We learned more about graphing and linear equation. W learned about different ways to find solutions to equations. They are Inspection, Substitution, and Elimination.

Inspection is only used when a system is very easy and understood. It is basically just eyeballing the question and guessing what the solution is. You can test your solution by plugging the numbers in the appropriate spots (x and y).

Substitution is another method to find solutions, but it uses algebra. You can choose one of the equations, and then choose on of the variables to isolate. Then you plug it into the correct spot on the other equation. You then use algebra to solve the equation, and then use the answer to plug in to find the other variable.

ex.

Elimination is used when there are no coefficients of 1. You start by adding the two equations together (you can subtract but it doesn’t work as well). You want to make a zero pair by making either the x or y cancel the other out. If they don’t do this from the start, you can multiply one of the equations or both of them to get one, and then do your adding. Once you have the answer to one variable, you plug it into the equations and find the other.

This week we further continued unit 7/8 by learning about point-slope form and general form. Both of which are put into a different form than slope formula. Point slope form is useful for doing quick algebra, and general form has no fractions and other “imperfections”.

Point-slope form looks like the following:

is the slope. is not connected to a point, but is. The same goes for and . You can take an equation or “hints” to make a point-slope formula.

ex.

To further turn this into slope formula and make it easier to understand, you use algebra and steps.

continuing with ex.

General form is the “pretty useless” form, pretty, but useless. It contains no fractions or decimals, but doesn’t tell you about the graph itself. The equation usually looks something like ax ± ny ± b = 0. the x is always first, y is always second, x is always positive, and everything is always on one side equalling zero.

You can change all forms/formulas into general form.

ex.

slope formula.

Point-slope form.

slope formula with fractions.

General form uses no fractions and follows several rules as listed in the paragraph above. It is used to take away fractions and “imperfect” parts of a formula, but it doesn’t tell you anything until it is changed into another form.

We researched several different chemical reactions, but eventually settled on The Black Snake. We looked at the components to make sure it wasn’t dangerous, or at least not too dangerous. The Black Snake uses powdered sugar and sodium bicarbonate (baking soda) along with rubbing alcohol. These chemicals aren’t inherently bad, but alcohol fumes can be dangerous. Another danger is when lighting it on fire to commence the reaction, because fire can obviously be dangerous.

To make The Black Snake, you take 4 parts baking soda, and 1 part powdered sugar, and mix it together. Make a vessel out of preferably tinfoil filled with sand. Make a divot in the sand, and pour the mixture into it. Put rubbing alcohol around the edges of the mixture, and a little bit throughout the middle. Use a barbecue lighter to begin the reaction. A snake made of what looks like ash emerges from the white powder mixture. The snake is very light and airy because of gases produced during the experiment. The snake can grow quite long, but doesn’t always.

What is happening is the sugar C_{12}H_{22}O_{11 }combusts and turns into carbon dioxide and water vapour, this decomposition forms the snake. The baking soda is added to help the experiment rise (2NaHCO_{3} → Na_{2}CO_{3} + H_{2}O + CO_{2}), just like how it is used in baking. Reactions:

Sugar combusts into water vapour and carbon dioxide: С_{12}H_{22}O_{11} + 12O_{2} → 12CO_{2} + 11H_{2}O

Decomposition into carbon and water vapour: С_{12}H_{22}O_{11} → 12C + 11H_{2}O

Baking soda decomposes into carbon dioxide, water vapour, and sodium carbonate: 2NaHCO_{3} → Na_{2}CO_{3} + CO_{2} + H_{2}O

The outcome should be a carbon, black snake. It should be light, look and feel like ash, and should be quite delicate. The snake is not edible, you can touch it but it is not recommended. It can be very hot after burning. Best to do in outdoors or under fume hood because of alcohol fumes produced. The snake and all components of experiment can be thrown out in a household garbage.

The experiment can seem magical because when it is growing, it looks as if it is alive and moving, like a snake. It could also possibly look like a plant growing. It’s like creating life because of its natural seeming movement, even though it is just burning, rising chemicals.

Bibliography:

Maric, Vladimir, and Teh Jun Yi. “How to Make a Fire Snake from Sugar & Baking Soda.” WonderHowTo, WonderHowTo, 18 Oct. 2017, food-hacks.wonderhowto.com/how-to/make-fire-snake-from-sugar-baking-soda-0164401/.

“Hooked on Science: ‘Black Snake’ Experiment.” SeMissourian.com, 3 July 2013, www.semissourian.com/story/1983035.html.

Experiments, Life Hacks &. “How to Make Fire Black Snake? Amazing Science Experiment.” YouTube, YouTube, 17 June 2018, www.youtube.com/watch?v=Y7snO0pA8Sk.

Although I was sick for half of this week, and wasn’t able to learn fully what the rest of that class did, or at least learned less “hands on”, I was still there for the first two days, and so what we learned then is what I understand most. We learned how mapping notation can be put into the form of function notation.

Mapping notation is where you use a math “sentence” to find an output with the use of an input. (went over semi-briefly on last blog post).

ex.

ƒ : x → 3x – 2

↑name ↑input ↑changes into ↑output

Function notation is generally the same thing, but like how functions are relations but relations aren’t always functions, functions notation is the same. Function notation is helpful when finding inputs and outputs of functions. They are written slightly differently as well.

ex.

↓name ↓input ↓changes into ↓output

ƒ (x) = 3x – 2

↑ “ƒ of x”

Both are used generally the same way, to find the output using an input. It is “ƒ of x” because the ƒ is the functions name, and the relation is a function.

Functions & Graphs

Using the inputs and outputs from mapping and function notation, you can plot points on a graph. The input is x, and the output is y. To get the output, you put the input in the correct spot on the opposite side.

This week we had our midterm and spent most of the week studying for it. But on Friday, we learned about functions, a kind of relation.

A function is a relation that is special and each input has one output, no more. A function is a kind of a relation but a relation is not a kind of function.

On a graph, if any of the points are on the same x axis, then it is not a function. Each point has to be in a different x coordinate.

ex.

A function is unique, and is often named a single letter (f, g, h, etc.), and followed by x, changing into blank.

ex.

ƒ:x →7x + 6

ƒ is its name, x is the input, the arrow signifies “changing into”, and the final numbers are the output.

This week we started our graphing and linear relations unit. One of the main things we learned was domain and range. The domain is all of the coordinates that the graph covers, and the range is all of the coordinates that are covered.

Domain and range can be shown in “curly brackets” such as in the following example.

{x|-4 ≤ x ≤ 7, x ∈ R}

Sometimes if the graph just contains a bunch of points, the domain and range can be given in specific numbers,

ex. D = {-2,0,1,4,7} or R = {1,3,4,9,12}

here’s what one of those graphs could look like:

But they can also be lines meaning their points can be anywhere on those lines,

ex. D = {x|-2 ≤ x ≤ 7, x ∈ R} or {y|1 ≤ y ≤ 12, y ∈ R}

here’s what one of those graphs could look like:

They can also be a line, but have no beginning and/or end. This graph would have lines with arrows to represent that it continues on.

ex {x|x ∈ R} or {y| y ≤ 12, y ∈ R}

here’s what one of those graphs could look like:

When writing in these curly brackets, especially with line graphs, you need to form a “sentence”. You start with the axis you are talking about (x/y), then the possible points, and then finish with x ∈ R or y ∈ R, which means x/y is an element of a real number.

This week was short, and on top of that, I missed a day because I was at a field trip for my science honours class. But even though it was a short week, I still learned more about trigonometry, specifically word problems and how to use them to find angles and side lengths, something we’ve been learning over the whole unit.

To find the angle of a triangle, you can use two side lengths for the equation (sin/cos/tan) xº , for example: sin xº . With that equation (using the example for the following), you find x by isolating it as in xº , and then you will have the value of xº.

To find a side length, you would use one side tenth, and an angle for the equation (sin/cos/tan) xº , For example: sin 31º . With that equation (using the example for the following), you find n by isolating it, in this case 15 sin 31º . If it were sin 31º , you would use the equation sin 31º = 15 meaning you would have to divide both sin 31º (cancelling it out) and 15 to find n.

I made a simple word problem to find the side length of a triangle based on the height of a person and the suns angle to find the length of the persons shadow:

This week we began our unit on trigonometry. One of the things I learned was SOH CAH TOA, abbreviations that help you to memorize how to find side lengths using angles and equations.

Each set of abbreviations begins with a letter that describes the angle equation to use on your calculator (S=sin C=cos T=tan). Depending on the angle of the triangle you are using and what side lengths are given to you, you can choose the correct angle to use.

Each triangle has 3 sides, and when a base angle is given to you, these sides are given names. The longest of them is the hypotenuse, and the side that the base angle sits on is called the adjacent side, the final one, the opposite side, sits on the opposite of the angle.

The other two letters in each abbreviation shows you what sides to use and in what order (OH=opposite/hypotenuse AH=adjacent/hypotenuse OA=opposite/adjacent. \

Even if you only have one side of the triangle and the angles (90 & other), you can find the side length you are looking for.

This week I learned how negative exponents work and how they effect their base’s. Negative exponents unlike positive exponents don’t increase the numbers size. Normal exponents multiply a number over and over (ex. ) and negative exponents turn the number into a fraction (ex ).

You can turn it into a normal number by following the next steps. I will use for this example.

First, if the exponent is negative, then you turn it into a fraction of .

Then you put the denominator on the bottom, therefor making the exponent positive.

Alternatively, I learned that if the negative was originally on the bottom, you would then move it to the top.

Then you find out what the product of the power is and put that underneath a 1, in this case

I learned that if it were for example , then only the and its exponent would be moved to the denominator as in

If it were , then both would move to the bottom.

if it were would equal because you multiply the exponent.