Problem Set for Previewing Unit 4

In the fourth unit of Math I students are focusing on data analysis and statistics. The one focus of this unit is for students to review the measures of central tendencies (mean, median, and mode) as well as learn about measures of dispersions (range and mean absolute deviation). However, the main focus of the unit is for students to have a better understanding of statistics. The unit begins by demonstrating to students the fundamental counting principle; we want students to have a better understanding of how to count possibilities when it comes to real life situations. In addition, using the counting principles we show to students the how to find the total number of possibilities whether it be a permutation or a combination; this will be useful when demonstrating to students how to find the probability of events whether, independent, dependent, or mutually exclusive.

Vocabulary

Counting Principle Probability Statistics Permutation Combination Independent Events Dependent Events Mutually Exclusive Events

Scenario

At the local National Council of Teachers of Mathematics (NCTM) town hall meeting 17 teachers met to discuss the best way to prepare students for the rigor of high school math classes. Each of these teachers shook hands with each other to introduce themselves. How many handshakes occurred at this meeting?

Investigation

On the Future Channel Website there is an activity for and short film for students that can be used for introducing the counting principle.

Additional Questions

1) In the activity we know how much of each piece of clothing we have. If we let x by the number of shirts and y be the number of pants; what is the number of different outfits possible?

2) If we know we have 5 more pants than shirts, how many total outfits can we create?

- x = number of shirts so x+5 = number of pants thus total number of outfits is x(x+5)

- We want students to recognize that the number of possibilities to choose 1^st is then multiplied by the number of possibilities that come 2^nd.

If possible, give each group of students a circular geoboard and ask the students to create a circle like the following:

We have 9 points on the circumference of the circle. Using these points how many unique chords can we make?

-Using the geoboard will help students explore this problem and work out a solution.

- Students will find a solution through addition 8+7+6+5+4+3+2+1 or through multiplication (9)(8)/2 this is done in the same manner as before we have 9 choices for the first point followed by 8 choices for the second point. Since we want unique chords we divide by two in order to remove repeats.

3) If we are given a circle with n+ 1 point how many unique chords are there?

a. Same idea as before: using addition (n+1) + n + (n-1) + (n-2) + … + 3 + 2 + 1 using multiplication n(n-1)/2

Discussion Questions

Here we can discuss with students the difference between a permutation and a combination. We can give various examples on the difference and how to find using the formulas. The same idea that was used before is in the formulas.

1) Now we can solve our handshake problem. If there are 17 people in a room, how many handshakes occur between them? Is this a permutation or a combination?

2) Let’s say you and 5 friends go to Six Flags, how many different ways can you all stand in line?

3) One of your friends doesn’t like to be on the end seat of a ride with 6 seats, how many different ways can you all sit together?