Download 7TH GRADE MATH MID YEAR STUDY GUIDE

7TH GRADE MATH MID YEAR STUDY GUIDE
4 STEPS OF PROBLEM SOLVING
1. EXPLORE-GENERAL UNDERSTANDING OF THE PROBLEM AS TO WHAT IS
GIVEN AND NEEDED
2. PLAN- STEPS NEEDED AND OPERATIONS NEEDED TO SOLVE
3. SOLVE- CARRY OUT THE METHOD
4. EXAMINE – DOES THE ANSWER MAKE SENSE COMPARE TO AN ESTIMATE
Order of Operations:
1.
2.
3.
4.
First do all operations that lie inside parentheses.
Next, do any work with exponents or radicals.
Working from left to right, do all multiplication and division.
Finally, working from left to right, do all addition and subtraction.
Property Laws
#1. Commutative properties
the commutative property of addition states that numbers can be added in any order and it will not change
the sum. The commutative property of multiplication states factors can be multiplied in any order without
changing the result.
Addition
5a + 4 = 4 + 5a
Multiplication
3 x 8 x 5b = 5b x 3 x 8
#2. Associative properties
The associative property of addition numbers can be grouped to form a sum in any
and still get the same answer. The associative property of multiplication numbers can
be grouped in any way to form a product and still get the same answer.
Addition
(4x + 2x) + 7x = 4x + (2x + 7x)
Multiplication
2x2(3y) = 3y(2x2)
#3. Distributive property
the distributive property involves both addition and multiplication. A longer name for
it is, "the distributive property of multiplication over addition." The term is multiplied
by terms in parenthesis, we need to "distribute" the multiplication over all the
terms inside.
2x(5 + y) = 10x + 2xy
#5. Identity property
the identity property zero added to any number is the number itself. Zero is called
the "additive identity." The identity property for multiplication, the number 1
multiplied times any number gives the number itself. The number 1 is called the
"multiplicative identity."
Addition
5y + 0 = 5y
Multiplication
2c × 1 = 2c
Additive Inverse – if add a positive and negative of the same number will cancel to zero
7 + (- 7) = 0
Multiplicative Inverse- if multiply a number by its flip will equal a positive one
1/4 x 4/1 = 4/4 =1
THE USE OF STATISTICS IN PROBLEM SOLVING
CENTRAL TENDENCY- THE CENTER OF THE DATA, THE MEAN MODE AND
MEDIAN WHICH IS USED TO MAKE PREDICTIONS
,
MEAN- THE AVERAGE THE EFFECT OF VERY HIGH AND LOW NUMBERS BALANCED TO A
CENTER POINT, ADD ALL THE NUMBERS FOR A TOTAL AND DIVIDE BY THE TOTAL
NUMBER OF NUMBERS
100 +90 +100 +80 = 370 DIVIDE BY 4 = 93.33
MODE- THE MOST FREQUENT REPEATED NUMBER IN A DATA SET
IF HAVE 2 CALLED BI-MODAL
MEDIAN – THE NUMBER IN THE CENTER OF A DATA SET , TO DETERMINE
COUNT THE TOTAL NUMBER OF NUMBERS IN THE LIST ADD ONE THEN
DIVIDE BY 2 THAT NUMBER POSITION WILL BE THE MEDIAN
RANGE- THE TOTAL SPAN OF THE NUMBERS
THE MEAN, MODE AND MEDIAN VALUES SHOULD ALL BE VERY CLOSE BECAUSE THESE VALUES
REFLECT THE CENTRAL TENDENCY OF THE DATA
FREQUENCY TABLE- A TABLE USED TO ORGANIZE DATA BY TALLIES TO DETEMINE HOW OFTEN
A NUMBER REPEATS
INTERVAL- A RANGE OF NUMBERS IN WHICH DATA COLLECTS IN A TALLY AND EVERY RANGE
GROUP HAS THE SAME SPAN, SUCH AS 0 TO 5, 5 TO 10
SCALE – IS THE RANGE OF THE INTERVALS 0 TO 5 HAS A SCALE OF 5
LINE PLOT- A NUMBER LINE THAT REPRESENTS EVERY NUMBER IN A DATA SET FROM LOWEST
TO HIGHEST WITH ALL NUMBERS IN BETWEEN USED TO SHOW FREQUENCY, GAPS, CLUSTERS,
MEDIAN AND MODE EVERY TIME A NUMBER REPEATS AN X IS PLACED OVER THAT NUMBER
GAPS- ON A GRAPH AN AREA IN WHICH NO DATA WAS PRESENT BUT WAS ON BOTH SIDES OF THIS
AREA
CLUSTER- AN AREA IN A LINE PLOT IN WHICH A CONCENTRATION OF POINTS COLLECTS WITH
FREQUENCY
STEM AND LEAF PLOT- A METHOD TO ORGANIZE AND COMPARE DATA BY GROUPING IN TENS
AND ORDERING BY ONES TO SHOW FREQUENCY, THE MODE AND MEDIAN
BAR GRAPH – A GRAPH MADE OF BARS THAT MAY OR MAY NOT TOUCH WITH EACH BAR
REPRESENTING AN AVERAGE OF A GROUP OF DATA TO COMPARE AVERAGES
HISTOGRAM- A BAR GRAPH IN WHICH EACH COLUMN HAS TO BE THE SAME WIDTH AND EACH
BAR TOUCHES, IN ORDER TO SHOW CLUSTERS AND GAPS, THE GRAPH SCALE IS IN INTERVALS
SCATTERPLOT- A GRAPH IN WHICH ONE SET OF DATA IS ON THE X AXIS AND ONE SET IS ON THE Y
AXIS, THE POINTS ARE PLOTTED BUT ARE NOT CONNECTED THE PURPOSE IS TO SEE IF THE DATA
HAS A CORRELATION THE CAUSE AND EFFECT
BOX AND WHISKER PLOT
A GRAPH THAT BREAKS DATA IN TO 4 EQUAL PARTS TO COMPARE THE DATA EACH
SECTION IS CALLED A QUARTILE
BOX AND WHISKER PLOT
A GRAPH THAT BREAKS DATA IN TO 4 EQUAL PARTS TO COMPARE THE DATA EACH
SECTION IS CALLED A QUARTILE
18 27 34 52 54 59 61 68 78 82 85 87 91 93 100

First find the median. The median is the value exactly in the middle of an ordered set of
numbers. *
68 is the median

Next, we consider only the values to the left of the median: 18 27 34 52 54 59 61. We now find
the median of this set of numbers. Remember, the median is the value exactly in the middle
of an ordered set of numbers. Thus 52 is the median of the scores less than the median of all
scores, and therefore is the lower quartile.
52 is the lower quartile

Now consider only the values to the right of the median: 78 82 85 87 91 93 100. We now find
the median of this set of numbers. The median 87 is therefore called the upper quartile.
87 is the upper quartile

You are now ready to find the interquartile range (IQR). The interquartile range is the
difference between the upper quartile and the lower quartile. In our case the IQR = 87 - 52 =
35. The IQR is a very useful measurement. It is useful because it is less influenced by
extreme values; it limits the range to the middle 50% of the values.
35 is the interquartile range
Now we begin to draw our graph.
HOW TO READ A BOX PLOT
1.
2.
3.
4.
50% ABOVE OR BELOW IS THE MEDIAN
THE CENTRAL TENDENCY IS THE BOX OR INTERQUARTILE RANGE
25 % BELOW IS THE Q 1 NUMBER
25 % ABOVE IS THE Q3 NUMBER
PROBABILITY: FAVORABLE OUTCOME/ POSSIBLE x 100 = THE PERCENT
ODDS: CHANCE WILL HAPPEN TO CHANCE WILL NOT
THEORETICAL PROBABILITY: ALL THE POSSIBLE OUTCOMES
EXPERIMENTAL PROBABILITY: THE OBSERVED OUTCOMES OF AN EVENT
SAMPLE SPACE: ALL THE OUTCOMES OF AN EVENT
COUNTING PRINCIPLE: MULTIPLY THE NUMBER OF WAYS EACH EVENT IN AN EXPERIMENT CAN
OCCUR TO FIND THE NUMBER OF POSSIBLE OUTCOMES IN A SAMPLE SPACE
COUNTING TREE- A DESIGN TO SHOW ALL THE POSSIBLE OUTCOMES
THE FACTS ABOUT INTEGERS
1. ABSOLUTE VALUE
The absolute value of x, denoted "| x |" (and which is read as "the absolute value of x"), is the
distance of x from zero. This is why absolute value is never negative; absolute value only
asks "how far?” not "in which direction?". This means not only that | 3 | = 3, because 3 is three
units to the right of zero, but also that | –3 | = 3, because –3 is three units to the left of zero.
2. If the absolute value has a negative sign in front of the absolute value bars it means the absolute
value is still positive but was multiplied by a negative one so the solution is a negative -| 3 | = -3
3. Adding Integers
If both numbers are the same sign add the total and keep the sign the same
7 + 7 = 14
(-2) + (-3) = -5
If one number is positive and the other is negative positives and negatives will cancel each other out and
the sign is the number with the largest absolute value
7 + (-4) = + + + + + + + plus - - - - If the positive and negatives cancel 3 positives are left and 7 has
a larger absolute value than -4
(+ + + +) + + + plus (- - - -) in () means canceled and leaves 3 positives
4. Multiplying and Dividing Integers
2 negatives equals a positive even signs = positive uneven sign = negative the
the number does not have any meaning
(-5) (-2) =10 (5) (-2) = -10
(-2) (5) = -10
(5) (2) =10
12/4 = 3
-12/4 = -3 12/-4 = -3 -12/-4 = 3
absolute value of
5. Subtracting Integers
If you have a positive and subtract a positive same as basic math operations
7–2=5
If you have a negative minus a negative 2 negatives make a positive changes the
addition
7 – (-2) = 7 + 2 = 9
operation to
If you have a -7 -3 = -10 had 7 negatives and took 3 more away = -10
Integers
To add integers having the same sign, keep the same sign and add the numbers
To add integers with different signs, positive and negatives will cancel
Watch out! The negative of a negative is the opposite positive number. That is, for
real numbers,
-(-a) = +a
Here's how to add two positive integers:
4+7=?
If you start at positive four on the number line and move seven units to the right,
you end up at positive eleven. Also, these integers have the same sign, so you can
just keep the sign and add their absolute values, to get the same answer, positive
eleven.
Here's how to add two negative integers:
-4 + (-8) = ?
If you start at negative four on the number line and move eight units to the left, you
end up at negative twelve. Also, these integers have the same sign, so you can just
keep the negative sign and add their absolute values, to get the same answer,
negative twelve.
Here's how to add a positive integer to a negative integer:
-3 + 6 = ?
If you start at negative three on the real number line and move six units to the right,
you end up at positive three. Also, these integers have different signs,
so keep the sign from the integer having the greatest absolute value and subtract
the smallest absolute value from the largest.
Subtract three from six and keep the positive sign, again giving positive three.
Here's how to add a negative integer to a positive integer:
5 + (-8) = ?
If you start at positive five on the real number line and move eight units to the left,
you end up at negative three. Also, these integers have different signs, so keep the
sign from the integer having the greatest absolute value and subtract the smallest
absolute value from the largest, or subtract five from eight and keep the negative
sign, again giving negative three.
To subtract a number, add its opposite:
5-8=?
Because they give the same result, you can see that subtracting eight from five is
equivalent to adding negative eight to positive five. The answer is - 3.
To subtract a number, add its opposite:
-3 - (-6) = ?
Because they give the same result, you can see that subtracting negative six from
negative three is equivalent to adding positive six to negative three. The answer is 3.
When you multiply two integers with the same signs, the result is always positive.
Positive x positive = positive
Negative x negative = positive
When you multiply two integers with different signs, the result is always negative.
Positive x negative = negative
Negative x positive = negative
When you divide two integers with the same sign, the result is always positive.
Positive ÷ positive = positive
Negative ÷ negative = positive
When you divide two integers with different signs, the result is always negative.
Positive ÷ negative = negative
Negative ÷ positive = negative
An integer is a whole number that can be either greater than 0, called positive, or
less than 0, called negative. Zero is neither positive nor negative.
Two integers that are the same distance from zero in opposite directions are called
opposites.
Every integer on the number line has an absolute value, which is its distance from
zero. Absolute Value is noted by the bars and is never negative unless a negative sign is in front of the
bars I 3 I = 3 I -3 I = 3 - I 3 I = -3
Powers and ExponentsExponent tells how many times the base number is used as a factor. A base of five
raised to the second power is called "five squared" and means "five times five." Five
raised to the third power is called "five cubed" and means "five times five times five."
Here are some simple rules to use with exponents.
1.
2.
3.
a1 = a
Any number raised to the power of one equals the number itself.
For any number a, except 0, a0 = 1
Any number raised to the power of zero, except zero, equals one.
For any numbers a, b, and c,
ab x ac = ab+c
This multiplication rule tells us that we can simply add the exponents when
multiplying two powers with the same base.
Product Rule
The exponent "product rule" when multiplying two powers that have the same base,
you can add the exponents. In this example, you can see how it works. Adding the
exponents is just a short cut!
Quotient Rule
The quotient rule can divide two powers with the same base by subtracting the
exponents.
Prime Factorization- identify the factors of a number which the only factor is that number and one the
numbers 2, 3 5 7, 11 will prime factor any number, one is not prime or composite, if the factors are grouped
with exponents called prime factorization, if the factors are listed as multiples called the prime factors, a
factor tree can be used to identify the factors
24
2
12
2
6
2
3
22 x 3 Prime Factorization
2 x 2 x 2 x 3 Prime Factors
Composite – more factors than the number and one
Rules of Fractions
Ordering – change all fractions to decimals by dividing the top (numerator) by the denominator (bottom)
and then multiply by 100 to see as a percent
Adding and subtracting must have a common denominator which is the same as the least common
multiple a number both denominators will divide into once the denominator is divided into the number the
number of times must be multiplied times the numerator
½ + ¾ = 2/4 + ¾ = 5/4
Multiplication – no common denominator numerator times numerator and denominator times denominator
3/5 x 6/8 = 18/40 =9/20
Division – no common denominator flip the second fraction then follow multiplication rules
Estimation – round to the nearest whole number and solve .5 go up below go down
23% of 100 =
25% of 100 rounded
Number Theory
Rational Numbers: Natural – 1,2,3,4,5,6,7,8,9
Whole – above plus 0 Integers- whole number
positives an negatives Rationals- fractions a/b b not equal to 0 decimals with an end and pattern whole
number square roots
Irrational- only 3 forms decimals no pattern or end, square roots in decimal form and the symbol for
pi
Scientific Notation- writing a very large or small number using a factor greater than 1 less than 10 times a
power of 10, all numbers less than 1 will have a negative power on the 10, to convert from scientific to
standard multiply the factors by 10 for the given exponent that many times to reverse divide by
2500000 ( 2.6 x 10 6 )
0.000045 (4.5 x 10 -5 )
EQUATION IS A MATHEMATICAL SENTENCE IN WHICH A VARIABLE IS A PART OF THE
SENTENCE THAT CAN BE REPLACED WITH ONLY ONE NUMBER THAT WILL MAKE THE
SENTENCE TRUE, TO FIND THIS NUMBER USE INVERSE OPERATIONS
EX. X + 5 =12 TO SOLVE SUBTRACT 5 FROM BOTH SIDES WHICH WILL RESULT IN X = 7
X – 10 = 8 TO SOLVE ADD 10 TO BOTH SIDES WHICH WILL RESULT IN X = 18
2X = 12 DIVIDE BOTH SIDES BY 2 X= 6
INEQUALITY – A MATHEMATICAL SENTENCE THAT USES AN ORDER SIGN OF > < OR > <
SOLVED THE SAME AS EQUATIONS BUT WILL RESULT IN MORE SOLUTIONS, THE
SOLUTION MAKE NOT BE USED FOR < > ONLY BUT MAY BE USED WITH > < TO GRPAH
THE ANSWER ON A NUMBER LINE FOR < > USE AN OPEN CIRCLE FOR > < USE A CLSED
CIRCLE TO SHOW CAN USE THAT NUMBER
IN SOLVING INEQUALTIES IF THE VARIABLE HAS TO BE MULTIPLIED OR DIVIDED BY A
NEGATIVE MUST FLIP THE ORDER SIGN
-2X > 12 BECOMES X < -6
GRAPHING LINEAR EQUATIONS
LINEAR EQUATION- ANY EQUATION IN WHICH THE X & Y VALUES OF A PAIR WHEN
REPLACED IN THE EQUATION BOTH SIDES ARE EQUAL AND EVERY PAIR THAT ALSO
MATCHES THE RULE WHEN CONNECTED ON A GRAPH CREATES A STRAIGHT LINE
SLOPE- THE CHANGE IN Y OVER THE CHANGE IN X THE PATTERN OF X & Y IN AN
ORDERED PAIR TABLE
The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, Given two
points (x1,y1) and (x2,y2) on a line, the slope m of the line is
If use the pairs (-4 -5) & (2 7) then 7- (-5)/ 2 – (-4) = 7 +5 /2 +4 = 12/6 = 2
So the slope is 2 a rise of 2 with a run of 1
EXPLORING THE LANGUAGE OF ALGEBRA
SOLVING 2 STEP EQUATIONS GOAL MOVE THE VARIABLES TO ONE SIDE AND NUMBERS
TO THE OTHER TO SOLVE FOR X WITH INVERSE OPERATIONS
METRIC SYSTEM OF MEASUREMENT- A system of measurement based on units of 10 in
which all measurements are multiples of ten (or a hundred, or one-tenth, etc) of each other in
order to covert between the various different sizes move the decimal point the correct number of
places.
The basic metric units are meters (for length), grams (for mass or weight), and liters (for volume)
The basic prefixes are : kilo-, Hecto-, Deka-, deci-, centi-, and milli-. To convert between the
various prefixes, and thus the variously-sized units, you just move up and down this list of
prefixes, moving the decimal point as you go.
In order to covert from large to small multiply because it takes a lot of little units to cover the same
space covered by a larger unit, to covert from small to large divide.
To remember the prefixes in order, you can use the following sentence:
King Henry Doesn't [Usually] Drink Chocolate Milk
The first letters of the words stand for the prefixes, with "Usually" in the middle standing for the
"unit", being meters, grams, or liters.
kilo- Hecto- Deka- [unit] deci- centi- milliSince each step is ten times or one-tenth as much as the step on either side, we have:
1 kilometer = 10 hectometers = 100 dekameters = 1000 meters
= 10 000 decimeters = 100 000 centimeters = 1 000 000 millimeters
1 meter = 100 cm 1000mm
1 milliliter = 0.1 centiliters = 0.01 deciliters = 0.001 liters = 0.000 1 dekaliters
= 0.000 01 hectoliters = 0.000 001 kiloliters

Convert 12.54 kilometers to centimeters.
How many jumps is it from "kilo-" to "centi-"? Five, to the right. So 12.5 kiolmeters move
the decimal 5 to the right which will require 3 zeros so the result is 1,254,000cm

Convert 457 mL to hL.
Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
How many jumps is it from "milli-" to "Hecto-"? Five, to the left.

So I move the decimal point five places to the left, filling in the empty spots after the
decimal point with zeroes: Therefore the answer would be .00457 hl
When converting units every number has an understood decimal at the end when run out of
numbers place zeros in those spaces
May also sole by ratio proportion of the known = to the unknown cross multiply and solve
1 meter /100 cm = 1500 meters/ x cm ( 1 x X) = (1500 x 100 ) X = 150,000 cm