The Objective for This Lesson on Functions Is, the Student Will Develop an Understanding

FUNCTIONS

INTRODUCTION

The objective for this lesson on Functions is, the student will develop an understanding of functions using equations, graphs, tables and ordered pairs.

The skills students should have in order to help them in this lesson include patterns and plotting points.

We will have three essential questions that will be guiding our lesson. Number one, what is a function? Number two, how can we determine if a set of ordered pairs is a function? Justify your thinking. And number three, how do we determine if the graph of a line is a function? Explain and defend your answer.

We will begin by completing the warm-up on completing patterns to prepare for functions in this lesson.

SOLVE PROBLEM – INTRODUCTION

The SOLVE problem for this lesson is, Sarah is babysitting to earn extra money during the summer. She babysits for three different families and charges six dollars and fifty cents per hour. When she babysits two hours, she makes thirteen dollars. When she babysits for four hours she makes twenty six dollars. How much money will she make babysitting for seven hours?

We will begin by Studying the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, how much money will she make babysitting for seven hours?

Now that we have identified the question, we want to put this question in our own words in the form of a statement. This problem is asking me to find the amount she earns in seven hours.

During this lesson we will learn how to identify functions using tables, graphs, ordered pairs and equations. We will use this knowledge to complete this SOLVE problem at the end of the lesson.

MODELING WITH THE FUNCTION MACHINE

Take a look at the drawing seen here. We will refer to this drawing as a function machine. What is the word at the top of the drawing? It is the word input. What is the meaning of the word input? An input can be something that is put into the machine. What is the word at the bottom of the drawing? It is the word output. What is the meaning of the word output? An output can be something that will come out of the bottom of the machine.

Do you have any experiences or knowledge of an assembly line? One example is an auto assembly line. If you have an assembly line where a car is being put together, each station has a particular job, such as putting the windshield on. With each car, you put the windshield on only at that station. A function machine is similar in that it does the exact same thing each time.

Take a look at the function machine now. What is the operation inside the function machine? It says we will add three. Explain the meaning of the function – add three. This means that the value put into the machine goes down into the function and a value of three is added. The number comes out of the function machine as an output.

What can we use to model the function of add three? Explain your answer. We can use three yellow tiles because they represent adding or positive three. Let’s place three yellow tiles inside of the machine to show the function of adding three.

Now what is the first input in the chart? Let’s take a look at the chart. The first input is three. Since the first input is three, how can we model the input of three? We can use three yellow unit times. Let’s represent the three yellow unit tiles for the input at the top of the function machine.

Now what can we do to model the addition of the three yellow unit tiles? We can push them together with the input. Let’s do that now. After we push together the unit tiles, what comes out of the function machine as our output? There are six yellow tiles, which is equal to positive six. Let’s include this information in our chart. When the input is three, the output is six.

Now what happens when we input two? How can we model the input of two? We can use two yellow unit tiles. Let’s represent two yellow unit tiles as our input for the function machine. What can we do to model the addition of the three yellow unit tiles to our input? We can push them together with the input. Let’s do this now. After we push together the unit tiles, what comes out of the function machine as our output? We have five yellow tiles, which is equal to positive five. Let’s record this information in our chart. When we have an input of two the output is five.

Now let’s look at what happens when we input one. How can we model the input of one? We can use one yellow unit tile. Let’s represent one yellow unit tile as our input on our function machine. What can we do to model the addition of the three yellow unit tiles inside the function machine? We can push them together with the input. Let’s do this now. After we push together the unit tiles, what comes out of the function machine as our output? This time the output is four yellow unit tiles, which is equal to positive four. Let’s record this information on our chart. With an input of one the output is four.

Now let’s look at what happens when the input is zero. How can we model the input of zero? We represent zero tiles as the input. What can we do to model the addition of three yellow unit tiles? We need to push them together with the input of no tiles. Let’s do this now. Since there is not an input we will move the three yellow unit tiles to the center of the function machine. After we push together the unit tiles, what comes out of the function machine as our output? We have three yellow tiles, which is equal to positive three. Let’s record this information in our chart. When the input is zero, the output is three.

Now our next input is negative one. What happens when there is an input of negative one? How can we model the input of negative one? Let’s model out input of negative one with one red tile at the top of our function machine. Now what can we do to model the addition of the three yellow unit tiles? As we have done with the previous examples we will push them together with the input. Let’s do this now. When we input one red tile and push it together with the three yellow unit tiles, what happens? We create a zero pair. What do we do with the zero pair that has been created? The value of the red and yellow unit tile is zero so when we take the red and yellow out of the machine, let’s do that now, it leaves an output of two yellow or positive two. Let’s record this in our chart. With an input of negative one, the output is positive two.

Now what happens when the input is negative two? Let’s see. How can we model the input of negative two? This time we will use two red tiles. Let’s represent the input of negative two in our function machine using two red tiles. What can we do to model the addition of the three yellow unit tiles? We push them together with the input. Let’s do this now. When we input two red tiles and push them together with three yellow unit tiles, what happens? We create two zero pairs. What do we do with the zero pairs that have been created? The value of the red unit tile and the yellow unit tile is zero so when we take the red and yellow out of the machine two times, let’s do this now, one, two zero pairs, it leaves an output of one yellow or positive one. Let’s record this in out chart. With an input of negative two, our output is one.

Next we have an input of negative three. Hmm. Take a look at these input values that are coming up next. What do you notice about the input values after negative two in the table? The value of negative three is listed three times. What do you predict about the output for each input of negative three? It will be the same each time. Let’s see if this is true. Let’s find out what happens when we input negative three into the function machine. How can we model the input of negative three? We can use three red tiles. Let’s do this now. What can we do to model the addition of the three yellow unit tiles? We push them together with the input. When we input three red tiles and push them together with the three yellow unit tiles, what happens? We create zero pairs. What do we do with the zero pairs that have been created? The value of the red unit tile and the yellow unit tile is zero so when we take the red and yellow out of the machine three times, let’s do this know, it leaves an output of zero. Let’s record this in our chart. With an input of negative three the output of the function is zero.

So let’s see what’s going to happen the next time that we input the value of negative three. How can we model the input of negative three? The same as we did the last time. We used three red tiles. Let’s do this now. What can we do to model the addition of the three yellow unit tiles? We can push them together with the input. When we input three red tiles and push them together with the three yellow unit tiles, what happens? We create zero pairs. What do we do with the zero pairs that have been created? The value of the red unit tile and the yellow unit tile is zero so when we take the red and yellow out of the machine three times, let’s do this now, it leaves an output of zero. Let’s record the output for the second time that we input negative three into the chart. When we input negative three into our function machine the second time we got an output of zero again.

Now we will try one more time to see if the get the same output again, when we input negative three into the function machine. How can we model the input of negative three? Three red unit tiles. What can we do to model the addition of the three yellow unit tiles? Push them together with the input. When we input three red tiles and push them together with the three yellow unit tiles, what happens? We create zero pairs. So what do we do with the zero pairs that have been created? The value of the red unit tile and the yellow unit tile is zero so when we take the red and yellow out of the machine three times, it leaves an output of zero. Once again, we input negative three into our function machine the output is zero.

So let’s check our prediction. Was the prediction correct? Yes, when the input was negative three the output was always zero.

Let’s take a look at the function machine in Problem two!

What is our function for Problem two? Our function is to add negative five.

How can we represent the operation of adding negative five? We can use five red unit tiles inside the function machine. Let’s do this now.

What is the first input in the chart for Problem two? Let’s take a look at the chart now. The first input is negative three. So how can we represent negative three as an input? We can use three red tiles. Let’s place three red tiles as the input on our function machine. How do model adding negative five? We push the tiles together. Let’s do this now. What is our output after we push the tiles together? We have a total of eight unit tiles or negative eight. So let’s record this in our chart. For an input of negative three we get an output of negative eight.

Complete the table by using the function machine and the input values. Go ahead and do this now.

Now let’s review what the output is for each of the inputs into this function machine. Using the function of adding negative five, with an input of negative two the output is negative seven. With an input of negative one the output is negative six. With an input of zero the output is negative five. With an input of one the output is negative four. With an input of two the output is negative three. With an input of three the output is negative two. If we input three again the output is again negative two. And if we input three one more time we will again get an output of negative two.

This completes the chart for the function in Problem two. We will refer back to this chart again later in the lesson.

FUNCTION MACHINE AND TABLES

Now let’s go back and take a look at the table from Problem one that we completed earlier in the lesson.

Explain the process we followed when using three as the input value for our function machine. We placed three yellow unit tiles in the function machine to represent the function of adding three. We then put three yellow unit tiles into the machine as the input and pushed them together with the three yellow tiles in the machine. This gave us an output of six yellow tiles.

Describe the mathematical process we followed. We added positive three and positive three for a sum of positive six.

If we go back to the table, what is our output for an input of three? It is six. Record the value in the table if you have not done so already.

If our input is positive two to the function of add three, what is the output? It is five.

Be sure to fill in the rest of the output values in the table if you did not do so in the previous section.

What do you notice about the outputs when the input was negative three? Each time the output was zero. Explain your thinking. The output was the same when the input was the same.

Now take a look at the table for Problem two. What do you notice about the outputs when the input was positive three? Each time the output was negative two. Explain your thinking. The output was the same when the input was the same.

So what did you discover about the function relationships? A function is a relationship where every input has a unique output.

What did we notice in the function table for Problem one? Each time we input a number, we added three and this gave us an output.

What if we want to create a rule or formula we can use to represent this explanation so that it would work for any number? We can use a variable to represent the input. Justify your response. A variable is a letter that we can use to represent any number. So we can use a variable to represent to input, since the input can be any number.

How can we represent this in our function table? We can write an x in the input column.

Let’s take a look at the table for Problem one. At the bottom of the column for input place the variable x, so that this would represent any number that we would use as the input to the function.

Explain what we did each time with the input value. We added three.

How could we write this as an output using the variable? We can write this as x plus three. Let’s record this in our table. When the input is x, the output is x plus three.

Now what letter represents our output in the table? The output is represented by the variable y. If our output is equal to the input plus three, how could we write that using the variables x and y? We can say that y is equal to x plus three.

Let’s go back to our table. The input is the x-value and the output is the y-value.

Now let’s talk about Problem two. What did we notice in the function table for Problem two? Each time we input a number, we added negative five and this gave us an output.

What if we want to create a rule or formula we can use to represent this explanation so that it would work for any number? We can use a variable to represent the input. Justify your response. A variable is a letter that we can use to represent any number.

So how can we represent this in our function table? We can write an x in the input column. This will tell us that we can use any value as an input to our function. At the bottom of the input column for our table we will record the variable x, to show we can have any value for the input into our function.

Now explain what we did each time with the input value. We added negative five.

So show could we write this as an output using the variable? We can say that we have x plus negative five. Let’s record this as the output for this function table. When the input is x, the output is x plus negative five.