What is sin(x)?
I first think about what x means in this scenario. In this case, x represents the value of an angle measure, usually in units of radians, where the angle has one ray positioned on the 3 o'clock position. Then, I imagine a circle around the vertex of my angle (it does not have to be the unit circle - it can be any circle). The second ray intersects this circle, and sin(x) represents the vertical distance above the horizontal diameter of this circle measured in radius lengths.
What is 3*(5/4), and why?
First, it's important to focus on the order of operations, which says that we need to first understand what 5/4 is. I imagine taking a strip of paper, and splitting that paper into four equally sized pieces. Then, I take one of those pieces, and make 5 copies of those pieces. I then put those 5 pieces together, and that is 5/4 of the strip. Now this says we want 3*(5/4). To understand this, we must first understanding multiplication. I think about 5*4 as taking 5 copies of 4 units. So, I think about 3*(5/4) as 3 copies of these 5 copies that we made earlier. So I imagine doing this process of making 5 pieces a total of three times, and then I end up with 15 copies of 1/4. So, I have 15 fourths, or 15/4.
What does it mean for the derivative to be positive?
When we refer to derivatives, we are describing a rate of change between two varying quantities. For example, imagine a bath tub filling up with water. What are some quantities that you can imagine in this scenario? Which of these quantities are varying? Can you attribute these quantities to values? Are those values changing? The concept of the derivative describes how quantities change with respect to each other. In this bath tub example, I can imagine the number of seconds elapsed since the bath tub began running, and I can represent the values of this quantity with the variable t. I can also imagine the total number of gallons of water in the bath tub, and I can represent the values of this quantity with the variable y. To understand the derivative, we must imagine one quantity varying, say time. As time increases, what happens to the other quantity? Well, the number of gallons increases. In this example, we would say that the derivative of quantity y with respect to quantity x would be positive.