A Free Tutoring Service: Preparing For the Future

 Lesson Review B10 July 24, 2002... Geometry: More of the same!

A new Student: Raquel, 15-years old, joined us today. She was given little time to adjust to a new learning environment. She didn't need any time, either: she joined in a lively lesson on geometry. Raquel is an aspiring actress and a great dancer.

There were approximately 2 hours of very intense tutoring. It took place on the front porch of my home. Raquel, Katrina and Jamise listened to my lecture on the geometric mean of a right triangle. They took it all in, and as a review, I covered the height or altitude of a right triangle as the geometric mean between two similar triangles. I used the dry erase board to describe the altitude and the division of the hypotenuse into 2 sections (part a and part b). The formula looks something like this when one sets it up to find a side of the triangle.

height/part a = part b/height

height² = (part a) (part b)

square root of height = square root of (part a)(part b)

height =square root of (part a) (part b)

It was a simple matter for the students to look at a right triangle, draw the altitude down to the hypotenuse, and assign part a and part b to the two sections of the hypotenuse that were created by the altitude line. They then proceeded to find unknown sides of other right triangles. This is Corollary One, according to the text book.

Next, we moved to the "leg" solution of proving two right triangles are similar. That is Corollary Two according to text book.

Again, we draw the altitude from the right angle to the hypotenuse. Next we assign letters to the two other legs of the right triangle. We select one of the legs as our geometric mean. Then the formula becomes:

leg "a"/whole hypotenuse = short side of hypotenuse closest to the leg"a" / leg "a"

Cross-multiplication yields, leg "a" squared = (short side closest to "a")(whole hypotenuse.

Then, the students followed their laws for finding square roots to solve for an unknown side.

As an added bonus, we reviewed the perfect squares from 1 to 144. It was not long before Katrina and Raquel were repeating them with rapidity. They are 1, 4, 9, 26, 25, 36, 49, 64, 81, 100, 121, 144. This is a mental exercise and the students did quite well. There were reminded of square roots and the need to reduce them to their lowest terms. The perfect squares are a good help in doing this.

We also discussed exponents and their laws. The laws of multiplication and division of exponents were thoroughly covered. The rules governing square roots were also covered. We used the scientific calculator often to find square roots of numbers.

We moved on to the 45 degree and the 30 degree right triangles. We learned the relationship between the sides of each. The 45 degree right triangle has equal legs and a hypotenuse that is the square root of 2 times one of the legs. That makes it a 1, 1, square root of two relationship; it is a simple matter to assign a value to a leg and multiply the value by the square root of 2 to find the hypotenuse. When the hypotenuse is given and the sides are not, then divide the square root of two into the length of the hypotenuse to find the sides. Very simple, huh?

The 30 degree right triangle is different in that the ratio of the sides and the hypotenuse is 1, 2, the square root of 3. That is to say that if the side that is opposite to the 30 degree angle is 1, then the hypotenuse is 2 times 1 or 2. Then the other side is equal to the square root of 3 times 1.

We used the Pythagorean Theory to find the other side of a triangle that had the hypotenuse as 2, and one leg as 1. The other leg turned out to be the square root of 3, just as we predicted. This could be a mental exercise and the students were encouraged to do the Pythagorean Theory in their heads.

Next, we moved to the scientific calculator and the sine, cosine and tangent function. We delved into arc sine, arc cosine and arc tangent, as well. We discussed "opposite" side, "adjacent" side and the hypotenuse of 30 degree and 45 degree right triangles. It was not long before the students learned that they could discard those scientific calculators and find sine, cosine and tangent in their heads.

Katrina will take a test on those items covered above on tomorrow. Good luck, Katrina at your Carson High School class.

Parents came to pick up Raquel and Katrina. Jamise went to retrieve our star geometry student, Jarrell,  from Banning High School. The time stood at 9:30 PM when they returned. Jarrell had completed another one of his 13-hour school days and looked a bit tired. He is determined to make all "A's" and "B'" in his 5 summer school classes. I believe he will!

At 9:30PM, our Volunteer Tutors, Medical Doctor Cheryl Browne-Ojei of Kaiser Hospital, Harbor City, and her husband Chudi, a Chief Accountant for the LAUSD, came to see me about a fundraiser for the tutoring service. It involved the selling of candle holders by the students. The money raised would be to support them when we take field trips.  Doctor Cheryl was extremely tired after having worked a 36-hour shift at the Hospital. Still, she found time to come and present the sample candleholders to me. Doctor Cheryl, as we affectionally call her, is a wonderful human being. She and her husband are avid supporters of the tutoring service, and stand ready to assist in any way that they can. God bless our Volunteer Tutors.

Doctor Cheryl, though extremely tired, explained how we could raise funds for the tutoring service. She talked through "tired eyes" until she decided to go home and get some much needed rest. She will probably be at it again on tomorrow. That is... assisting the students of "A Free Tutoring Service" Preparing for the Future".

Jarrell and I will arise at 5:00AM on tomorrow to study geometry. We will continue his climb to the top. We will continue "Jarrell's Journey"!

My wife, Eddie Gayle, had been busy at the Church Vacation Bible School. She served food to all of the attendees. She came home just as tired as the rest of us. She is a wonderful wife to assist with the tutoring duties as she does. We are truly blessed!