Essential Question: If a square has an area of 100, why are the side lengths not 25 inches?
Notes: Perfect Squares from 2-15 and Perfect Cubes from 2-6
√ - square root
∛ - cube root
∜ - fourth root
Negative Signs with Radicals
∛ -125 = -5
∛ -64 = -4
-√ 4 = -4
-√ 100 = -10
√ -100 = Doesn't exist (imaginary number)
√( 125-25) = 10
√ 125 - √ 25 = √ 125 - 5
Square roots can have two solutions...
√ 4 = +2, -2
√ 9 = +3, -3
√ 144 = +12, -12
1. The product of a number and negative 6 is equal to 42. What is the number?
-6x = 42 x = -7
2. A taxi costs an initial fee of $2.50 and then $0.50 per mile. If the total taxi
fare was $17.50, how many miles did the taxi drive?
2.50 + 0.50x = 17.50 x = 30
3. A video game and a CD cost $48.00 together. The video game costs three
times as much as the CD. What is the price of the CD?
3c + c = 48 c = 12
4. During a basketball season, Veronica scored 39 points. She scored 5 fewer
than twice the number Rochelle scored. How many points did Rochelle score?
39 = 2r - 5 r = 22
5. The maximum area of a square is 100. What is the largest that the perimeter can be?
√100 * 4 = P P = 40 or x^2 = 100 x = 10 P = 4x P =40
6. A cube’s volume is 125. What is the side length of each side?
x^3 = 125 x = 5
Essential Question: Why is a repeating decimal a rational number?
1) Set the decimal equal to x. In this example, that means that x = 1.833333 repeating.
2) Create two equations using x to represent the decimal so that the repeating part of the decimal lines up in both equations. For this example, my two equations can be 10x = 18.333 and 100x = 183.333. Why can't we use x = 1.8333? (The 8 in the decimal will not line up properly with another 8 in any other equation. Only the 3's can line up with one another.)
3) Subtract the equations from one another. Make sure that like terms are put together. 100x - 10x = 183.333 - 18.333 will equal 90x = 165.
4) Solve the equation by doing the inverse operation to isolate the variable. Here we would divide by 90 on both sides and simplify by dividing by 15 to get 11/6, which is 1 5/6.
1) Set the decimal equal to x. x = 1.267267267
2) Create two equations so the repeating part of the decimal lines up correctly (this helps us when we need to subtract so there is no leftover in the decimal place values - if there were a decimal left there, we would have a weird looking fraction after solving the equation). Here I can use x = 1.267267 and 1000x = 1267.267267 because the part after the decimal is the same for both.
3) Subtract 1000x - x = 1267.267267 - 1.267267 to get 999x = 1266 (Hopefully you realize that the decimal part should always subtract to equal 0, leaving a whole number to work with.)
4) Solve the equation and simplify it from 1266/999 to 422/333, which is 1 89/333.
Using 1.8333 repeating
1) Take the non-repeating part of the decimal and subtract it from the entire decimal. Put the answer as the numerator. 83 - 8 = 75 for the numerator.
2) Count the repeating part of the decimal until it repeats itself again. In this example, we need only 1 place value before it repeats again. This tells you how many 9's to put as your denominator so we should have one 9 in the denominator.
3) Before you are finished, count how many numbers are NOT repeating in the decimal, and add a 0 to the denominator for each one. We have one digit that does not repeat so we have one 0 to add to the 9 in the denominator.
4) All done - your answer will be 75/90.
From the Quizizz:
1. 2x = -5
2. a/-2 = 5
3. x+(-3) = 4
5. -4m + (-12) = -8
6. 2p + 5 = -17
7. (-3x+2)/-5 = 9
8. -17 + 4 + 3x = -2x - (-5) - 4x
9. 20-7k+3(k + 4)=4k
10. -5+2(y - 4)=12-3y
Answers: 1. 2.5 2. -10 3. 7 4. -1/2 5. -1
6. -11 7. -15 2/3 8. 2 9. 4 10. 5
This Jeopardy game was played in class, and it's a good way to check yourself.
Answers: 1) x = 1 2) x = -5 3) x = -6 4) x = -8
5) x= -6 6) x= 2 7) x = 1
There are more practice problems here.
Reflections should be completed after you present.
Your reflection must answer every question below using full sentences. Your reflection should be between ⅓ of a page and 1 page long.
Variables on Both Sides
Below is a great video found by a group that presented on multi-step equations.
Combining Like Terms
Solve equations with more than one like term the same way that you simplified expressions with more than one like term.
x + 3 + 4 + 5x - 6 + (-8x) = 1 + -9 Look for all the terms that are the same and pay
attention to the sign in front of each one.
x + 5x + -8x can go together to make -2x
3 + 4 - 6 can go together to make 1
1 + -9 can go together to make -8.
Notice that you can't combine terms that are on opposite sides of the equation because the equal sign is separating those terms. You can only combine those terms by using the inverse operation.
We simplified the equation to look like -2x + 1 = -8
Following the order of operations, we would multiply 2 and x together first, then subtract 1, so to solve the equation we work backwards using the inverse operation for each step. This means we will add 1 first, then divide x by 2.
-2x + 1 = -8
-2x = -9
x = 4.5 or 4 1/2
A video from a student explains this concept perfectly.
This type of equation can be confusing if you don't understand that we are working with the order of operations (PEMDAS) in reverse. This means that whatever you would do first when you simplify the expression would now come last as its inverse. Below are a couple examples and a very helpful video.
Notes: An equation is a problem that uses an equal sign to show that both sides of the equation are balanced. In a 2-step equation, you have exactly 2 operations that are affecting the variable. In order to solve a 2-step equation, you must undo these 2 things by doing the inverse operation in order to isolate the variable, or get it by itself, to one side of the equation. When you undo the operations on one side, you must do the same thing on the other side so the equation stays balanced. With two-step equations, you have to undo addition or subtraction first, then undo multiplication or division because you are working backwards with the order of operations.
This powerpoint explains 2-step equations step by step and includes several practice problems.
***Working with -x ***
If the variable is negative with no number beside it, it means that the variable has been multiplied by a -1. Remember that we don't write 1 beside the variable because it is understood that x is the same as 1x. The same thing is true for -x and -1x. To solve an equation with a negative variable, undo addition or subtraction first, then undo the negative coefficient by dividing by the negative. These are the same steps as usual.
Example: -x + 6 = -10
-x = -16
x = 16
Essential Question: Why do we use inverse operations when solving an equation?
Provide the inverse of each operation.
A 1-step equation is an equation that involves one operation. In order to solve the equation, you must use the inverse operation, or do the opposite of the operation in the equation. For example, the inverse of addition is subtraction. When you undo the operations by using the inverse, you will isolate the variable to get it by itself on one side of the equation. When you perform the inverse operation on one side of the equation, you must do it to the other side so both sides remain balanced and equal.
Solving 1-step equations video
More notes below.
What can an equation be compared to? What is an equation made up of?
In the real world, when might you use an equation? Come up with an example to justify your answer.
This site compares and contrasts expressions and equations.