Grade level: eighth grade
Specific area of Math: Algebra: adding algebraic functions
Time allowed: 45 minutes(12)
Materials needed (9): graphing calculators (class set)
graphing calculator for the overhead projector
overhead projector
class handouts of instructions for generative activity
writing utensils
Prerequisites: Knowledge of algebraic functions and equations is necessary. The students should have some knowledge of how to add algebraic functions and be able to recognize that "x" terms can not be added to whole numbers(2).
Class arrangement: The students will work on the generative activity individually. A graphing calculator will be given to each student. After the students have completed a portion of the activity, the students should be allowed to demonstrate their examples on the overhead graphing calculator for the class to see.
Generative activity: (11)
(6)To get the class started and interested in the lesson, the students should be given graphing calculators at the beginning of the lesson. The instructions should be to turn on the calculator by pressing the [ON ] key. The idea of the lesson should be explained stating that we are going to experiment with using calculators to show graphs of algebraic functions. The next instructions should be to press the [Y=] key. The students can experiment with entering in functions and graphing them. The first function will be given to them. Ex. Enter 3x. Press the [3] key then the [X, T, 0]. Then press [enter]. Press [graph] and they will see the graph of the function entered. The students could then enter a couple of their own functions to get used to the idea of graphing algebraic equations on the calculator(10).
Discussion: What kinds of graphs did you see? Did any of the functions you entered end up on the same line? If yes, then what functions made up the same line?
The students should be told that we have discussed functions that are equivalent of each other, such as: 2x= x+x. We have also discussed how to add functions. Now we will put these two concepts together.
The students will be given a function that they can practice with that is not too difficult.
For example, what two functions added together would give you 4x?(1)
Possible solutions could be: 2x +2x or 3x + x, etc.(7)
Then the students will enter their findings on the calculators.
Press [Y=]. Then press clear. All of the "Y=" should not have anything entered. Now enter 2x + 2x for "Y1". Then enter 4x for "Y2". Press [graph]. What do you observe? The two functions should form one line. Why is this so? The two functions are equivalent of each other.
Draw the graph.
After the students have experimented with an easy function, they will be given a more difficult problem.
Find two functions that when added are the same as 2x + 3(1).
The students should work on their own paper to find two functions that add up to 2x + 3. Then after they have two functions, they can plug these into the calculator to visibly see if the two graphs will again match.
Ex. 4x + 1 and -2x + 2. These two will add up to equal 2x + 3(7).
The graphs are equal when plugged into the computer.
Draw the graph.
Find three more pairs that add up to equal 2x =3. This time try using negative numbers or even fractions.(15)
The students should again practice on their calculator to see if the functions match.
They should draw what they see.
The class should have the opportunity to go up in front of the class and show their two functions that would add up to equal 2x+3(4). There should only be a limited number because of time(5).
Discussion: What do we notice about algebraic functions?(8) Is the calculator a good way to test your results?(5)
Extension: The students will make
up another algebraic function. They will find three functions that can add
up to equal their chosen function. Then they will give their function to
a partner and vice-versa. The partner will find three functions that will
equal the given function and test it on a graphing calculator(13).(3)