Wednesday, December 21, 2011
Friday, December 9, 2011
Math Skills 12-6 Compass and clock
Degrees around a compass circle, clockwise from 12 o'clock North
12:00 0 degrees 0/8 north
1:30 45 degrees 1/8 northeast
3:00 90 degrees 1/4 east
4:30 135 degrees 3/8 southeast
6:00 180 degrees 1/2 south
7:30 225 degrees 5/8 southwest
9:00 270 degrees 3/4 west
10:30 315 degrees 7/8 northwest
12:00 360 degrees 8/8 north
Degrees around a clock circle, clockwise from 12 o'clock
12:00 0 degrees 0/8 midnight or noon
1:00 30 degrees 1/12
2:00 60 degrees 1/6
3:00 90 degrees 1/4
4:00 120 degrees 1/3
5:00 150 degrees 5/12
6:00 180 degrees 1/2
7:00 210 degrees 7/12
8:00 240 degrees 2/3
9:00 270 degrees 3/4
10:00 300 degrees 5/6
11:00 330 degrees 11/12
12:00 360 degrees 12/12 noon or midnight
12:00 0 degrees 0/8 north
1:30 45 degrees 1/8 northeast
3:00 90 degrees 1/4 east
4:30 135 degrees 3/8 southeast
6:00 180 degrees 1/2 south
7:30 225 degrees 5/8 southwest
9:00 270 degrees 3/4 west
10:30 315 degrees 7/8 northwest
12:00 360 degrees 8/8 north
Degrees around a clock circle, clockwise from 12 o'clock
12:00 0 degrees 0/8 midnight or noon
1:00 30 degrees 1/12
2:00 60 degrees 1/6
3:00 90 degrees 1/4
4:00 120 degrees 1/3
5:00 150 degrees 5/12
6:00 180 degrees 1/2
7:00 210 degrees 7/12
8:00 240 degrees 2/3
9:00 270 degrees 3/4
10:00 300 degrees 5/6
11:00 330 degrees 11/12
12:00 360 degrees 12/12 noon or midnight
Thursday, December 8, 2011
12-8 Math Skills - Numerical pitch of Billie Jean notes
The bass riff to Billie Jean graphed as frequencies in Hz on the y-axis, and an internal field trip two floors down to investigate the pitches of piano notes.
Wednesday, December 7, 2011
12-2 notes - Point-slope form
Point-slope form is
y - ysub1 = m ( x - xsub1) where x and y are variables, and xsub1 and ysub1 are values from a given ordered pair, from a given point.
After substituting, distribute on the right and move -ysub1 from the left side to the right side. This isolates y on the left and leaves a slope times x term and a y-intercept term on the right.
y - ysub1 = m ( x - xsub1) where x and y are variables, and xsub1 and ysub1 are values from a given ordered pair, from a given point.
After substituting, distribute on the right and move -ysub1 from the left side to the right side. This isolates y on the left and leaves a slope times x term and a y-intercept term on the right.
12-6 Class notes - Standard form of a linear equation
( , ) indicates an ordered pair, contains an x and a y, suggests the shape of a point
( , ), ( , ) indicates two ordered pairs, has the four values needed to find a slope, suggests a line
m indicates a rate, a slope, a measure of slant, a rise over a run; m stands next to the x
b indicates the height above the origin, it's the y-intercept, it gives the point (0, b)
The slope-intercept form y = mx + b has the variable y by itself on one side of the = sign.
The standard form Ax + By = C has the constant C (a number) by itself on one side of the = sign.
A is the coefficient of x (a number that multiplies x)
B is the coefficeint of y (a number that multiplies y)
To change from slope-intercept form to standard form, isolate b, the y-intercept. Move the mx term to the other side of the equal sign.
To change from standard form to slope-intercept form, move the Ax term away from the By term. Divide both sides by B. m, the slope, will be -A/B and b, the y-intercept, will be C/B.
( , ), ( , ) indicates two ordered pairs, has the four values needed to find a slope, suggests a line
m indicates a rate, a slope, a measure of slant, a rise over a run; m stands next to the x
b indicates the height above the origin, it's the y-intercept, it gives the point (0, b)
The slope-intercept form y = mx + b has the variable y by itself on one side of the = sign.
The standard form Ax + By = C has the constant C (a number) by itself on one side of the = sign.
A is the coefficient of x (a number that multiplies x)
B is the coefficeint of y (a number that multiplies y)
To change from slope-intercept form to standard form, isolate b, the y-intercept. Move the mx term to the other side of the equal sign.
To change from standard form to slope-intercept form, move the Ax term away from the By term. Divide both sides by B. m, the slope, will be -A/B and b, the y-intercept, will be C/B.
Wednesday, November 30, 2011
11-29 Finding a slope from a table of values; finding y-intercept from a graph
Finding a slope from a table of values.
From one line to the next,
See the change in y and see the change in x.
Make a ratio.
Finding y-intercept from a graph.
Walk to the origin. Look up or look down for the line that is the graph.
From one line to the next,
See the change in y and see the change in x.
Make a ratio.
Finding y-intercept from a graph.
Walk to the origin. Look up or look down for the line that is the graph.
Tuesday, November 29, 2011
11-29 Math Skills notes
Factoring turns a number or expression into a multiplication problem.
72 = 9 times 4 times 2
6x^2 = 6 times x times x
If a number's digits add up to 9, then 9 is a factor of that number.
If a number ends in 5 or 0, then 5 is a factor of that number.
If a number ends in 0, then 10, 5, and 2 are factors of that number.
If a number is even, then 2 is a factor of that number.
Factoring into pairs results in a multiplication problem with two pieces. There may be more than one pair for an answer.
72 = 9 x 8
72 = 6 x 12
72 = 36 x 2
Factoring completely means factoring the factors until only prime number factors remain.
72 = 6 times 12
factor the 6 and factor the 12
= 2 x 3 times 3 x 4
factor the 4
= 2 x 3 times 3 x ( 2 x 2 )
= 2 x 3 x 3 x 2 x 2
A problem that involves factoring into possible pairs may require a specific pair for the answer because another condition, or "catch", is placed on the relationship between the factors.
Given
72 = 9 x 8
72 = 6 x 12
72 = 36 x 2
an additional condition might be that the difference between factors is 6,
or one factor minus the other factor = 6.
The only pair of factors that make this true is 6 and 12, because 12 - 6 = 6, the catch.
72 = 9 times 4 times 2
6x^2 = 6 times x times x
If a number's digits add up to 9, then 9 is a factor of that number.
If a number ends in 5 or 0, then 5 is a factor of that number.
If a number ends in 0, then 10, 5, and 2 are factors of that number.
If a number is even, then 2 is a factor of that number.
Factoring into pairs results in a multiplication problem with two pieces. There may be more than one pair for an answer.
72 = 9 x 8
72 = 6 x 12
72 = 36 x 2
Factoring completely means factoring the factors until only prime number factors remain.
72 = 6 times 12
factor the 6 and factor the 12
= 2 x 3 times 3 x 4
factor the 4
= 2 x 3 times 3 x ( 2 x 2 )
= 2 x 3 x 3 x 2 x 2
A problem that involves factoring into possible pairs may require a specific pair for the answer because another condition, or "catch", is placed on the relationship between the factors.
Given
72 = 9 x 8
72 = 6 x 12
72 = 36 x 2
an additional condition might be that the difference between factors is 6,
or one factor minus the other factor = 6.
The only pair of factors that make this true is 6 and 12, because 12 - 6 = 6, the catch.
Monday, November 28, 2011
11-28 Class notes - Page 113 Plan A or Plan B for messaging
Plan A function: y = .04x + 4.00
The rate is 4 cents per message, the variable x is the number of messages, the height above the origin is 4 dollars.
Plan B function: y = .05x
The rate is 5 cents per message, the variable x is the number of messages, the height above the origin is zero.
If you plan to send 360 messages, the cost for Plan A is $18.40. (plug 360 into the function)
If you plan to send 360 messages, the cost for Plan B is $18.00.
Choose Plan B because it's cheaper.
If you plan to send 500 messages, the cost for Plan A is $24.00. (plug 500 into the function)
If you plan to send 500 messages, the cost for Plan B is $25.00.
Choose Plan A because it's cheaper.
The domain for both plans is x > 0 and x is a whole number.
The range for Plan A is y > 4.00
The range for Plan B is y > 0
The graph of Plan B shows direct variation because the line passes through (0,0).
The graph of Plan a does not show direct variation.
The rate is 4 cents per message, the variable x is the number of messages, the height above the origin is 4 dollars.
Plan B function: y = .05x
The rate is 5 cents per message, the variable x is the number of messages, the height above the origin is zero.
If you plan to send 360 messages, the cost for Plan A is $18.40. (plug 360 into the function)
If you plan to send 360 messages, the cost for Plan B is $18.00.
Choose Plan B because it's cheaper.
If you plan to send 500 messages, the cost for Plan A is $24.00. (plug 500 into the function)
If you plan to send 500 messages, the cost for Plan B is $25.00.
Choose Plan A because it's cheaper.
The domain for both plans is x > 0 and x is a whole number.
The range for Plan A is y > 4.00
The range for Plan B is y > 0
The graph of Plan B shows direct variation because the line passes through (0,0).
The graph of Plan a does not show direct variation.
Friday, November 18, 2011
11-18 Class notes - What type of variation. Function equation from a table.
If a problem indicates inverse variation, you are looking for a relationship where y = k/x or xy = k.
X appears in the denominator. K is divided by x.
If a problem indicates direct variation, you are looking for a relationship where y = kx.
K is multiplied by x.
If, for the (x,y) ordered pairs of a function, y1 / x1 = y2 / x2 = y3 / x3 = . . . = the same constant k, then k is the coefficient of direct variation. The shape of a direct variation graph is a straight line, and y / x is analogous to the slope, change in y divided by change in x.
If, for the (x,y) ordered pairs of a function, x1 times y1 = x2 times y2 = x3 times y3 = . . . = the same constant k, then k is the coefficient of inverse variation.
X appears in the denominator. K is divided by x.
If a problem indicates direct variation, you are looking for a relationship where y = kx.
K is multiplied by x.
If, for the (x,y) ordered pairs of a function, y1 / x1 = y2 / x2 = y3 / x3 = . . . = the same constant k, then k is the coefficient of direct variation. The shape of a direct variation graph is a straight line, and y / x is analogous to the slope, change in y divided by change in x.
If, for the (x,y) ordered pairs of a function, x1 times y1 = x2 times y2 = x3 times y3 = . . . = the same constant k, then k is the coefficient of inverse variation.
Thursday, November 17, 2011
11-17 Direct and Inverse Variation
A graph indicates direct variation if it is a straight line and it passes through (0,0) the origin.
If the graph does not pass through the origin (0,0) is does not represent direct variation.
In a table of values, look at the relationship between x and y. If you use the same number everytime to multiply x to get a 7, that's direct variation.
In a linear function, if there is no y-intercept, that is direct variation. If there is a y-intercept, that's not direct variation. A linear equation y = mx + b with no y-intercept becomes y = mx which has the same structure as the general equation for direct variation, y = kx.
A graph indicates inverse variation if it is a curved line that approaches but never touches the x- and y- axes.
The general equation for inverse variation is y = k/x.
In direct variation, the constant of variation k multiplies the x.
In inverse variation, the constant of variation k is divided by x.
If the graph does not pass through the origin (0,0) is does not represent direct variation.
In a table of values, look at the relationship between x and y. If you use the same number everytime to multiply x to get a 7, that's direct variation.
In a linear function, if there is no y-intercept, that is direct variation. If there is a y-intercept, that's not direct variation. A linear equation y = mx + b with no y-intercept becomes y = mx which has the same structure as the general equation for direct variation, y = kx.
A graph indicates inverse variation if it is a curved line that approaches but never touches the x- and y- axes.
The general equation for inverse variation is y = k/x.
In direct variation, the constant of variation k multiplies the x.
In inverse variation, the constant of variation k is divided by x.
Wednesday, November 16, 2011
11-15 Class notes Direct Variation
Direct variation describes linear functions whose graphs pass through the origin (0,0).
The constant of variation k in y = kx is the coefficient standing next to the variable, so it is the rate of change in y as x changes. It is also the slope of the graph.
In our box-stacking example, the height h changes by 10cm each time the number of boxes n increases by 1. The constant of variation k is 10. The rate of change is 10cm per box, or 10 cm/box.
Solving for k by isolating k in the equation y = kx (divide both sides by x) gives k = y/x .
If you think of k as a slope, and slope as change in y divided by change in x, that's consistent with k being equal to y divided by x.
The constant of variation k in y = kx is the coefficient standing next to the variable, so it is the rate of change in y as x changes. It is also the slope of the graph.
In our box-stacking example, the height h changes by 10cm each time the number of boxes n increases by 1. The constant of variation k is 10. The rate of change is 10cm per box, or 10 cm/box.
Solving for k by isolating k in the equation y = kx (divide both sides by x) gives k = y/x .
If you think of k as a slope, and slope as change in y divided by change in x, that's consistent with k being equal to y divided by x.
Thursday, November 10, 2011
11-9 Class notes
In the measuring stacks of cups activity, the height of the stack changed by some number of units (for example, one cm or 1/2 cm) every time a cup was added.
When we plotted points to graph our results, the pattern of dots lined up to suggest a slope. Drawing vertical lines to show the change in y (stack height) and horizontal lines to show the change in x (number of cups) created the "up and over" stairsteps that are associated with slope.
The slope as a number tells us about the steepness of the graph.
The slope written as rise units in the numerator and run units in the denominator represents the rate of change.
A rate is a ratio which means how the y units change every time the x unit changes by 1.
In our cup stacking the rate of change is, for example, 0.5 cm per cup, or 0.5 / 1.
Other examples of rate of change:
7 cents per minute of phone usage.
Gasoline costs $3.75 per gallon.
Driving at 60 miles per hour
Counting 16 ounces each time you weigh a pound.
Rates in word problems are indicated by language prepositions: per, for every, each time.
When we plotted points to graph our results, the pattern of dots lined up to suggest a slope. Drawing vertical lines to show the change in y (stack height) and horizontal lines to show the change in x (number of cups) created the "up and over" stairsteps that are associated with slope.
The slope as a number tells us about the steepness of the graph.
The slope written as rise units in the numerator and run units in the denominator represents the rate of change.
A rate is a ratio which means how the y units change every time the x unit changes by 1.
In our cup stacking the rate of change is, for example, 0.5 cm per cup, or 0.5 / 1.
Other examples of rate of change:
7 cents per minute of phone usage.
Gasoline costs $3.75 per gallon.
Driving at 60 miles per hour
Counting 16 ounces each time you weigh a pound.
Rates in word problems are indicated by language prepositions: per, for every, each time.
Tuesday, November 8, 2011
11-8 Slope quiz follow-up
We will continue to review the slope concepts. We need to practice drawing and reading graphs on the coordinate plane (grid paper).
We need to be able to interpret graphs for vertical change and horizontal change.
We need to be able to calculate slope by moving from point A to point B and by counting grid squares.
A horizontal line has a slope of zero. A slope can be positive (uphill) or negative (downhill)
Slopes are a measure of steepness. A wheelchair ramp with a large slope makes it hard to push the chair up. A wheelchair ramp with a shallow slopemay not get you high enough, or it may require a longer or more expensive ramp.
We need to be able to interpret graphs for vertical change and horizontal change.
We need to be able to calculate slope by moving from point A to point B and by counting grid squares.
A horizontal line has a slope of zero. A slope can be positive (uphill) or negative (downhill)
Slopes are a measure of steepness. A wheelchair ramp with a large slope makes it hard to push the chair up. A wheelchair ramp with a shallow slopemay not get you high enough, or it may require a longer or more expensive ramp.
Friday, November 4, 2011
Prepare for Homework quiz on rise and run, 11-7 or 11-8
Know the difference between calculating slope by starting at a left point and moving to a right point, and by starting at a right point and moving to a left point. We covered this in today's (11-4) class notes.
Know how the slope changes as you move along the graph of a straight line.
Write correct mathematical expressions using words or symbols for "change in", "movement in", x, y, "rise",
"run", "vertical", "horizontal", "up", "over".
Today in class we looked at the diagram for the wheelchair ramp. We converted 28 inches to 2.33 ft, and we used the rise of 2.33 ft with the run of 31.9 ft to get a slope of .073.
The table on page 86 shows recommended slopes of 1/16 and 1/12, which are equivalent to .0625 and .0833. Our slope of .073 falls between these values, so we can use our plans to build a ramp.
Know how the slope changes as you move along the graph of a straight line.
Write correct mathematical expressions using words or symbols for "change in", "movement in", x, y, "rise",
"run", "vertical", "horizontal", "up", "over".
Today in class we looked at the diagram for the wheelchair ramp. We converted 28 inches to 2.33 ft, and we used the rise of 2.33 ft with the run of 31.9 ft to get a slope of .073.
The table on page 86 shows recommended slopes of 1/16 and 1/12, which are equivalent to .0625 and .0833. Our slope of .073 falls between these values, so we can use our plans to build a ramp.
Thursday, November 3, 2011
10-31 Class notes - Restricted domains
When listing or describing the domain of a function, see if you have to exclude any x's because of these rules regarding real numbers:
Division by zero is undefined.
The square root of a negative number is not defined.
If an x appears in a denominator or underneath a square root sign, set that expression equal to zero and solve for x to see which x's should be excluded from the domain. Describe the domain as real numbers with some x's excluded or omitted.
When listing or describing the raange of a function, see how far the graph goes along the y-axis. For example, functions containing an x-squared term have a curved part that changes direction so that some y's never appear on the graph. Describe the range as an inquality in y.
Division by zero is undefined.
The square root of a negative number is not defined.
If an x appears in a denominator or underneath a square root sign, set that expression equal to zero and solve for x to see which x's should be excluded from the domain. Describe the domain as real numbers with some x's excluded or omitted.
When listing or describing the raange of a function, see how far the graph goes along the y-axis. For example, functions containing an x-squared term have a curved part that changes direction so that some y's never appear on the graph. Describe the range as an inquality in y.
11-3 Class notes
Homework for Friday 10-4: page 86, items 10, 11, 12, 13, 14
Today in class we investigated the changes in x (left and right movement) and the changes in y (up and down movement) as we trace along the graph of a straight line. We did the exercises on pages 84 and 85.
Slope is vertical change divided by horizontal change.
Slope is change in y divided by change in x.
Slope is rise divided by run.
Slope is "up" divided by "over.
Slope is a ratio that can be an integer, a fraction or a decimal.
The slope of a straight line is the same everywhere along the line.
Slope is a measure of the steepness, the uphillness or downhillness, of a line.
Since a slope is a ratio, its units are the numerator units divided by the denominator units.
Slope is the ratio of output to a given input.
Today in class we investigated the changes in x (left and right movement) and the changes in y (up and down movement) as we trace along the graph of a straight line. We did the exercises on pages 84 and 85.
Slope is vertical change divided by horizontal change.
Slope is change in y divided by change in x.
Slope is rise divided by run.
Slope is "up" divided by "over.
Slope is a ratio that can be an integer, a fraction or a decimal.
The slope of a straight line is the same everywhere along the line.
Slope is a measure of the steepness, the uphillness or downhillness, of a line.
Since a slope is a ratio, its units are the numerator units divided by the denominator units.
Slope is the ratio of output to a given input.
Monday, October 31, 2011
10-31 Vertical line test
The graph of a function has no place where a vertical line intersects (touches) the graph more than once.
If a vertical line intersects (touches) the graph of a relation more than once, the graph fails the vertical line test and it is not a function.
If there is no place on the graph of a relation where a vertical line intersects (touches) the graph more than once, the graph passes the vertical line test and it is a function.
If a vertical line intersects (touches) the graph of a relation more than once, the graph fails the vertical line test and it is not a function.
If there is no place on the graph of a relation where a vertical line intersects (touches) the graph more than once, the graph passes the vertical line test and it is a function.
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