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3.14 × d
2 × 3.14 × r

Problem: What is the circumference of the circle if the diameter is 8 in?

Answer:
3.14 × 8
25.12 in

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Answer:
Rectangle 1=48 in²
Rectangle 2=15 in²
Final area=63 in²

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Problem: What is the area of this composite shape?

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Problem: What would happen to the area if both dimensions increased by 5?

Answer: The area would increase by 10.
ORIGINAL: 48 in²
NEW: 480 in²

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Rectangle
If the length doubles but the width stays the same, the area is doubled. Basically, if one dimension changes, the area will change by the same scale factor as the dimension. If you change both dimensions, you take the scale factor that the dimensions changed by from each dimension and multiply them. That is the scale factor for the area, what it will change by.

Circle
For the circle, if the radius doubles, the area quadruples. If the radius triples, the area is increased by 9, because you multiply by how much your increasing twice, and so on.

Problem: What would happen to the area if the radius is quadrupled?

Answer: The area would increase by 16.
ORIGINAL: 153.86 in
NEW: 2461.76 in

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To find the area of an inscribed shape, you need to find the area of the whole shape, the finding the area of the unshaded region. Then, you will subtract the area of the unshaded region from the whole shape, to find the area of the shaded region. To find the area of a composite shape, you need to break up the shape into a familiar shape (ex. rectangle),  and combine the area's of each shape.

Problem: What is the area of the shaded region?

Answer:
Square=100 in²
Circle=78.5 in²
Final area=21.5 in²

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2(lw + wh +lh)

Problem: Sarah wants to wrap a present. The box is 3 in length, 4 in in width, and 6 in height. How much wrapping paper will she need?

Answer:
2(3 × 4 + 3 × 6 + 4 × 6)
108 in²

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bh + H(side 1 + side 2 + side 3)

Problem: Find the surface area of this triangular prism.

Answer:
4 x 5 + 11(5+3+3)
141in²

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2 x 3.14 x radius

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1/3 × l × w × h

Problem: Find the volume of a square pyramid with sides of 6 ft and a slant height of 7 ft.

Answer:
1/3 x 6 × 6 × 7
84 in³

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s² + 2(bh)

Problem: Find the surface area of a square pyramid with sides of 3 in and a slant height of 5 in.

Answer:
3² + 2(3×5)
39 in²

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2 × 3.14 × r² + 2 × 3.14 × d x H

Problem: Find the area of a cylinder with a radius of 2 and a height of 7.

Answer:
2 x 3.14 x 2² + 3.14 x 4 x 7
113.04 in²

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3.14 × r² × H × 1/3

Problem: Find the volume of a cone with a radius of 2 in and a height of 6 in.

Answer:
3.14 x  2² x 6 x 1/3
25.12 in³

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3.14 × r² × H

Problem: Find the volume of a cylinder with a radius of 3 and a height of 8.

Answer:
3.14 x 3² x 8
226.08 in³

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4/3 × 3.14 × r³

Problem: Find the volume of a sphere with a radius of 4 in.

Answer:
4/3 x 3.14 x 4³
267.94 in3

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1/2 × b × h × H

Problem: Find the volume of this triangular prism.

Answer:
1/2 x 6 x 5 x 10
150 in³

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3.14 × r²

Problem: You have just bought a pizza, and want to find the area of it. It has a radius of 7. What is the area?

Answer:
3.14 x 7²
153.86 in²

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h(base 1 + base 2)/2

Problem: Find the area of this trapizoid with a base 1 of 5, base 2 of 8, and a height of 3.

Answer:
3(5+8)/2
39/2
19.5 ft²

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bh

Problem: A parallelogram has a base of 9 in and a height of 5 in. What is the area?

Answer:
5 x 9
45 in²

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lw

Problem:
A rectangle has a length of 7 and a width of 5. What is the area?

Answer:
7 x 5
35 ft²

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1/2bh

Problem:
A triangle has a base of 5 and a height of 3. What is the area?

Answer:
1/2 x 5 x 3
7.5 in²

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Vertical angles is when there are two lines cross and two angles are opposite from each other. Both angles share the same vertex.

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Adjacent angles are when there are two angles that share a vertex and a ray.

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Complementary angles are when there are two angles that add up to 90 degrees.

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The Triangle Sum Theorem states that three interior angles for every triangle must equal to 180 degrees.

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Supplementary angles are when two angles in a figure equal to 180 degrees.

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The Triangle Inequality Theorem states that of  two sides of a triangle must be greater than the third side of the triangle. If the total measurement of the two sides is not greater than the third side, then the figure is not classified as a triangle.

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Definition

Two angles that are opposite from each other when two lines cross. Share the same vertex.

Sample Problem
Find the missing angles.

ANSWER
Using the 120 degrees already given, thst helps solve for the rest of the angles. The angle below 120 must also equal 120 because it is corresponding. 120+120=240, and 360-240=120. 120÷2=60, which is the remaining missing angle for both the left and right.

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Definition
Two angles that share a vertex and one ray.

Sample Problem
Which of these examples of angles are adjacent?

ANSWER:
They are all adjacent, because they each share a vertex, and one ray.

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Definition
Two angles in which the sum is 90 degrees.

Sample Problem
Find the missing angle.

ANSWER:
Since a complementary angle is when there are two angles that make up 90 degrees, using the angle measurement already given helps solve for the missing angle.

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Definition
Three interior angles of any triangle add up to 180 degrees.

Sample Problem
Find the missing angle.

ANSWER:
Since a triangle must have angles that equal 180 degrees, using the two angles already given, that helps find the missing angle.

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Definition
When there are two angles which add up to 180 degrees.

Sample Problem
Which of these angles are supplementary?

ANSWER:
Both A and B are supplementary angles, since both have two angles that equal 180 degrees. C is a complementary angle, meaning two angles make up only 90 degrees.

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Definition
The sum of any two sides of a triangle must be greater than the remaining side (third side).

Sample Problem
Determine if you can make a triangle based on the side measurements.

4 in, 7 in, 2 in

ANSWER:
Why? It's because 4+2=6, and it is less than 7.

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Proportional Reasoning and Linear Equations Grade - 99%

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Proportional Reasoning and Linear Equations Grade - 99%

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- x=4 is an example of an equation for undefined slope not the slope value. 22/0 or -4/0 would be examples.

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Constant of proportionality can also be written as y = kx, where "k" is the constant. Two quantities such as x and y are proportional if yx is constant.

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Indirect measurement is when you use proportions to find the distance / length in similar figures. If sides are given, you could use the available sides to help you find the unknown sides.

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Proportion is comparing two numbers, and the numbers can represent anything (people, objects, etc.). You write a proportion with a colon (3:6), or a equivalent fraction (3/6).

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Unit rate is when you take the ratio and reduce it to its lowest term or singular unit.  For example,  you work 49 hours in a 7-day week, so the unit rate would be 7 hours a day (7/1). You reduce 49/7 by dividing 49 by 7, which is 7.

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What are the different parts of a linear equation?

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What are the different parts of a linear equation?

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The formula for a linear equation is y = mx + b. You find the "y" as the first variable, and it stands for the y coordinate. Next is "m", which is the slope, or the rise/run. The "x" then stands for the x coordinate. Last is the "b", which determines the y-intersept.

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The 4 types are positive, negative, zero, and undefined slope. An example slope for a positive slope would be y = 3x + 2, or m = 3/1. For negative, an example would be y = -5x + 3, or m = -5/3. An example for a zero slope would be y = 0/7, 0/7, or y = 7. And finally, an example for a undefined slope would be x = 4

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Explain what has to be true for two figures to be similar.

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Explain what has to be true for two figures to be similar.

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Both figures must be the same shape, but in different size. The shapes must also have the same angles and the corresponding sides must be proportional.

Example:

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Definition
The constant value of a ratio of two proportions.

E.g. Constant is × 6 (multiplying 6).

Sample Problem
What is the constant of proportionality?

ANSWER:

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Definition
Using proportion to find the distance / length of similar figures.

E.g. Use shadows or mirrors to find length.

How it Relates
Indirect measurement is a method of using proportion to find unknown sides of similar figures. You can use shadow lengths to help find the lengths / height / distance. You can also use already known sides to help find the unknown sides.

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Definition
When two ratios / fractions are equal, or a comparison of two numbers.

E.g. 1/2 = 4/8

Sample Problem
What could make these fractions proportional?

6/8 =?/24

ANSWER:

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Definition
When the second rate is 1 unit / quantity.

E.g. 14 apples in 2 baskets, unit rate is 7 apples in one basket.

Sample Problem
Find the unit rate of 65 students in 5 buses.

ANSWER:

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Kelly has less than 8 coins in her jar.

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There are at least 7 birds on the tree.

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The theater has a seat capacity of 50 people.

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Angelina has more than 14 pencils in her pencil case.

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• minimum
• at least
• profit

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• maximum
• at most
• up to

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• above
• more than
• over

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• fewer than
• below
• under

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You flip the inequality when you multiply / divide with a negative number on both sides. You solve the problem like normal, but just flip the sign when you multiply / divide with a negative number.  For example:

-6x - 9 ≥ 15

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Definition
A set of values that satisfy a set of an equation / inequality.

E.g.
{4, 3, 2, 1..}

How the word relates
The solution set is the result of an inequality. A solution set can be best expressed in a number line, to help get the idea of the solution.

Sample problem
Six less than the product of 3 and a number is greater than or equal to 15. What is the solution set?

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Definition
Uses symbols for less than, greater than, less than or equal to, and greater than or equal to.

< (less than)
> (greater than)
≥ (greater than or equal to)
≤ (less than or equal to)

How it relates
An inequality can technically have more than one answer. This results in a solution set; a set of multiple solutions. Such as {3, 4, 5 , 6...}.

Sample problem
7x + 4 ≥ 25

ANSWER:

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And irrational number is a number that can't turn into a fraction. Pi is a popular example, as it keeps going on forever with different numbers, and it's not possible to convert it into a fraction.

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A square root is a number timed by itself to create a number. In the image below, it shows that the square root of 25 is 5, as 5x5 is 25.

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As the definition above says, a cube root is a number timed by itself 3 times to get a cubic unit.

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A repeating decimal is a decimal that goes on forever; keeps on repeating.

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Definition:
A number multiplied by itself to produce a number / square root.

E.g.
The square root of 25 is 5.
(5x5-25)

Sample problem:
What is the square root of 49?

ANSWER:
(Also using a form of estimation)

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Definition:
A number multiplied by itself 3 times to get a cubic value.

E.g.
The cube root of 64 is 4.
(4x4x4=64)

Sample problem:
What is the cube root of 27?

ANSWER:
(Using estimation at the start)

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Definition:
A number that can't be expressed as a fraction.

E.g.
π (Pi)

"Pi" is infinite decimal with many different numbers, can't be expressed ad a fraction.

Example / Non example

E.g. -  π (Pi)
"Pi" can't be converted into a fraction.  Therefore it is a irrational number.

Non E.g. - Any rational number, such as repeating decimals (.5555...)

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Definition:
A decimal that consists of some numbers that repeat infinitely, there's no end.

E.g.
4.333333...

The "3" is the repeating decimal, it keeps repeating.

Sample problem:
Convert this repeating decimal into a fraction.
(.7777...)

ANSWER:

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How to estimate the square root of a number.

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How to estimate the square root of a number.

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First, it's a good idea to have the perfect squares from 1 to 100 memorized. You should then find two perfect square roots that your number is between. For example, if you need the square root for 31, the square root of 25 is 5, and the square root of 36 is 6. Because 31 is 6 away from 25, and 5 away from 36, the square root is about 5.6.

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Find the fraction of a repeating decimal.

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Find the fraction of a repeating decimal.

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First, you have "X" represent the repeating decimal.  For example, in .1424242... "42" is the repeating decimal, so we would put X for 42.  Next, you would need to move the decimal point BEFORE the repeating decimal, resulting in the number 1.424242...  The decimal point moved 1 space, which equals 10x.  Then, we would need to put the decimal point AFTER the repeating decimal, resulting in the number 142.424242... The decimal moved 3 spaces, which equals 1000x.  Line up the equations, putting the larger equation first.

1000x= 142.424242...
- 10x=1.424242...

Solve for X, meaning subtracting the X's, resulting in 990.  You ignore the repeating decimals, as they are cancelling each other out.  You then subtract the remaining whole numbers, in this case, 142-1, which equals 141.  You then "divide" both sides by 990, giving you your fraction 141/990, simplfied to 47/330.

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Equivalent Fractions are fractions that share the same amount. In the image below, 1/2 and 4/8 are both the same amount. 4/8 can be simplified to 1/2.

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A Mixed Number is a whole number alongside with a fraction. The picture below shows the whole number (4), combined with the fraction 5/8.

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An Improper Fraction is when the numerator is either greater or the same as the denominator.

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A Common Denominator is when two or more fractions have the same denominator. The image below shows that 5/8 and 7/8 may have different numerators, but both fractions have the same denominators.

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Definition:

A whole number combined with a fraction.

E.g.

2 1/2

In this example, the whole number is 2, and is combined with the fraction 1/2.

Sample problem:

Change the improper fraction into a mixed number.

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Definition:

When the numerator overpowers the denominator.

E.g.

8/7

In this case, 8 > 7. The numerator must be less than the denominator in order to it to be a proper fraction.

Sample problem:

Improper or proper?

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Definition:

Fractions that have the same amount, even if they look different.

E.g.

2/ 3 = 4/6

These two fractions show the same amount. We can simplify 4/6 by dividing both the numerator and the denominator by 2, since both numbers can go into 2. This gives us the answer of 2/3.

Sample problem:

Are these fractions equivalent?

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Definition:

When two or more fractions have the same denominator.

E.g.

4/6 and  5/6

The "6" is the common denominator.

Sample problem:

Simplify both fractions to get the least common denominator.