This week in math we started our new unit on Polynomials. We learned names of equations we see like a binomial, monomial, trinomial. We also learned How to multiply and add equations to make sure we get the right answer.

Here is an example:

In this question we are told to use distributive property in order to figure out the answer. Distributive property is when you take everything in our first set of brackets and multiply it with everything in the second brackets.

In this photo i placed lines on our question to indicate what i’m multiplying together. All i did next was put the answers i got down below but that’s not our final step. Our final step is to take our “like terms” and add them together. In our equation we can add -5a and +21a together because they have the same variable. We can’t add the 3a squared simply because the variable has a 2 and is different from the ones without.

My final step was the take out like terms, add/subtract them together and write down our final answer.

One final thing we learned about was area diagrams. It gives u the same answer but it helps you visually see what’s happening.

Example:

This week in math we finished up our trigonometry unit.

We did lots of work on word problems making sure we understood things such as angle of depression, elevation, and making sure we knew where to place the numbers that were given to us.

Here is an example:

Our first step of course is to draw out our right triangle.

Our next step is to fully analyze the word problem and start to pick out our angles and our sides.

In the word problem they tell are we have a flag pole which is 5ft high so of course we know where to place our 5 on our triangle which is going to be our vertical side. We are also given a shadow (3.2 ft)which the flag pole makes. When we look at a shadow in real life we see it straight on the ground so in a word problem whenever we see the word shadow it will be our horizontal side.

In this image I placed the sides and i also add the angle. I knew that the angle went there because in the word problem it’s asking up for the angle of elevation of the sun and the word elevation means the angle is at the bottom across from the right angle. X is our place holder since we are trying to figure out that angle.

In this final photo i solved the equation. The first thing i did was label our sides. 5 being our opposite side since it’s across from our angle and 3.2 being our adjacent side since our hypotenuse is always our longest side. The next step is to figure out which ratio we have to use to get the correct answer. Our options are Tangent, Sine, or Cosine. As you can see i chose tangent because we are given our opposite and adjacent sides. Next we plug our numbers into our equation which is shown on the left side of my image. Tan X= 5 over 3.2 (opposite over adjacent)

All we have left is to switch our equation around so we can plug the numbers into our calculator leaving our final answer to be X= 57.4 We need to put a dot over our equal sign to show that we rounded our answer.

This  week in math we learned more about trigonometry. We learned about the shortened form of sine cosine and tangent which is SOH CAH TOA. This helps us figure out which formula we are going to use. We also learned how to figure out a given angle as learned to use our scientific calculators.

Here is a example question:

Here we are shown where our angle is and we are given two sides. Our first step is to label our sides so we know which equation we are going to use.

As you can see, I labelled our hypotenuse (65m) our opposite side and our adjacent side (16m). Since we are given two sides we can figure out which equation to use which will be the cosine ratio or “CAH”. The reason we use this ratio is because we are given the A and the H in “CAH”.

In this photo what I did was write out our ratio (CAH) and I wrote our our given sides. After that i plugged those numbers into our equation which is our ratio, angle, then it equals our sides over each other. We have 16 over 65 because in our ratio (CAH), A comes before H so we know that A goes on top of H. (We also know that H will always be on the bottom of our equation based off previous trials).

Our final step was to figure out the answer to our equation. In order to do that we have to isolate our angle. As you can see I move our ratio (cos) to the opposite side and changed its exponent which was 1 to a negative and i added brackets to our fraction. After that everything is simple. We just have to type that equation into our calculator making sure our cosine ratio is in the negative and we get our answer of 75.749… but since our question is asking us to round that long decimal to the tenths we can just leave it as 75.7.

All I did in this picture was write our our final answer making sure to put a tiny dot over our equal sign which indicates that we rounded our decimal.

This week we learned about exponents both positive and negative and we also learned about laws and rules of exponents.

An example that we learned.

In this problem, we are trying to figure out the answer while having a positive exponent. As you can see our base is X and when both our bases are the same we can move them together which is why X is remaining. We aren’t done there though.

We now have to deal with our exponents. We can go ahead and do the equation which leaves us with a remainder of 5.

Our answer is X to the power of 5.

Now these exponents won’t always work out this easily. When we are multiplying we add or our exponents because “exponents are lazy”. If we have a division question we will be subtracting our exponents. If we end up with an answer that has a negative exponents is when things get a bit tough. We then have to flip our equation in order to get the exponent to a positive.

Here is an example.

This week in math, we did some review with integers as well as linear equations. After refreshing our minds on how to solve equations correctly we moved on to prime factorization. With prime factorization we learned how to figure out the prime numbers in any number with the help of factor trees and division tables.

Lets say we have the number 1911 and we want to express the prime factorization of this number. Since this is a big number we will try a division table. We will first start off with figuring out what the factors of this number are. By adding all of the digits in this equation and dividing it by 3 is how I figured out you can divide this number by 3.

The next step is to figure out what 3 divided by 1911 is. In order to do this in a quick and easy way we will need to put 3 into each number within this number. For example we start off with 3 going into 1. Obviously we know we cant put 3 into 1 so what we are going to do is move up a digit. Instead of doing 3 into 1 we will do 3 into 19. Of course 3 doesn’t perfectly go into 19 but it does go in 6 times. 6 multiplied by 3 is 18 which leave us with a remainder of 1, since our number was 19. So, we can’t just leave this number with the 1 hanging around, we have to put the 1 somewhere. We are going to move it up and make the next number (which is 1) a double digit number, 11. I put the 1 we moved, in red so we know that is how we got 11. All we have to do is continue this process. 3 goes into 11, 3 times with a remainder of 2. We move that 2 up to make the 1 a 21. 3 then goes into 21 exactly 7 times which leaves us with our number 637. You continue to do this process with 637, figuring out first what its divisible by and then putting that number into each number.

Our final step is to make sure we have our final number. In order to do that we just make sure that our number is a prime number. A prime number is one that is only divisible by 1 and itself. As you can see we have 13 and 13 is a prime number so we know that it’s our final. All we have left is to put all of the prime factors in order from least to greatest. When we have multiple of the same prime factor, like the two 7’s we have, we can use exponents to show that there are multiple. We use the dot in between each number to show multiplication because using X can confuse us.