Thursday, May 17, 2012
Math 6H ( Period 1, 2, & 3)
Triangles 4-4
A triangle is the figure formed when three points, not on a line are jointed by segments.
Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC
Each of the angles angle A. angle B. and angle C is called an angle of ΔABC
In any triangle-
The sum of the lengths of any two sides is greater than the length of the third side
The sum of the measures of the angles is 180
There are several ways to name triangles. One way is by angles
Acute Triangle
3 acute angles
Right Triangle
1 right angle
Obtuse Triangle
1 obtuse angle
Triangles can be classified by their sides
Scalene Triangle
no 2 sides congruent
Isosceles Triangle
at least 2 sides congruent
Equilateral Triangle
all 3 sides congruent
The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.
Wednesday, May 16, 2012
Algebra Honors (Period 6 & 7)
Inequalities in One Variable
Solving Problems Involving Inequalities 10-3
For practice, we went through the examples in our textbook on Page 469
We discovered that reading and re-reading the problem was critical to make sure we answered the exact question. As noted in Example 1: the question asked what is the minimum total distance, to the nearest mile, that she will have to travel...?" The critical part, was "to the nearest mile." Please read the problem and then realize why re arrived at the following:
Let d = the distance fro the sign to home
d - 16 > 25
solving that open sentence we get
d > 41
Since the distance needs to be greater than 41, the next whole number is 42 so the answer is
The minimum distance she will travel is 42 miles.
To translate phrases such as "is at least" and "is no less than" you will need ≥
...think I want at least $200 when going to Disneyland. I obviously want more.. but I will be happy with $200.
To translate phrases such as "is at most" and "is no more than" you will need ≤
...think I want at most 7 problems of homework. I really want fewer than 7 but I'll be okay with 7.
Algebra Honors (Period 6 & 7)
Inequalities in One Variable
Solving Inequalities 10-2
Property of Comparison
For all real numbers a and b, one and only one of the following statements is true:
a < b, a = b, or a >; b
Transitive Property of Order
For all real numbers a, b, and c :
- If a < b and b < c , then a <; c
- If a > b, and b > c, then a > c
Addition Property of Order
For all real numbers a, b, and c :
- If a < b, then a + c < b + c
- If a > b, then a + c > b + c
Multiplication Property of Order
For all real numbers a, b, and c such that
c > 0 ( c is positive)
- If a < b, then ac < bc
- If a > b then ac > bc
c < 0 ( c is negative)
- If a < b, then ac > bc
- If a > b, then ac < bc
Multiplying each side of an inequality by a negative number reverses the direction or order of the inequality.
These properties guarantee that the following transformations of a given inequality will always produce an equivalent inequality
- Substituting for either side of the inequality an expression equivalent to that side
- Adding to (or subtracting from) each side of the inequality by the same real number
- Multiplying (or dividing) each side of the inequality by the same positive number.
- Multiplying or dividing) each side of the inequality by the same negative number AND reversing the direction of the inequality.
To solve an inequality you use the same steps used to solve equations
- Simplify each side of the inequality as needed
- Use the inverse operations to undo any additions or subtractions
- Use the inverse operations to undo any multiplications or divisions
Remember when graphing these inequalities a closed dot indicates that the number is included while an open dot indicates a less than or a greater than.
When working with inequalities it is important to read both the symbols and the words slowly and carefully!!
Word problems involving inequalities also require very careful reading. An important step to solving this type of problem is determining which inequality symbol to use. Before you write your inequality you should be certain that you will use the correct symbol.
Math 6H ( Period 1, 2, & 3)
Angles and Angle Measure 4-3
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol from the textbook.
To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.
To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.
We often label angels with their measures. When angles have equal measures we can write m angle A = m angle B
We say that angle A and angle B are congruent angles
If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”
Angles formed by perpendicular lines each have measure of 90° . A 90° angle is called a right angle. A small square is often used to indicate a right angle in a diagram
An acute angle is an angle with measure less than 90°. An obtuse angle has measure between 90° and 180°
Two angles are complementary if the sum of the measures is 90°
Two angles are supplementary if the sum of their measures is 180°
Tuesday, May 15, 2012
Algebra Honors (Period 6 & 7)
Inequalities in One Variable
Order of Real Numbers 10-1
This sections should be review... you have been working with less than and greater than symbols since 6th grade. A number line shows order relationships among all real numbers. The value of a variable may be unknown but you may know that is either greater than or equal to another number.
For example,
x ≥ 5 is read "x is greater than or equal to 5."
x ≥ 5 is another way of writing " x >5 or x = 5"
Translating statements into symbols is a critical concept. Practice these as review
-3 is greater than -5. -3 > -5
and
x is less than or equal to 8 x ≤ 8
To show that x is between -4 and 2 you write
-4 < x < 2
which is read
"-4 is less than x and x is less than 2."
or you could read it as
" x is greater than -4 AND less than 2."
The same comparisons are stated in the sentence 2 > x > -4
When all the numbers are know you can classify the statement as true or false.
Thus,
-4 < 1 < 2 is true
but
-4 < 8 < 2 is false
An inequality is formed by placing an inequality symbol ( > , < , ≤ , or ≥) between numerical pr variable expressions-- called the SIDES of the inequality
You solve an inequality by finding the values from the domain of the variable which make the inequality a true statement. Such values are called the solutions of the inequality. All the solutions make up the solution set of the inequality.
Order of Real Numbers 10-1
This sections should be review... you have been working with less than and greater than symbols since 6th grade. A number line shows order relationships among all real numbers. The value of a variable may be unknown but you may know that is either greater than or equal to another number.
For example,
x ≥ 5 is read "x is greater than or equal to 5."
x ≥ 5 is another way of writing " x >5 or x = 5"
Translating statements into symbols is a critical concept. Practice these as review
-3 is greater than -5. -3 > -5
and
x is less than or equal to 8 x ≤ 8
To show that x is between -4 and 2 you write
-4 < x < 2
which is read
"-4 is less than x and x is less than 2."
or you could read it as
" x is greater than -4 AND less than 2."
The same comparisons are stated in the sentence 2 > x > -4
When all the numbers are know you can classify the statement as true or false.
Thus,
-4 < 1 < 2 is true
but
-4 < 8 < 2 is false
An inequality is formed by placing an inequality symbol ( > , < , ≤ , or ≥) between numerical pr variable expressions-- called the SIDES of the inequality
You solve an inequality by finding the values from the domain of the variable which make the inequality a true statement. Such values are called the solutions of the inequality. All the solutions make up the solution set of the inequality.
Math 6H ( Period 1, 2, & 3)
Points, Lines, Planes 4-1
We can describe but CANNOT DEFINE point, line or plane in Geometry
We use a single small dot to represent a point and in class we labeled with a P and we called it Point P
A straight line in Geometry is usually just called a line...
Two points DETERMINE exactly ONE LINE
we connected Point P with Point Q and created Line PQ
We placed this type of arrow ↔ over PQ to show a line
↔
PQ
that would represent the line PQ but we found we could write
↔
QP
and mean the SAME line!!
You can name ANY line with ANY TWO points that fall on that line!! USE only TWO points to name a line!!
Three or more points on the same line are called collinear. Notice the word "line" in collinear.
collinear
Points NOT on a same line are called noncollinear.
We have a RAY if we have an endpoint and it extends through other points. We name the rame by Naming the endpoint FIRST
→
PQ is RAY PQ and it begins at P and goes through Q. It is NOT the SAME as
→
QP which is Ray QP, which begins at Q and goes through P
Segments are parts of lines with TWO ENDPOINTS.
⎯
PQ is a segment with endpoints Point P and POint Q
We then looked at the drawing from page 105 and the class named all of the names for the line in the drawing, all of the rays that existed in the figure as well as the segments. We found that there were just 3 collinear points: A, X, B but that we could name 3 sets of non collinear points
X, Y, B and A, X, Y, AND A, B, Y
Three non-collinear points determine a flat surface called a plane.
We name a Plane by using three of its non collinear points!! Plane ABC was out example.
Lines in the same plane that do not intersect are PARALLEL lines. Two segments or rays are parallel if they are parts of parallel lines.
↔
AB is parallel to
↔
CD
may be written
↔ ↔
ABllCD
using two straight lines to indicate parallel
Parallel lines DO NOT intersect.
Intersecting lines intersect in a single point!!
Planes that do not intersect are called parallel planes... we looked around the room and found examples of parts of planes.. noticing which ones were parallel!! (the floor and ceiling were a great example)
Then we drew the box from PAge 106 and identified parallel segments and lines from that box.
Two non parallel lines that do not intersect are called SKEW LINES.
Tuesday, May 1, 2012
Algebra Honors (Period 6 & 7)
Star Prep
Conjunctions & Disjunctions
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
Solve the equation twice - Once with the solution positive and once with it negative
I2x - 4I = 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
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 up between them
I3xI < 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
I3xI >15 is equal to 3x < -15 and 3x > 15
Conjunctions & Disjunctions
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
Solve the equation twice - Once with the solution positive and once with it negative
I2x - 4I = 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
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 up between them
I3xI < 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
I3xI >15 is equal to 3x < -15 and 3x > 15
Math 6 Honors ( Periods 1, 2, & 3)
Polygons 4-5
A polygon is a closed figure formed by joining segments ( the sides of the polygon) at their endpoints ( the vertices of the polygon). Polygons are named for the number of sides they have.
Triangle- 3 sides
Quadrilateral- 4 sides
Pentagon- 5 sides
Hexagon- 6 sides
Octagon- 8 sides
Decagon- 10 sides
A polygon is regular if ALL of its sides are congruent and ALL of its angles are congruent.
To name the polygon we name its consecutive vertices IN ORDER.
A diagonal of a polygon is a segment joining two nonconsecutive vertices.
Look at the quadrilateral on Page 123 and notice the two segments that represent the diagonals of the quadrilateral PQRS.
Certain quadrilaterals have special names.
A parallelogram has it opposite sides parallel and congruent.
A trapezoid has just one pair of parallel sides.
Certain parallelograms also have special names
A rhombus ( rhombii plural) has all it sides congruent.
A square has congruent sides and congruent angles
A rectangle has all its angles congruent.
Thus a square is a rectangle.. but a rectangle isn't necessarily a square!!
TH\he perimeter of a figure is the distance (think fence) around it. Thus the perimeter of a polygon is the sum of the lengths of its sides.
Always label your perimeters. If the figure provides a specific measure, such as meters (m), centimeters (cm), feet (ft), inches (in.)-- make sure to use that label.
If no unit of measure is given, always include "units"
Circles 4-6
A circle is the set of all points in a plan at a given distance from a given point O (called the center).
A segment joining the center to a point on the circle is called a radius ( plural: radii) of the circle. All radii of a given circle have the same length and the length is called the radius of the circle.
A segment joining two points on a circle is called a chord... and a chord passing through the center is a diameter of the circle. the ends of the diameter divide the circle into two semicircles. The length of a diameter is called the diameter of the circle.
Two radii equal one diameter-- a fact we will use in the formulas below
The perimeter of a circle is called the circumference and the quotient
circumference ÷ diameter is the same for all circles--> regardless of size
This quotient is denoted by the Greek letter ∏ ( pronounced "pie")
No decimal gives ∏ exactly
No fraction gives ∏ exactly, either
A fairly good approximation is either 3.14 or 22/7
If we denote the circumference by C and the diameter by d we can write
C ÷ d = ∏
This formula can be put into several useful forms.
Let C = circumference d = diameter and r = radius
Then:
C = ∏d
d = C/∏
C = 2∏r
and
r = C/(2∏)
We tried a few examples.
Using ∏≈ 3.14 and rounding to three digits, as described by our textbook.
The diameter of a circle is 6 cm. Find the circumference.
WE are given d and are asked to find C.
WE use the formula
C = ∏d
C ≈ 3.14(6) = 18.84
C ≈ 18.8
So, the circumference is approximately 18.8 cm
The circumference of a circle is 20 feet. Find the radius.
To find the radius, use the formula
r = C/(2∏)
r = 20/2∏
Simplify first
r = 10/∏
r ≈ 10/3.14
r ≈ 3.1847
Since the third digit from the left is in the hundredths' place, round to the nearest hundredth.
r ≈ 3.18
The radius is approximately 3.18 feet
A polygon is inscribed in a circle if all of its vertices are on the circle. Check on the diagram in our textbook on page 129-- we added that to our notes as well.
Three noncollinear points (not on a line) determine one and only one circle that passes through the three given points.
A polygon is a closed figure formed by joining segments ( the sides of the polygon) at their endpoints ( the vertices of the polygon). Polygons are named for the number of sides they have.
Triangle- 3 sides
Quadrilateral- 4 sides
Pentagon- 5 sides
Hexagon- 6 sides
Octagon- 8 sides
Decagon- 10 sides
A polygon is regular if ALL of its sides are congruent and ALL of its angles are congruent.
To name the polygon we name its consecutive vertices IN ORDER.
A diagonal of a polygon is a segment joining two nonconsecutive vertices.
Look at the quadrilateral on Page 123 and notice the two segments that represent the diagonals of the quadrilateral PQRS.
Certain quadrilaterals have special names.
A parallelogram has it opposite sides parallel and congruent.
A trapezoid has just one pair of parallel sides.
Certain parallelograms also have special names
A rhombus ( rhombii plural) has all it sides congruent.
A square has congruent sides and congruent angles
A rectangle has all its angles congruent.
Thus a square is a rectangle.. but a rectangle isn't necessarily a square!!
TH\he perimeter of a figure is the distance (think fence) around it. Thus the perimeter of a polygon is the sum of the lengths of its sides.
Always label your perimeters. If the figure provides a specific measure, such as meters (m), centimeters (cm), feet (ft), inches (in.)-- make sure to use that label.
If no unit of measure is given, always include "units"
Circles 4-6
A circle is the set of all points in a plan at a given distance from a given point O (called the center).
A segment joining the center to a point on the circle is called a radius ( plural: radii) of the circle. All radii of a given circle have the same length and the length is called the radius of the circle.
A segment joining two points on a circle is called a chord... and a chord passing through the center is a diameter of the circle. the ends of the diameter divide the circle into two semicircles. The length of a diameter is called the diameter of the circle.
Two radii equal one diameter-- a fact we will use in the formulas below
The perimeter of a circle is called the circumference and the quotient
circumference ÷ diameter is the same for all circles--> regardless of size
This quotient is denoted by the Greek letter ∏ ( pronounced "pie")
No decimal gives ∏ exactly
No fraction gives ∏ exactly, either
A fairly good approximation is either 3.14 or 22/7
If we denote the circumference by C and the diameter by d we can write
C ÷ d = ∏
This formula can be put into several useful forms.
Let C = circumference d = diameter and r = radius
Then:
C = ∏d
d = C/∏
C = 2∏r
and
r = C/(2∏)
We tried a few examples.
Using ∏≈ 3.14 and rounding to three digits, as described by our textbook.
The diameter of a circle is 6 cm. Find the circumference.
WE are given d and are asked to find C.
WE use the formula
C = ∏d
C ≈ 3.14(6) = 18.84
C ≈ 18.8
So, the circumference is approximately 18.8 cm
The circumference of a circle is 20 feet. Find the radius.
To find the radius, use the formula
r = C/(2∏)
r = 20/2∏
Simplify first
r = 10/∏
r ≈ 10/3.14
r ≈ 3.1847
Since the third digit from the left is in the hundredths' place, round to the nearest hundredth.
r ≈ 3.18
The radius is approximately 3.18 feet
A polygon is inscribed in a circle if all of its vertices are on the circle. Check on the diagram in our textbook on page 129-- we added that to our notes as well.
Three noncollinear points (not on a line) determine one and only one circle that passes through the three given points.
Preparing for the STAR
This was made over two years ago... but it is still fun to watch..
Larry & Sarah.. and the CA STAR
Larry & Sarah.. and the CA STAR
Algebra Honors (Period 6 & 7)
Puzzle Problems 9-7
This includes digit, age, and fraction problems
Example 1: Digit problem
The sum of the digits in a 2 digit number is 12. The new number obtained when the digits are reversed is 36 more than the original number. Find the original number.
Let t = the ten's digit of the original number
Let u = the unit's digit of the original number
then the quantity relationship is t + u = 12
The value expression of the original number is 10t +u so when the digits are reversed the value would be 10u+ t
SO, the value equation becomes
10u + t = 10t + u + 36 or
9u - 9t = 36 Now, divide by 9
u - t = 4
So the two equations are
u - t = 4
u + t = 12
2u = 16
u = 8
If u = 8 then t= 4
and the original number is {48}
EXAMPLE 2: Age Problem
Mini is 4 years older than Ronald
5 years ago she was twice as old as he was. Find their ages now.
Let m = Mimi's age now
Let r = ronald's age now
The two equations created from this information are the following:
m = r + 4
and
m-5 = 2(r-5)
m -5 = 2r - 10 -->I would use substitution with this one so
(r+4) - 5 = 2r - 10
r-1 = 2r - 10
9 = r
Since Ronald is 9 years old
Mimi is 13 years old
Example 3: Fraction Problem
The numerator of a fraction is 3 less than the denominator
If the numerator and the denominator are each increased by 1, the value of the resulting fraction is 3/4 What was the original fraction?
Let n = the numerator
Let d = the denominator
n/d is the fraction
n = d -3
which can be translated to n - d = -3
and
n+1=3
d+1 4
4(n+1) = 3(d+1)
4n + 4 = 3d+ 3
4n-3d = -1
n - d = -3
Using the addition/subtraction method and multiplying the 2nd equation by (-4)
4n-3d = -1
-4n - (-4)d = -3(-4)
4n-3d = -1
-4n+4d = 12
d = 1
Since d = 11
n = 8
and the fraction is 8/11
{8/11}
We then turned to Page 447 and completed several even problems # 8-16 even.
Here are a few of the problems worked out. TUrn to our textbook to read the actual wording of each problem.
#8. Let t = ten's digit
Let u = unit's digit
t = u-5
t + u = (1/3)(10t + u)
3t + 3u = 10t + u
-7t + 2u = 0
Using substitution
-7(u-5) + 2u + 0
-7u + 35 + 2u = 0
-5u = -35
u = 7
Since u = 7 t = 2 and the number is 27
{27}
10.
Let t = ten's digit
Let u = unit's digit
The original number would be 10t + u and the reverse would be 10u + t
so 10u + t = 10t + u -54
9u -9t = -54
divide that by 9
u - t = -6
so
u - t = -6
u + t = 8
2u =2
u = 1
so the number must be 71
{71}
12.
Cecelia is 24 years younger than Joe
Let c = Cecelia's age NOW
:et j = Joe's age now
c + 24 = j
c - 6 = (1/2)(j - 6)
2c - 12 = j- 6
2c - j = 6
c - j = -24
Becomes
2c - j = 6
- c + j = 24
c = 30
So Cecelia's age is 30 years old and Joe is 54 years old
This includes digit, age, and fraction problems
Example 1: Digit problem
The sum of the digits in a 2 digit number is 12. The new number obtained when the digits are reversed is 36 more than the original number. Find the original number.
Let t = the ten's digit of the original number
Let u = the unit's digit of the original number
then the quantity relationship is t + u = 12
The value expression of the original number is 10t +u so when the digits are reversed the value would be 10u+ t
SO, the value equation becomes
10u + t = 10t + u + 36 or
9u - 9t = 36 Now, divide by 9
u - t = 4
So the two equations are
u - t = 4
u + t = 12
2u = 16
u = 8
If u = 8 then t= 4
and the original number is {48}
EXAMPLE 2: Age Problem
Mini is 4 years older than Ronald
5 years ago she was twice as old as he was. Find their ages now.
Let m = Mimi's age now
Let r = ronald's age now
The two equations created from this information are the following:
m = r + 4
and
m-5 = 2(r-5)
m -5 = 2r - 10 -->I would use substitution with this one so
(r+4) - 5 = 2r - 10
r-1 = 2r - 10
9 = r
Since Ronald is 9 years old
Mimi is 13 years old
Example 3: Fraction Problem
The numerator of a fraction is 3 less than the denominator
If the numerator and the denominator are each increased by 1, the value of the resulting fraction is 3/4 What was the original fraction?
Let n = the numerator
Let d = the denominator
n/d is the fraction
n = d -3
which can be translated to n - d = -3
and
n+1=3
d+1 4
4(n+1) = 3(d+1)
4n + 4 = 3d+ 3
4n-3d = -1
n - d = -3
Using the addition/subtraction method and multiplying the 2nd equation by (-4)
4n-3d = -1
-4n - (-4)d = -3(-4)
4n-3d = -1
-4n+4d = 12
d = 1
Since d = 11
n = 8
and the fraction is 8/11
{8/11}
We then turned to Page 447 and completed several even problems # 8-16 even.
Here are a few of the problems worked out. TUrn to our textbook to read the actual wording of each problem.
#8. Let t = ten's digit
Let u = unit's digit
t = u-5
t + u = (1/3)(10t + u)
3t + 3u = 10t + u
-7t + 2u = 0
Using substitution
-7(u-5) + 2u + 0
-7u + 35 + 2u = 0
-5u = -35
u = 7
Since u = 7 t = 2 and the number is 27
{27}
10.
Let t = ten's digit
Let u = unit's digit
The original number would be 10t + u and the reverse would be 10u + t
so 10u + t = 10t + u -54
9u -9t = -54
divide that by 9
u - t = -6
so
u - t = -6
u + t = 8
2u =2
u = 1
so the number must be 71
{71}
12.
Cecelia is 24 years younger than Joe
Let c = Cecelia's age NOW
:et j = Joe's age now
c + 24 = j
c - 6 = (1/2)(j - 6)
2c - 12 = j- 6
2c - j = 6
c - j = -24
Becomes
2c - j = 6
- c + j = 24
c = 30
So Cecelia's age is 30 years old and Joe is 54 years old
Monday, April 30, 2012
Math 6 Honors ( Periods 1, 2, & 3)
Prep for STAR
Angles and Angle Measure 4-3
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol from the textbook.
To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.
To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.
We often label angels with their measures. When angles have equal measures we can write m angle A = m angle B
We say that angle A and angle B are congruent angles
If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”
Angles formed by perpendicular lines each have measure of 90° . A 90° angle is called a right angle. A small square is often used to indicate a right angle in a diagram
An acute angle is an angle with measure less than 90°. An obtuse angle has measure between 90° and 180°
Two angles are complementary if the sum of the measures is 90°
Two angles are supplementary if the sum of their measures is 180°
Triangles 4-4
A triangle is the figure formed when three points, not on a line are jointed by segments.
Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC
Each of the angles angle A. angle B. and angle C is called an angle of ΔABC
In any triangle-
The sum of the lengths of any two sides is greater than the length of the third side
The sum of the measures of the angles is 180
There are several ways to name triangles. One way is by angles
Acute Triangle
3 acute angles
Right Triangle
1 right angle
Obtuse Triangle
1 obtuse angle
Triangles can be classified by their sides
Scalene Triangle
no 2 sides congruent
Isosceles Triangle
at least 2 sides congruent
Equilateral Triangle
all 3 sides congruent
The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.
Angles and Angle Measure 4-3
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol from the textbook.
To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.
To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.
We often label angels with their measures. When angles have equal measures we can write m angle A = m angle B
We say that angle A and angle B are congruent angles
If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”
Angles formed by perpendicular lines each have measure of 90° . A 90° angle is called a right angle. A small square is often used to indicate a right angle in a diagram
An acute angle is an angle with measure less than 90°. An obtuse angle has measure between 90° and 180°
Two angles are complementary if the sum of the measures is 90°
Two angles are supplementary if the sum of their measures is 180°
Triangles 4-4
A triangle is the figure formed when three points, not on a line are jointed by segments.
Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC
Each of the angles angle A. angle B. and angle C is called an angle of ΔABC
In any triangle-
The sum of the lengths of any two sides is greater than the length of the third side
The sum of the measures of the angles is 180
There are several ways to name triangles. One way is by angles
Acute Triangle
3 acute angles
Right Triangle
1 right angle
Obtuse Triangle
1 obtuse angle
Triangles can be classified by their sides
Scalene Triangle
no 2 sides congruent
Isosceles Triangle
at least 2 sides congruent
Equilateral Triangle
all 3 sides congruent
The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.
Thursday, April 26, 2012
Math 6 Honors ( Periods 1, 2, & 3)
Simple Interest 9-7
When you borrow money you pay the lender INTEREST for the use of the money. The amount of interest you pay is usually a percent of the amount borrowed figured on a yearly basis. This percent is called the annual rate.
When interest is computed year by year we call it
SIMPLE INTEREST
The formula is I= Prt
Let I = simple interest charges
P = principal ( amount borrowed)
r= annual rate
t = time in years
I = Prt
simple interest is calculated just on the principal.
Let's work through a few examples
How much simple interest would you owe if you borrowed $640 for 3 years at atan annual rate of 15%?
I = Prt
I = (640)(.15)(3)
I = (1920)(.15)
I = 288
$288 in interest
Sarah borrowed $3650 for 4 years at 16% How much must she repay.
Remember she will oe the amount she borrowed as well as the the interest.
I = Prt
I = (3650)(.16)(4)
I =14600(.16)
I = 2336.
Principal + Interest = 3650 + 2336
Sarah owes $5986.
$150 borrowed at 12% annual rate for 1 year
I = Prt
I = (150)(.12)(1)
I = 18
so you would owe $18 in interest after 1 year.
The total due would be $150 + 18 = $168
What if instead you borrowed the same amount but for 2 years... nothing was due until the end of two years
I = Prt
I = 150(.12)(2) = 36
You would owe $36 in interest .. so the total due was 150 + 36 = $ 186.
What if you borrowed the same amount for 3 years...
I = 150(.12)(3) = 54 or $54 in interest.
You would owe 150 + 54 = $ 204 after three years...
However, let's say you could only borrow that amount for 6 months...
I = Prt
I = (150)(.12)(.5)
Why 0.5? that is 1/2 a year.
Now you can always multiply by 1/2 as well.. in fact, sometimes that is easier
I = 150(.12)(1/2) = 9 or $ 9.00
After 6 months you would owe $159.
Dylan paid $375 in interest on a loan of $1500 principal at 12.5% interest.
What was the length of time?
Look at what it is asking and see which of the variables you have...
I= Prt
We have the interest paid, the principal and the annual rate so
375= (1500)(.125)(t)
375 = 187.5t
solve this one step equation by dividing both sides by 187.5
375 = 187.5t
187.5 187.5
t = 2
so 2 years
divide carefully...
Alexis paid $ 585 simple interest on a $6500 loan for 6 months.
what was the annual rate?
What do we know?
I = 585
P = 6500
t= 6 months ( which is 0.5 or 1/2)
I = Prt
585 = 6500 (r)(.5)
585 = 3250r
divide both sides by 3250
585 = 3250r
3250 3250
r = 0.18
which means 18%
annual--> once a year
6 months --> 1/2 or 0.5
4 months--> 1/3
3 months --> 1/4 or 0.25
8 month --> 2/3
(We did not get to this yet... but I thought I would post it... read this..it is interesting to see the difference. We will go over this after STAR testing)
Compound Interest 9-8
Compound interest is ALWAYS more than simple interest.
interest is compounded on the interest!!
$100 savings earning $10 interest/ annual.. [this only happens NOW if your dad is the one paying you... :)]
I = Prt
at the end of the first year
I = 100(.10)(1) = 10 or $10
add that to the 100
$110.
Now for the 2nd year,
$110 is your principal
so
I = Prt
I = 110(.10)(1) = 11 or $11
so at the end of 2 years you have $110 + 11 or $121
Now for the 3rd year
I = Prt
I = 121(10)(1) = $12.10
So at the end of three years you have $121 + 12.10 = $133.10
What if you had $500 at 8% compounded quarterly for one year.
quarterly means 1/4 or .25
I = Prt
I = 500(.08) (1/4)
calculate the 08(1/4) because that will be the constant you will multiply your principal by each time
(.08)(1/4) = .02
so I = 500(.02) = 10
after the first quarter it is 510
I = Prt for the 2nd quarter
I = 510 (.02) = 10.20
so after the 2nd quarter $510 + 10.20 = $520.20
I = Prt for the third quarter
I = 520.20 (0.02) = about $10.40 ( round to the nearest penny)
so after the third quarter
$520.20 + 10.40 = $530.60
I = Prt
I = 530.60(.02) = about $10.61
So at the end of 4 quarters -- or one year
530.60 + 10.61 = $541.21
compounding terms:
annually--> once a year
semiannually --> twice a year
quarterly--> four times a year
monthly--> 12 times a year
daily--> 365 times a year
When you borrow money you pay the lender INTEREST for the use of the money. The amount of interest you pay is usually a percent of the amount borrowed figured on a yearly basis. This percent is called the annual rate.
When interest is computed year by year we call it
SIMPLE INTEREST
The formula is I= Prt
Let I = simple interest charges
P = principal ( amount borrowed)
r= annual rate
t = time in years
I = Prt
simple interest is calculated just on the principal.
Let's work through a few examples
How much simple interest would you owe if you borrowed $640 for 3 years at atan annual rate of 15%?
I = Prt
I = (640)(.15)(3)
I = (1920)(.15)
I = 288
$288 in interest
Sarah borrowed $3650 for 4 years at 16% How much must she repay.
Remember she will oe the amount she borrowed as well as the the interest.
I = Prt
I = (3650)(.16)(4)
I =14600(.16)
I = 2336.
Principal + Interest = 3650 + 2336
Sarah owes $5986.
$150 borrowed at 12% annual rate for 1 year
I = Prt
I = (150)(.12)(1)
I = 18
so you would owe $18 in interest after 1 year.
The total due would be $150 + 18 = $168
What if instead you borrowed the same amount but for 2 years... nothing was due until the end of two years
I = Prt
I = 150(.12)(2) = 36
You would owe $36 in interest .. so the total due was 150 + 36 = $ 186.
What if you borrowed the same amount for 3 years...
I = 150(.12)(3) = 54 or $54 in interest.
You would owe 150 + 54 = $ 204 after three years...
However, let's say you could only borrow that amount for 6 months...
I = Prt
I = (150)(.12)(.5)
Why 0.5? that is 1/2 a year.
Now you can always multiply by 1/2 as well.. in fact, sometimes that is easier
I = 150(.12)(1/2) = 9 or $ 9.00
After 6 months you would owe $159.
Dylan paid $375 in interest on a loan of $1500 principal at 12.5% interest.
What was the length of time?
Look at what it is asking and see which of the variables you have...
I= Prt
We have the interest paid, the principal and the annual rate so
375= (1500)(.125)(t)
375 = 187.5t
solve this one step equation by dividing both sides by 187.5
375 = 187.5t
187.5 187.5
t = 2
so 2 years
divide carefully...
Alexis paid $ 585 simple interest on a $6500 loan for 6 months.
what was the annual rate?
What do we know?
I = 585
P = 6500
t= 6 months ( which is 0.5 or 1/2)
I = Prt
585 = 6500 (r)(.5)
585 = 3250r
divide both sides by 3250
585 = 3250r
3250 3250
r = 0.18
which means 18%
annual--> once a year
6 months --> 1/2 or 0.5
4 months--> 1/3
3 months --> 1/4 or 0.25
8 month --> 2/3
(We did not get to this yet... but I thought I would post it... read this..it is interesting to see the difference. We will go over this after STAR testing)
Compound Interest 9-8
Compound interest is ALWAYS more than simple interest.
interest is compounded on the interest!!
$100 savings earning $10 interest/ annual.. [this only happens NOW if your dad is the one paying you... :)]
I = Prt
at the end of the first year
I = 100(.10)(1) = 10 or $10
add that to the 100
$110.
Now for the 2nd year,
$110 is your principal
so
I = Prt
I = 110(.10)(1) = 11 or $11
so at the end of 2 years you have $110 + 11 or $121
Now for the 3rd year
I = Prt
I = 121(10)(1) = $12.10
So at the end of three years you have $121 + 12.10 = $133.10
What if you had $500 at 8% compounded quarterly for one year.
quarterly means 1/4 or .25
I = Prt
I = 500(.08) (1/4)
calculate the 08(1/4) because that will be the constant you will multiply your principal by each time
(.08)(1/4) = .02
so I = 500(.02) = 10
after the first quarter it is 510
I = Prt for the 2nd quarter
I = 510 (.02) = 10.20
so after the 2nd quarter $510 + 10.20 = $520.20
I = Prt for the third quarter
I = 520.20 (0.02) = about $10.40 ( round to the nearest penny)
so after the third quarter
$520.20 + 10.40 = $530.60
I = Prt
I = 530.60(.02) = about $10.61
So at the end of 4 quarters -- or one year
530.60 + 10.61 = $541.21
compounding terms:
annually--> once a year
semiannually --> twice a year
quarterly--> four times a year
monthly--> 12 times a year
daily--> 365 times a year
Math 6 Honors ( Periods 1, 2, & 3)
Commission and Profit 9-6
Some sales jobs pay an amount based on how much you sell. This amount is called a commission.
Like a discount, the commission can be expressed as a percent or as an amount of money.
amount of commission = percent of commission X total sales.
Using the examples from our textbook,
Maria sold $42,000 word of insurance in January. If her commission is 3% of the total sales, what was the amount of her commission in January?
amount of commission = percent X total sales
0.03 X 42,000 = 1260
Her commission was $1,260.
Profit is the difference between total income and total operating costs.
profit = total income – total costs
The percent of profit is the percent of total income that is profit
percent of profit = profit/total income
A shoe store had an income of $8600 and operating costs of $7310. What percent of the store's income was profit?
profit= income- total costs = 8600 -7310 = 1290
percent of profit = profit/total income = 1290/8600 = 0.15
So the percent of profit was 15%.
Practice finding 10%-- its easy--- just move the decimal over one place.
We practiced finding 20%. Just double what you got for 10%.
MATH AT WORK:
Caterer
A caterer provides food for parties, weddings, bar/bat mitzvahs, and other events. Caterers plan the menu, buy the ingredients, and cook the food. Often they provide seating and music as well. For each event, a caterer determines the cost per guest. The catering business requires a thorough knowledge of ratios, proportions, and percents.
Some sales jobs pay an amount based on how much you sell. This amount is called a commission.
Like a discount, the commission can be expressed as a percent or as an amount of money.
amount of commission = percent of commission X total sales.
Using the examples from our textbook,
Maria sold $42,000 word of insurance in January. If her commission is 3% of the total sales, what was the amount of her commission in January?
amount of commission = percent X total sales
0.03 X 42,000 = 1260
Her commission was $1,260.
Profit is the difference between total income and total operating costs.
profit = total income – total costs
The percent of profit is the percent of total income that is profit
percent of profit = profit/total income
A shoe store had an income of $8600 and operating costs of $7310. What percent of the store's income was profit?
profit= income- total costs = 8600 -7310 = 1290
percent of profit = profit/total income = 1290/8600 = 0.15
So the percent of profit was 15%.
Practice finding 10%-- its easy--- just move the decimal over one place.
We practiced finding 20%. Just double what you got for 10%.
MATH AT WORK:
Caterer
A caterer provides food for parties, weddings, bar/bat mitzvahs, and other events. Caterers plan the menu, buy the ingredients, and cook the food. Often they provide seating and music as well. For each event, a caterer determines the cost per guest. The catering business requires a thorough knowledge of ratios, proportions, and percents.
Tuesday, April 24, 2012
Math 6 Honors ( Periods 1, 2, & 3)
Discount and Markup 9-5
A discount is a decrease in the price of an item. A markup is an increase in the price of an item. Both of these changes can be expressed as an amount of money or as a percent of the original price of the item. A store may announce a discount of $3 off the original price of $30 basketball, or a discount of 10%
A warm-up suit that sold for $42.50 is on sale at a 12% discount. What is the sale price?
Method 1: Use the formula
amount of change = percent of change X original amount
= 12% X $42.50
therefore the discount is 0.12 X 42.50 or 5.10
The amount of discount is $5.10
The sale price is 42.50 – 5.10 = $37.40
Method 2: Since the discount is 12%, the sale price is 100% - 12% = 88%.
The sale price is 0.88 X 42.50 = $ 37.40
When you know the amount of discount you subtract to find the new price. When dealing with a markup you add to find the new price.
The price of a new car model was marked up 6% over the previous year’s model. If the previous year’s model sold for $7800, what is the cost of the new car? {and what kind of a car could that be?}
Method 1: Use the formula
amount of change = percent of change X original amount
= 6% X 7800
Therefore the markup is 0.06 X7800= $468
The new price is 7800 + 468 = $8268
Method 2: Since the markup is 6% the new price is 100% + 6% or 106% of the original price. so the new price is 1.06 X7800 = $8268
This year a pair of ice skates sells for $46 after a 15% mark up over last year’s price. What was last year’s price?
This year’s price is 100 + 15 or 115% of last year’s price. Let n present last year’s price
46 = (115/100)n
46 = 1.15n
46/.15 = 1.15n/1.115
40 = n
So last year’s price was $40.
A department store advertised eclectic shavers at a sale price of $36.
If this is a 20% discount, what was the original price?
The sale price is 100 - 20 or 80% of the original price. Let n represent the original price.
36 = (80/100)n
36 = .8n
36/.8 = .8n/.8
45 = n
The original price was $45.
Check to see that your answers are logical and reasonable.
Try these: A service station (that’s gas station, now—they no longer provide service!!) give cash customers a 5% discount on the price of gasoline. If gasoline regularly sells for $3.00 a gallon, what is the discounted price?
A store marks up the price of a $5 item to $12. What is the percent of markup?
A discount is a decrease in the price of an item. A markup is an increase in the price of an item. Both of these changes can be expressed as an amount of money or as a percent of the original price of the item. A store may announce a discount of $3 off the original price of $30 basketball, or a discount of 10%
A warm-up suit that sold for $42.50 is on sale at a 12% discount. What is the sale price?
Method 1: Use the formula
amount of change = percent of change X original amount
= 12% X $42.50
therefore the discount is 0.12 X 42.50 or 5.10
The amount of discount is $5.10
The sale price is 42.50 – 5.10 = $37.40
Method 2: Since the discount is 12%, the sale price is 100% - 12% = 88%.
The sale price is 0.88 X 42.50 = $ 37.40
When you know the amount of discount you subtract to find the new price. When dealing with a markup you add to find the new price.
The price of a new car model was marked up 6% over the previous year’s model. If the previous year’s model sold for $7800, what is the cost of the new car? {and what kind of a car could that be?}
Method 1: Use the formula
amount of change = percent of change X original amount
= 6% X 7800
Therefore the markup is 0.06 X7800= $468
The new price is 7800 + 468 = $8268
Method 2: Since the markup is 6% the new price is 100% + 6% or 106% of the original price. so the new price is 1.06 X7800 = $8268
This year a pair of ice skates sells for $46 after a 15% mark up over last year’s price. What was last year’s price?
This year’s price is 100 + 15 or 115% of last year’s price. Let n present last year’s price
46 = (115/100)n
46 = 1.15n
46/.15 = 1.15n/1.115
40 = n
So last year’s price was $40.
A department store advertised eclectic shavers at a sale price of $36.
If this is a 20% discount, what was the original price?
The sale price is 100 - 20 or 80% of the original price. Let n represent the original price.
36 = (80/100)n
36 = .8n
36/.8 = .8n/.8
45 = n
The original price was $45.
Check to see that your answers are logical and reasonable.
Try these: A service station (that’s gas station, now—they no longer provide service!!) give cash customers a 5% discount on the price of gasoline. If gasoline regularly sells for $3.00 a gallon, what is the discounted price?
A store marks up the price of a $5 item to $12. What is the percent of markup?
Subscribe to:
Posts (Atom)
