Things you should know in my math class...
C.C. 1 Rubric
When you substitute for a variable...put it in parenthesis. 5g + 2 when g= 3 5(3) + 2
Two thing right next to each other means...MULTIPLY. 7(4) is the same as 7 x 4 or 28
Two things on top of each other means...DIVIDE. so 16/2 is the same as 16 divided by 2 or 8
Squared is to the 2nd power. Cubed is to the 3rd power.
In 7 to 4th power (7 is the BASE and 4 is the EXPONENT) there is NO 4 in the problem...it is all 7s. 7 to the 4th power is the math problem 7 x 7 x 7 x 7.
EVERY problem should go through the ORDER OF OPERATIONS.
1st Parenthesis/Grouping Symbols
2nd Exponents
3rd Multiplication and Division AT THE SAME TIME LEFT TO RIGHT
4th Addition and Subtraction AT THE SAME TIME LEFT TO RIGHT
How to convert between Fraction - Decimal - Percent (See the skills sheet below)
C.C. 2 Rubric
1. To Add (bring decimal straight down into your answer), Subtract (bring decimal straight down into your
answer), and Compare (Compare from the left) DECIMALS you LINE UP THE DECIMALS.
2. To Multiply Decimals (no need to line up the decimals) you simply multiply the numbers...then count the number of digits to the RIGHT of the decimal in the problem...then move the decimal THAT same number of places to the left in the quotient.
3. To Divide Decimals...move the decimal to the end (outside the box)...then move the decimal THAT same number of places (inside the box)...then move the decimal straight up into your answer...then divide.
4. To Round Decimals look at the digit to the right of the place value you want to round to. If the digit is 0,1,2,3, or 4 you leave the digit in the place value you want to round to alone and change all digits to the right of that place to zeros. Then get rid of any UNNECESSARY zeros. If the digit is a 5,6,7,8, or 9 you add one to the place value you want to round to and change all digits to the right of that place to zeros. Then get rid of any UNNECESSARY zeros. Example: 31.714 to the nearest hundredth. I look to the right of the hundredth place at the 4. This makes the 1 in the hundredths place STAY THE SAME...so it rounds to 31.710 and I get rid of the unnecessary zero...so 31.71. Example 2: 578.31 to the nearest ten. I look to the digit to the right of the ten place at the 8. This makes the 7 in the tens place GO UP...so it rounds to 580.00 and I get rid of the unnecessary zeros...so 580 (the zero that remains is necessary to keep the 8 in the tens place and the 5 in the hundreds place).
C.C. 3 Rubric Top number = numerator Bottom number = denominator
1. To change from improper to mixed numbers - divide the TOP number by the bottom number until you
get to the decimal (this answer is the whole number) Then put your remainder over your divisor (this
answer is your fraction). Put the whole number and fraction together to make a mixed number.
2. To change from mixed to improper -(M.A.D., B.L.T. Horseshoe Method) - multiply the bottom number by
the whole number...then add the top number...and put it over the original bottom number.
3. To Add or Subtract - COMMON DENOMINATOR (whatever you do to the bottom...you do to the top).
Then add/subtract the tops and LEAVE the bottoms.
4. To Multiply - IMPROPER FRACTION...See if you can cross simplify (reduce)...Multiply the tops
across...then multiply the bottoms across.
5. To Divide - IMPROPER FRACTION...Flip the 2nd fraction to its reciprocal...Change the problem to
multiplication...then cross simplify...multiply the tops across and the bottoms across.
C.C. 4 Rubric Integers - all whole numbers (0,1,2,3...) and their opposites (-1,-2,-3...) so...{...-4,-3,-2,-1,0,1,2,3,4...}
Integers do not include partial fractions or decimals.
Hints: 1. Number lines are very useful for comparing and understanding integers.
2. Use your integer rules (See Skills Sheet below) and SAY them to yourself as you work each problem.
3. Be Careful! Integer rules are a common place to make errors.
C.C. 5 Rubric Algebraic Expressions and Equations
1. Several Number Properties are useful to know for this section. The Commutative Property of + and x states that
order does not matter in addition or multiplication so...5+3 = 3 + 5 and 7 x 9 = 9 x7. The Associative Property of +
and x states that grouping does not matter in addition or multiplication so... (9 + 3) + 7 = 9 = (3 + 7) and
4 x (5 x 7) = (4 x 5) x 7. The Identity Property of x states that anything x 1 is itself...that is its identity so...9 x 1 = 9
and -3h x 1 = -3h. The Identity Property of + states that anything + 0 is itself...that is its identity so... 13h + 0 = 13h
and -5 + 0 = -5. The Distributive Property is a way to break things out of a grouping symbol by multiplying by all of
the terms in the grouping symbol. For example 3(4b + 7) is the same as (3 x 4b) + (3 x 7) or 12b +21.
2. Terms are "chunks of math" that are added together. They may be numbers, variables, or a combination of numbers
and variables. Constants are numbers by themselves. Coefficients are numbers that are connected to a variable
(by multiplication). So in 7f + 3, 7f and 3 are terms. 3 is a constant and 7 is a coefficient.
3. Like terms are terms that have the exact same variable part to the exact same power such as 5 and -9 or 2m and 5m
or 3b squared and -8b squared. Like terms can be combined so in 5b + 2 + 7b + 15 + 3b squared... the 5b and 7b can
be added to equal 12b the 2 and the 15 can be added to make 17 and the 3b squared can not combine with either.
You are left with 3b squared + 12b + 17 after combining like terms.
4. Algebraic equations contain a variable (the letter in a math problem) and an equals sign (=). In an algebraic
equation the work has already been done so we need to work backwards and undo what has been done to find the
value of the variable. (To see worked out examples with explanations see multiple Skills Sheets below.)
In short the steps are...
1. Leave Change Opposite
2. Distributive Property
3. Combine LIKE TERMS on both sides of the equals sign
4. If there are variables on both sides of the equals sign...subtract one set of variables from both sides to
get all variables on the same sign.
5. ***You are in a 2-step algebraic equation!
6. Solve it using U-turn or Cancellation.
7. Plug your answer back into the original equation (for the variables) to make check your answer.
C.C. 6 Rubric Ratios, Rates, Proportions, Percents
1. A ratio is a comparison of two numbers. It can be written with a colon (:), the word to, or as what looks like a
fraction. Ratios can be simplified just like fractions with one exception - Do not simplify them to whole numbers or
mixed numbers. It is perfectly fine to leave a ratio as what looks like an improper fraction. 5:2 5 to 2 or 5
2
2. Rates are ratios of two different quantities that have different labels. Driving 210 miles in 3.5 hours could be set up
as a rate. Unit rates are often used to compare a unit rate is a rate in which the denominator is 1. In the above
example driving 210 miles in 3.5 hours can be written as a unit rate of 60 miles per hour where the hour is an
understood 1 hour. A unit rate can be found by dividing the top number by the bottom number. (Rates and Unit
Rates must contain labels.)
Proportions a = c
b d
3. There are 7 things you should know about proportions. Proportions are a ratio = ratio, they are read a is to b as c is
to d, they can not CAN NOT be cross-simplified, ORDER MATTERS, they have special connections, you should use
labels to set them up, and their cross products are always equal. (For how to solve - see skill sheet below.)
4. Percent problems can be easily solved by setting up a proportion using the formula below. (See skill sheet below.)
is = % or part = %
of 100 whole 100
C.C. 7 Rubric Geometry See CC7 Skills Sheet under the Skills Sheet tab above.
Vocabulary
parallel - lines, rays, or segments that are always the same distance apart. (For the entire length of the lines, rays, or
segments...they stay the same distance apart.)
perpendicular - lines, rays, or segments that intersect (meet) to form one or more right angles (90 degrees).
congruent - having the same exact measure. A 43 degree angle is congruent to another 43 degree angle. A 9 cm line segment
is congruent to another 9 cm line segment. Two triangles that are the exact same shape and the exact same sign
are congruent triangles.
vertical angles are the angles opposite each other when two lines, rays, or segments cross each other. Vertical angles are
congruent.
adjacent angles - are 2 angles that share a common vertex and a common side and DO NOT overlap
Complementary angles - are 2 angles whose measures add up to 90 degrees (example: a 60 degree angle and a 30 degree
angle or a 4 degree angle and an 86 degree angle)
Supplementary angles - are 2 angles whose measures add up to be 180 degrees (example: a 125 degree angle and a 55
degree angle.)
Perimeter - the distance around an object (measured in units such as inches, cm, feet, m, miles, etc.)
Circumference - the distance around a circle
Radius - the distance from the center of a circle to a point on the circle (half way across a circle)
Diameter - the distance across a circle through its center
Area - the number of square units it takes to cover an object (measured in square cm, square feet, square miles, square inches,
etc.)
Volume - the amount of space a 3d solid takes up or the number of cubic units it takes to fill a 3d solid (measured in cubic cm,
cubic ft, cubic inches, etc.)
Surface Area - the number of square units it takes to cover all sides of a 3d solid (measured in square cm, square feet, square
miles, square inches, etc.
C.C. 8 Rubric Central Tendencies, Counting Principle, and Probability
Mean - (average) Add all the numbers. Divide this SUM by the number of numbers.
Median - (middle) Line up the numbers from least to greatest. Count in from both ends to find the middle data value. If there
are 2 data values in the middle...add them and divide by two.
Mode - (MOst) The data value that occurs the most. There may be multiple modes or no mode at all.
Range - (Difference between the high and low) Highest data value minus the Lowest data value equals the range.
Interquartile Range - (Difference between the upper quartile and the lower quartile) Upper quartile (median of the top 50% of the data) minus the Lower quartile (median of the bottom 50%) equals the Interquartile Range.
Mean Absolute Deviation - the average deviation (variability) of a data set from its center (measured as either as the mean or median). Subtract each data value from either the mean or median of the data set. Add all the absolute values of these differences and then divide by the number of numbers.
Counting Principle - If there are a ways for one activity to occur, and b ways for a second activity to occur, then there are
a • b ways for both to occur.
Probability - the chances (likelihood) that an event will happen
Theoretical Probability - the probability that is calculated using math formulas. This is the probability based on math
theory. (What SHOULD happen.)
Experimental Probability - calculated when the actual situation or problem is performed as an experiment. In this case,
you would perform the experiment, and use the actual results to determine the probability.
(What DOES happen.)
Simple Probability - number of ways the event CAN happen / total possibilities
Integers do not include partial fractions or decimals.
Hints: 1. Number lines are very useful for comparing and understanding integers.
2. Use your integer rules (See Skills Sheet below) and SAY them to yourself as you work each problem.
3. Be Careful! Integer rules are a common place to make errors.
C.C. 5 Rubric Algebraic Expressions and Equations
1. Several Number Properties are useful to know for this section. The Commutative Property of + and x states that
order does not matter in addition or multiplication so...5+3 = 3 + 5 and 7 x 9 = 9 x7. The Associative Property of +
and x states that grouping does not matter in addition or multiplication so... (9 + 3) + 7 = 9 = (3 + 7) and
4 x (5 x 7) = (4 x 5) x 7. The Identity Property of x states that anything x 1 is itself...that is its identity so...9 x 1 = 9
and -3h x 1 = -3h. The Identity Property of + states that anything + 0 is itself...that is its identity so... 13h + 0 = 13h
and -5 + 0 = -5. The Distributive Property is a way to break things out of a grouping symbol by multiplying by all of
the terms in the grouping symbol. For example 3(4b + 7) is the same as (3 x 4b) + (3 x 7) or 12b +21.
2. Terms are "chunks of math" that are added together. They may be numbers, variables, or a combination of numbers
and variables. Constants are numbers by themselves. Coefficients are numbers that are connected to a variable
(by multiplication). So in 7f + 3, 7f and 3 are terms. 3 is a constant and 7 is a coefficient.
3. Like terms are terms that have the exact same variable part to the exact same power such as 5 and -9 or 2m and 5m
or 3b squared and -8b squared. Like terms can be combined so in 5b + 2 + 7b + 15 + 3b squared... the 5b and 7b can
be added to equal 12b the 2 and the 15 can be added to make 17 and the 3b squared can not combine with either.
You are left with 3b squared + 12b + 17 after combining like terms.
4. Algebraic equations contain a variable (the letter in a math problem) and an equals sign (=). In an algebraic
equation the work has already been done so we need to work backwards and undo what has been done to find the
value of the variable. (To see worked out examples with explanations see multiple Skills Sheets below.)
In short the steps are...
1. Leave Change Opposite
2. Distributive Property
3. Combine LIKE TERMS on both sides of the equals sign
4. If there are variables on both sides of the equals sign...subtract one set of variables from both sides to
get all variables on the same sign.
5. ***You are in a 2-step algebraic equation!
6. Solve it using U-turn or Cancellation.
7. Plug your answer back into the original equation (for the variables) to make check your answer.
C.C. 6 Rubric Ratios, Rates, Proportions, Percents
1. A ratio is a comparison of two numbers. It can be written with a colon (:), the word to, or as what looks like a
fraction. Ratios can be simplified just like fractions with one exception - Do not simplify them to whole numbers or
mixed numbers. It is perfectly fine to leave a ratio as what looks like an improper fraction. 5:2 5 to 2 or 5
2
2. Rates are ratios of two different quantities that have different labels. Driving 210 miles in 3.5 hours could be set up
as a rate. Unit rates are often used to compare a unit rate is a rate in which the denominator is 1. In the above
example driving 210 miles in 3.5 hours can be written as a unit rate of 60 miles per hour where the hour is an
understood 1 hour. A unit rate can be found by dividing the top number by the bottom number. (Rates and Unit
Rates must contain labels.)
Proportions a = c
b d
3. There are 7 things you should know about proportions. Proportions are a ratio = ratio, they are read a is to b as c is
to d, they can not CAN NOT be cross-simplified, ORDER MATTERS, they have special connections, you should use
labels to set them up, and their cross products are always equal. (For how to solve - see skill sheet below.)
4. Percent problems can be easily solved by setting up a proportion using the formula below. (See skill sheet below.)
is = % or part = %
of 100 whole 100
C.C. 7 Rubric Geometry See CC7 Skills Sheet under the Skills Sheet tab above.
Vocabulary
parallel - lines, rays, or segments that are always the same distance apart. (For the entire length of the lines, rays, or
segments...they stay the same distance apart.)
perpendicular - lines, rays, or segments that intersect (meet) to form one or more right angles (90 degrees).
congruent - having the same exact measure. A 43 degree angle is congruent to another 43 degree angle. A 9 cm line segment
is congruent to another 9 cm line segment. Two triangles that are the exact same shape and the exact same sign
are congruent triangles.
vertical angles are the angles opposite each other when two lines, rays, or segments cross each other. Vertical angles are
congruent.
adjacent angles - are 2 angles that share a common vertex and a common side and DO NOT overlap
Complementary angles - are 2 angles whose measures add up to 90 degrees (example: a 60 degree angle and a 30 degree
angle or a 4 degree angle and an 86 degree angle)
Supplementary angles - are 2 angles whose measures add up to be 180 degrees (example: a 125 degree angle and a 55
degree angle.)
Perimeter - the distance around an object (measured in units such as inches, cm, feet, m, miles, etc.)
Circumference - the distance around a circle
Radius - the distance from the center of a circle to a point on the circle (half way across a circle)
Diameter - the distance across a circle through its center
Area - the number of square units it takes to cover an object (measured in square cm, square feet, square miles, square inches,
etc.)
Volume - the amount of space a 3d solid takes up or the number of cubic units it takes to fill a 3d solid (measured in cubic cm,
cubic ft, cubic inches, etc.)
Surface Area - the number of square units it takes to cover all sides of a 3d solid (measured in square cm, square feet, square
miles, square inches, etc.
C.C. 8 Rubric Central Tendencies, Counting Principle, and Probability
Mean - (average) Add all the numbers. Divide this SUM by the number of numbers.
Median - (middle) Line up the numbers from least to greatest. Count in from both ends to find the middle data value. If there
are 2 data values in the middle...add them and divide by two.
Mode - (MOst) The data value that occurs the most. There may be multiple modes or no mode at all.
Range - (Difference between the high and low) Highest data value minus the Lowest data value equals the range.
Interquartile Range - (Difference between the upper quartile and the lower quartile) Upper quartile (median of the top 50% of the data) minus the Lower quartile (median of the bottom 50%) equals the Interquartile Range.
Mean Absolute Deviation - the average deviation (variability) of a data set from its center (measured as either as the mean or median). Subtract each data value from either the mean or median of the data set. Add all the absolute values of these differences and then divide by the number of numbers.
Counting Principle - If there are a ways for one activity to occur, and b ways for a second activity to occur, then there are
a • b ways for both to occur.
Probability - the chances (likelihood) that an event will happen
Theoretical Probability - the probability that is calculated using math formulas. This is the probability based on math
theory. (What SHOULD happen.)
Experimental Probability - calculated when the actual situation or problem is performed as an experiment. In this case,
you would perform the experiment, and use the actual results to determine the probability.
(What DOES happen.)
Simple Probability - number of ways the event CAN happen / total possibilities
order_of_operations.pdf | |
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fractions_rules.pdf | |
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fd.pdf | |
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fraction_flowchart.pdf | |
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proportions_pdf.pdf | |
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percent_skills_sheet.pdf | |
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cc7_skills.pdf | |
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cc8_skills.pdf | |
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equations_pdf.pdf | |
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state_assessment_skills_study_guide.pdf | |
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