Curriculum  /  Math  /  9th Grade  /  Unit 1: Functions, Graphs and Features  /  Lesson 7

Functions, Graphs and Features

Lesson 7

Math

Unit 1

9th Grade

Lesson 7 of 11

Objective

Analyze the key features of a contextual situation and model these graphically.

Common Core Standards

Core Standards

  • N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.
  • F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
  • F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
  • F.IF.B.6 — Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

  • 8.F.B.5

Criteria for Success

  1. Translate features presented in a contextual situation by identifying important points in the situation. 
  2. Use the rate of change over an interval to graph a situation. Note descriptions such as fast, slow, constantly, increasing, and decreasing. 
  3. Interpret a graph in the context of a situation, and use sketches of graphs to illustrate contextual situations. 

Tips for Teachers

  • In terms of pacing, this lesson may be spread over two days. The Desmos activities “Function Carnival” and “Function Carnival Part Deux” (noted in the Problem Set Guidance) are fabulous for wrapping up the ideas of modeling situations based on what you observe about the features of a situation. There is a teacher demo of Function Carnival here.
  • The following resources may be helpful in determining the structure you use for Function Carnival, if you choose to use it. 
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Anchor Problems

25-30 minutes

Problem 1

The figure shows the graph of $$T$$, the temperature (in degrees Fahrenheit) over one particular 20-hour period in Santa Elena, as a function of time,  $$t$$. 

  1. Estimate $$T(14)$$. 
  2. If  $$t=0$$ corresponds to midnight, interpret what we mean by  $$T(14)$$ in words. 
  3. Estimate the highest temperature during this period from the graph. 
  4. When was the temperature decreasing? 
  5. If Anya wants to go for a two-hour hike and return before the temperature gets over 80 degrees, when should she leave? 

Guiding Questions

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References

Illustrative Mathematics Warming and Cooling

Warming and Cooling, accessed on June 22, 2017, 1:11 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem 2

Below is an illustration of an aquarium. When the faucet is on, the water flows into the aquarium at a constant rate. When the plug is pulled out, the water drains at a constant rate (but slower than the faucet’s rate). At various times, some events happen that affect the water level and/or the rate at which the water level changes. 


Draw a graph of the water level as a function of time for each of the following situations: 

  1. The aquarium is initially empty with the plug in, and water flows in at a constant rate for 10 minutes. 
  2. The aquarium is initially half full with the plug in. Nothing happens for 5 minutes, then somebody pulls the plug. 
  3. The aquarium is half full and then a bucket of water is dumped into the aquarium. 
  4. There is a rock in the half-filled aquarium for the first 7 minutes, then it is removed. 
  5. The aquarium is initially empty and then the water faucet is turned on with the plug out.

Guiding Questions

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References

Illustrative Mathematics The Aquarium — Part 1

The Aquarium, accessed on June 22, 2017, 1:09 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem Set

15-20 minutes

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

5-10 minutes

Below is an illustration of an aquarium. There are four possibilities of changes that affect the water level: addition/removal of a rock, insertion/removal of a plug, turn on/off the faucet, use the bucket to add/remove water. 

The graph below shows the height of the water in the aquarium over a 17-minute time interval. Write a story from the point of view of someone watching what is going on with the aquarium to produce the given graph. 

The graph given above is a simplified version of reality. What simplifying assumptions did we make when we created the graph? 

References

Illustrative Mathematics The Aquarium — Parts 2 and 3

The Aquarium, accessed on June 22, 2017, 1:09 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Next

Draw quadratic functions represented contextually. Identify key features of the graph and relate to context.

Lesson 8
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Features of Functions

Topic B: Nonlinear Functions

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