Math 6 Honors Periods 1, 6 & 7

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Pre Algebra Period 2

Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!

Algebra Period 3

Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!

Math 6 Honors Periods 6 & 7 (Wednesday)

Volumes of Cylinders 10-7

A cylinder is a space figure that has two circular bases and one curved surface. The perpendicular distance between the bases is the height (h) of the cylinder.
If the base radius is r, then the base area, B is

πr²

Volume of cylinder = Base area X height
V = BH
V = πr² h
Find the volume of a cylinder having a base radius of 6 cm and a height of 8 cm.
B = πr² so B = π6²

36π

≈

36 X 3.14 113 ( rounded to three digits as our book requests)

V = Bh

113 X 8 = 904 904 cubic cm.

The volume of a cylinder or box is often called its capacity. For containers of liquids, capacity is usually measured in liters (L) or milliliters (mL)

Note: 1 L = 1000 cm³ and 1 mL = 1 cm³.


Note: Certain mathematics may have different meanings than they have in ordinary usage. For example, in everyday language, base refers to the bottom of an object and height refers to how tall an object is. In mathematical usage these terms have special meanings,

Math 6 Honors Periods 6 & 7 (Tuesday )

Polyhedrons 10-5

A polyhedron is a figure formed of polygonal parts of planes that enclose a region of space.

A prism is a polyhedron that has two congruent regions called bases that are parallel. Prisms are named according to their bases. Thus, a box is a rectangular prism. Take a look at Page 339 for some more examples of prisms.

A pyramid has only one base and a vertex. It is also named by the shape of its base. A triangular pyramid is also called a tetrahedron.
A regular polyhedron has all of its faces bounded by congruent regular polygons. There are only five such polyhedrons having 4, 6, 8, 12, and 20 faces.
Regular tetrahedron, cube, regular octahedron, regular dodecahedron, and regular icosahedron.

Check out Page 340 in our textbook for a picture of these.



Volumes Of Prisms 10-6

A polyhedron together with the region inside it is called a solid. The measure of the space occupied by a solid is called the colume ofthe solid.
Volume of prism = Base area X height
V = Bh where B = the Base area or the area of the base

Math 6 Honors Periods 6 & 7

Area of Circles 10-3

Recall that there are two formulas for the circumference C of a circle. If the diameter of the circle is denoted by d and the radius by r, then

C =πd and C = 2πr

Two approximations for the number π are 3.14 and

The part of the plane enclosed by a circle is called the area of the circle.

Formula

Area of circle = π · (radius)²

A = πr²

Find the area of the shaded regions. Use π ≈ 3.14

Recall that we give answers to only three digits when we use the approximation π≈ 3.14. In fact, sometimes to avoid approximation we give the answer in terms of π.

Find the area of the shaded region. Leave your answer in terms of π

(Look at your textbook for examples)

Area of shaded region = Area of large circle – Area of small circle.

We must first find the area of the small circle.

A = πr² = π2² = 4π

We then find the area of the large circle. Since the radius of the large circle is the same as the diameter of the small circle we know that the radius of the circle must be 4 m so A = πr² = π4²= 16π

Thus, the area of the shaded region is equal to 16π - 4π = 12π

Math 6 Honors Periods 6 & 7 (Wednesday)

Areas of Triangles and Trapezoids 10-2

Any side of a triangle can be considered to be the base. The height is then the perpendicular distance from the opposite vertex to the base line.

Let us find the area of a triangle having base b and height h. The triangle and a congruent copy of it can be put together to form a parallelogram

Since the area of the parallelogram is bh—from yesterday’s lesson--the area of the triangle is half the area of the parallelogram so we have the following

Formula

Area of triangle = ½ · base · height

A = ½ bh

Find the area of each triangle

Look at the pictures in your textbook Page 325 and practice a few

Note that in the second example, the lengths of the sides of the right angle of the triangle were used as the base and the height. this can be done for any right triangle.

Trapezoids

The height of a trapezoid is the perpendicular distance between the parallel sides. These parallel sides are called the bases of the trapezoid. The method used to find the formula for the area of a triangle can be used to find the formula for the area of a trapezoid having bases b₁ and b₂ and height h. The trapezoid and a congruent copy of it can be put together to form a parallelogram. The area of the parallelogram is (b₁ + b₂)h and the area of the original trapezoid is half the area of the parallelogram.

Formula

Area of trapezoid = ½ · (sum of bases) · height

A = ½ (b₁ +b₂)h

Algebra Period 3 (Tuesday)

Quadratic Equations Review: Chapter 13


Finding the x intercepts of a parabola:
x intercepts = roots = solutions = zeros of a quadratic
You can find these one of two ways:
1) Read them from the graph

2) Set y or f(x) = 0 and then solve

Where the graph crosses the x axis is/are the x intercepts.

(Remember, y = 0 here!)
The x intercepts are the two solutions or roots of the quadratic.
When we factored in Chapter 6 and set each piece equal to zero,
We were finding the x value when y was zero.
That means we were finding these two roots!

You can find these roots (solutions, x intercepts, zeros) by several different methods.
You already know the following 2 methods:

1) Graphing and see where the graph crosses the x axis
2) Factoring, then using the zero products property to solve for x for each factor

ANOTHER METHOD:
Chapter 13-2 Solving Quadratics Using Square Roots -
You did this in Ch 11 for Pythagorean Theorem!
If there is no middle x term, it's easiest to just square root both sides to solve!

BUT THE DIFFERENCE FROM PYTHAGOREAN--NOT LOOKING FOR JUST THE PRINCIPAL SQUARE ROOT ANY MORE.

YOU NEED THE + OR - SYMBOL!!

EXAMPLE:
3x² = 18
divide both sides by 3 and get: x² = 6
square root each side and get x = + or - SQRT of 6

HARDER EXAMPLE:
(x - 5)² = 9
SQRT each side and get: x - 5 = + or - 3
+ 5 to both sides: x = 5 + 3 or x = 5 - 3
So, the 2 roots are x = 8 or x = 2

HARDEST EXAMPLE:
(x + 2)² = 7
SQRT each side and get: x + 2 = + or - SQRT of 7
-2 to both sides: x = -2+ SQRT 7 or x = -2 - SQRT 7

THE QUADRATIC FORMULA 13-4

-b plus or minus the square root of b squared minus 4ac all over 2a

Notice how the first part is the x value of the vertex -b/2a
The plus or minus square root of b squared minus 4ac represents
how far away the two x intercepts (or roots) are from the vertex!!!!

Very few real world quadratics can be solved by factoring or square rooting each side.
And completing the square always works, but it long and cumbersome!

All quadratics can be solved by using the QUADRATIC FORMULA.

(you will find out that some quadratics have NO REAL solutions, which means that there are no x intercepts - the parabola does not cross the x axis! Think about what kinds of parabolas would do this....ones that are smiles that have a vertex above the x or ones that are frowns that have a vertex below the x axis. You will find out in Algebra II that these parabolas have IMAGINARY roots)

So now you know 5 ways that you know to find the roots:

1. graph
2. factor if possible
3. square root each side
4. complete the square - that's what the quadratic formula is based on!
5. plug and chug in the Quadratic Formula -
This method always works if there's a REAL solution!

DON'T FORGET TO PUT THE QUADRATIC IN STANDARD FORM BEFORE PLUGGING THE VALUES INTO THE QUADRATIC FORMULA!

ax² + bx + c = 0

DISCRIMINANTS - a part of the Quadratic Formula that helps you to understand the graph of the parabola even before you graph it!
the discriminant is b² - 4ac

(the radicand in the Quadratic Formula, but without the SQRT)

Depending on the value of the radicand, you will know
HOW MANY REAL ROOTS IT HAS

1) Some quadratics have 2 real roots (x intercepts or solutions) - Graph crosses x axis twice
2) Some have 1 real root (x intercept or solution) - Vertex is sitting on the x axis
3) Some have NO real roots (no x intercepts or solutions) - vertex either is above the x axis and is a smiley face (a coefficient is positive) or
the vertex is below the x axis and is a frown face (a coefficient is negative)

In both of these cases, the parabola will NEVER CROSS (intercept) the x axis!

b² -4ac > 0 That is, if it's positive, then there are 2 roots
b² -4ac = 0 That is, if it's zero , then there is 1 root
b² -4ac <0>

Algebra Period 3 (Tuesday)

Review of how to graph a quadratic

Find the vertex ( x = -b/2a, then plug in to the equation to find the y coordinate)

Find the axis (line) of symmetry. Just use the x value of the vertex that is
x = -b/2a

Draw the line of symmetry using a dotted or dashed line

Find 2 more points in an x y table (go either left or right of the vertex and use 2 points close to the line of symmetry for the x, then plug into the equation to find the y values)

Find the 2 shadow (mirror) points by counting over from the axis at the exact same y value

Draw a U (not a V) through the points. Extend it past and put arrows at the end
Remember to label the x and y axes and put arrows on the axes.

Function vs. Relations: Functions are special relations where there is a unique x for each y
Therefore, there will never be 2 x’s that will repeat and if you use a vertical line on the graph, it will only hit one point on the graph.

Domain vs. Range: the domain for any quadratic function is ALL REAL NUMBERS
The range depends on where the vertex is and whether the quadratic is a smile or a frown.
Generally, it will be of the form:

y such that y is either greater than or equal to (≥ ) or less than or equal to (≤) the y value of the vertex. {y l y≥ n} or {y l y ≤ n}

f(x) is just a different way of saying y. What is better about it? Without seeing the work before the solution, you can actually tell the x value as well as the y value in the solution.

For example, if the solution is f(-3) = 12 you know that the point is (-3, 12)
Compare that to the solution y = 12. For that solution, you would not know the x value unless you look back in the problem.

Math 6 Honors Periods 6 & 7 (Tuesday )

Areas of Rectangles and Parallelograms 10-1

In Chapter 4 (so long ago) we measured lengths of segments and found perimeters of various polygons. Now we will measure the part of the plane enclosed by a polygon. We call this measure the area of the polygon.

In the metric system the unit area often use is the square centimeter. cm²

Notice that the area of each rectangle is the product of the lengths of the two consecutive sides. These sides are called the length and the width of the rectangle The length names the longer side and the width names the shorter side. These sides are also named the base and height.

Formula

Area of rectangle = length · width

A= lw

The length and the width of the rectangle are called its dimensions

In the case of a parallelogram, we may consider either pair of parallel sides to the bases. (the word base is also used to denote the length of the base) the height is the perpendicular distance between the bases.

Formula

Area of parallelogram = base · height

A= bh

The unit areas used are square meters (m²), square millimeters (mm²) and square centimeters (cm²). For very large regions, such as the State of California, or entire countries (Canada), we could use square kilometers (km²)

Sometimes we work with an unspecified unit of length. Then the unit of area is simply called a square unit. It is vital, however, that you indicate the area is in square units.

Algebra Period 3

Compound Sentences 9-2

Conjunction = and - the graph is an intersection ("yo") and inequality looks like our domains and ranges on our projects EXAMPLE: 5 < x < 10
open dots; between 5 and 10 is colored in

Disjunction = or - graph will go opposite ways ("dorky" dancer) and inequality looks like this:
x < -2 OR x > 4
open dots; one arrow goes right at 4 and the other arrow goes left at -2

Equations and Absolute Value - 1 VARIABLE 9-3
Solve the equation twice - Once with the solution positive and once with it negative in this setting [ ] represents absolute value.
[2x - 4] = 10
Solve it twice:
2x - 4 = 10 or 2x - 4 = -10
x = 7 or x = -3

If there is a term on the same side of the equation as the absolute value, move that to the other side of the equation first (just like we did with radical equations!) Then solve twice.

REMEMBER THAT THE SOLUTION GIVEN CANNOT BE NEGATIVE (the null set)

Inequalities and Absolute Value - 1 VARIABLE 9-4
There are 2 possible types of inequalities - less than and greater than
For less thAND:
These are conjunctions and so you solve it twice and the solution ends us between them
[3x] < 15 is equal to -15<3x<15
so x is greater than -5 and less than 5

For greatOR than:
These are disjunctions and are solved twice with the solution infinitely in different directions
[3x] >15 is equal to 3x < -15 and 3x > 15

INEQUALITIES IN 2 VARIABLES 9-5
You will shade an x y graph to find the side of a linear equation that fits the solution
EXAMPLE: x + y > 5
You graph the line with DOTTED line because it cannot be equal to 5
You pick an easy point on one side of the line and substitute to see if that side is a solution.
If that does not work, pick an easy point on the other side to see if that side checks.
I USUALLY USE (0,0) as my first point!
If there is more than 1 Inequality (a system of inequalities), follow the same procedure and where the 2 shadings overlap each other is called the solution of the system of inequalities.

SYSTEMS OF INEQUALITIES IN TWO VARIABLES 9-6
If there is more than 1 Inequality (a system of inequalities),
1. Follow the same procedure as above for one equation
but you will need to do it for each equation.
2. Use a different type of shading for each so you won't get confused
(Ex: Use slanted lines one way and then slanted lines the other way. Use different colors if possible. Make one set of lines wavy and the other set straight)
3. Where the 2 shadings overlap each other is called the solution of the system of inequalities.
(any point in the overlap should work in BOTH inequalities - make sure you check!!!!!)

Pre Algebra Periods 1, 2, & 4

Graphing Review - make sure to look at your book as you review these notes. Look at the examples

Box and Whisker Plots

Suppose you have a large set of data and you want a display that gives a general idea of how the data clusters together. A box-and-whisker plot displays the median, the quartiles, and outliers of a set of data but does not display any other specific values.

To make a box-and-whisker plot:

Write the data in order from least to greatest

Draw a number lime that can show the data in equal intervals – make sure to have intervals that include the least and the greatest

Find the median – Mark it with a dot below ( or above the number line)

Find the upper quartile ( the median of the numbers above the actual median) Mark it with a dot below ( or above the number line)

Find the lower quartile ( the median of the numbers below the actual median) Mark it with a dot below ( or above the number line)

Mark with a dot the upper extreme – the greatest number

Mark with a dot the lower extreme – the lowest number

Draw a box between the lower quartile and upper quartile. Split the box by drawing a vertical line through the median.

Draw two ‘whiskers’ from the quartiles to the extremes.

50% of all the data will be within the box. 25% will be below and 25% will be above.

Frequency Table

A frequency table is a way to show how often an item, a number or a range of numbers occurs.

number	1	2	3	4
frequency	4	0	5	8

The range is the difference between the highest number and the lowest. In this case 4-1 = 3

Line Plots help show the spread of the data. When you look at a line plot you can easily see the range, the mode, and any outliers in the data,

Draw a horizontal line segment on grid paper

Make a scale of numbers below the line. The numbers should include the greatest value and the least value of the set of data.

For each piece of data, draw an X above the corresponding number.

Stem-And-Leaf Plots

Stem and Leaf Plots allow you to easily see the greatest, least, and median values in a set of data.

For example—from class

As of 1997 the following are the ages, in chronological order, at which US Presidents were inaugurated

57,61, 57, 57, 58, 57, 61, 54, 68, 51, 49, 64, 50, 48, 65, 52, 56, 46, 54, 49, 50, 47, 55, 55, 54, 42, 51, 56, 55, 51, 54, 51, 60, 62, 43, 55, 56, 61, 52, 69, 64, 46

To make a stem-and-leaf plot:

Write the data in order from least to greatest

Find the least and greatest values

Choose stem values that will include the extreme values. For this graph, it makes sense to use tens.

Write the tens vertically from least to greatest. Draw a vertical line to the right of the stem values.

Separate each number into stems (tens – in this case) and leaves (ones- in this case). Write each leaf to the right of its stem in order from least to greatest.

Write a key that explains how to read the steams and leaves.

Scatter Plots

Suppose you want to analyze two sets of data to see how closely they are related. On a scatter plot you plot corresponding numbers from two sets of data as order pairs (x,y). You then decide if they are related by determining how close they come to forming a straight line.

For example, here is a frequency table of the number of hours studied and grades of a student

study hours	1.5	1	3	2.5	1.5	4	3.5
grade on test	75	71	88	86	80	97	92

To make the scatter plot

Decide which set of numbers you will plot on each axis and label the axis. In this case, The study time ( in hours) would be the X-axis and the Grade would be the Y-axis.

Choose a scale for each axis.

Plot corresponding numbers as ordered pairs. For example (1.5, 75) and (1, 71) are the first two from the table above.

The dots on the scatter plot are close to forming a straight line ( going up) so this is a strong positive correlation.

If the line formed was going down—it would be a negative correlation

and if you could not determine any line- it would be no correlation.

Math 6 Honors Periods 6 & 7

Percent and Fractions 9-1

The word “percent” is derived from the Latin “per centum” meaning “per hundred” or “out of one hundred” so 28% means 28 out of 100

A percent is a ratio that compares a number to 100. Therefore you can write a percent as a fraction with a denominator of 100, so 28% is also 28/100

Our book’s example is as follows;

During basketball season, Alice made 17 out of 25 free throws, while Nina made 7 out of 10. To see who did better, we compare the fractions representing each girl’s successful free throws.

17/27 or 7/10 or

In comparing fractions it is often convenient to use the common denominator 100, even if 100 is not the LCD of the fractions.

17. 4 = 68 thithithiett7.10=70
24^t4th100thiethithei10t10t100

Since Alice makes 68 free throws per 100 and Nina makes 70 per hundred, Nina is the better free throw shooter.

the ratio of a number to 100 is called a percent. We write percents by using the symbol %

so

17/25=68% and 7/10= 70%

Rule

To express the fraction a/b as a percent, solve the equation

n/100 = a/b

for the variable n and write n%

Just set up as a ratio and compare your fraction to 100. You know three out of the four numbers. Express 17/40 as a percent

Use the proportion method or just divide 17 by 40 multiply both sides by 100

n= 42½

Therefore, 17/40 = 42.5%

Rule

To express n% as a fraction, write the fraction

n/100 in lowest terms

Express 7 1/2 % as a fraction in lowest terms

7 ½ % = 7.5% = 7.5/100 How do we get rid of the decimal? multiply numerator and denominator by ten

7.5 . 10 = 75 simplify
100 t10 t1000

=3/40

Since a percent is the ratio of a number to 100, we can have percents that are greater than or equal to 100%

100 = 100%
100

165 =165%
100

Write 250% as a mixed number in simple form

250% =

250=
100

2 50/100 = 2 1/2

Percents and Decimals 9-2

By looking at the following examples, you will be able to see a general relationship between decimals and percents

51% = 47/100 = 0.57 113% = 113/100 = 1.13

0.79 = 79/100 = 79% .06 = 6/100 = 6%

Rules

To express a percent as a decimal, move the decimal point two places to the left and remove the percent sign

57% = 0.57 thith 113% = 1.13

To express a decimal as a percent, move the decimal point two places to the right and add a percent sign

0.79 = 79% thithith0.06 = 6%

Express each percent as a decimal ( remember- the decimal is so sad to see the % leave that it runs away – two places to the left—from where it was)

1. 83.5% = 0.835

2. 450% = 4.5

3. .25% = 0.0025

Express each decimal as a percent [careful] ( remember- the decimal is so happy to see the % that it runs ( 2 place to the right) towards the %)

1. 10.5 = 1050.%

2. 0.0062 = 0.62%

3. 0.574= 57.4%

In 9-1 you learned one method of changing a fraction into a percent. Here is an alternative method

Rule

To express a fraction as a percent, first express the fraction as a decimal

and then as a percent

Express 7/8 as a percent

Divide 7 by 8

7/8

= 0.875 = 87.5%

Express 1/3 as a percent to the nearest tenth of a percent

divide 1 by 3

1/3

since this becomes a repeating decimal

.33333333….

or .3 with a vinculum, we write 1/3 as

33 1/3%

Pre Algebra Periods 1, 2, & 4

SQUARE ROOTS & PYTHAGOREAN THEOREM 11-1 & 11-2

Square root undoes squaring!
So you're looking for the number/variable that was squared to get the radicand
1) RADICAL sign: The root sign, which looks like a check mark.
If there is no little number on the radical, you assume it's the square root
But many times there will be a number there and then you are finding the root that the number says.
For example, if there is a 3 in the "check mark," you are finding the cubed root.
One more example: The square root of 64 is 8. The cubed root of 64 is 4. The 6th root of 64 is 2.
2)RADICAND : Whatever is under the RADICAL sign
In the example above, 64 was the radicand in every case.
3) ROOT (the answer): the number/variable that was squared (cubed, raised to a power)
to get the RADICAND (whatever is under the radical sign)
4) SQUARE ROOTS: The number that is squared to get to the radicand. Every POSITIVE number has 2 square roots - one positive and one negative.
Example: The square root of 25 means what number squared = 25
Answer: Either positive 5 squared OR negative 5 squared

You can estimate nonperfect square roots by guess and check
Find the 2 numbers that it is between
Example: Square root of 52
It's between the 2 perfect squares: 49 and 64
So the square root is between 7 and 8
Since 52 is only 3 away from 49 and 12 away from 64, the square root will be closer to 7
Guess: 7.2 Square this: (7.2)(7.2) = 51.84 (adjust your estimate as necessary)

REAL NUMBER SYSTEM
2 PARTS: RATIONAL AND IRRATIONAL (both real)

Review LEAP FROG number systems
I: RATIONAL NUMBERS (definitions of different number systems):
Natural = counting = 1, 2, 3, . . .
Whole = natural + 0 = 0, 1, 2, 3, . . .
Integers = whole + opposites = -3, -2, -1, 0, 1, 2, 3, . . .
Rational = integers and all the fractions/decimals in between - terminating and repeating decimals

II: IRRATIONAL - numbers like pi and square root of 3 - never repeat or terminate - round!

PYTHAGOREAN THEOREM
FOR RIGHT TRIANGLES ONLY!
2 legs - make the right angle - called a and b
(doesn't matter which is which because you will add them and adding is COMMUTATIVE!)
hypotenuse - longest side across from the right angle - called c
You can find the third side of a right triangle as long as you know the other two sides:
a² + b² = c²
After squaring the two sides that you know, you'll need to find the square root of that number to find the length of the missing side (that's why it's in this chapter!)
EASIEST - FIND THE HYPOTENUSE (c)
Example #1 from p. 510
8² + 15² = c²
64 + 225 = c²
289 = c²
c = 17

A LITTLE HARDER - FIND A MISSING LEG (Either a or b)
Example #5 from p. 510
5² + b² = 13²
25 + b² = 169
b² = 169 - 25
b² = 144
b = 12

CONVERSE OF PYTHAGOREAN THEOREM
If you add the squares of the legs and that sum EQUALS the square of the longest side, it's a RIGHT TRIANGLE.
If you add the squares of the 2 smallest sides and that sum is GREATER THAN the square of the longest side, you have an ACUTE TRIANGLE.
If you add the squares of the 2 smallest sides and that sum is LESS THAN the square of the longest side, you have an OBTUSE TRIANGLE.