Students are expected to build on their knowledge from the previous year, at a higher level of complexity. The depth of concepts, mathematical procedures and processes will be expanded each year. More homework, harder tests, and tough reading assignments—middle school and junior high is a big transition for your child. Tutoring is developed on an individualized plan that builds the academic skills, good habits and positive attitudes your child needs to succeed in middle school and beyond. Prerequisite: Math 7

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= 2^{3}

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= 3^{5}

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= b^{3}

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= c^{5}

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× 10

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× 10

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Make a common denominator in order to compare. Use the lowest common multiple (LCM) of 6 & 4: 12...

Make a common denominator (with the lowest common multiple, LCM)

Multiples of 3: 3, 6, 9, 12,__15__, ...

Multiples of 5: 5, 10,__15__, ...

Therefore

Multiples of 3: 3, 6, 9, 12,

Multiples of 5: 5, 10,

Therefore

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2.5

These are *multiples* of 22. Multiples multiply, going up higher each time... There are infinite amount of multiples...

(There are only a certain amount of factors... for 22 it's: 1, 2, 11, 22)

(There are only a certain amount of factors... for 22 it's: 1, 2, 11, 22)

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36 is a factor of 36 because you can divide 36 by 36.

$

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20 [steps not shown here]

Fraction | Percent |

50 % | |

75 % | |

10% |

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%

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Fraction | Percent |

50 % | |

75 % | |

40 % | |

10% |

Decimal | Percent |

0.3 | 30% |

0.25 | 25% |

5% | |

0.01 | 1% |

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Decimal | Percent |

0.3 | 30% |

0.25 | 25% |

0.05 | 5% |

0.01 | 1% |

$

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$

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If you are a bit advanced by now, and know algebra, then you can solve simply,
Otherwise use the slower, guess and check method (continually trying different numbers using estimation)...

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%

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If you are a bit advanced by now, and know algebra, then you can solve simply,
Otherwise use the slower, guess and check method (continually trying different numbers using estimation)...

$

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$

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%

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hours

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Make algebra expressions for her hours worked at job 'A' and at job 'B'.
Let (x) be her hours worked at job 'A'

Let (2x - 12) be her hours worked at job 'B' Then extend the earnings equation to account for both jobs. [This could also be done with separate equations (not shown).]

Let (2x - 12) be her hours worked at job 'B' Then extend the earnings equation to account for both jobs. [This could also be done with separate equations (not shown).]

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Make the denominator with the lowest common multiple (LCM)...

4 → 8 → 12 → 16 →

5 → 10 → 15 →

4 → 8 → 12 → 16 →

5 → 10 → 15 →

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Make the denominator with the lowest common multiple (LCM)...

3 → 6 → 9 →

2 → 4 → 6 → 8 → 10 →

4 → 8 →

3 → 6 → 9 →

2 → 4 → 6 → 8 → 10 →

4 → 8 →

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You don't need a common denominator for multiplying fractions, look it's easy!

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With multiplication of fractions you can cancel something once on the top and bottom...

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With division of fractions... multiply by the reciprocal of the fraction after the division sign ÷...

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With division of fractions... multiply by the reciprocal of the fraction after the division sign ÷ ... And cancel across division...

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Use PEMDAS/BEDMAS...

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Use PEMDAS/BEDMAS...

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Use PEMDAS/BEDMAS...

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Append 4 zeros to the end

You start with 25.0 and add 4 zeros to the end (move the decimal 4 places, right).

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You start with 123456.0 and move the decimal 3 places, left, to make the number smaller.

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This is trick because the combination of '÷' with a negative exponents, is like a double negative, in that you end up moving the decimal place to the __right__. It's equivalent to 200 × 10^{5}.

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A square root cancels out the 'perfect square' number.

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to

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kg

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This is a directly proportional relationship, where the density is constant, so increasing one property (volume) by a certain amount will increase the other property (mass) by the same factor (× 10). You can determine this by guess-and-check, or some algebra...

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Circumference:

Area:

Area:

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Circumference,
Area,

You can multiply the top (numerator) and bottom (denominator) by the same amount... 11

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cm^{2}

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Area = Length × Width

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m^{2}

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Know that the place value difference between m^{2} and cm^{2} is 4 places...

Convert to meters... move the decimal 4 places left. Or, convert the lengths to meters first, and then calculate the area:

Convert to meters... move the decimal 4 places left. Or, convert the lengths to meters first, and then calculate the area:

- 10 cm --> 0.1 m
- 20 cm --> 0.2 m

- 1 m
^{2}= 10,000 cm^{2} - 1 m
^{3}= 1,000,000 cm^{3}

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cm^{3}

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m^{3}

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m^{3}

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Convert to m^{3}

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cm^{2}

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Radius = Diameter ÷ 2

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cm^{2}

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First, calculate the radius, this uses algebra, which some students may learn later
Next, use this radius to calculate the area.

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cm^{2}

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Given h_{1} = 5 cm, then...
With the area of the bigger triangle
Now, with b_{2} = 40 cm, then...
Now with the area of the smaller triangle

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cm^{2}

See that triangle 'xhb' and triangle 'abc' share the same height 'h'.

With area of triangle 'abc'
Use Pythagorean theorem to find length 'x'
Area of triangle 'xhb'

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cm^{2}
%

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The area missed is the area of a square, minus the area of one quarter of a circle. Also, make sure that your units 'agree' all in cm...

Make a sketch to see that the area that cannot be reached by the vacuum is the area of a square, minus the area of one quarter of a circle.
Radius 'r' = 10 cm
There are four corners, so the total area that cannot be reached by the vacuum is,
And the percent of the (200 cm × 300 cm) room that gets cleaned:

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cm

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First find side length 's' of the square, given the area
Next, see that the side length 's' of the square is equal to the diameter 'd' of the semicircle.
The perimeter of the semicircle, is half of this circle,

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cm^{2}

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Given the perimeter of the smaller circle, determine its diameter 'd'
Determine the radius 'r' of both the larger, and smaller circles, based on the diameter 'd' of the smaller circle,
Calculate the shaded area,

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cm

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Use the given length 'L' of semicircle, to calculate the radius. A semicircle is *half* the circumference of the whole circle.
Calculate the radius 'r'

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cm

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Previously, the radius 'r' was calculated, as 6 cm. The height of the rectangle is equal to the diameter of the circles,
The width of the rectangle is equal to two circle halves (radii), plus one whole diameter.
Calculate the perimeter,

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cm^{2}

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See that the two semicircles, put together, make a whole. Plus, the other whole circle, so,

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cm^{2}

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units^{2}

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Split the kite into compound shapes: 2, identical triangles
Calculate the area
Yes, there is also an equation for kite, but this was an experiment in breaking down into compound shapes.

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units^{2}

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Split the trapezoid into compound shapes: rectangle + square
Calculate the area
Yes, there is also an equation for trapezoid, but this was an experiment in breaking down into compound shapes.

length =

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The statement says the circle and rectangle have the *same area*:

Therefore the rectangle has a larger perimeter.

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times larger

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Remember this is area... proportional to the *square* of the radius. Doubling the radius or diameter is always equivalent to quadrupling (4×) the area, which is the amount of pizza that you get.

For example, Hopefully you can see that the area of the largest pizza is 4x the area of the smallest pizza,

For example, Hopefully you can see that the area of the largest pizza is 4x the area of the smallest pizza,

is the formula for surface area of a cylinder. Think of the sum of the compound shapes.

h =

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cm

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cm

Open ended pipes don't get painted on the ends, meaning the area of the two circles at both ends is not included in the calculation for surface area.

The surface area of the open-ended cylinder does not include the two circles:
Make sure to convert to the same units [5 cm = 0.05 m] Now substitute the known values into the equation, and solve,
Therefore a gallon of paint can paint approximately 73.2 cm of that pipe.

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cm

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First, determine the radius, given the circumference of the circle, where 0.30 m = 30 cm,
Now see that the surface area of the open-top vase is the sum of one circle (the base) plus the rectangular curved area,

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cm^{3}

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First, convert 15 mm to 1.5 cm [There are 10 mm in 1 cm]

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m

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First convert 800 cm to 8 m [There are 100 cm in 1 m... See that it is easier to convert cm to m, rather than convert m^{3} to cm^{3}]

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cm

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If the "the height is *triple* the radius",
Now use these values in the equation for volume of a cylinder,
Now, depending on people's abilities, 'r' can be solved either with guess-and-check (left), or with cube roots (right)...

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cm

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If the "the radius is *triple* the height",
Now use these values in the equation for volume of a cylinder,
Now, depending on people's abilities, 'r' can be solved either with guess-and-check (left), or with cube roots (right)...

Length (cm) | Width (cm) | Area (cm^{2}) |

1 | ___ | ___ |

2 | ___ | ___ |

3 | ___ | ___ |

5 | ___ | ___ |

6 | ___ | ___ |

9 | ___ | ___ |

10 | ___ | ___ |

15 | ___ | ___ |

18 | ___ | ___ |

30 | ___ | ___ |

45 | ___ | ___ |

The 30cm × 30cm = 900cm^{2}.
By inspection, you see that the 30cm × 30cm enclosure has the maximum area, 900cm^{2}.

The volume of a cube, rectangular prism, cylinder, and triangular prism are calculated with the formula:

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cm^{3}

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Volume = (Area of Base) × (Height)

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cm

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Calculate the length of each side of the triangular base, given its perimeter,
Find the height 'h' using pythagorean theorem. Use the right-angled triangle in the base with sides: 'h' 'b' and 'b/2'
Given the surface area (one triangle plus three rectangles), you can find the height of the prism 'h_{2}'

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Calculate the length of each side of the triangular base, given its perimeter,
Find the height 'h' using pythagorean theorem. Use the right-angled triangle in the base with sides: 'h' 'b' and 'b/2'
Given the surface area (one triangle plus three rectangles), you can find the height of the prism 'h_{2}'
Now find volume of the prism,

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cm^{3}

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First, determine the height of the equilateral triangle, using Pythagorean theorem,
Then, determine the area of the base, which consists of 6 equilateral triangles,
Then, determine the volume of the prism,

s =

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cm

Based on what is given, you don't need height 'h' to solve for the side length 's'

For the triangular prism, the equation for volume is typical, but the surface area (SA) of the open-topped shape includes the ends plus *two* rectangular sides, instead of the usual three.
First, you can find the area of the triangle, because you have the volume and length
Now that you have the area of the triangle, you can substitute this into the equation for the surface area,

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units^{2}

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:

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Earlier, you determined the area of...

Then calculate the area of the whole (larger) trapezoid, Then subtract the area of the parallelogram from the area of the trapezoid, So the ratio of the shaded trapezoid area to the whole trapezoid area is... [And remember to reduce fully!]

Then calculate the area of the whole (larger) trapezoid, Then subtract the area of the parallelogram from the area of the trapezoid, So the ratio of the shaded trapezoid area to the whole trapezoid area is... [And remember to reduce fully!]

h =

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cm

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cm^{3}

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Given the area of the shaded triangle, determine the base 'b'
Now you can calculate the area of the whole base of the hexagonal prism. See that a hexagon consists of compound shapes: two triangles, plus a rectangle,
Now that you have everything you need, calculate the volume of the hexagonal prism,

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cm

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Any two of the following with same sides and angles: equilateral triangle, square, pentagon, hexagon, ...

Regular polygons have all side lengths and angles equal.

Regular polygons have all side lengths and angles equal.

All are 4-sided:

- Quadrilateral
- Rectangle
- Rhombus
- Square
- Trapezoid
- Parallelogram
- Kite

- Square (all sides equal, all right angles)
- Quadrilateral (4 sides)
- Rectangle (all right angles)
- Polygon (closed, 2D, straight sides)
- Rhombus (all sides equal)
- Parallelogram (2 parallel sides)
- Kite (two different pairs of equal, adjacent sides)
- Trapezoid (at least one pair of parallel sides)

This is the general order...
Google 'em!

A **rhombus** is a parallelogram with equal side lengths.

However this can only include squares.

However this can only include squares.

The circle has the greatest ratio of:
[Calculations not shown]

BAC references an angle in the triangle with the middle letter, 'A' representing the angle, I.

Corresponding angles are also known as the 'F' angles.

Other corresponding angles are C and Y

or A and W...

Other corresponding angles are C and Y

or A and W...

Alternate angles are also known as the 'Z' angles.

Other alternate angles are C and X.

Other alternate angles are C and X.

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The sum of angles on a line = 180˚

As you can see in the diagram, the area of the square on the longest side (hypotenuse) is 5^{2} = 25.

The sum of the areas of the squares on the two shorter sides is 3^{2} + 4^{2} = 9 + 16 = 25.

∴ The area of the larger square is always equal to the sum of the two smaller areas.

The sum of the areas of the squares on the two shorter sides is 3

∴ The area of the larger square is always equal to the sum of the two smaller areas.

c =

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z =

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c = 12.8

- Left: (x - h, y)
- Right: (x + h, y)
- Up: (x, y + k)
- Down: (x, y - k)

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( , )

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( , )

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So, the original point is (17, -4)

- a(x, y) = (a × x, a × y)

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( , )

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- Mirror across the 'x'-axis: (x, y) → (x, -y)
- Mirror across the 'y'-axis: (x, y) → (-x, y)
- Mirror across the line 'y = x': (x, y) → (y, x)

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( , )

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Rotation | 90˚ | 180˚ | 270˚ |

Clockwise (CW) | (y, -x) | (-x, -y) | (-y, x) |

Counterclockwise (CCW) | (-y, x) | (-x, -y) | (y, -x) |

True, see the table above.

True, see the table above.

True, see the table above.

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Any of the above. [coordinate translation/rotation work not shown]

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Side Length (cm) | Area (cm^{2}) |

2 | 4 |

5 | 25 |

7 | |

10 | |

144 |

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Side Length (cm) | Area (cm^{2}) |

2 | 4 |

5 | 25 |

7 | 49 |

10 | 100 |

12 | 144 |

Number of White Roses | Cost of Flower Arrangement |

1 | $3.00 |

4 | $12.00 |

8 | $24.00 |

12 | $36.00 |

16 | $48.00 |

24 | $65.00 |

Make points on the axis that have equally spaced intervals, like 5, 10, 15, 20, 25, 30, etc... Don't use 1, 4, 8, 12, 16, 24 on your x-axis because those points have uneven spacing!

Make sure to label the axes and make a title of the graph.

Make sure to label the axes and make a title of the graph.

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Know the terms,

- Strong vs. Weak
- Positive vs. Negative

$

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~$30

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x | y |

-2 | |

-1 | |

0 | |

1 | |

2 |

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Substitute the 'x'-values into the equation, to determine the outputs ('y'-values)...

x | y |

-2 | -5(-2) + 3 = 10 + 3 = 13 |

-1 | -5(-1) + 3 = 5 + 3 = 8 |

0 | -5(0) + 3 = 0 + 3 = 3 |

1 | -5(1) + 3 = -5 + 3 = -2 |

2 | -5(2) + 3 = -10 + 3 = -7 |

C =

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Notice that this is an equation in the form *y = mx + b* ...

slope (m) is the hourly rate, and the y-intercept (b) is the fixed amount.

slope (m) is the hourly rate, and the y-intercept (b) is the fixed amount.

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hours

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C = $465

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The cost is higher, $660 for the same amount of hours, so it is not a better deal.

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( , 20)

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units

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Length AB:
Length DC:
Use Pythagorean theorem to determine the slant lengths...
Using Pythagorean theorem, length AD:
Using Pythagorean theorem, length BC:
So, the perimeter is:

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units^{2}

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Scale the 'x' and 'y' values by a factor of 2...
Base length:
Height:
Calculate,

True, the perimeter and area does not change for rotations or reflections.

y =

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The 'base' length of the triangle can be calculated with,
Then substitute the givens into the equation for area of a triangle,
Then the 'y'-value could be, either above or below the 'base' at y = -1,
The point 'C' could be anywhere on the lines y = -8, or y = 6. For instance C(1, 6)

Area =

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u^{2}

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Area = u^{2}

Perimeter = u

Perimeter = u

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- (0, 0) --> (0 × 3, 0 × 3) --> (0, 0)
- (4, 0) --> (4 × 3, 0 × 3) --> (12, 0)
- (4, 2) --> (4 × 3, 2 × 3) --> (12, 6)
- (0, 2) --> (0 × 3, 2 × 3) --> (0, 6)

+12

+ 4n

- 3n

x | y |

1 | 2 |

2 | 6 |

3 | 14 |

4 | 30 |

5 | 62 |

Find out what it multiplies by each time: × 4.
The pattern rule is multiply by 4...

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n = 60

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m

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cm

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m

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m

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sandwiches

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Cost = $3.50(number) + $45

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x =

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y =

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x =

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Conventionally, these increases in numbers are shown with a scatter plot, with time on the x-axis, and total numbers on the y-axis. The trend can be extended to predict population in future years using a line of best fit.

These relative, fraction or percentage comparisons are best displayed with a circle (a.k.a. pie) graph.

Typically, a bar chart would be used to show the number (on the y-axis) of each category (on the x-axis). A histogram would not commonly be used because histograms must have sequential numerical values on the x-axis.

Conventionally, a histogram is used for this, with sequential intervals on the x-axis, and frequency count on the y-axis. Histograms have number values on the x-axis (vs. bar graphs that can have numbers, or category names).

Population includes the entire set of a study, while a sample is the subset of the population used to represent the population. A larger sample size is more representative of the entire population.

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Sample are used to extrapolate and understand trends about populations. It is far less expensive, and less time consuming to collect data from a smaller sample, rather than the entire population. Often, if the sample is chosen well, it is a good representation of the properties of a population.

Mode is the measurement of the 'most common' data point, in this case the type of pizza that most students want, should be ordered.

Homeroom Classroom | Pepperoni | BBQ Chicken | No Preference |

7A | 10 | 12 | 8 |

8D | 11 | 15 | 5 |

9F | 9 | 9 | 9 |

9C | 6 | 8 | 10 |

10G | 13 | 13 | 0 |

10B | 14 | 15 | 1 |

11T | 21 | 1 | 3 |

11P | 6 | 16 | 6 |

12G | 4 | 8 | 14 |

12D | 3 | 11 | 18 |

Out of the 269 responses, 97 want pepperoni, and 108 want BBQ chicken.

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Mean is the average

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Median is the middle of the ordered data set

Order ascending:
1 2 2 2 __2 2__ 2 3 3 3
For an odd number of values, the median is the middle value. For an even number of values, the median is the mean (average) of the middle two, which just equals 2.

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You don't need to order the numbers, but it makes it easier to pick out the most common value, the mode.
1 __2 2 2 2 2 2__ 3 3 3
The most common value, the mode, is 2.

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You know the equation for calculating the average. In this case you can work backwards, because you are given two values (average, and number of values), and you can calculate the third (the missing ranking).

Depending on your ability (some people are still just learning algebra), you can solve this with guess-and-check or algebraic [shown here] methods... Let 'x' be the missing ranking.

The quickest way to check is to assume the highest score (100%) for the sixth class, and see what their average would be. Of course you can also work backwards, calculating the data point, given 6 terms, and the 90% average [not shown].
No the student will not receive the scholarship.

If dropped lowest mark,
Take an extra class,
Do one capstone,
Do two capstones,
Therefore, dropping the lowest mark will give the student the highest possible final average.

It shouldn't matter what class the student gets the extra percent in, because the sum and number of values will not differ. For each case the resulting calculation will be:

Loading and rendering MathJax, please wait...

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