New Unit!

We started our lessons for Unit 4 today. This unit included probability and statistics concepts. Today's lesson was on the Fundamental Counting Principle. There are two parts to this principle. First the multiplication counting principle shows that to find the total number of outcomes from a certain sequence of events can be found by multiplying the number of ways each event can occur.

Example: Find the total number of ways to order a dinner when selecting one entree from 10 options, one drink from 5 options, and one dessert from 6 options.

Answer: 10 entrees * 5 drinks * 6 dessert = 300 different dinners

The addition counting principle indicated that the sums from events can be grouped in such a way that the groupings do not share possible outcomes, then the total number of ways each group can happen should then be added together.

Example: Find the total number of ways a 3 symbol code made of digits or letters can be made if at least one symbol must be a letter.

Answer: There are 26 different letters and 10 different digits

1-letter and 2-digit code (A # #): 26 * 10 * 10 = 2600 A## codes.

But, the letter does not have to be in the first spot.

It can occur in any of the 3 spots, so we multiply the total by 3.

26o0 * 3 = 7800 different 1-letter 2-digit codes

2-letters and 1 digit code (A A #): 26 * 26 * 10 = 6760 AA# codes

But, the digit does not have to be in last spot.

It can occur in ay of the 3 spots, so we multiply the total by 3.

6760 * 3 = 20,280 different 2-letter 1-digit codes

3-letters (A A A): 26 * 26* 26 = 17,576 AAA codes

Now, add together the total from each group:

7800 + 20,280 + 17576 = 45,656 different 3-symbol codes

(with at least 1 letter)

There are some typical modifications to the above problem, such as limiting the digits to only evens (5 options), or not allowing a symbol to repeat. For the latter modification, the number of symbols available decreases for one that is already used.

Example: How many 3 letter codes can be made if no letter can be repeated?

Answer: 3-letters (A A A): 26 * 25 * 24 = 15,600 different 3-letter codes

(with no repeated letters)

There will be a quiz on Friday covering the lessons from this week: counting principles, permutations (Tuesday), and combinations (Wednesday).

Homework: Page 340, #2 - 10 even, #11; Page 341, #2 - 8a even (omit 8b)

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Catching up!

Sorry to have missed so many days. It has been a busy January!!

Today's notes finished up our "special segments" in triangles with medians and altitudes. Medians are segments that connect a vertex of a triangle with the midpoint of the opposite side. Every triangle has 3 medians. The medians will always all meet at a single point inside the triangle called the centroid as shown below.

The centroid has two properties:

1. It is known as the "balancing" point of the triangle.


2. It occurs exactly 2-thirds of the distance from the vertex to the midpoint.

An altitude is the perpendicular segment to a side of the triangle and ending at the opposite vertex. This segment is also known as the height of the triangle. Every triangle has 3 altitudes. The altitudes will always meet at a single point called the orthocenter. This point can occur inside, on, or outside the triangle depending on whether it is acute, right, or obtuse, respectively.

Acute Triangle example:

Obtuse Triangle example:

Homework: Page 280, #1 - 6; #10 - 12; and #14 - 26 even

There is a review session tomorrow morning in room 5207, tomorrow afternoon in room 5202, and Friday morning in room 4303. Students will get their quizzes back tomorrow and a review packet for Friday's test.

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Quadrilaterals Day 3

Today's lesson began with a discussion and an activity investigating parallelograms with more properties: Rectangle, Rhombus, and Square. These are parallelograms, so they still have the 5 properties specific to parallelogram and the 2 properties specific to all quadrilaterals. In addition to them, we added more.

For Rectangles:

1. Definition: a parallelogram with 4 congruent angles (all 90º)
2. diagonals are congruent

For Rhombuses:

1. Definition: a parallelogram with 4 congruent sides (think of a diamond shape)
2. diagonals are perpendicular
3. each diagonal bisects a pair of opposite angles

For Squares:

1. Definition: a parallelogram with 4 congruent sides AND 4 congruent angles
2. Because a square is BOTH a rhombus and a rectangle, it has ALL properties previously mentioned!

Homework: Page 319, #2 - 18 even; Page 321, #2 - 18 even, #28

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Quadrilaterals Day 2

After reviewing the 5 properties specific to parallelograms, we turned those properties around. If those properties are true about a quadrilateral, then the quadrilateral MUST be a parallelogram. So, if a figure of a quadrilateral shows that ...

1. both pairs of opposite sides are parallel,
2. both pairs of opposite sides are congruent,
3. both pairs of opposite angles are congruent,
4. one angle is supplementary with both of its consecutive angles,
5. diagonals bisect each other,
6. or, one pair of opposite sides is BOTH parallel and congruent

... then the quadrilateral is a parallelogram.

Again, students are expected to be able to identify which property is needed as well as to create and solve necessary equations, much like Day 1.

Homework: Page 313, #2 - 12; Page 315, #1 - 14, #19

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Quadrilaterals Day 1

On Tuesday, our first day of the new semester, we reviewed the properties of Quadrilaterals:

1. Definition: 4 sided polygon
2. the sum of the 4 interior angles is 360º

After that short review, we investigated the properties specific to parallelograms by measuring the angles, side lenghts, and diagonal lenghts of a few parallelograms. We found and discussed the following properties specific to parallelograms:

1. Definition: a quadrilateral with both pairs of opposite sides parallel
2. both pairs of opposite sides are congruent
3. both pairs of opposite angles are congruent
4. consecutive angles are supplementary
5. diagonals bisect each other

From these properties, the students are expected be able to identify which one is shown in a figure and be able to use the relationships to create and solve equations. When approaching these problems, students should first identify which part of the parallelogram the problem measures. Then, apply the property for those parts. Remember, congruent means the measures/lengths are equal. Supplementary means that the angle measures sum to 180º.

Homework: Page 306, #2 - 26 even; Page 308, #2 - 20 even (omit #10)

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