Converting Units

Video Link

Slope Rap

Video Here

Alternative:

Video Here

Slope Games

Save the Zogs

This game will help you understand linear equations better.  Use the sliders at the upper left to help guide your answers.

Halo Slope 

This game uses images similar to the famous video game.  Slide the sliders from left to right to control the character and line up the "shot".

Destroy the Roaches

This game requires you to know what slope is and how to manipulate it.  Play it after you're comfortable with the previous games.

Mean Average Deviation

Mean Average Deviation:

The average distance each number is from the mean.

Mean:

The middle found by adding all the numbers of a data set together and then dividing by the amount of numbers.

Distance:

The positive difference between two numbers on a number line.


How do I find mean average deviation?

1.  First find the mean by adding all the numbers and dividing the sum by the amount of numbers.
2.  Subtract the mean from each number in the data set.
3.  Use absolute value to represent the distance of each difference.  This makes them all positive.
4.  Add all those numbers together and then divide by the amount of numbers in the data set.

Sales Tax

1.  What is a sales tax?

2.  What does the sales tax of Texas pay for?

3.  What does the sales tax for San Antonio pay for?

4.  Pretend you are an adult.  You would like to treat your partner to a meal.  How much will it cost?  What should you pay for tax? What should you pay for tip?  What is the total cost?

Scientific Notation Uses

We use scientific notation to deal with really big and really small numbers, especially when they are measurements.  The precision (how close the number is to the actual measurement) is based on how many numbers are represented in the notation.

So a number with many decimal places times a power of ten is more precise than a number with only a few decimal places times a power of ten.  2.9 times 10 to the 8th power is less precise than 2.98334 times 10 to the 8th power.

We use a trailing zero to indicate that there are no further numbers: 3.0 times 10 to the 3rd power is 3000, whereas 3.238 times 10 to the 3rd power is 3238...etc.

Big Problems

Pick a social problem that is currently affecting you or that you care about.


1. Gather information. Define the problem. List 3 facts about the problem.

2. Analyze information. What issues are related to this problem? What challenges need to be overcome before the problem will be solved?

3. Make a Plan. What steps need to be taken to solve the problem?

4. Determine the Solution. What is the outcome of following each of the steps?

5. Justify the Solution. Will this solution partially or completely solve the problem? Explain why.

6. Evaluate for reasonableness. How much work would it take to implement your solution?

Net Worth Project

Answer the following questions: 1. Define Net Worth 2. Pick a famous person/company who is alive. 3. What are they famous for? 4. What are their costs? 5. What is their net worth? 6. How did they make the money?

Ideas about Money

For this writing exercise, I want you to write a paragraph about the following question:


If you were an adult living on your own and had a paycheck, what kinds of things would you spend it on?

Types of Numbers

In this post, I will describe types of numbers we will be working with.  You may use this information to fill out your Frayer model graphic organizers. 

Natural Numbers

The counting numbers starting with 1.  These numbers are used to count physical objects. 

Whole Numbers

These numbers are counting numbers and zero.  Zero is an important concept because it allows you to completely reduce a quantity.  We also use zero to hold places, such as representing the number 10 as one ten and zero ones. 


0 and 1 are both very important numbers, due to some information we will learn about later.

Integers

The positive and negative counting numbers and zero.  The negative numbers allow you to represent missing quantities, or take large quantities away from small ones but still represent that some were there.


But what about numbers that represent partial objects?

Rational Numbers

This set includes ratios, fractions and decimals that end (this is called truncate).  This set allows for numbers to represent parts of objects. 

Real (Irrational) Numbers

This set includes decimals that don't form repeating patterns.  Square roots of numbers that are not even, and pi, e, and tau go here.  Pi, e, and tau are numbers that you will learn about in future math classes.


Source: http://jamesbrennan.org/algebra/numbers/real_number_system.htm

Math As Language

Your first assignment for me is to answer this question in paragraph form. (complete sentences, and correct grammar and punctuation please)


If Math is a language, what does it say?

Binders Full of...Math!

Hello my students.  This is a reminder that you do indeed need a 1" binder for my class.  If you are not in class when we start to build our notebook, I will refer you to this post, where I will explain how to build your binder.

The binder will help you keep track of what assignments you may have missed due to being out sick or on a trip, as well as help you use your notes.  It is important that the binder stay organized and that it is taken care of.  Please do not throw the binder into the bins, as this will tear them up.  Please do not touch other people's binders or shove them to the side to get your binder.  I will have someone who comes in early pass out the binders.

To assemble the binder (in case you are new or have misplaced or damaged your binder), you will need either dividers or different colored construction paper and a 3 hole punch.  You need sections for notes, practice, quizzes and tests.

The first page of your binder needs to be your grading page.  I will provide copies of a blank grading page which you will fill out as we are in class.  I will post an updated one at the start of each week that will show all the assignments we've already done as well as the new assignments for the week.

Thus it will be up to you to check for yourself if you have missing assignments.  If you have missed something, there will be a binder with blank copies of all of the notes or assignments (except for tests and quizzes) that we will be working on. Quizzes and Tests will have their own binders.  If you miss one, you will receive an alternate version of the quiz everyone else took.

Scientific Notation Games

Here is another explanation of scientific notation.

Here is a site that has some practice scientific notation.

Here is another site that has more practice on this concept.

To My 7th Graders

I know we've been working hard lately in preparation for the test.  Now that the test is behind us, we will go back to normal schedule.  This does not let you off the hook for math class.  We do have new things to learn and we will cover new topics.  These will be previews of math you will be expected to be better at next year.  Unfortunately, the State standards we've been trying to meet are changing, and requiring yet more of you, so next year will be somewhat difficult.  If you were ok this year, you should be ok next year, too.

We will be covering some more detailed volume ideas, something called surface area, and we will graph lines. If we have time, we may briefly touch on Pythagorean Theorem.  We will also discuss math's role in your everyday lives, (or perhaps the lives of your parents), with regards to taxes, commission sales, leaving tips, shipping, and mark-up/discount.  We will also calculate interest and learn a bit more about why interest can be really bad, as well as really good.

So keep heading onward!

Advance To Algebra

For my 8th Graders, we will explore what skills we need to be successful in 9th grade algebra class.  Algebra has been called the gateway to 21st century jobs.  By starting early, we have the ability to preview content to identify areas of difficulty, relate our current understanding to what skills we will need in the upcoming class, and be aware of what skills we need to retain or brush up on during the summer.  My blog will be active during the summer and will be a great place to practice your math skills.  I will try to have some extra games to bridge the gap between 8th grade and 9th grade.  You can already play Shuttle Mission, Save the Zogs, and Save Our Dumb Planet.  Mastering these three games will help a great deal when dealing with linear functions.

We will also be working in Pre-Algebra Demystified.  We should be starting to reach a point in that book that we have yet to cover, so new information will be good.

Also, we will be using this website as a tentative guide for what we will be studying.

To those of you who think school is "done" now that your tests are complete, I want to ask you a simple question.  Do you want to have a whole year extra of math just because you don't want to work now?

If yes, just please try not to disturb the people who answered no.

Fractions Practice

Subtracting Fractions Practice
Shoot the correct fruit to win.

Multiply Fractions Millionaire


How to divide fractions
This page explains how to divide fractions.

Division Practice
A soccer game to practice dividing fractions.

Proportions Games

Equivalent Fractions Dirtbike


Dunk Tank

Scale Factor and Area

Most of you understand that the scale factor is the scale length over the original length.  But what happens if you're looking for the area and don't know the length?  To find out, let's first use scale factor to find area of a new shape.

This rectangle has an area of 3 * 8 or 24.  If we dilate it with a scale factor of 2, the new sides will be multiplied by two.  But is the area also multiplied by 2?  Let's find out if we get the same thing.  24 * 2 = 48.   But 3 * 2 = 6 and 8 * 2 = 16.  Multiplying 6 * 16 gives us 96!  96 does not equal 48, so something else must have happened.  Let's compare 96, the scale area to 24, the original.  If we divide 96 by 24 we get 4, not 2.  So something happened to the scale factor when we calculated area.  It got squared!  2 * 2 = 4.

This means that any time we are working with area only and we have a scale factor to find new area, we need to square it.  Also, we know that area is two-dimensional and that units that are two dimensional are square.  Therefore it makes sense to square it.  It also leads to the next thought.

What do we do with scale factor and volume?

Proportions in Nutrition

http://threeacts.mrmeyer.com/sugarpackets/

Work through the entire thing.

Here are some foods and drinks you like.  How many sugar packets are in each?  Don't forget to multiply the calories by the servings if you drink the whole can.

Here's the information for a bag of M&M's.

Using Probability to Learn Statistics

Probability tests using number cubes or spinners are a good way to create a set of data.  The numbers have variance without being overly large, a mode is present, the range is also not very large, and you can relate the mean to probability through the "Law of Averages".  You can also bring up the bell curve.  The data set created can be used to create circle graphs by showing experimental vs theoretical probability, bar graphs by breaking it down per probability device, histograms and line plots to show frequency of results, scatter plots to show trends between the spinner and the number cube, and box and whisker graphs to show range, median, and quartiles.  This will require a bit more care with the data sheet, as carelessness will lead to the initial part failing.

Graphing Linear Equations

For those of you who are ready, here's the next step from what we were doing with charts:

Graphing Linear Equations!

It actually gets kinda fun once you get the hang of it.

Here's Cool Math's explanation.

Here's a game for solving them.

You may have seen this one before but it is also about graphing linear equations.

Probability

How to solve probability problems:

Each of the letters in the word
SUPREMACY are on separate cards
face down on the table. If you pick a
card at random, what is the
probability that the letter on it will fall
within the range of "H" and "U" in the
alphabet?

Probability problems are usually about creating ratios, then using them to answer questions related to them.  To find the ratio, find out the total things that can happen. This is the SECOND  or BOTTOM number of the ratio.  Then find the things the problem desires to happen.  This is the FIRST or TOP number of the ratio.  In this way, the desired outcome is set to the total outcomes.  This gives you a fraction.  You can then simplify the fraction and/or convert it to percent or decimal depending on what the problem asks.  The percent triangle can help you with that.

Let's solve the example problem: The problem says there are cards for each letter in the word.  So count them up.  Supremacy has 9 letters.  This is our second or bottom number.  

Then they have defined a range of letters.  Any time a problem defines a PART of the set, they are wanting you to create a FRACTION, which is the ratio of part to whole.  So Figure out which cards are between H and U, including H and U.  We leave out A, C, and E because they are before H and thus out.  We leave out Y because it is after U.  This leaves us with 5 letters.  This is the first or top number.

So our ratio is 5 / 9.

Things to be careful about:  In this set of problems, the word "or" usually means add the probabilities together.  So if they wanted to know the probability of getting an "S" or getting a "U", you would add those probabilities together:  Each is somewhat unlikely though, as getting an "S" has a 1 in 9 chance.  The "or" increases that by 1 in 9.  So 1/9 + 1/9 = 2/9.

Also, please note that this is just ONE event.  If the problem asks for you to draw again, YOU'RE NOT DONE with the math.

The problem would have stated "If you pick a card at random, replace it, and then draw again..." You would multiply the probability of getting the first with the probability of getting the second. So with the "H" to "U", we would have 5/9 * 5/9.  This gives us 25/81.  Note this number is SMALLER, because the chances of it happening GO DOWN.

Other than that, you need to practice these.  Here are some probability games and here are some practice problems.

This description of probability can be displayed in Spanish..

One Reason to Understand Math...

http://www.youtube.com/watch?v=8jkri0AeZWQ

Decimals Games

Scooter Quest
Identify the correct place value.

Bubble Burst
This game deals with adding decimals.

Multiply and Divide by 10s
Practice this one before doing decimal division.

Decimal Division
This is practice for dividing decimals.


Snake Math:  Basic    Advanced
This is a classic game with math about fractions and decimals added to it.  Advanced is faster than basic.

City Defense
This game involves adding, subtracting and multiplying decimals.

Making Mistakes

You’re afraid of making mistakes. Don’t be. Mistakes can be profited by. Man, when I was young I shoved my ignorance in people’s faces. They beat me with sticks. By the time I was forty my blunt instrument had been honed to a fine cutting point for me. If you hide your ignorance, no one will hit you and you’ll never learn.- Ray Bradbury

Keep It Cloud It Junk It (Math Version)

A few weeks ago, Mrs. Parker demonstrated a reading strategy for us.  I've been thinking about a way to change it into a math strategy and I think I've come up with it.

Her strategy was a focus question, then ideas would be set into 3 categories, 1 lead to group discussion, 1 was certain, and the other was extraneous information.  So Cloud It, Keep It, Junk It.  I think this might be applied to word problems like so: Solve It, Use It, Junk It.

Word problems typically have details about the problem and extra information in them just like reading a passage for a purpose.

So for example, look at the following question:

6.  You can win $10 if you can successfully do one of the following given one shot:

I Roll a 6 on a single number cube
II Roll a sum of seven on two number cubes
III Roll a number greater than 4 on a single number cube

Which of these three choices should you have the best chance of getting the $10?
(note, I am not making this offer)

F. I only
G. II and III
H. III only
J. I, II, and III

There's a lot of information here.  Let's split up these things into Solve It, Use It, Junk It.

Solve It                           Use It                        Junk It
P(6)                                best                           $10
P(sum 7)                         number cube              win
P(5 & 6)                         one shot
                                       sum
                                       greater than
Remember that P(6) is shorthand for the probability of getting a 6.  As you can see 3 things are in the solve it category.

The $10 is extra information and doesn't help you solve the problem, so is junked.

Then in Use It, we see that we want to compare the probabilities from the word "best", that they all involve number cubes because no other probability device was used, and that they are not multiple events (one shot).  We also see the word sum (which means add) and greater than.  These are details that modify the probability needed.

After that we solve it by creating ratios.  P(6) = 1/6.  P(sum 7) = 6/36 and P(5 & 6) = 1/3.

Might need to think about this some more.

Coordinate Grid Games

Geometry Game
Practice translations, reflections and rotations around a point on a shape with this game.  (note: anticlockwise is how the Brits and Aussies say counterclockwise)

Dilations Game
Practice dilations in this game.

Rotation around the Origin
This little game shows you what happens to points when you rotate them about the origin.

Integer Operations Games

Here are some games I found to help you practice integer (negative numbers) operations.  You may have to reload the page after playing them.

Here is a Jeopardy game with the rules for how to do integer operations.
http://www.math-play.com/Integers-Jeopardy/Integers-Jeopardy.html

Here is a game where you subtract integers
http://www.math-play.com/math-racing-subtracting-integers-game/math-racing-subtracting-integers-game.html
(You must reload after each play to play again)

Here is a similar one with adding integers
http://www.math-play.com/math-racing-adding-integers-game/math-racing-adding-integers-game.html

Here is a game about dividing and multiplying integers
http://www.math-play.com/multiplying-and-dividing-integers-game.html

Probability Vocabulary

Here is a list of vocabulary words for probability.

outcome - what happens as the result of a choice or event, for example, heads on a coin, or 6 on a number cube

desired outcome - the outcome you or the problem wants, for example, heads on a coin, or 6 on a number cube

total outcomes - the entire set of outcomes possible from the choice or event, for example, heads or tails on a coin, 1-6 on a number cube

sample space - another name for total outcomes, which usually holds outcomes for more than one event

event - the action that decides the outcome, for example, flipping a coin, rolling a number cube

probability - the chance the desired outcome has of happening, usually written in the theoretical by a ratio of desired outcomes to total outcomes

independent event - an event that is not changed by another event, such as flipping a coin or rolling a number cube

dependent event - an event that is changed by another event, such as having to reroll a number cube if you get a 1, or drawing from a bag of marbles after one has been taken out.  Dependent events typically have different total outcomes than independent events.

multiple independent events - when an event happens more than once, we multiply its probability.  So rolling a number cube, then rolling again, will have a 1/36 chance of getting 6 and another 6.

complement - outcomes other than a desired or specific outcome.  Anytime the word NOT is used, look for that outcome's complement.

experiment - causing an event to check the outcome.  Sometimes this is done to test the validity of the ratio for deviation (such as loaded dice).

percent chance - writing the ratio as a percent.

spinner - a probability device, usually a circle that has been divided into different colors or regions, with a needle attached that, when flicked, spins around the circle until it points to a region.  If the needle points to a dividing line, it is typically flicked again.

number cube - a cube with its faces labeled 1, 2, 3, 4, 5, and 6.  It is hidden from view, shufffled in the hand and then tossed onto the table.

Probability Games

How Spinners work:
http://shodor.org/interactivate/activities/BasicSpinner/

Guessing Game:
What's your probability of getting the same number as the computer?
http://www.prongo.com/guess/index.html

How a pair of dice (number cubes) work
http://www.shodor.org/interactivate/activities/ExpProbability/

An actual game using probability knowledge:
http://www.pearsonschool.com/live/images/custom/envisionmath_ca/games/pond.html

Instructions:

1. Spinners:
a. Try the spinner a few times.  Look at the chances of getting each color on the right.
b. Push the plus 1 button.  What happened to the spinner?
c. Why is the probability of getting a color going down?
d. Are these spinners fair?  Why or why not?

2.  What is the probability of getting the same number as the computer?
a.  Why is it difficult?
b.  What would the probability be if you could only pick 0-10?

3.  Pair of dice:
a.  What is the sample space for adding both dice together?
b. Why is it larger than the sum of both dice?
c.  Try the dice roller a few times.  Why can you get the same result more than once?

4. Pond game
a.  Finish all the tasks at the menu.
b. Write down their names and if you had trouble with any of them.

8th Grade Quiz: Unit 8

1. What is the probability of rolling an even number on one number cube, and an odd number on the other?

A 1/2
B 3/4
C 1/36
D 1/4

2. Maria is playing war and is down to 8 cards.  She has all 4 aces and all 4 twos.  Draw the sample space. She cannot replace the cards until all are drawn.  She first drew an ace of spades.  Then she drew a 2 of diamonds.  What is the probability that her third draw will be a heart?

A 33%
B 30%
C 6%
D 3%

3.  If Maria replaces all the cards, what is the probability that that exact sequence will happen again?

A 1/168
B 1/100
C 1/8
D 1/3

4.  A spinner has equal sections, labeled A, B, C, D, and E.   If I spin the spinner two times, what is the probability it will land on A each time?

A 1/5
B 1/10
C 1/25
D 2/5

5. There are three yellow discs, two blue discs and two green discs.  What is the probability of drawing a blue disc, not replacing it, then drawing a yellow disc?

A 1/3
B 2/7
C 1/7
D not possible

6.  You can win $10 if you can successfully do one of the following given one shot:

I Roll a 6 on a single number cube
II Roll a sum of seven on two number cubes
III Roll a number greater than 4 on a single number cube

Which of these three choices should you have the best chance of getting the $10?
(note, I am not making this offer)

F. I only
G. II and III
H. III only
J. I, II, and III

7th Grade Quiz: Unit 10

1.  What is the sample space for getting 3 "heads" when flipping a coin?

A: HHH, TTT
B: HHH, HHT, HTH, THH, HTT, TTH, THT, TTT
C: 1/8
D: 1/2

2.  If 1 number cube is rolled 18 times, and the desired outcome is 2, find P(2) and then multiply the probability by the events in the experiment.

F: P(2) is 1/9, 2 times
G: P(2) is 1/6, 3 times
H: P(2) is 1/4, 4 times
J: P(2) is 1/3, 6 times

3. Your mom wants to invite some friends over for a picnic.  She has made sandwiches: half are ham, and half are roast beef.  She also bought an equal amount of cheetos and funyuns.

She had your younger sibling help her put them in boxes, who put them in randomly.  What is the probability of a box containing a ham sandwich and cheetos?

A P(ham, cheetos) 1/2
B P(ham, cheetos)  1/3
C P(ham, cheetos) 1/4
D P(ham, cheetos) 2/3

4. Jorge was playing basketball and attempted 50 shots but only made 35 of them.  5 shots were 3 pointers.  What is the percent of shots that were made that were 3 pointers?

A 10%
B 14%
C 86%
D 90%

5. Gary went to the store to purchase a bag of apples.  Out of 12 apples, 8 were green and the others were red.  What is the number of red apples as a percent?

A 16%
B 25%
C 33%
D 66%

6. You forgot your diary lock combination.  It is a 3 digit non-repeating number.

You think a good place to start would be picking 3 numbers whose sum is 9.  You think you remember that they could be multiplied together to get 24. Create a sample space that includes these numbers.

Rotation

First, identify what counterclockwise means.  The way to remember is that counter means against.

Next, identify the points.  U is at (2,-2), A is at (3, -2), G is at (1, -5), then N is at (2, -5). The next part is a bit tricky and uses your knowledge about the quadrants.  So to go 90 degrees, I need to switch X and Y. and then because I went from Quadrant IV to Quadrant I, I turn all the negatives into positives. To go 180 degrees, I need to go from Quadrant IV to Quadrant II, so I need to switch their signs.  To go 270 degrees, I need to switch their signs and then switch X and Y.

In this case, I only want to go 90 degrees, so lets switch X and Y and turn them all positive.  U becomes U' (2,2).  A becomes A' (2, 3).  N becomes N' (5, 2).  Finally G becomes G' at (5, 1)

All that's left is to draw in the lines.

Translations

First identify the points of the shape.  2 of them are in quadrant 1 and 1 is in quadrant 4.  So D is at 1 left and 3 up.  So D (1, 3) is the top of the triangle.  Then U is 4 left 0 up, so U (4, 0).  Then C is 0 left and 2 down: C (0, -2).

Next, subtract 5 from the x and add 2 to the y.  So D' (1 - 5, 3 + 2)  or (-4, 5).  U' would be (4 - 5, 0 + 2) or (-1, 2).  C' would be (0 -5, -2 + 2) or (-5, 0).

Once the new points are plotted, draw the lines in of the shape.

Geometry Basics

Coordinate Plane

Two number lines set at 90 degrees from each other create a coordinate plane.

http://www.math.com/school/subject2/images/S2U4L1DP.gif


x-axis

The x-axis is the number line going left to right.  Positive numbers are on the right, and negative numbers are on the left.

y-axis

The y-axis is the number line going up and down.  Positive numbers are on the top, and negative numbers are on the bottom.

point

A point is a specific spot on the graph.

origin

The point where the x-axis and y-axis intersect.  It's ordered pair is (0,0).

ordered pair

The distance from the origin showing first to the left or right (x) then up or down (y).  The point (4,2) is right 4, up 2.

Line

A line is when two or more points are in a row.  Most people can draw a line based on two points, but in order to find other points on the line without using the graph, you need to understand the formula, which is y = mx + b.  Y is the dependent variable, X is the independent variable.  This means that all the X will have one and only one Y.  The other two letters help define the line.  b is called the y-intercept and shifts the graph up or down by that number.  m is called the slope and defines how fast the graph goes up.

Slope

To find the slope, you get two points.  Then you find the difference between the first and second in height, then the difference between the first and second in distance left or right from 0.

On this graph here, you can identify the following points: (-2, 1 ) (-1, 2) (0, 3) (1, 4) (2, 5)

To find the slope identify the distance between Y, then divide it by the distance between X.  So for (-2, 1) to (-1, 2), the distance between 1 and 2 is 1.  Then we divide it by the distance between -2 and -1 and we get 1.  1 divided by 1 is 1, so m = 1.  Then we identify where the line is when X is zero.  (0, 3).  So b = 3.  The line is y = 1x + 3.

8th Grade Test Corrections

This problem is not terribly difficult if you know what to do.

All you need to do is find the scale factor.  The scale factor is the number being multiplied by the original shape to make the scale shape.  In this case, the original shape is Triangle ABC.  So you create the following equations: 10 * x = 4 and 7.5 * x = 3  Then solve.  10 * x /10 = 4/10, so x = 0.4  Do the same for 7.5 * x /7.5 = 3/7.5

Dividing 3 by 7.5 is the same as 30/75.  Simplification leads to 6/15 then 2/5.


Look at the next one:

First you should have seen that a semicircle was half a circle.  Glass is on the inside of the shape, so you need area, NOT circumference.  Area of a circle is pi * radius squared and 15^2 is 225.  So 225 * 3.14 = 706.5 but DON'T STOP THERE.  That's Area of a whole circle.  Half of 706.5 is 353.25.

This one is also multi-step.

The words that tell you what to do are at the end of the paragraph.  Price per square foot is a ratio.  The word same means that the booths are proportional.  So the steps are to find the area of each booth and then find the price of the second.  10-foot by 12-foot is a rectangle unless otherwise stated in the problem, so the first booth is 10 * 20 = 120 square feet.  The second booth is 12 * 14 = 168.00  So your proportion should be:

Then your cross products are 12096 = 120 x   Then divide.  Remember you're dividing by a multiple of ten so you can simplify the division.  And you get 100.8

This one is multi-step, too (seeing a pattern? Most of the STAAR is also multi-step), but easy.  I'll show you why.

So in this chart there is one complete ratio of width to length, which are the dimensions.  Everyone look in the problem to see that the dimensions should be proportional, which means set up a proportion. You should use 4 width and 6 length first.  I recommend using 1 after that because it will make things easy.

Working it out you should get 6 = 4x.  Then you divide 6 by 4 and get 1.5 = x.  You can then use the ratio of 1 to 1.5 much easier on the last two numbers, 8 to 12 and 10 to 15.  All that's left is that pesky 2.  x over 2 = 1 over 1.5 results in 1 1/3 = x.

This one is not so pretty.  It's doable, but you should use fractions.

Reflections

Here's how to do reflections, step by step.

Here's the problem.

First: Identify the Line you must reflect across.  In this case it's the y-axis, which is the line when all x are zero.

 Next, locate the points on the shape.  Point F is at (3, 5); V (3, 2) and X (5, 2).  Knowing this, you can use the distance from the y-axis to define the next points.  Each point must be the same distance from the line of reflection as its prime (').

So, F' is at (-3, 5); V (-3, 2) and X at (-5,2)  Notice how each point is the same distance away from the y-axis.  F' is 3 away, and F is 3 away too.

Finally draw in your lines.

Misconceptions:

1.  A reflection does not change any of the shapes angles or the lengths of any segments forming the shape.

2.  A reflection is based on a line, not a point.  The line can have any value, not just on the x-axis or y-axis.  Finding the line of reflection is based on how far the points are from each other, then plotting points halfway between until a line is formed.  This means the distance from the two shapes divided by 2.

3.  The shape should not "turn".  It is reflected, not translated and then rotated.

Solving 2-Step Equations

Solving equations needs the properties of basic operations and equality.  A quick review can be found here.

These problems typically look like:

Things to note: Any number next to the x is being multiplied by x.  Watch for addition and subtraction.  One of the rules about adding variables is that if you try to divide first, you have to divide anything you're adding or subtracting, so it is typically a good idea to remove that part first.  In this case, 20 is being added to 3 times a number to get 41.  So to isolate the number, we would need to understand and use the identity property of addition and the inverse property of addition, which both have to do with adding or subtracting to get zero and then adding zero to get the same number.  So in order to get just 3 times a number, we need to have it as 3 times a number plus zero.  Using the inverse property, 20 + -20 = 0.  This gives us:

We must subtract 20 from both sides to keep equality.  If you could take some only from one side, then it would indicate that they weren't equal to begin with, which would make the problem wrong from the beginning.  This gives us:

Again to get the unknown number by itself, we need to use identity and inverse properties of multiplication.  This means finding a number that we can multiply by 3 to get one.  Some of you remember from learning fractions that any time you divide fractions, you can multiply by their reciprocal.  The reciprocal is the opposite part/whole ratio to your first fraction.You might also remember that any whole number can be written as a fraction by putting it over one.  So 3 becomes 3/1.  To make a fraction equal one, we multiply by its reciprocal.  3/1 * 1/3 = 1  Then we use the identity property 1 * x = x.  Like above, it would be wrong to only multiply one side by 1/3.  Thus:

This gives us x = 21/3 or:

There is the full example of how to work 2 step equations.  Common issues are as follows:

1.  Trying to multiply or divide first.  This can work but often leads to trouble.  In our example above, you would be able to solve it still, but the problem would be more difficult.  By increasing complexity, you increase your chance of a math error.  Multiplying first would lead to x + 20/3 = 41/3  If you have a strong dislike of fractions this could increase your distress.

2.  Picking the wrong number to turn to zero.  Subtracting 41 from both sides would give us x - 21 = 0.  Not the worst thing that can happen but it added an extra step where it was not needed.  Now you have to add 21 to both sides after.  Trying to multiply by zero is an exceedingly bad idea as that gives you 0x + 0 = 0.  x becomes all numbers, so instead of solving it, you've made it unsolvable!

3. Not making addition or subtraction zero:  This simply increases the number of steps needed and can increase your frustration level.  By subtracting 5 from both sides, you don't get very far with 3x + 15 = 36.  You could divide next but you'd still need to shift more to the other side.  Likewise with -10 from both sides.

4. Not making multiplication by 1.  Without the identity, you still have to deal with another step.  Just like not making addition/subtraction not zero, not multiplying by 1 gives you more steps .

5.  Not using the inverse properly.  The opposite of 20 is -20, not +20.  Likewise the reciprocal of 3 is 1/3, not 3 or some other number.

6.  Dividing by a number that is added or subtracted.  Like 3, this still creates problems.  In our problem, dividing by 20 would do very little.  3x/20 + 1 = 41/20 looks really ugly and difficult to solve.