#4193. Graphing and Deriving Linear Equations using dice

Mathematics, level: Middle
Posted Mon May 5 11:31:22 PDT 2008 by Andrew Jahan (Andrew Jahan).
Temple City, USA
Materials Required: Graph paper, Dice, Pencils
Activity Time: 1 hour
Concepts Taught: Linear Equations, Coordinate Planes, Prediction, Probability

Title: Graphing and Deriving Linear Equations using dice.
Subject: Algebra
Standards: Algebra I
6.0 Students graph a linear equation.
7.0 Students are able to derive linear equations.

Materials:
Graph Paper
Dice

Set-up:
On the board create a coordinate plane with x- and y- axis.
On the x- axis include 12 ticks; (2 to the left of 0) and (10 to the right of 0)
On the y- axis include 15 ticks; (1 below 0) and (14 above 0)
Also create a tally chart numbered 1, 2, 3, 4, 5, and 6. This will be used for tallying what
you roll during guided practice.

Guided Practice:
1. Explain to the students that today we are going to continue working with graphing linear equations.
2. Ask for a volunteer to explain what they think a linear equation is.
a. A linear equation is an equation whose graph forms a straight line.
3. Write the student's definition on the board.

4. Next, pull out a die from your pocket.
5. Confirm that every student knows that you are holding a die and that the die has 6 sides, numbered by black dots 1 through 6.
6. Tell the students that you will roll the die but before you do you'd like to know what the odds of rolling a 6 are.
a. There is a 1 out of 6 chance of rolling a 6.
7. Roll the die and tally on the board the number that you rolled.

8. At this point re-emphasize that having a 1 out of 6 chance of rolling a 6 is the same as saying if I roll the die 6 times I should get a 6 one time.
9. Roll the die another 5 times and tally each result on the board.

10. Did we roll at least one 6?
11. If not, ask the students why they think we didn't?
a. Well our prediction might not be perfect but if we continued to roll the die over and over again, say 60 times, we could realistically predict that we would roll a 6 approximately 10 times.
b. 10 times / 60 rolls = 1/6

12. On the board, number the x- axis from -12 to +60 and the y- axis from -1 to +14
13. Then label the x- axis as number of rolls, and the y- axis as number of 6's rolled.
14. Ask students to think about what the graph would look like if we were to get perfectly proportional results.
15. What would the ordered pairs be?
16. For example, if we rolled the die 6 times, how many times could we expect to roll a 6?
a. 1 time because there is a 1 in 6 chance of rolling a 6.
b. The ordered pair would be (6 rolls, 1 time rolling a 6 ) or (6, 1)
17. How about if we rolled the die 12 times? 18 times? 60 times? (12, 2) (18,3) (60, 10)
18. From this information derive a graph that shows the probability of rolling a 6 after 6, 12, 18, 24, 30, 36, 42, 48, 54, and 60 rolls.
Independent Practice:
19. Instruct students to copy on their graph paper the graph we constructed together on the board in pencil.
20. Then, have the students pair off with one other classmate and give each group a die.
21. Each group will roll their die a total of 60 times, making a tally of how many 6's they rolled for every 6 rolls.
22. As they are rolling their die and tallying their totals tell them to mark their results on their graph in red pen.
a. Make them aware that each tick on their graph's x-axis is equal to 6 rolls.
b. After every 6 rolls they will mark a point on the y-axis that equates to the number of times they rolled a 6 at that stage of the experiment.

Assessment:
23. When the students are finished have them compare their results graph with the predicted results graph.
a. Ask for volunteers to explain their findings.
b. Did anyone notice any trends?
c. How did their results compare with the predicted results?
d. What can we conclude about how things work in the real world as opposed to the lab?
e. Can anyone explain how linear graphs can help us in the real world despite the fact that they aren't always perfect.
f. What are some other ways that we can use graphs?

24. Assessment for this lesson will consist of informal observation and class participation and a grade for each group's graph and tally paper.
a. Each groups will receive 20 points for copying the prediction graph, keeping track of their information using tally marks, and for graphing their results correctly.
b. If one of these items is missing the group grade will drop to 15 points.
c. If two of these items are missing the group grade will drop to 10 points.
d. If more than two of these items are missing then the group grade will drop to 5 points.
e. All groups will receive 5 points for participating.