Shirley Bassey — History Repeating

Pre-Calculus Assignment — Book 3.R

Pre-Calculus Assignment — Book 3.CR

Lesson Plans – November 2 – 6, 2009

Lesson Plans – November 2 – 6, 2009

Pre-Calculus

Standards: Models for Real World Phenomena; Algebraic Functions; Trigonometric Functions; Sequences and Series
Monday, November 2: TEST Chapter 3 Part 1
Tuesday, November 3: TEST Chapter 3 Part 1
Wednesday, November 4: 3.6 Solve polynomial inequalities and rational inequalities algebraically and graphically
Thursday, November 5: 3.4 and 3.5 Find the domain of a rational function; Determine the vertical asymptotes of a rational function; Determine the horizontal or oblique asymptotes of a rational function; Analyze the graph of a rational function; Solve applied problems involving rational functions
Friday, November 6: 3.4 and 3.5 Find the domain of a rational function; Determine the vertical asymptotes of a rational function; Determine the horizontal or oblique asymptotes of a rational function; Analyze the graph of a rational function; Solve applied problems involving rational functions

Geometry

Standards: Number and Operations; Algebra; Geometry; Measurement; Data Analysis and Probability

Monday, November 2: Review Chapter 4
Tuesday, November 3: TEST Chapter 4
Wednesday, November 4: TEST Chapter 4
Thursday, November 5: 5.2 and 5.3 Identify and use properties of medians and altitudes of triangles; Identify and use properties of perpendicular bisectors and angle bisectors of triangles
Friday, November 6: 5.1 Use properties of midsegments to solve problems

Statistics

Standards: Experimental Design; Data Analysis

Monday, November 2: 4.1 Distinguish between discrete random variables and continuous random variables; Construct a discrete probability distribution and its graph; Determine if a distribution is a probability distribution; Find the mean, variance, and standard deviation of a discrete probability distribution; Find the expected value of a discrete probability distribution
Tuesday, November 3: 4.1 Distinguish between discrete random variables and continuous random variables; Construct a discrete probability distribution and its graph; Determine if a distribution is a probability distribution; Find the mean, variance, and standard deviation of a discrete probability distribution; Find the expected value of a discrete probability distribution
Wednesday, November 4: 4.2 Determine if a probability experiment is a binomial experiment; Find binomial probabilities using the binomial probability formula; Construct a binomial distribution and its graph; Find the mean, variance, and standard deviation of a binomial probability experiment
Thursday, November 5: 4.2 Determine if a probability experiment is a binomial experiment; Find binomial probabilities using the binomial probability formula; Construct a binomial distribution and its graph; Find the mean, variance, and standard deviation of a binomial probability experiment
Friday, November 6: 4.2 Determine if a probability experiment is a binomial experiment; Find binomial probabilities using the binomial probability formula; Construct a binomial distribution and its graph; Find the mean, variance, and standard deviation of a binomial probability experiment

For all classes.
Activities: Lecture, Board work, Classroom participation
Material: Book problems and Teacher made handouts
Assessment: Daily assignments graded for accuracy. Unit test planned.

Pre-Calculus Assignment — Book 3.8

Pre-Calculus Assignment — Book 3.7

Geometry Notes — Proving Triangles Congruent 4 — Right Triangles — HL, HA, LL, LA

If you have the right three pairs of corresponding congruent parts from two triangles, that is all you need to prove that two triangles are congruent.

If the two triangles you are working with are right triangles, you already have one of those three parts.

To prove that two right triangles are congruent, consider these combinations

HL – Hypotenuse Leg
HA – Hypotenuse Acute angle
LL – Leg Leg
LA – Leg Acute angle

Of these four, only HL needs to mastered.

Why?

1. The Hypotenuse-Acute Angle Theorem is a rule specially designed for use with right triangles. It states if the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, the two triangles are congruent.

Look at the two right triangles and the congruence marks that are given. Before the hypotenuse-acute angle theorem was ever mentioned, you already had the means to declare these two triangles congruent. You would have said the two triangles were congruent because of Angle-Angle-Side.

2. The Leg-Leg Theorem is a rule specially designed for use with right triangles. It states if the legs of one right triangle are congruent to the legs of another right triangle, the two right triangles are congruent.

Again, look at the two right triangles and the congruence marks that are given. You would have said the two triangles were congruent because of Side-Angle-Side.

3. The Leg-Acute Angle Theorem is a rule specially designed for use with right triangles. It states if a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.

Look at the two right triangles and the congruence marks that are given. Before you were lucky today by being around me today, you already had the wonderful knowledge of Angle-Side-Angle.

So instead of learning more about HA, LL, and LA, we will continue to use the rules AAS, SAS, and ASA. These last three are more useful because they work for any triangle. HA, LL, and LA are right triangle specific.

The Hypotenuse Leg Theorem

Hypotenuse Leg Theorem
(HL)  If a leg and the hypotenuse of one right triangle are congruent to a leg and the hypotenuse of a second right triangle, then the two right triangles are congruent.

This is the one case where SSA is valid. The angle must be a right angle, thus the triangles must be right triangles.

Since ΔBOW and ΔMAN are right triangle,  BO ≅ MA and OW ≅ AN , then ΔBOW ≅ ΔMAN by HL.

Congruent Triangles – Hypotenuse and leg of a right triangle (HL): Two right triangles are congruent if the hypotenuse and one corresponding leg are congruent in both triangles.

A combination of two lengths – the hypotenuse and a leg – are highlighted to indicate they are the parts being used to test for congruence.

Here’s a reason why this works. Since we know the hypotenuse and one other side, the third side can be accurately determined, due to the Pythagorean Theorem.  So this is really a version of the SSS case. (side-side-side).

What does this mean? Because the triangles are congruent:
1. The remaining third sides are congruent (PQ ≅ LM)
2. The other two angles are congruent (∠R ≅ ∠N and ∠Q ≅ ∠M)

Pre-Calculus Notes — Video — Parabolic Arch Problem

ACT — Preparing for the ACT 2009/2010

Geometry Notes — CPCTC — Corresponding Parts of Congruent Triangles are Congruent

Define congruent triangles: Triangles with a correspondence between their vertices such that each pair of corresponding sides and each pair of corresponding angles are congruent.

Written as a conditional: If two triangles are congruent, then each pair of corresponding sides and each pair of corresponding angles from the triangles are congruent.

Written as a converse: If each pair of corresponding sides and each pair of corresponding angles from two triangles are congruent, then the two triangles are congruent.

Written as a biconditional: Two triangles are congruent if and only if each pair of corresponding sides and each pair of corresponding angles from the triangles are congruent.

If we have two triangles and we know they are congruent, then we have information about six pairs of corresponding parts — 3 pairs of sides and 3 pairs of angles.

If ΔBOW is congruent to ΔMAN we know

B ⟷ M, O ⟷ A, and W ⟷ N

BO ≅ MA , OW ≅ AN, and BW ≅ MN

∠B ≅ ∠M, ∠O ≅ ∠A, and ∠W ≅ ∠N

When we started proving whether two triangles were congruent, we used the converse. However instead of proving that each pair of corresponding sides and each pair of corresponding angles from the two triangles were congruent, we tried to prove certain combinations of 3 pairs of sides or angles. Combinations like Side Side Side for instance.

Since BO ≅ MA and BW ≅ MN and OW ≅ AN, we can conclude that ΔBOW ≅ ΔMAN because of SSS.

CPCTC

CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. It is statement developed from the definition of congruent triangles. It allows us to prove things about the remaining unproven parts of the triangles that we have just proven congruent. It allows us to state correctly after two triangles are congruent, then corresponding parts that were not previously known to be congruent are now allowed to be considered congruent.

Example

Since BO ≅ MA and BW ≅ MN and OW ≅ AN, we can conclude that ΔBOW ≅ ΔMAN because of SSS.

Now we can say ∠B ≅ ∠M, ∠O ≅ ∠A, and ∠W ≅ ∠N because of CPCTC. Since the two triangle were proven congruent, we can now correctly assume that corresponding parts that we knew nothing about, are now congruent.

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Another example

Since BO ≅ MA and OW ≅ AN and ∠O ≅ ∠A, then ΔBOW ≅ ΔMAN by SAS.

Now we can say BW ≅ MN, ∠B ≅ ∠M, and ∠W ≅ ∠N because of CPCTC.

.

Another example

Since BO ≅ MA and ∠B ≅ ∠M and ∠O ≅ ∠A, we can conclude that ΔBOW ≅ ΔMAN because of ASA.

Now we can say ∠W ≅ ∠N, BW ≅ MN and OW ≅ AN because of CPCTC.

.

Another example

Since BO ≅ MA and ∠O ≅ ∠A and ∠W ≅ ∠N, we can conclude that ΔBOW ≅ ΔMAN because of AAS.

Now we can say ∠B ≅ ∠M, BW ≅ MN and OW ≅ AN because of CPCTC.

.

Another example

Since ΔBOW and ΔMAN are right triangle,  BO ≅ MA and OW ≅ AN , then ΔBOW ≅ ΔMAN by HL.

Now we can say ∠B ≅ ∠M, ∠O ≅ ∠A, and BW ≅ MN because of CPCTC.

.

Remember, we use CPCTC to prove parts are congruent after we have proven triangles congruent. CPCTC is used after SSS, or SAS, or ASA, or AAS, or HL, never before. First we prove that two triangles are congruent. Then if we haven’t already proven that a desired pair of corresponding sides or angles are congruent, we can now do so using CPCTC.