Math 58 Class Notes
Notes for the term.
January 12, 2021
Tuesday
Page 23, Question 6 of Class Work for Lesson 1-2.
Algebra is a branch of math where variables are used. Variables are letters used to represent physical measurements and to represent numbers that vary.
An example of variables, formula for area of a rectangle.
A = L x W, where A stands for area, L stands for length, and W stands for width.
We can stick in values for variables or we can add and subtract variable expressions.
Looking at Page 23, Question 6 of Class Work for Lesson 1-2, we use P to stand for grams of protein.
6P + 9P + 3P + 1 P + 6P + 0P
2
In algebra you can add or subtract items with the same variable because they represent the same type of items.
Add or subtract numbers in front of each variable and these numbers are called coefficients.
We have 24.5P or 24.5 grams of protein.
Question 7.
24 g sugar in her yogurt and 2 mg cholesterol in her yogurt. Can we add these??? NO, NO, NO, they represent different items.
We call terms with the same variables, LIKE TERMS. You can only add or subtract like terms.
Question 8.
Adding 3000mg of carbs to 68.5 g of carbs ?? NO, adding carbs but one is in grams and one in milligrams.
Question 9.
Convert 3000 mg of carbs to 3 g of carbs because 1000 mg in one g. We now add 3g + 68.5g = 72.5g.
January 14, 2021
Thursday
Material:
- You need a book. You can buy a paperback book, you can buy a PDF copy of the book, or you can rent a book. We will use the book for content and problems.
- You need a Connect Math license and then go to website connectmath.com. To do the work. Some purchases of the book and/or Connect Math give you the license and an ebook.
- Notebook either on the computer or spiral, binder, etc.
- Working computer to use Zoom.
How to turn in work:
- Most days we will have an Applications assignment and a Technology assignment.
- Each Lesson of the book has an Applications assignment. You can simply use your own paper and write complete sentences, and put the math required on each question, or graphs, or charts. You can also tear out pages of a paperback book and fill out the answers on the pages from the book. You can also copy and scan pages from the book and make a electronic copy. I may provide a fillable PDF or Word form of the Applications assignment.
- If you do Applications on paper or need to do a graph or chart on paper, take pictures, upload pictures and put a Word document, Google Doc, Excel and then make a PDF file. There is also a phone application called AdobeScan to take pictures and it will email a PDF file to you.
- Once you have a PDF file for your Applications assignment go to D2L to the appropriate Applications number and submit your file.
- The other assignment due most days is a Technology assignment. The Technology assignments are almost always an Excel spreadsheet and you can submit the spreadsheet (.xlsx or .xls ) file to D2L.
When to write or use a computer ?? In class when we do the Class work in the book or when you do your Applications assignment, you can generally use old fashioned pencil math, use a calculator or Excel, or other tools, just put the information in your notes, on the Applications assignment or on paper copies of Class work.
In Lesson 1-2 we learned the concept of a variable which is a letter that represents numbers that can vary. Letters or variable are used in formulas or mathematical expressions.
Example: A = L x W. Area of a rectangle is equal to the length times the width.
Variables can be used in expressions and they can be added, subtracted, multiplied and divided without substituting in numbers for the variables.
We only add the same variable terms together. These are called like terms.
Like terms have the same variable or variables and the exponents on the variables are the same.
Examples:
Like terms: 3x + 5x
Not like terms: 3x + 5y, [Not like terms because one term has x and the other term has y.]
Like terms: 3x2 + 5x2
Not like term: 3x2 + 5x [Not like terms because one term has an exponent of 2, and the other term does not have the exponent.]
A math term has a number called a coefficient, a variable and possibly and exponent.
Example: 5x2 has coefficient of 5, a variable of x, and an exponent on the variable as 2.
To add or subtract like terms, only add or subtract the coefficients.
Example:
3x + 6x + 9x = 18x
Lesson 1-3:
Addition and multiplication is commutative; add or multiply in a different order and still get the same answer.
Move terms, or commuting terms, does not change answer when adding or multiplying.
Examples:
2 x 3 = 6 or 3 x 2 = 6.
5 + 6 = 11 or 6 + 5 = 11.
Multiplication is repeated addition of the same number.
Example: 10 + 10 + 10 + 10 is the same thing as 4 x 10.
Class work is on page 38. 1-3 Class
January 19, 2021
Tuesday
Excel is a spreadsheet with rows and columns.
The rows are numbered 1, 2, 3, etc.
The columns have letters, A, B, C, D, etc. After 26 columns, the naming changes to double letters, AA, AB, AC, until AZ, and then BA, BB, BC, etc.
The intersection of any row and column is called a cell. A cell is named using the letter or letters on the top of the column and the row number on the left. For example, the very first cell is A1. If you stay in the first column and go down 10 rows, the cell is A10.
On the upper left of Excel is a Name Box. You can see the location of your cell. In the Name Box you can also type in a cell name and the cursor will pop over to that cell.
You can enter numbers, letters, pictures shapes and many other items into a cell. Generally only number or letters making up words are put into cells.
Excel is good for making a chart or table, but the real power comes in using math. Numbers in cells can be calculated in your own equations or in many provided equations. To do math, type in an = sign, and then put in your values and operations or a provided function. You type in a number or a cell name. You can see the math in the upper right in the Formula Bar.
To add use a +, to subtract use a -, to multiply us an *, to divide us a /, use ^ for an exponent, use ( ) for grouping and doing operations in a particular order.
If an operation cannot be performed you will get a #VALUE! in a cell.
Excel has a fill feature by dragging a cell or cells. Move your cursor to the to lower right and your mouse tracking symbol will change to a skinny-black-plus sign from a wide-white-plus sign. Once you see the small black sign, push the left button on mouse, hold it down and then drag in any direction.
You can also clean up Excel by hitting CTRL-A and then hit the Delete key.
When dragging and filling cells, formulas with cell names will also be copied and changed.
You can make graphs or charts by selecting cells and going to Insert charts.
You can insert common formulas by going to Formulas.
Math Items, exponents, square roots, order of operations, distributive property.
New topic: Exponents
Exponents or power are used to show repeated multiplication of the same number. There are two parts of an exponential expression the base and the exponent. Example:
23: 2 is the base and 3 is the exponent.
An exponential expression is simplified by using the base as a factor. You have as many factors as the exponent.
23: Means the base, 2, is used as a factor three times, 2 times 2 times 2.
A factor is a number multiplied by another number to obtain a product. For example:
(2)(3) = 6, 2 is a factor, 3 is a factor and 6 is the product.
Be careful identifying the base when there are negative signs.
Examples:
Base is (-3). When a there is an exponent outside of a ( ), the base is the entire ( ).
(-3)2 = (-3)(-3)
= 9
Base is just 3. When there is a negative or minus in front of a number and there is an exponent on the number and there is no ( ), the base is just the number without the negative sign.
-32 = – 3 ∙ 3
= – 9
Why ? What if -32 was used in a bigger problem such as:
20 – 32, We would not want to lose the minus sign and thus,
20 – 32 = 20 – 9
= 11
In the case of 20 – 32, there is still a minus when evaluating the exponent, and if – 32 is alone it is still a minus.
Another example with an exponent.
25 = (2)(2)(2)(2)(2)
= 4(2)(2)(2)
= 8(2)(2)
= 16(2)
= 32
A square root is the inverse operation of a base to the second power. A base to the second power is also called “squared.” For example 32 can be called 3 to the second power, or just 3 to the second, or 3 squared.
We use the term, “squared” because when we are multiplying for an area we are measuring how many squares. Area is calculated by multiplying base times height. We are multiplying two numbers.
The symbol for a square root looks like a long division symbol with a V in front.
A square root of a number means to find a new number such that when the new number multiplied by itself gives you the starting number.
The square root of 16 means what new number times itself gives you 16. The new number would be 4, because 4 times 4 = 16.
Another way to look at the answer in a square root, is what number squared gives me the starting number. For example looking at the square root of 16, , what number squared gives me 16 and the answer is 4.
Order of Operations:
Many real world problems have math equations where there are multiple operations. If the order is not done properly, there are a variety of answers.
Steps to Simplify a Problem Using Order of Operations (Section 1.4)
- Write original problem.
- Use the acronym PEMDAS or the verse, Please Excuse My Dear Aunt Sally to help remember the order.
- P: Means to work inside of parenthesis ( ), [ ], { }, | |, square roots, or other grouping symbols. If there are nested symbols, start with innermost set. Once there is just one number inside, leave the ( ) alone. It will be cleared on another step.
- E: Means exponents should be simplified. Go off to the side with the base and exponent, expand, multiply and find result. Put result back into the problem. If base was a ( ), put result back in problem with a ( ).
- M or D: Means multiply or divide and the order is which one comes first working left to right. This step is commonly where ( ) are cleared.
- A or S: Means add or subtract in order from left to right.
- Other notes:
- If there is an absolute value, | |, simplify inside, evaluate and put result in a ( ).
- If there is a square root, , simplify inside, evaluate and put result in a ( ).
- In general only do one operation per line of math.
January 21, 2021
Thursday
Examples of Order of Operations
Simplify or evaluate:
- 2000 + 4 x 10
(We have two operations multiply and add, the priority is multiply)
= 2000 + 40
= 2040
- 100(1.2)3
(We have two operations, multiply and an exponent, the priority is the exponent)
(Go to side and expand)
(1.2)2 = (1.2)(1.2)(1.2)
= 1.44(1.2)
= 1.728
= 100(1.728)
= 172.8
Watch ( ) and negative signs when expanding exponents.
If there is an exponent outside of a ( ), the base is the entire ( ) and contents.
(-5)2 = (-5)(-5)
= 25
If there is a minus sign or negative sign and no ( ) and a number with an exponent, the base is just the number and not the negative sign.
.. – 52 = – 5 . 5
= – 25
Section 1-4
A percent means per cent, or per 100, or divided by 100.
8%, means 8 per 100, or .
= .08
That is why 8% = 0.08.
When money is borrowed or invested usually interest is owed or is earned. There are two main types of how interest is calculated.
Simple interest and compound interest.
Simple interest take the starting amount times the annual interest rate times the number of years plus the starting amount.
A = Prt + P
Or
A = P + Prt
Where,
A = amount have t years.
P = principal or starting amount
r = annual interest rate expressed as a decimal
t = time in years.
Compound interest is calculated differently in that the interest each year is added to the principal and then the next year there is more principal and the interest is added again.
Start with $1000 and a rate of 5%. After the first year, take $1000 times 0.05 and the interest will be $50.00 and we have an amount of $1050.00.
At the start of the second year the principal is now $1,050. We take $1,050 times 0.05 and the interest will be $52.50. Add $52.50 + $1,050.00 and after two years the amount will be $1,102.50.
Compound Interest Formula:
A = P(1 + r)t
Where,
A = amount have t years.
P = principal or starting amount
r = annual interest rate expressed as a decimal
t = time in years.
…
Distributive property allows you to multiply a number times terms in a ( ). This is especially useful when we cannot add the terms in the ( ).
Ex 2(5a + 6b) = 2(5a) + 2(6b)
= 10a + 12b
Simple interest written as A = Prt + P is a specific type of equation. You should notice the time is multiplied by the a number made up from P times r.
The type of equation is a linear equation. A linear equation using x and y is:
y = mx + b
or
y = b + mx
Compound interest is a type of an equation called exponential equation. The time is used as a power.
In conclusion, linear uses variable times and exponential uses variable as a power or as an exponent.
In class, we did not get to finish a few problems from 1-4 Class problems. Please see the following video:
https://www.viddler.com/embed/85bcd25e/.