August 4, 2010

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In this lesson, students are introduced to the concepts of relations, scatterplots, and dependent and independent variables. They work with each of these concepts, ultimately creating their own scatterplots.

- Mathematics > General
- Mathematics > Algebra
- Mathematics > Applied Mathematics
- Mathematics > Data Analysis & Probability
- Mathematics > Equations
- Mathematics > Graphing
- Mathematics > Problem Solving

- Grade 6
- Grade 7
- Grade 8

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**Materials needed for this particular lesson:**

1. Student family portfolios.

2. Graph paper, colored pencils, and rulers.

3. TI-83+ or equivalent graphing calculator (or graphing program) and means to project it on the board (digital projector, TV connection to calculator, etc.).

4. *Example Scatterplots* Excel File (see Lesson 4 Resources folder).

5. *Creating a Scatterplot of Account Balances *worksheet (see Lesson 4 Resources folder).

6. *Example Tables and Scatterplot Examples* Powerpoint (see Lesson 4 Resources folder).

**Procedure:**

**(The following is a suggested script for the presentation of the concepts of relation, scatterplot, relationship, and function. The script is in normal type and instructions to the teacher are given in italics. All of the data mentioned in the script can be found in the Microsoft Excel file associated with this lesson and the tables and associated scatterplots can be found in the Powerpoint file associated with this lesson.)**

Today, we will be learning a new way to express certain relationships between numbers (awww, how sweet!). We will actually be exploring *relations*, which can simply be defined as the *relationship between two sets of numbers*. A set of numbers is exactly what it sounds like, *a set of numbers*. When two different sets are compared, they can be matched up into what is called a relation. For instance, the sets of a (hypothetical) class's shoe sizes and IQs follows (project this slide from the *Example Tables* Powerpoint):

Student | Shoe Size | IQ |

A | 8.5 | 120 |

B | 11 | 115 |

C | 9.5 | 80 |

D | 10 | 100 |

E | 11 | 75 |

In this hypothetical class, the shoe sizes and IQs can be expressed separately as two *sets* of numbers:

Shoe Size (S) = {8.5, 11, 9.5, 10, 11}

IQ (I) = {120, 115, 80, 100, 75}

In looking at the set of data this way, we can just focus on IQs or shoe sizes, or both. Do you see any potential relationship between shoe size and IQ? For instance, does a larger shoe size imply a higher IQ, or *vice versa*? How can you tell?

*(Lead a discussion in which you elicit the fact that their can't be any real relationship between shoe size and IQ because there are two people with the same shoe size and wildly different IQs.)*

If we wanted a graphical way to present this data, we could use what we call a *scatterplot*. This is simply a way of looking at a relation graphically. Essentially, we use a "dot" to represent each person, in terms of their attributes (in this case shoe size and IQ). If we were to plot this data set on a scatterplot (using an appropriate scale), it would look like this *(project the corresponding scatterplot from the Example Tables powerpoint on the board).*

Notice that each dot represents a person, and that there seems to be no discernible rhyme or reason as to where the dots will land on the plot.

Next, let's consider the following relation (still with the same class): *(project the Table #2 from the Example Tables powerpoint)*

Student | Height (in.) | Arm Span (in.) |

A | 72 | 70 |

B | 60 | 61 |

C | 58 | 59 |

D | 68 | 64 |

E | 64 | 67 |

Graphed (with appropriate scales on the axes), the scatterplot looks like this *(project the corresponding scatterplot).*

Notice how the dots move up in a diagonal line. What does this mean? *(Elicit that this implies that as a student's height increases, so does their arm span.)* This is called a *positive* relationship - that is, there is a general relationship or pattern that holds between people's height and their arm span.

Finally, let's consider one more mythical scatter plot for this same class. This time, we are going to show a relation where one set of numbers represents the number of minutes each student spent on social networking websites the night before the last test, and the other set of numbers represents their numerical grade for that test. What would you expect? *(Elicit that they should expect that the more time they spent socializing (and thus NOT studying), the lower their grade. Try to elicit what they expect the shape of that graph would look like.)*

Here are the data and the scatter plot. See if your predictions were correct! *(Project Table #3.)*

Student | # min. spent on social websites | Grade on last test |

A | 120 | 71 |

B | 25 | 98 |

C | 45 | 89 |

D | 70 | 82 |

E | 90 | 75 |

And the scatterplot: *(Project the corresponding scatterplot.)*

Notice that the general trend is *downward*. This is called a *negative* relationship. That is, as one set *increases *(say the amount of time NOT spent studying), the other set *decreases*.

These three scatterplots demonstrate the three different possible types of relations: *no relationship, positive relationship, and negative relationship.*

In each of these relations, there was a set that gave the values for the X-axis and a set that gave the values for the Y-axis. Let's look more closely at this:

In the first graph:

X-axis: Shoe Size; Y-axis: IQ

In the second graph:

X-axis: Height; Y-axis: Arm Span

In the third graph:

X-axis: # of minutes NOT spent studying; Y-axis: Grade

For each of these, the X-axis is the quantity that seems to *determine* the Y-quantity in some way. This is why X-axes always represent *independent variables*. That is, this is the set of data that you expect to influence the other. This set is called the *domain* of the relation. Can you name, then, the three domains for the three different relations we've looked at? (*Answer: shoe size, height, # min. spent NOT studying.)*

Notice, too, that the Y-axes for all of the relations represent sets that seem to have been influenced in some way or another by that relation's X-axis. That is, this set, the *range*, is *dependent* on the domain. That's why the quantity represented by the Y-axis is always called the *dependent variable.* Can you name the three different ranges for the three scatterplots we've looked at? *(Answer: IQ, arm span, and grade.)*

Now, we're going to create a scatterplot for some of the data from this class's families. We're going to look at the relation between families gross and net incomes. Before we begin, however, let's make some predictions. Should this be a positive or negative relationship? How can you tell? What should we make the independent variable? What should be the independent variable? How can you tell? *(Use these questions to lead a brief discussion in which you draw out that the gross income should be the independent variable because it dictates the net income. They should expect the relation to be positive. That is, the more the gross income is, the more the net income should be.)*

*(Use the class data to create a relation for the two sets of Monthly Gross Income and Monthly Net Income for the students' families. Highlight which set would be the domain and which would be the range. Generate a scatterplot and analyze it for the patterns covered in this lesson.)*

*(Next, have the students go through their family checkbooks to find the checking account balance after they deposited each paycheck. They are going to create a relation between time (the month's number in the calendar, i.e. January = 1, February = 2, etc.) and the balance after that deposit. Although this will be less clearly positive (because of the various events in the family's year), this should a general positive trend. Have this be the students' homework: creating a scatterplot that demonstrates the relation between the time and their account balance. This is described in the "Creating a Scatterplot" worksheet associated with this lesson.)*

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