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.