedron.
Conversing: MLR2 Collect and Display. Collect the language that students use to describe how the three Platonic solids are alike and different. Display words and phrases such as "vertices," "edges," and "faces." During the synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.
Advances: Conversing, Reading
Advances: Conversing, Reading
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of the table describing the platonic solids. Terms may include "vertices," "edges," and "faces."
Supports accessibility for: Conceptual Processing, Language
Supports accessibility for: Conceptual Processing, Language
2.3
Activity
Instructional Routines
MLR1: Stronger and Clearer Each TimeMaterials
NoneActivity Narrative
In this activity, students write equations to represent quantities and relationships in two situations. In each situation, students express the same relationship multiple times: initially using numbers and variables and later using only variables. The progression helps students see that quantities can be known or unknown, and can stay the same or vary, but both kinds of quantities can be expressed with numbers or letters.
Launch
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to the final question. Invite listeners to ask questions and give feedback that will help their partners clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive.
Advances: Writing, Speaking, Listening
Advances: Writing, Speaking, Listening
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 3–4 situations to work with. Chunking this task into more manageable parts may also benefit students who benefit from additional processing time.
Supports accessibility for: Organization, Attention, Social-Emotional Functioning
Supports accessibility for: Organization, Attention, Social-Emotional Functioning
2.4
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
This activity gives students another opportunity to represent a relationship with numbers and letters, to reason repeatedly, and to see more clearly that equations are helpful for generalizing a relationship.
From the given descriptions, students are aware that there are four relevant quantities in the car purchase situations. In each situation, the value of at least one quantity is not given, creating a need for students to name it, either using a word or a phrase (for instance, "total price" or "original price") or to use a variable (for example, or ). Some students might choose to use a symbol. Monitor for the different ways students represent these quantities.
Lesson Synthesis
To help students synthesize their work in the lesson, consider asking them to write a response to one or both of the following prompts:
- We could use numbers or letters to represent the quantities in a situation. When might it make sense to use only numbers? When might it make sense to use letters?
- You've heard the phrases "a quantity that varies" and "a quantity that stays constant" in this lesson. Describe what they mean in your own words. If possible, give an example of a situation that has a quantity that varies and a quantity that stays constant.
Student Lesson Summary
Suppose your class is planning a trip to a museum. The cost of admission is \$7 per person,
and the cost of renting a bus for the day is \$180.
- If 24 students and 3 teachers are going, we know the cost will be ,
or , dollars. - If 30 students and 4 teachers are going, the cost will be dollars.
Notice that the numbers of students and teachers can vary. This means that the cost of admission and the total cost of the trip can also vary, because they depend on how many people are going.
Letters are helpful for representing quantities that vary. If represents the number of students who are going, represents the number of teachers, and represents the total cost, we can model the quantities and constraints by writing
Some quantities may be fixed. In this example, the bus rental costs \$180 regardless of how many students and teachers are going (assuming only one bus is needed).
Letters can also be used to represent quantities that are constant. We might do this when we don’t know what the value is, or when we want to understand the relationship between quantities (rather than the specific values).
For instance, if the bus rental is dollars, we can express the total cost of the trip as . No matter how many teachers or students are going on the trip,
dollars need to be added to the cost of admission.