a1tD0000003mBioIAE

Course: High School Math with NAO
Linear, Quadratic, and Exponential Functions

  • 9-12 grade
  • Intermediate

Lesson Description:

In this lesson, students will be exposed to techniques for evaluating a relationship between variables as linear, quadratic and exponential.  Students will get practice in determining if a relationship is linear or quadratic within the context of a racing game.
 


 

Standards Covered

CCSS.MATH.CONTENT.HSF.LE.A

Construct and compare linear, quadratic, and exponential models and solve problems.

image description

Lesson Modules


Teaching Tips:

Act 1 (15 minutes): Teacher Modeling

Concept


Introduce two NAO robots, of one them is called ‘Linear’ and other is called ‘Quad’.  One of them is placed along a ‘race-track’, 3-4 feet from the finish line. Explain that both robots get ‘fueled’ by answering their challenge. Their challenge is an x value and the answer should be according to a linear or quadratic equation respectively.  For each correct answer, the robot will make forward progress. 
 
The key material conveyed in this act is that the output of a linear function changes an amount proportional to the change of the input.  For example, if a set of (Input, Output) pairs are (1, 2),  (2, 4), (3,6), we can see that for a 1 unit change in input (1 à 2 à 3), we always get the same change in output, in this case, 2 (2à4à6).  In other words, the ‘first difference” of the output remains the same.  This makes this is a linear function.  This is shown in the table below:
 
 

xf(x)1st difference
12
242
362
482

 
 
For a quadratic function, we look at the ‘second difference’.  This is the change of the change, and if that is the same, then the function is quadratic.  An example is shown below:
 

xf(x)1st difference2nd difference
26

4104
61773
827103
1040133

 

Robot Demo

The initial value is always (1,1) for both functions. To demonstrate, the first robot calls out a number (i.e., 2).  The teacher responds with 3.  The robot makes forward progress.  The robot then calls out another number (i.e., 3).  The teacher then responds with 5 and the robot makes forward progress again.  The robot then says another number (i.e., 4).  The teacher responds with 6 – which is incorrect – (the correct answer is 7) and the robot makes a buzzing sound and does not move.
 
Explain that the responses have to match a linear or quadratic equation, and explains the material.
 
 

Robots are going to race!

image source from https://static.nascar.com/content/dam/nascar/articles/2017/2/24/main/daytona-sellout.jpg/jcr:content/renditions/original


The idea is that all robots make the same progress when their question is answered correctly.  This is similar to the horse game in the fair, where participants throw balls into bins and the higher the bin the more progress the horse makes.  In this case, all progress is the same (so there are no advantages to linear/expo etc.) but the robots only make one step progress when the user responds to their query according to what function they are patterned after.
 

Your teacher will introduce two NAO robots, of one them is called ‘Linear’ and other is called ‘Quad’.  One of them is placed along a ‘race-track’, 3-4 feet from the finish line. Both robots get ‘fueled’ by answering their challenge.  Their challenge is an x value and the answer should be according to a linear or quadratic equation respectively.  For each correct answer, the robot will make forward progress.  
 

 
Observe your teacher's demonstration.
 
 


Teaching Tips:

Act 2 (15 minutes) : Practice


The students are asked to identify this function as either Linear, Quadratic or Exponential.  They can do this by calculating first differences (i.e., 12-8 = 4,  16-12 = 4, 20-16 = 4).  Since the first differences are the same, this is a linear function.  If it was not linear, the students can calculate the second difference, and if they are equal, then the function is quadratic.  Otherwise, it’s exponential.
 
The system goes through several iterations of examples allowing students to practice.
 
 

With your team members, pick a strategy that works best for you and share what you have found with the class.


x 2 3 4 5
y 8 12 16 20


Use the data and figure out if the given function is linear, quadratic or exponential. You can find the 1st and 2nd differences or plot the graph. Use this Math tool as needed.



Identify this function as either Linear, Quadratic or Exponential. 
  • Linear
  • Quadratic
  • Exponential
  • None of above

Now, look at this set of data. Is this function linear, quadratic, or exponential?


x 4 6 8 10
y -1 0 2 4

Identify this function as either Linear, Quadratic or Exponential. 
  • Linear
  • Quadratic
  • Exponential
  • None of above

 


Try one more time with the data below.


x -4 -2 0 1 2
y 0.06 0.25 1 2 4


Identify this function as either Linear, Quadratic or Exponential. 
  • Linear
  • Quadratic
  • Exponential
  • None of above

 


Teaching Tips:

Act 3 (20 minutes): Race
In this activity, two robots are set side by side.  The Linear one expects responses to follow a linear equation and the Quad one expects responses to follow a quadratic equation.  For every correct answer, the robot makes forward progress.  The robot to first cross the finish line wins.
 
The robots start by sounding out the initial X and Y (i.e., 2, 5).  Then, for each challenge they sound out (which represents sequential X values), the student ‘driver’ has to respond with an answer that maintains the function type.  The 1st answer will always be correct but will also establish the first difference.  For a quadratic function, the second answer will also establish the second difference.
 
To help the students keep track, they can create a small sheet like the one below:
 
X        
Y        
1st difference        
2nd difference        
 
 
 
In this activity, two robots are set side by side.  The Linear one expects responses to follow a linear equation and the Quad one expects responses to follow a quadratic equation.  For every correct answer, the robot makes forward progress.  The robot to first cross the finish line wins.
 
The robots start by sounding out the initial X and Y (i.e., 2, 5).  Then, for each challenge they sound out (which represents sequential X values), the student ‘driver’ has to respond with an answer that maintains the function type.  The 1st answer will always be correct but will also establish the first difference.  For a quadratic function, the second answer will also establish the second difference.
 
To help you keep track, you can create a small sheet like the one below:
 
X        
Y        
1st difference        
2nd difference        
 
 
 

Teaching Tips:

 Close: (5 mins)   
Students will respond to the question explaining how to determine if an equation is either linear or quadratic

Explaining how to determine if an equation is either linear or quadratic.