### Algebra

Utilizing projects and experiments, you will be introduced to the real life applications of Algebra. Build upon your fundamental knowledge of variables, constants and coefficients with advanced topics such as dimensional analysis, graphing, linear and non-linear functions, quadratics, radicals, using graphing calculators and more.

### Relationships between Quantities and Reasoning with Equations: How to talk your way out of being arrested as a spy

We started class by making sure we all meant the same thing when we used the words equation, expression, variable, constant, coefficient, and speed.

What is the difference between an equation and an expression?

An expression is a math "phrase" that is any combination of numbers, letters (usually standing in for a number we do not yet know) and operations such as adding, subtracting, multiplying, dividing and exponents. Some examples are: a + b - c and 7 + 3x.

An equation is a math "sentence" which uses two math expressions joined by an equal sign. The equal sign shows that the two expressions are equivalent or have equal value. For example, the equation: x + 2 = 6 says the expression on the left, x + 2, is equal to the expression on the right, 6. So an equation is like a statement "this equals that."

A variable is (usually) a letter or other symbol that represents an unknown number or value. For example, in the equation 2x + 5 = 10, the variable is x. In the equation, 7y - 10 = 24, the variable is y. In the expression 8a + 5b, both a and b are variables.

A constant is a number on its own, or sometimes a letter (usually early on in the alphabet) such as a, b or c that stands for a fixed number. In the equation x + 5 = 9, 5 and 9 are constants. The number pi is also a constant since its value never changes. (Sorry, this editor is not sophisticated enough to have greek letters so I can't use the greek letter, pi.)

A coefficient is the number used to multiply a variable. In other words, coefficients are the number part of terms with variables. In the expression, 3x + 2y - 7xy, 3 is the coefficient of the first term, 2 is the coefficient of the second term, and 7 is the coefficient of the third term.

Speed is the distance covered is a certain unit of time. For example, the speed of a car is 50 miles per hour. This means that for every hour that the car travels, it will go 50 miles. Speed is a rate of travel, since it describes how fast. We can calculate a speed by dividing how far we go by how long it takes us.

We played a game called Equation Mania where we simplified 12 different expressions. Remember that an expression is fully simplified when all like terms have been combined. A like term is a term that has matching letter(s) and matching exponents, for example 3a and - 5a are like terms. 16xy and -2xy and 7xy are all like terms. (Unfortunately, it looks like we will have to use the ^ symbol for exponents, as this editor does not do superscripts, so x^2 means x squared and x^3 means x cubed, so 5x^2 and 7x^3 would not be like terms since the letters may match, but the exponents do not.)

We then calculated how long it would take someone to count to 1 billion if it takes one second to count each number. Of course the answer is one billion seconds. But how long is one billion seconds? Days, months, or years? What is your best guess?

Now let's change units until that big number one billion is more human sized, say between 1 and 1,000. The method for changing units that we will explore today is called dimensional analysis, which can sound a bit intimidating but it is a very powerful method which I will illustrate with a story a little later which will explain why this forum started with a picture of a nuclear explosion...

You can take any quantity like 1,000,000,000 seconds and multiply it by a conversion factor like there are 60 seconds in a minute as long as you make your conversion factor into a fraction and make sure your seconds part of your conversion is on the bottom of your fraction so that the seconds in our one billion seconds cancels with the seconds of the 60 seconds in our conversion factor. See picture of my paper showing the conversions we did to the left. This gives more than 16 million minutes, still a big number!

So let's change to hours by now applying the conversion that there is 60 minutes in an hour, making sure that the minutes part of the conversion is on the bottom to cancel with the minutes. Now we have over 277 thousand hours which is getting smaller but not small enough yet. If we convert from hours to days using 24 hours in a day, that gives us over 11 thousand days. We have to convert to years to get a number under a thousand! Using 365 days in a year we find that one billion seconds is 31.7 years! That is counting 24 hours a day without taking any breaks!

In college my thermodynamics professor told a great story about how a scientist used this dimensional analysis technique to calculate the energy released by the first nuclear bomb test from seeing two pictures of the explosion taken at two times which were given. Since the amount of energy in the explosion was top secret classified information, they started to arrest the scientist until he explained how he used dimensional analysis, the times on the two photos and the size of the explosions shown to calculate the amount of energy released in the explosion. This is one case where being able to show your work saved someone's life!

Our first project was to use unit conversion relationships to figure out which of four given speeds, 55 mi/hr, 640 m/min, 448 cm/sec, 721 ft/sec belong to a runner, a biker, a car and an airplane. We successfully used the method shown above to convert all the speeds to miles per hour and figure out which speed belonged to each item.

Next class we will measure our walking and running speeds, make a graph and look at the linear relationship between distance and time. If we graph the time on the x axis and the distance on the y axis, the slope of the line connecting points that we measure is the speed at which we were moving.

### Computing and Graphing Walking and Running Speeds

As a warm-up we played the Equation Mania game, simplifying some algebraic expressions. We then marked off ten meters and measured the time for each student and Dr Lynne to walk and then run 10 meters. We did two trials and averaged the two times measured to get an average time. We now had the average time for each student and Dr Lynne to walk and run 10 meters. We used these average times to calculate the walking and running speeds using speed = distance/time with the distance of 10 meters. We first calculated our speeds in meters/second then we used our dimensional analysis method from last week to change our speeds to miles/hour (see the bottom of the data table for the conversion). We now graphed the distance (y-axis) in meters (10 m) versus time (x-axis) in seconds for walking and running for each of three students and Dr Lynne. The slope of these lines are the speed of the movement. Slope = rise/run or (change in y(dist))/(change in x(time)). You can remember this by: you need to rise before you run to the bathroom in the morning! The steeper the line the faster the speed because you are going more distance in a smaller amount of time. So looking at the graph, Jasmine ran the fastest and Dr Lynne the slowest while for walking Celin walked the fastest and Jasmine walked the slowest.

We reviewed the relationship between distance, time and speed, speed = distance/time. We solved this relationship for time by multiplying both sides by time, speed*time = distance, then isolating time by dividing both sides by speed. time = distance/speed. Now we can easily compute, if we know our speed, how long it will take us to walk say, from home to Homeroom Education. Students used maps printed out from Google Maps to calculate the distance (as the crow flies, which is not exactly the route we would take) from home to Homeroom Education. Optional homework - compute the time to walk to Homeroom Education and the time to run to Homeroom Education using the distance from Google Maps and walking and running speeds measured in our experiment.

Next week we will be doing a project where students will create equations of various forms for each of the letters of the alphabet, ABCs and 123s. See you next week!

### Many Forms of Equations for Finding a Variable!

We started by practicing simplifying expressions by playing our Equations Mania game. Please see picture at left/top to remind you of the rules of signed multiplication. You can work on a number line - say you have 6 * 9, that is six groups of nine, so you move six spaces to the right on the number line (positive direction) and do it nine times for 54. If you have 6 * (-9) then you can go nine to the left of zero on the number line and do it six times for -54, or have -6 * 9 you travel 9 spaces at a time, but go to the left of zero to create a negative 6 groups for -54. If you have -6 * (-9) -6 is six spaces to the left of zero, but to make -9 groups we have to go the opposite direction and move six spaces to the right of zero nine times for 54. Then we reminded ourselves of the definitions of expression, equation, etc. that we worked on the first class.

Our warm-up was to identify y - 19 = 12 as an expression or an equation? Why?

y - 19 = 12 is an equation because it has the equals sign. How many values can y have?

Only one value. y only has one solution! If you add 19 to both sides, we find that y = 31.

y - 19 is an expression. So is 12. y - 19 can have many values depending upon what you choose y to be. Without the equation, we are not as restricted.

We now created an example equation for j = 10. We made it have the form coefficient/variable+/-constant = constant. See photo at left/top. Our project is to create an equation for each of the letters of the alphabet with a = 1, b = 2, and so on through z = 26 making an equal number of each of 8 equation type with the last two your choice. Half the coefficients and constants should be negative and the other half be positive. I handed out instructions and a rubric for checking over your results. When all of your equations have been made, you will create an artistic display of all 26 equations. This is a chance for you to experience a technique that teachers often use to create quiz and test questions, working backwards from the answer that you want to get. It gives you another perspective on solving equations!

### Project ABC's and 123's Work Day!

All students finished designing their 26 equations, one for each letter of the alphabet, and began using the supplies in the MakerSpace to make an artistic display of their creation. Each letter is assigned a number, a = 1, b = 2, and so on up to z = 26. Nine forms of linear and nonlinear equations were given:

o Variable +/- constant = constant (Ex. z – 14 = 12 )

o (Coefficient) * (Variable) = constant (Ex. 3z = 78)

o (Coefficient) * (Variable) +/- constant = constant (Ex. 2z + 2 = 54)

o (Coefficient) /(Variable) = constant (Ex. -156 / z = -6)

o (Coefficient) / (Variable) +/- constant = constant (Ex. 26 / z + 21 = 22)

o (Variable) / (Coefficient) = constant (Ex. z / 13 = 2)

o (Variable) / (Coefficient) +/- constant = constant (Ex. z / 2 - 21 = -8)

o (Variable)^2 = constant (Ex. z^2 = 676)

o (Variable)^2 +/- constant = constant (Ex. z^2 – 376 = 300)

Equations were to be evenly divided between the nine types given, 2 - 3 of each type. At least half of the constants should be negative and at least half the coefficients should be negative. A rubric was given as a guideline for success. Variables, variable value and equations were to be recorded in a table where the correctness of the project will be checked. The design will only be assessed for creativity and neatness. I saw some amazing and creative displays today!

### Expressions in Real Life!

Once we did a quick warm-up, we made sure we all understood the vocabulary for this section, monomial, binomial, trinomial and polynomial. Mono- as a prefix means one, bi- means two and tri- means three when we are counting terms in an expression. A term is all the coefficients and variables that happen until you get to an addition or subtraction sign. Each time you come to an addition or subtraction sign, you start a new term. So 9a + 6b - 10c has three terms. We can call this expression a trinomial or a polynomial, since one or more terms is a polynomial (poly- meaning many).

We discussed why algebraic expressions can be useful. If we have a problem in which we need to compute something many times, say the cost of a number of packages of pencils, it is useful to figure out how much p packages of pencils cost given each package costs $1.19. One package would cost $1.19, two would cost 2 x $1.19 = $2.38 and three would cost 3 x $1.19 = $3.57, so p packages would cost $1.19p. We could say cost = $1.19p. If we buy 5 packages, we would substitute in 5 for p, getting cost = $1.19 x 5 = $5.95. One equation, cost = $1.19p can help us compute the cost of any number of packages of pencils.

In the expression $1.19p, what can p be? Any natural number, meaning 0, 1, 2, 3, 4, 5, etc. What are the possible values of $1.19p? Multiples of $1.19.

We played a game, It's In the Cards, where we created expressions based on cards pulled from a deck, then simplified the expressions as much as we could, either combining any like terms and/or factoring out any common factors in all the terms of the expression. When you encounter an algebraic expression, make sure you see how many terms there are, note if there are any common factors, see what the coefficients are, look for like terms (only like terms if the letters and exponents match exactly). When you think you might be done, make sure you do not have any like terms you could combine, and all the terms no longer share any common factors! Then, you have simplified fully.

Next we introduced our next project, creating a poster or PowerPoint presentation about a formula that you choose. I demonstrated the process with the expression that will change a temperature in degrees Fahrenheit, F into degrees Celsius: 5/9(F - 32).

You are to locate an expression in a formula used in the real world.

Identify the terms, factors and coefficients

Explain why the expression can or cannot be simplified

Discuss the meaning of the expression for its value in the real world

Evaluate the expression given a value for any variables in the expression by me

Discuss any limits on values of the variable and of the expression placed by the real world context

Example:

5/9(F - 32) converts F a temperature in degrees Fahrenheit into degrees Celsius.

There are two terms. It has a factor of 5/9. If you apply the distributive property of multiplication over addition/subtraction, you get 5/9 F - 160/9. Again two terms but now we have a coefficient of 5/9 and a constant of -160/9.

The expression could not be simplified from the first form given because the two terms are not like terms, one having a F and the other not having an F. Also the only common factor between the two terms had already been factored out.

In the original form, the 32 is the number of degrees that Fahrenheit was off of the freezing point of pure water, because he did not know it was important not to have salt in the water when measuring the freezing temperature. By the time Celsius developed his temperature scale, we had figured this out and he made sure to use pure water for his experiments. The 5/9 is needed because Celsius also chose the size of his degrees so that there are exactly 100 degrees between where pure water freezes, 0 degrees Celsius, and where pure water boils, 100 degrees Celsius, whereas Fahrenheit used salty water where that froze at 0 degrees Fahrenheit and pure water boils at 212 degrees Fahrenheit. The salt in the water depressed the freezing point of the water which made the two scales offset by 32 degrees Fahrenheit but people who live where it snows are grateful for this effect every winter! Only because of this effect are we able to salt roads and melt the ice so it is safer to drive when it has snowed until it gets colder that 0 degrees Fahrenheit.

Temperatures can go as high as you might imagine but can only go as low as "absolute zero," -459.67 Fahrenheit degrees or -273.15 degrees Celsius.

Some students now had time to complete their ABCs and 123s project. When they finish they can work on their Formula project.

### Check back soon!

Notes will be added within 48-hours after class. If not, please send an email to Admin@HomeroomEducation.com to request this Forum be updated.

### Check back soon!

Notes will be added within 48-hours after class. If not, please send an email to Admin@HomeroomEducation.com to request this Forum be updated.

### Check back soon!

Notes will be added within 48-hours after class. If not, please send an email to Admin@HomeroomEducation.com to request this Forum be updated.

### Check back soon!

### Check back soon!

### Heading Goes Here

You can edit text on your website by double clicking on a text box on your website. Alternatively, when you select a text box a settings menu will appear. your website by double clicking on a text box on your website. Alternatively, when you select a text box.