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When dividing radicals, the denominator can never be a radical. So in order to solve, the denominator must first be made into a rational number.

First, use a difference of squares to rationalize the denominator. But since whatever you do to the bottom you must also do to the top, multiply both the numerator and denominator.

now add like terms to get a more simplified expression.

now you can just simplify the expression as far as you possibly can! It’s okay for the expression to not equal a single number.

The absolute value of a real number is defined as the principle square root of the square of a number.

for instance the absolute value of $\mid 4 \mid$ is 4

when a number is between the brackets, it must be brought out of the bracket like this:

$\mid 4 \mid$

$=4^2$

$=\sqrt {16}$

$=4$

Whenever a number is between brackets, the answer must be a positive number. However, after the number is out of the brackets it can become negative in an equation.

We also learned about roots and radicals. Numbers with square roots can never be negative, like $\sqrt {-25}$ because it cannot be calculated and results in an error when put into a calculator. On the other hand, a cubed number can be negative. Roots with an index that’s even cannot be negative, but roots that have an uneven index can be negative.

when simplifying roots that are not perfect, a different approach must be taken.

for example:

$\sqrt {24}$

$=\sqrt {4*6}$ find two multiples of a number, one being a perfect square

$=\sqrt {4} * \sqrt {6}$ separate the roots

$=2 \sqrt {6}$ square the perfect square and leave the radical as a square root

this applies to other index’s as well, just find perfect cubes instead of perfect squares, etc.

This week we learned about geometric sequences and series. A Geometric sequence differs from an arithmetic sequence because instead of adding a common difference, a common ratio is multiplied by the term to get the next term.

the equation used to find a term in a sequence is

$t_n=ar^{n-1}$

in this equation $t_1$ is represented by $a$

for example: 5,10,20,40… find $t_{10}$

$t_{10}=5(2)^{10-1}$

$t_{10}=5(2)^9$

$t_{10}=5(512)$

$t_{10}=2560$

A finite geometric series can be found using the equation

4, 9, 14, 19, 24

$d=5$, $t_1=4$

$t_n=t_1+(n-1)d$

$t_n=4+(n-1)5$

$t_n=4+5n-5$

$t_n=5n-1$

using this information, we can find any value of $t$ using the equation $t_n=5n-1$. We can plug in any term into $n$ to find $t_n$

this equation will work for any term within the sequence, as long as the starting term and common difference are constant. If either of these change it is no longer an arithmetic sequence.

To find $S_{50}$, the sum of all 50 terms, we must use the equation $S_n={n}{2}(a+1)$

In this equation, $n=$ number of terms(50) and $a=$ the first term(4)

$S_{50}={50}{2}(4+1)$

$S_{50}=25(5)$

$S_{50}=125$

1. Were you able to be an active listener and support & encourage others? Give an example.

Yes i was able to be an active listener and support & encourage others. An example would be when we were planning the layout of our project, and worked together to decide. I asked people what their opinions were and how they would go about doing certain things. At one point, we weren’t sure as to what the layout of the strand of mandalas. One of the girls that didn’t talk very often looked like she had an idea, but was too shy to say it. So i asked her what her thoughts were, and she came up with the layout we chose.

2. Were you able to hear different points of view & disagree respectfully? Give an example.

Yes i was able to here different points of view & disagree respectfully. I did so by listening to what others had to say, and if i didn’t agree i would suggest a different idea that would appeal to everyone. Usually involving a bit of what the person had suggested.

3. Reflect on how you were able to give, receive & act on feedback.

During the project, we had to following specific themes. Sometimes, someone would do something that didn’t work with the idea, and i would ask them to change something or work around it to make it better. Often, i would look around the table at my group mates’ work and find inspiration. If that failed, i would ask the person sitting beside me if they had any suggestions. I never got angry if someone suggested a way to improve my work.

4. How successful were you in working with others to achieve a common goal?

I found we were quite successful in working with others to achieve a common goal.There were a couple times that things were misinterpreted and we had to work around it, but as a group we made a finished project that was satisfactory to us. We successfully stayed within a common theme and colour scheme. We also collaborated on two-person mandalas that involved the both of us working together to make a single mandala.

5. What roles & responsibilities did you take on/in the group?

I took on a leadership role in my group, as i was with three international students. I was difficult to get a couple of them to talk, but it made decision making a little easier. I encouraged people to talk and made the larger decisions for the group. I talked for the group to the class, and got down to business when needed. I urged people to do their best and stay on task.

1. For my unit 8 math journal, i did question 7 on page 567.The question gives us a starting function, as well as a blank cartesian plane. The question provides the student with a base point to start the question off with, before building on that as the question progresses.

when i first read the problem, i had to think about it. i new what my starting point was, but needed to figure out where to go from there. The formula y=mx+b was a key component in the problem, as well as knowing how to find the y-intercept and x-intercept.

Knowing what to do during the question was one thing, but realising how to apply it later was another. Part of the question asked for the domain and range of the equation, which made me think back to last unit where finding the domain and range was important. I remembered that if a ordered pair or equation has no x-intercept, it’s a zero slope, and the line runs across the graph horizontal without touching the x-axis

2) When you look for the x-intercept given a equation, you set y = 0 then solve for x. This is because where the point intercepts either the y or x axis, the other will be at zero on a graph. An x-intercept is a point on the graph where y is zero, and a y-intercept is a point on the graph where x is zero.

3) For my unit 9 math journal, i did question 8 on page 605. The question gives you a scenario where after 7 days of heavy rain, the water level of a river is 2.85m above regular level. After four more days, the water dropped to 2.25m above regular level. The questions asks that you create a function h(t) for the height of the river above regular level as a function of time, assuming t=0 at peak water level.

This question forced me to think a bit. Because there was so much information being given at once, I didn’t know where to start. When i figured it out, I knew the first thing i had to do was find the slope of the equation, and then plug it into the function.

I used the knowledge I already possessed from previous lessons to figure out how to do this question. I was on the right track, but i had to check the answer to figure out what my final answer should look like. The question confused me a bit, but overall I felt okay doing the question. I feel like if I hadn’t overthought the question, i would have been able to complete it without help.

For my math journal I’ll be using question #3 from the Relations and Functions unit. In questions 3, over the course of 10 years a high school collected data surrounding how much hot chocolate sold during different temperatures. We were given the formula N(t) = 150 – 10t, where N(t) is the number of hot chocolates sold during an average daily temperature of 3 degrees Celsius

When i first read the problem, i understood the concept and what the data was, but wasn’t positive what to do with it. But as i read the data and compared it to the question, it was clearer. Everything i needed to solve was included in the question, i just needed to put what i knew to work. I’m still a little foggy about the vocabulary but as i tried different ways i thought the question worked, it made more sense as to what the correct one was.

I’m a thinker, being that the longer i look at something the faster I’ll figure it out. If if i need a little help, I ask my friend what they did. As you read the question and understand how each piece of data corresponds with the question, plugging in the data only took a small amount of time.

This week we learned about exponents. We learned things like powers with whole number exponents, combining the exponent laws, integral exponents, and rational exponents. This unit we built off the numbers unit and added exponents to the numbers and radicals.

Exponent laws are used to add, multiply, or subtract exponents from each other. It depends on what the equation looks like

^^^ These are the exponent laws! They’re used to deal with specific exponent types in equations.

^^^ I feel pretty confident about this unit, but this is an example of the questions i did have trouble with. I know how to do them now though!