# Tutor profile: Ben V.

## Questions

### Subject: Trigonometry

What is one of the interior angles of a right triangle with leg lengths 3 and 5? The 90 degree angle doesn't count.

[This problem would be aided much by a drawing, but that's not possible in this format. In an actual tutoring session, I would use drawings heavily for this problem, as it clears things up a lot I think]. We know what trig functions are for. Basically, given an angle, we can find the ratio between two side lengths in a right triangle. But we aren't given an angle. Instead, we are given the ratio between two side lengths in a right triangle (3/5 or 5/3), and we need to find the angle. It looks like we need the opposite of what our trig functions can do. And that's where we can use our inverse trig functions. They do everything that a normal trig function would do, but in reverse. We can input the ratio of two sides, and it will give us an angle. But which trig function should we use? If we assume our angle is not the right angle, we are given an opposite and adjacent side. Using SOHCAHTOA, we can see that when we have opposite and adjacent sides (OA) we use tangent (T in TOA). Since we want to go in reverse, we would use the inverse tangent. So, the inverse tangent of (3/5), or $$ \tan ^{-1}(3/5) $$, equals 30.96 degrees. And that's our answer! If we wanted the other angle, we could do $$ \tan ^{-1}(5/3) $$, which equals 59.04 degrees.

### Subject: Pre-Algebra

What is $$ \sqrt49 $$ ?

We're asked what is the square root of 49. First, let's think about what a square root is. In a sense, a square root is the opposite of a square. Let's think about this in terms of a different problem, with 5 as our starting point. When you square a number, like 5, you are multiplying that number by itself: $$ 5^2 = 5 \cdot 5 = 25 $$. [Assuming that the student has already learned and understands squaring a number]. Let's think about this problem in reverse. Let's say we have the number 25, and we want to know what number, multiplied by itself, equals 25. Well, we just saw that the answer is 5! So, the square root of 25, or $$ \sqrt 25 $$, equals 5. And that's what a square root does. Basically, the square root just means "take this bigger number under this square root sign (49 for our final problem). What smaller number times itself equals our bigger number?". For our final problem, let's think. What number, times itself, equals 49? The answer is 7. [Assuming that the student has memorized their squares. If not, we could go through each whole number square to guess and check, or, as a more reliable method, we could do a prime factorization using a visual tree. Also, for this whole question, I would have the student be doing most of the work, with me guiding them as much as needed with tips and advice, in order to keep them engaged.]

### Subject: Basic Math

What is 4 times 100?

Well, first of all, let's think about what multiplication is. Basically, multiplication is just repeated addition. It seems more complicated when we use big numbers and fancy calculation techniques, but it is really nothing more complicated than repeated addition. So, let's try and solve this problem intuitively, without relying on a formula, which will allow you to understand the problem better. Logically, let's think it through based on what we know: we know that multiplication is repeated addition, and that we have 4 times 100. And, let's think about it in terms of an object, such as candy. What 4 times 100 really means, based on what we know, is that we add together 4 groups of 100 candies. [Here I would draw a picture of four 1 x 100 lines of candies, to demonstrate our different groupings]. So, let's do this one step at a time. Adding together two groups, we get 200 candies. Add another group, and we get 300. And adding the final group, we get to 400. [Assuming they know how to add hundreds in their head! Otherwise, we would slow down here]. And that's our answer! [Here, I would draw a 4 x 100 grid by combining 4 of the 1 x 100 lines from before]. Do you see how this grid represents our problem? We have four different lines of 100 candies, which adds to 400 candies in total. [Also, in a real tutoring setting, the whole way through the problem I would be asking questions to make the student do as much as possible. That way, they can't just zone out, and are actively engaged in their learning].

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