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.
Friday, October 28, 2011
Prepare for Quiz on Tuesday 11-1
Given a function with a table of values, write the ordered pairs and draw the mapping.
Given a function with a mapping, write the ordered pairs and write the table.
Given a function with ordered pairs, write the table of values and draw the mapping.
Given the graph of a function or relation, write the table of values, the ordered pairs, or draw the mapping.
Given the graph of a function, write the domain and range using set notation with { }
Know when to use a description or a list in set notation.
Match the terms domain, input, output, range, x, and y.
Explain the difference between a function and a non-function. You may use the boy-girl rule.
Given a function and a domain that is a list, evaluate the function.
Given a function with a mapping, write the ordered pairs and write the table.
Given a function with ordered pairs, write the table of values and draw the mapping.
Given the graph of a function or relation, write the table of values, the ordered pairs, or draw the mapping.
Given the graph of a function, write the domain and range using set notation with { }
Know when to use a description or a list in set notation.
Match the terms domain, input, output, range, x, and y.
Explain the difference between a function and a non-function. You may use the boy-girl rule.
Given a function and a domain that is a list, evaluate the function.
10-28 Class notes
Set notation: If you are asked to describe or list a domain or range, use curly brackets { }
For the example we did in class from page 76, item 3a, the domain is { -5, 0, 4 }
For the example on page 78, item 1a, the domain is { x > -5 and x < 5 }.
The range is { y > -4 and y < 4 }.
There will be a quiz on Tuesday 11-1.
For the example we did in class from page 76, item 3a, the domain is { -5, 0, 4 }
For the example on page 78, item 1a, the domain is { x > -5 and x < 5 }.
The range is { y > -4 and y < 4 }.
There will be a quiz on Tuesday 11-1.
Wednesday, October 26, 2011
Class notes 10-26
To illustrate the difference between a function and a non-function, we used the example of a rule for boys asking girls to go out.
A boy can only ask ("------>" symbol in a mapping diagram) one girl to go out. That's a function.
It's OK for one girl to be asked out by more than one boy.
If a boy asks ("------>" symbol in a mapping diagram) more than one girl to go out, that's not a function.
It's not OK for two different girls to be asked out by the same boy.
Homework for Friday 10-28 - because of the trip to see the play - is page 72, item 15, all four parts.
You can remember the proper sequence of pairs of terms by looking at the alphabeticl order of the first letters.
Domain first, Range second. D comes before R in the alphabet
x first, y second. X comes before Y in the alphabet
Input first, Output second. I comes before O in the alphabet
A boy can only ask ("------>" symbol in a mapping diagram) one girl to go out. That's a function.
It's OK for one girl to be asked out by more than one boy.
If a boy asks ("------>" symbol in a mapping diagram) more than one girl to go out, that's not a function.
It's not OK for two different girls to be asked out by the same boy.
Homework for Friday 10-28 - because of the trip to see the play - is page 72, item 15, all four parts.
You can remember the proper sequence of pairs of terms by looking at the alphabeticl order of the first letters.
Domain first, Range second. D comes before R in the alphabet
x first, y second. X comes before Y in the alphabet
Input first, Output second. I comes before O in the alphabet
Tuesday, October 18, 2011
10-14 Absolute value
To evaluate a number inside absolute value bars, such as | -4 |,
If what's inside the bars is positive, you take away the bars and the sign stays positive.
If what's inside the bars is negative, you take away the bars and the sign is changed to positive.
In this case, what's inside the bars is negative, so you change it to positive 4 when you remove the bars.
| -4 | = 4
If you don't know the sign of what's inside the bars because there's a variable (letter) you don't know,
You have to consider the two cases where what's inside could be positive and what's inside could be negative, depending on the value of the variable.
| x | = x, positive x, if x is positive
| x | = -x, the opposite of x, if x is negative
| x + 3 | = x + 3, if x > -3
| x + 3 | = -x - 3, the opposite, if x < -3
This means that when you are solving absolute value equations or inequalities, once you remove the bars you will have to work on two parts of the problem. The final answer will be the joining or junction of two graphs. It could be two dots, two split arrows, or the overlapping tails of two arrows.
If what's inside the bars is positive, you take away the bars and the sign stays positive.
If what's inside the bars is negative, you take away the bars and the sign is changed to positive.
In this case, what's inside the bars is negative, so you change it to positive 4 when you remove the bars.
| -4 | = 4
If you don't know the sign of what's inside the bars because there's a variable (letter) you don't know,
You have to consider the two cases where what's inside could be positive and what's inside could be negative, depending on the value of the variable.
| x | = x, positive x, if x is positive
| x | = -x, the opposite of x, if x is negative
| x + 3 | = x + 3, if x > -3
| x + 3 | = -x - 3, the opposite, if x < -3
This means that when you are solving absolute value equations or inequalities, once you remove the bars you will have to work on two parts of the problem. The final answer will be the joining or junction of two graphs. It could be two dots, two split arrows, or the overlapping tails of two arrows.
10-18 Graping an absolute value inequality
Homework for Wednesday and Thursday is to do review problems, page 59, 60, 61;
Items 7 through 49 odds. Do the work on a separate sheet with your name on it, and hand it in Thursday.
Today bell-ringer and inclass: Solve and graph |x-3| < 2
When a problem has absolute value bars | |, you need to remove them and follow the rules for solving two parts of the problem.
Remove the absolute value bars and write what's left twice:
Items 7 through 49 odds. Do the work on a separate sheet with your name on it, and hand it in Thursday.
Today bell-ringer and inclass: Solve and graph |x-3| < 2
When a problem has absolute value bars | |, you need to remove them and follow the rules for solving two parts of the problem.
Remove the absolute value bars and write what's left twice:
x-3 < 2 x-3 < 2
One of these is the original and you leave it alone. For the other, introduce a negative sign (to create the opposite) AND change the direction of the inequality:
(original) x-3 < 2 (opposite) x-3 > -2
add 3 to each side add 3 to each side
x < 5 x > 1
position is 5 position is 1
dot is closed dot is closed
thick arrow points left thick arrow points right
Combine the graphs into one segment from 1 to 5 where the arrow tails overlap.
Wednesday, October 12, 2011
10-11 Compound inequalities - class notes
A compound inequalitiy is two inequality statements joined by the words "and" or "or".
Refer to the notes on page 47, item 20
x > -1 and x < 4
x < 3 or x > 6
The numbers that satisfy the compound inequality and the graph of the compound inequality are
the junction of two sets.
Conjunction goes with the word "and". It means the numbers must satisfy both inequalities. The graph is where the tails of the two graph arrows overlap. The graph looks like a segment. The graph could have open or closed dots at its ends.
Disjunction goes with the word "or". It means the numbers only have to satisfy one of the inequalities. The graph is two arrows pointing in opposite directions. The graph has a gap between the arrows. The arrows are split.
Homework for 10-12: page 48, items a, b, c, d, e.
Hint: Some exercises are counted only with whole numbers. Use the idea of "between" to form the graphs.
Refer to the notes on page 47, item 20
x > -1 and x < 4
x < 3 or x > 6
The numbers that satisfy the compound inequality and the graph of the compound inequality are
the junction of two sets.
Conjunction goes with the word "and". It means the numbers must satisfy both inequalities. The graph is where the tails of the two graph arrows overlap. The graph looks like a segment. The graph could have open or closed dots at its ends.
x > -1 and x < 4
Disjunction goes with the word "or". It means the numbers only have to satisfy one of the inequalities. The graph is two arrows pointing in opposite directions. The graph has a gap between the arrows. The arrows are split.
x < 3 or x > 6
Homework for 10-12: page 48, items a, b, c, d, e.
Hint: Some exercises are counted only with whole numbers. Use the idea of "between" to form the graphs.
Thursday, October 6, 2011
10-6 Class notes
To solve an inequality, simplifiy using the properties of equality (using the same operation on both sides of the inequality sign) until you get the variable by itself.
2x < 12, then divide both sides by 2 to get
x < 6
This gives us three things to graph: A position (6), a dot (open because there's no equality line under the "less than" symbol), and a direction (to the left, the same way that the "less than" sign points).
When you are solving an inequality and you multiply or divide by a negative number, the negative sign causes the position and the direction to become opposites. Multiplying or dividing by the negative sign "flips" the sign of the number and it also "flips" the direction of the thick bar and arrow in the graph.
-2x < 12, then divide both sides by -2 to get
x > 6, having changed the direction of the inequality from pointing left to pointing right.
2x < 12, then divide both sides by 2 to get
x < 6
This gives us three things to graph: A position (6), a dot (open because there's no equality line under the "less than" symbol), and a direction (to the left, the same way that the "less than" sign points).
When you are solving an inequality and you multiply or divide by a negative number, the negative sign causes the position and the direction to become opposites. Multiplying or dividing by the negative sign "flips" the sign of the number and it also "flips" the direction of the thick bar and arrow in the graph.
-2x < 12, then divide both sides by -2 to get
x > 6, having changed the direction of the inequality from pointing left to pointing right.
Wednesday, October 5, 2011
10-5 Homework quiz
Wednesday 10-5 homework quiz
p37 11a, 11b, 12b
p38 1, 5
p44 a, b
Homework for Thursday 10-7
page 46; items a, b, c
p37 11a, 11b, 12b
p38 1, 5
p44 a, b
Homework for Thursday 10-7
page 46; items a, b, c
Friday, September 30, 2011
9-30 Class notes
Quiz on Monday, 10-3. Some problems open notes; some problems from homework.
Homework for Monday, 10-3: page 43; 9a, 9b and page 44; a, b, c.
Steps for graphing a one-variable inequality:
Simplify until you get the variable isolated on one side and a value isolated on the other side.
The value is the position on the number line.
Place an open dot at that position if there was no equal sign combined with the > or < symbol.
Place a closed dot if the inequality was > or < .
Shade a bar in the direction of the > (right) or < (left).
Position, type of dot, direction.
Homework for Monday, 10-3: page 43; 9a, 9b and page 44; a, b, c.
Steps for graphing a one-variable inequality:
Simplify until you get the variable isolated on one side and a value isolated on the other side.
The value is the position on the number line.
Place an open dot at that position if there was no equal sign combined with the > or < symbol.
Place a closed dot if the inequality was > or < .
Shade a bar in the direction of the > (right) or < (left).
Position, type of dot, direction.
Tuesday, September 27, 2011
9-27 Solving multi-step equations
Answers to the homework from the weekend, page 37:
11a -11/2
11b 10/8
11c 9/10
11d 5
12a z = 0
In-class problems and homework for 9-27 and 9-28:
page 37; items 12b, 12c
page 38; items 1, 2, 3, 4, 5, 6, 7, 8
11a -11/2
11b 10/8
11c 9/10
11d 5
12a z = 0
In-class problems and homework for 9-27 and 9-28:
page 37; items 12b, 12c
page 38; items 1, 2, 3, 4, 5, 6, 7, 8
Friday, September 23, 2011
9-23 Multi-step equations
There will be a homework quiz on Monday. You may use your textbook and your notes to answer the questions. The questions will be taken from the class notes, the homework, and the blog.
Answers for today's homework:
10a. x = -(11/15)
10b. x = 8
10c. no solution
Homework for Monday 9-26: page 37; 11a, 11b, 11c, 11d, 12a
Hint for 12c: Try to get a common denominator or try to clear the denominators (get rid of fractions)
After you simplify an equation, you may get to a statement that doesn't make sense. If you have done the proper undoing, balancing, and operating equally on both sides, and you get a statement that is false,
such as 0 = 5, it means there is no solution to the equation. There is nothing you can put in for x that will make the equation true.
If you simplify an equation and end up with a statement that is always true regardless of x, such as
5 = 5 or 2x = 2x, means the equation has infinite solutions. No matter what you put in for x, the statement will be true.
Answers for today's homework:
10a. x = -(11/15)
10b. x = 8
10c. no solution
Homework for Monday 9-26: page 37; 11a, 11b, 11c, 11d, 12a
Hint for 12c: Try to get a common denominator or try to clear the denominators (get rid of fractions)
After you simplify an equation, you may get to a statement that doesn't make sense. If you have done the proper undoing, balancing, and operating equally on both sides, and you get a statement that is false,
such as 0 = 5, it means there is no solution to the equation. There is nothing you can put in for x that will make the equation true.
If you simplify an equation and end up with a statement that is always true regardless of x, such as
5 = 5 or 2x = 2x, means the equation has infinite solutions. No matter what you put in for x, the statement will be true.
Thursday, September 22, 2011
Class notes 9-21 profit, other buttons, field trip
Homework for 9-22 is page 35, item 9, frequent visitor program
Tuesday, September 20, 2011
Breakeven 9-20 class notes
In the button making story, we found that when you order 200 buttons, neither company gives you a better deal. 200 buttons cost $155 whether you get them from PB or from BFY. If you get one more button over 200 from PB, you pay 15 cents more for a total of $155.15. If you get one more button over 200 from BFY, you pay 40 cents more for a total of $155.40. So, for an order greater than 200 buttons, PB gives you an better deal, and that is what you recommend.
In the bellringer we solved
In the bellringer we solved
200x = 12500 +15x
The money you get from selling x buttons at 200 cents each = the cost of buying those buttons from PB.
x = 68
68 is the number of buttons that makes the money coming in equal the money going out.
$136 = $136
68 is the breakeven number of buttons.
If you sell more than the breakeven number, you make money because revenue > cost.
For 80 buttons, 200(80) = $160 and 12500 + 15(80) = $137, so you make 160 - 137 = $23
For 80 buttons, 200(80) = $160 and 12500 + 15(80) = $137, so you make 160 - 137 = $23
If you sell fewer than the breakeven number, you lose money because cost > revenue.
For 40 buttons, 200(40) = $80 and 12500 + 15(40) = $131, so you make 90 - 131 = negative $41 (loss)
Profit is the money you make, calculated by subtracting cost from revenue.
Homework for 9-21 is page 34, items 6 and 7.
Friday, September 16, 2011
Homework for 9-19 page 32 items 1, 2, 3, 4, 5, 6
Homework for 9-19 page 32 items 1, 2, 3, 4, 5, 6
Use any method, show your work. You solved some of these problems in class.
Use any method, show your work. You solved some of these problems in class.
Thursday, September 15, 2011
Homework and class notes
Homework for Friday 9-16: From page 31, items b, c, d, e, f, and page 32, items a, b, c, d, e, f, choose three equations to solve. Be prepared to show your work and explain your answer in class.
Today we worked an example of solving an equation with pictures of square and rectangle shapes ("tiles"). Pairs of positive and negative tiles were erased to simplify the picture. In our example, one rectangle (t) equals seven squares (7).
Today students presented their small dry-erase solutions to equations.
Today we worked an example of solving an equation with pictures of square and rectangle shapes ("tiles"). Pairs of positive and negative tiles were erased to simplify the picture. In our example, one rectangle (t) equals seven squares (7).
Today students presented their small dry-erase solutions to equations.
Wednesday, September 14, 2011
Class notes 9-14 "UNDOing" to solve equations
Undoing an equation means using inverse operations to get back to the variable (mystery number, Who Am I?)
Look at the way an equation was built. The last operation that built the equation is the first operation to be undone. In most cases this means using PEMDAS in the other direction. We use SADMRP to remember this.
Work from outer to inner, towards the grouping symbols.
First, look for Subtractions and Additions to undo. Use addition and subtraction, the inverses.
Next, look for Divisions and Multiplications to undo. Use multiplication and division, the inverses.
Next, look for Exponents to undo. We use "R" in SADMRP because the inverse of exponentiation is Root (taking the nth root, finding the nth root).
P means it's time to work inside the Parentheses (grouping symbols).
Homework is page 29, items b, d, f.
Be prepared to talk about the steps of UNDOing, similar to the right-to-left arrows and ovals on page 29.
Look at the way an equation was built. The last operation that built the equation is the first operation to be undone. In most cases this means using PEMDAS in the other direction. We use SADMRP to remember this.
Work from outer to inner, towards the grouping symbols.
First, look for Subtractions and Additions to undo. Use addition and subtraction, the inverses.
Next, look for Divisions and Multiplications to undo. Use multiplication and division, the inverses.
Next, look for Exponents to undo. We use "R" in SADMRP because the inverse of exponentiation is Root (taking the nth root, finding the nth root).
P means it's time to work inside the Parentheses (grouping symbols).
Homework is page 29, items b, d, f.
Be prepared to talk about the steps of UNDOing, similar to the right-to-left arrows and ovals on page 29.
Tuesday, September 13, 2011
Class notes 9-13
Building an equation from an unknown number ("me", "what number am I?") uses the words closest to the number to suggest the first operations and grouping. Each word phrase adds another operation or grouping. Finally, you translate "am I" or "I am" into an equal sign.
Solving an equation for an unknown number looks at the operation farthest away from the number so you can undo it. Remember that when you undo something, such as when typing at a computer, you affect the last thing you did. When you undo in math, you affect the last operation used to build the equation.
To undo a math operation, you use its inverse. Multiplication and division are inverses. Addition and subtraction are inverses. If the equation's last piece was adding 3 (" + 3 ") to undo it so you can make the equation simpler you should subtract 3 (" - 3") from both sides.
To evaluate an expression or to build an equation, you use the order of operations PEMDAS.
To solve an equation, you use the order of undoing operations SADMRP.
We use R instead of E because Root is the inverse of Exponent. If raising to a power/raising to an exponent was used to build an equation, tkaing the root/finding the root is used to undo the equation
Solving an equation for an unknown number looks at the operation farthest away from the number so you can undo it. Remember that when you undo something, such as when typing at a computer, you affect the last thing you did. When you undo in math, you affect the last operation used to build the equation.
To undo a math operation, you use its inverse. Multiplication and division are inverses. Addition and subtraction are inverses. If the equation's last piece was adding 3 (" + 3 ") to undo it so you can make the equation simpler you should subtract 3 (" - 3") from both sides.
To evaluate an expression or to build an equation, you use the order of operations PEMDAS.
To solve an equation, you use the order of undoing operations SADMRP.
We use R instead of E because Root is the inverse of Exponent. If raising to a power/raising to an exponent was used to build an equation, tkaing the root/finding the root is used to undo the equation
Prepare for a quiz on 9-14
Wednesday's quiz will be on multiple representations of data. We have studied tables, graphs, and expressions as ways of representing information. You will demonstrate your ability to create different representations from an initial set of numbers, and you will be able to convert from table, to graph, to an expression and vice versa.
You will be able to see the pattern in a table of numbers.
You will be able to write an expression for a pattern of numbers.
You will be able to draw a graph from a table.
You will be able to make a table from a graph.
You will be able to write the linear equation of a straight graph.
You will be able to draw a graph from a linear equation.
You will be able to discuss differences between shapes of graphs.
You will be able to see the pattern in a table of numbers.
You will be able to write an expression for a pattern of numbers.
You will be able to draw a graph from a table.
You will be able to make a table from a graph.
You will be able to write the linear equation of a straight graph.
You will be able to draw a graph from a linear equation.
You will be able to discuss differences between shapes of graphs.
Friday, September 9, 2011
After the 9-8 quiz
Given the number 23, write an equation which relates it to its reciprocal and the identity element for multiplication.
The number is 23, the reciprocal is 1/23, the identity element for multiplication is 1, because 23 * 1 = 23, the identical number.
.
23 times 1/23 is 1
23 * (1/23) = 1
1/23 is the reciprocal or multiplicative inverse of 23
Given the number 23, write an equation which relates it to its opposite and the identity element for addition.
The number is 23, the opposite is -23, the identity element for addition is 0, because 23 + 0 = 23, the identical number.
23 added to -23 is zero
23 + (-23) = 0
-23 is the opposite or additive inverse of 23
The number is 23, the reciprocal is 1/23, the identity element for multiplication is 1, because 23 * 1 = 23, the identical number.
.
23 times 1/23 is 1
23 * (1/23) = 1
1/23 is the reciprocal or multiplicative inverse of 23
Given the number 23, write an equation which relates it to its opposite and the identity element for addition.
The number is 23, the opposite is -23, the identity element for addition is 0, because 23 + 0 = 23, the identical number.
23 added to -23 is zero
23 + (-23) = 0
-23 is the opposite or additive inverse of 23
Wednesday, September 7, 2011
Algebra Properties 9-7-2011
Transitive - If a = b and b = c, then a = c
If two things are equal to a middle thing, they are equal to each other.
Substitute a for b into b = c. This gives a = c.
Reflexive - A thing is equal to itself
a = a
Symmetric - You can write an equality in either order
if a = b then b = a
Identity elements -
0 is what you add to a number to get the identical number
1 is what you multiply by a number to get the identical number
To get from a number to the identity element for addition, add the opposite of the number
To get from a number to zero, add the negative of the number
To get from 7 to 0, add -7
7 + (-7) = 0
To get from a number to the identity element for multiplication, multiply by the reciprocal of the number
To get from a number to 1, multiply by the reciprocal of the number
To get from 7 to 1, multiply by 1/7
7 * (1/7) = 1
If two things are equal to a middle thing, they are equal to each other.
Substitute a for b into b = c. This gives a = c.
Reflexive - A thing is equal to itself
a = a
Symmetric - You can write an equality in either order
if a = b then b = a
Identity elements -
0 is what you add to a number to get the identical number
1 is what you multiply by a number to get the identical number
To get from a number to the identity element for addition, add the opposite of the number
To get from a number to zero, add the negative of the number
To get from 7 to 0, add -7
7 + (-7) = 0
To get from a number to the identity element for multiplication, multiply by the reciprocal of the number
To get from a number to 1, multiply by the reciprocal of the number
To get from 7 to 1, multiply by 1/7
7 * (1/7) = 1
Study hints for Math 9 Quiz 9-8
Math 9 Quiz Thursday 9-8 - Study hints
Given some numbers, tell which numbers are in a category.
Which numbers are irrational?
2 pi 2.22 2/4 square root of 2
Which numbers are rational?
2 pi 2.22 2/4 square root of 2
Which numbers are whole?
2 -2 2/4 -4/2 6/2
Be able to define real numbers with words or with a diagram
.
Be able to place numbers correctly into an N-W-I-R / Irrational and Real diagram
Be able to give an example of an algebra property (Commutative, Associative, Distributive, Reflexive, Symmetric, Transitive)
Be able to place parentheses properly into an Associative equation.
From a list of examples, be able to identify an algebra property (Commutative, Associative, Distributive, Reflexive, Symmetric, Transitive)
Explain the difference between the identity element for addition and the identity element for multiplication.
Given a number, write its opposite and its reciprocal.
Identify or give an example of the Reflexive, Symmetric, and Transitive Properties of Equality
Use Substitution to explain the Transitive Property.
Tuesday, September 6, 2011
Inversesand identity elements
Identity means "that which stays the same". If you move, or grow, many things about you stay the same even though you address or your height may change. Those things stay identical.
If you get something that's identical, you get something that's the same. An extra picture can be identical to a first picture.
In math an identity element is a number that, when used with a stated operation, gets you the same thing.
Zero is the identity element for addition because when you add 0 to any number, you get the same thing:
97 + 0 = 97 and 2z + 0 = 2z
One is the identity element for multiplication becasue when you multiply any number by 1, you get the same thing: 86 times 1 = 86 and 7b times 1 = 7b
An additive inverse, when added to a first number, gets you 0, the identity element for addition
4 plus -4 = 0, so -4 is the additive inverse of 4
A multiplicative inverse, when multiplied by a first number, gets you 1, the identity element for multiplication
4 times 1/4 = 1, so 1/4 is the multiplicative inverse of 4
An additive inverse is a also an opposite, since you change sign to get it
4 becomes its additive inverse, or opposite, by changing sign to -4
A multiplicative inverse is also a reciprocal,since you flip a numerator and a denominator to get it
4 becomes its multiplicative inverse, or reciprocal by flipping from 4/1 to 1/4
If you get something that's identical, you get something that's the same. An extra picture can be identical to a first picture.
In math an identity element is a number that, when used with a stated operation, gets you the same thing.
Zero is the identity element for addition because when you add 0 to any number, you get the same thing:
97 + 0 = 97 and 2z + 0 = 2z
One is the identity element for multiplication becasue when you multiply any number by 1, you get the same thing: 86 times 1 = 86 and 7b times 1 = 7b
An additive inverse, when added to a first number, gets you 0, the identity element for addition
4 plus -4 = 0, so -4 is the additive inverse of 4
A multiplicative inverse, when multiplied by a first number, gets you 1, the identity element for multiplication
4 times 1/4 = 1, so 1/4 is the multiplicative inverse of 4
An additive inverse is a also an opposite, since you change sign to get it
4 becomes its additive inverse, or opposite, by changing sign to -4
A multiplicative inverse is also a reciprocal,since you flip a numerator and a denominator to get it
4 becomes its multiplicative inverse, or reciprocal by flipping from 4/1 to 1/4
Commutative, Associative and Distributive Properties 9-1
Commutative, Associative and Distributive Properties for addition and multiplication.
(not for subtraction and division)
a + b = b + a
a times b = b times a
When you add or multiply two things, you can do them in either order.
(a + b) + c = a + (b + c)
(a x b) x c = a x (b x c)
When you add or multiply three things, you can start with the first two or the last two.
a times ( b + c ) = a times b + a times c
a x ( b + c ) = a x b + a x c
When you multiply a sum, the multiplication takes place with each of the things in the sum.
Homework for 9-6: page 20-21; items 7, 8, 9, 10
(not for subtraction and division)
a + b = b + a
a times b = b times a
When you add or multiply two things, you can do them in either order.
(a + b) + c = a + (b + c)
(a x b) x c = a x (b x c)
When you add or multiply three things, you can start with the first two or the last two.
a times ( b + c ) = a times b + a times c
a x ( b + c ) = a x b + a x c
When you multiply a sum, the multiplication takes place with each of the things in the sum.
Homework for 9-6: page 20-21; items 7, 8, 9, 10
Wednesday, August 31, 2011
Homework for 9-1 page 18, items 1 through 9 on the bottom
HW for 9-1 page 18, items 1 through 9 on the bottom
8-31 Math 9 Cornell notes – Rationals, Irrationals, and Reals
8-31 Math 9 Cornell notes – Rationals, Irrationals, and Reals
Today each table studied statements such as pi = 3.14 and 1/3 = .3
Then they gave presentations on which statements were true. Some of the statements had the dotted equal sign which means “approximately equal to” or “almost equal to.”
(drawing of N, W, I circles inside the Rational rectangle) | The decimal equivalents of rational numbers - Can terminate (end), such as ½ = .5 and 3/8 = .375 - Can repeat, such as 2/3 = .66666 . . . and 1/13 = .076923076923. . . - The “. . .” symbol is an ellipsis. It means the idea keeps going. Some square roots, such as those of perfect squares, are rational, such a SQRT (36) = 6 |
(add the Irrational rectangle to the N, W, I, R drawing) | The decimal equivalents of irrational numbers - Never end - Never repeat For example, pi = 3.141592654 . . . And SQRT(2) = 1.414213562. . . Example: SQRT (3) = 1.732050808. . . Example: e = 2.71828. . . |
(drawing of the large Real rectangle which includes the Rational and Irrational rectangles) | Rational numbers combined with Irrational numbers are the Set of Real numbers. None of the N, W, or I numbers is in the Irrational rectangle |
Write an Irrational number | Take all the odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, . . . Strike out the perfect squares: 3, 5, 7, , 11, 13, 15, 17, 19, 21 23, , 27. . . Take the square root of any of the remaining numbers. Example: SQRT (23) = 4.795831523. . . is irrational. Example: SQRT (7) = 2.645751311. . . is irrational. |
Tuesday, August 30, 2011
hw for 8-31, page 14 and 15, items 6, 7, 8, 9, 10
hw for 8-31, page 14 and 15, items 6, 7, 8, 9, 10
In the NWIR diagram we did in class, none of the fences crossed. Some Venn diagrams have fences that cross, this makes it possible for some things to be inside two different fences or categories at the same time.
In the NWIR diagram we did in class, none of the fences crossed. Some Venn diagrams have fences that cross, this makes it possible for some things to be inside two different fences or categories at the same time.
Homework due 8-30, page12, items 1, 2, 3, 4, 5
Something like item 3 will appear on a future homework quiz, where you have to look at a pattern to put the proper label at the top of the column.
heading for 1st column: Level (follow the example of the table on page 4)
heading for 2nd column: Number of black squares
heading for 3rd column: Perimeter
heading for 1st column: Level (follow the example of the table on page 4)
heading for 2nd column: Number of black squares
heading for 3rd column: Perimeter
Math Skills 9 Estimation
When you are estimating with mixed fractions, how do you tell whether to round down or to round up? What's the difference between a low fraction and a high fraction?
Consider the numbers (7 and 2/11) and (7 and 7/11). Here are the simple fractions that have 11 in the denominator:
1/11, 2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 9/11, 10/11.
Divide this set into halves, a lower half and a higher half:
low: 1/11, 2/11, 3/11, 4/11, 5/11
If one of these is the fraction part of the number, drop the fraction and round down to 7.
high: 6/11, 7/11, 8/11, 9/11, 10/11
If one of these is the fraction part of the number, round up to 8.
Consider the numbers (7 and 2/11) and (7 and 7/11). Here are the simple fractions that have 11 in the denominator:
1/11, 2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 9/11, 10/11.
Divide this set into halves, a lower half and a higher half:
low: 1/11, 2/11, 3/11, 4/11, 5/11
If one of these is the fraction part of the number, drop the fraction and round down to 7.
high: 6/11, 7/11, 8/11, 9/11, 10/11
If one of these is the fraction part of the number, round up to 8.
Monday, August 29, 2011
Math 9 homework for 8-30
We all need to make an adjustment to the Springboard textbook. In conventional textbooks, the homework problems are found gathered together on one or two pages after a section. In our Springboard book, homework problems are not found in a separate section. Homework problems can be assigned from the activity items in the text. It is important that you write down both the page and item numbers of homework assignments, and that you not get the two confused.
Homework for 8-30: page 12; items 1, 2, 3, 4, 5, 6
Hint: Put labels into the blank spaces at the tops of the columns.
Hint for item 5: The graph shown does not start at (0, 0).
Notes on the hw for today, 8-29: page 11, items 17a, 17b, 18
17a. Graph b reflects the intended design model for earning levels. The upward curve shows that you don't just get "more" for getting to the next level, it shows you get "more on top of more", or an increasing, instead of constant, reward.
17b. From the table on page 4, the columns "# of squares added" and "perimeter" look like 17a, straight graph a. The column "total area" looks like the curved graph b in 17a.
18. When playing a game, when you beat a level you are rewarded for recognizing a pattern, To beat the next level, you have to recognize a more difficult pattern. In a math problem, you may have two or more steps which are like levels. You may have to recognize ("beat") increasingly more difficult patterns, from the expression level to the equation level, to the system of equations level, to the feasible/practical level, in order to solve the problem.
Homework for 8-30: page 12; items 1, 2, 3, 4, 5, 6
Hint: Put labels into the blank spaces at the tops of the columns.
Hint for item 5: The graph shown does not start at (0, 0).
Notes on the hw for today, 8-29: page 11, items 17a, 17b, 18
17a. Graph b reflects the intended design model for earning levels. The upward curve shows that you don't just get "more" for getting to the next level, it shows you get "more on top of more", or an increasing, instead of constant, reward.
17b. From the table on page 4, the columns "# of squares added" and "perimeter" look like 17a, straight graph a. The column "total area" looks like the curved graph b in 17a.
18. When playing a game, when you beat a level you are rewarded for recognizing a pattern, To beat the next level, you have to recognize a more difficult pattern. In a math problem, you may have two or more steps which are like levels. You may have to recognize ("beat") increasingly more difficult patterns, from the expression level to the equation level, to the system of equations level, to the feasible/practical level, in order to solve the problem.
Friday, August 26, 2011
Math 9 assignment for Monday 8-29; notes from SBT pages 6-11
Homework for 8-29: page 11, items 17 and 18
9b. total area is L^2 which means L squared
10a. the graph is a line with a slope of 2
10b. the graph a line with a slope of 4, and you run out of room at the top
10c. the graph is not a straight line
13., 14., 15., 16:
a square is missing a smaller square
area of this level (the current level) is L squared
area of the previous level uses the L from the previous level
in math, "previous" means minus 1
if I am talking about level 42, the previous level is 42 minus 1 = level 41
if I am talking about level q, the previous level is q minus 1 = q - 1
area of the previous level is ( L minus 1 ) squared = ( L - 1 )^2
this level's area minus previous level's area = L^2 minus (L - 1)^2
the number of squares addes = L^2 - (L - 1)^2
9b. total area is L^2 which means L squared
10a. the graph is a line with a slope of 2
10b. the graph a line with a slope of 4, and you run out of room at the top
10c. the graph is not a straight line
13., 14., 15., 16:
a square is missing a smaller square
area of this level (the current level) is L squared
area of the previous level uses the L from the previous level
in math, "previous" means minus 1
if I am talking about level 42, the previous level is 42 minus 1 = level 41
if I am talking about level q, the previous level is q minus 1 = q - 1
area of the previous level is ( L minus 1 ) squared = ( L - 1 )^2
this level's area minus previous level's area = L^2 minus (L - 1)^2
the number of squares addes = L^2 - (L - 1)^2
Thursday, August 25, 2011
Math 9 homework assignment for 8-26
Springboard textbook, page 7, item 10, pick one of the three graphing activities a., b., or c. and plot the points required. You will need a completed table from page 4, item 1.
Reflect on our discussion of why we use a blog:
To save _________ .
To actually use _______________ in the classroom and at home.
To build a study __________ that we can share.
So the teacher can include things we talked about that day in class for our ___________ ________ .
To model the way studying and note-taking will be in _____________ .
Reflect on our discussion of why we use a blog:
To save _________ .
To actually use _______________ in the classroom and at home.
To build a study __________ that we can share.
So the teacher can include things we talked about that day in class for our ___________ ________ .
To model the way studying and note-taking will be in _____________ .
Word wall 8-25
educated guess - attempting to find an answer by applying additional knowledge, mathematics, and research.; better than a blind guess or a flip the coin guess.
expense - the cost of doing something or getting something.
extensive - done with more detail, more preparation, more resources, or better tools. Remodeling a kitchen is more extensive work than fixing a faucet.
hypothesis - an educated guess that is designed to be tested.
incidentals - small expenses that are unexpected or that are not planned for, such as bus fare, snacks, tips.
linear programming - the "programming" is a rigorous procedure for mathematically allocating resources; the "linear" refers to modeling processes and relationships with equations of the form y = mx + b or Ax + By = C
predict - to make an educated guess about a quantity or a behavior as you extend a table of data or extend a time frame.
expense - the cost of doing something or getting something.
extensive - done with more detail, more preparation, more resources, or better tools. Remodeling a kitchen is more extensive work than fixing a faucet.
hypothesis - an educated guess that is designed to be tested.
incidentals - small expenses that are unexpected or that are not planned for, such as bus fare, snacks, tips.
linear programming - the "programming" is a rigorous procedure for mathematically allocating resources; the "linear" refers to modeling processes and relationships with equations of the form y = mx + b or Ax + By = C
predict - to make an educated guess about a quantity or a behavior as you extend a table of data or extend a time frame.
Math Skills fraction and decimal equivalents
Recognition of common fraction and decimal equivalents
½ = .5 | 2/2 = 1.00 |
1/3 = .333 . . . repeats | 2/3 = .666 . . . repeats | 3/3 = 1.00 |
¼ = .25 | 2/4 = .5 | ¾ = .75 | 4/4 = 1.00 |
1/5 = .2 | 2/5 = .4 | 3/5 = .6 | 4/5 = .8 | 5/5 = 1.0 |
1/6 = .166 . . . | 2/6 = .333 . . . | 3/6 = .5 | 4/6 = .666 . . . | 5/6 = .833 . . . | 6/6 = 1.0 |
1/8 = .125 | 2/8 = .25 | 3/8 = .375 | 4/8 = .5 | 5/8 = .625 | 6/8 = .75 | 7/8 = .875 | 8/8 = 1.0 |
.1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 | 1.0 |
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