USING SPREADSHEETS TO DISCOVER LINEAR PATTERNS



Roberta Jones, Central High School, Champaign, Illinois


Level: Pre-algebra or Algebra Students

INTRODUCTION



When introducing linear functions in a beginning class, it is important to emphasize to students that these types of functions all have a commonality--a constant rate of change. This concept is difficult for some students to understand.
Using a spreadsheet, this lesson asks the students look at tables and scattergrams which exhibit this property and tables and scattergrams which don't. They will be asked to try various sets of ordered pairs and check to see if they form a straight line. Then they will be asked to see if they can discover what kinds of ordered pairs form straight lines and which kind don't. This experience will be useful when the more formal discussion of linear functions begins.

By establishing this notion before introducing the formal concept, students who have difficulty understanding functions should be able to better make connections. Also, by trying different ordered pairs themselves, the idea should make more sense to them. Making them verbalize the rule will give them practice in communication of mathematical concepts.

Link to the NCTM Standards: This project is linked to three of the standards--
  1. Mathematics as communication
    a. The students will reflect upon and clarify their thinking about mathematical ideas and relationships
    b. The students will express a generalization discovered through investigation.
  2. Mathematics as reasoning
    The students will make and test conjectures
  3. Mathematical connections
    The students will recognize equivalent representations of the same concept

Link to the State Standards: State Standard # 8

Identify and describe patterns and relationships in actual data, as well as solve problems and predict results using algebraic methods and symbols, tables, graphs, calculators and computers.

A. Identify numerical relationships using variables and patterns.

B. Analyze and describe numerical relationships using a variety of
representations. 8. Patterns and relationships

Technology used: Microsoft Excel for the MacIntosh


TEACHER INFORMATION:



This lesson can be done in one class period if students are relatively familiar with using spreadsheets. The worksheet should successfully lead most students through the exercise. This can be done by having students work together or individually, depending on computer access.


To create the spreadsheet file:

1. Open a Microsoft Excel file. Choose two columns. Label one x and one y. Under the x column enter simple integers like 1 - 5. Under the y column, enter another set of numbers which have a constant difference.

2. Select the two columns of numbers and then select Chart Wizard. This will give you a sequence of options.
i. Make sure your range and domain are correct.
ii. Choose scattergram
iii. Choose scattergram with a grid
iv. Leave these options alone
v. You may enter a title for the x and y axes and the scattergram.

3. Now you are ready to have your students use the scattergram. Make sure they do not save any changes which they make when they exit.


Extensions: You can use this same setup to talk about different rates of change, ( positive versus negative, large versus small, etc.) Another extension would be to look for patterns other than
linear patterns. You could also present a real world problem with real data and discuss lines of best fit in an intuitive way.


WORKSHEET



LOOKING FOR PATTERNS



Objective: To discover patterns using tables and graphs on a spreadsheet.

Please follow the instructions and answer the questions as you work through this lesson.

1. Open scattergram 1 on Microsoft Excel. You will see a table with 2 columns. The columns of numbers represent ordered pairs (x, y). The graph next to the columns has plotted the ordered pairs. Check the graph so that you can tell which point represents each ordered
pair.

Question 1: Do the points lie in a straight line? __________( If you're not sure, use a
straightedge to check the points)

Your answer should have been "yes".

2. Now change the table. Change the x values. Move the cursor to the first number. Change the number to 2. Use the down arrow on your computer to go to the second number and
change it to 4. Change the next numbers to 6, 8, and 10. Did you notice that the graph
has now changed to show the new ordered pairs?

Question 2: Do these points lie in a straight line? _______

3. Use the same method to look at the following order pairs. Determine whether or not each set of points lies in a straight line.




Line? _________ _______ _______






Line? _________ _______ _______




Question 3: By now you should have some "yes" answers and some "no" answers.

Can you see any differences between the "no"s and the "yes"s? If so,
what differences do you see?





4. If you think you have solved the pattern puzzle, try to come up with some tables of ordered pairs that lie in a straight line yourself and see if you are right. Write them here.




Line? _________ _______ _______






5. Now come up with some tables of ordered pairs which do not lie in a straight line.
Write them here.







6. A harder problem:

Can you find three tables of ordered pairs which lie in a straight line and in which all ten numbers are negative? Use the spreadsheet to help you.




7. An even harder problem:

Can you find three tables of ordered pairs which lie in a straight line and in which at least five of the not integers? Use the spreadsheet to help you.