Download Science 6

LOCKPORT CITY SCHOOL DISTRICT
PROGRESS MONITORING
MATHEMATICS
GRADE 7.
Students will: 1) Understand the concepts of and become proficient with the skills of mathematics, 2)
Communicate and reason mathematically, 3) Become problem solvers by using appropriate tools and
strategies through the integrated study of number sense and operations, algebra, geometry, measurement,
and statistics and probability.
STUDY GUIDE 2
Students will study Algebra, Probability, and Statistics.
SIMPLIFYING EXPRESSIONS
What is an expression? A mathematical representation containing numbers, variables, and operation
symbols; an expression does not include an equality or inequality symbol.
 Examples:
o (5+2) – 27 ÷ 3
(arithmetic/numerical expression)
o 2a + 3b
(algebraic/symbolic expression)
What is the order of operations? A specified sequence in which mathematical operations are expected to
be performed; an arithmetic expression is evaluated by following these ordered steps:
(1) simplify within grouping symbols such as parentheses or brackets, starting with the innermost
(2) apply exponents – powers and roots
(3) perform all multiplications and divisions in order from left to right
(4) perform all additions and subtractions in order from left to right
 Examples:
o Solve 4 + 6(5 – 2) ÷ 3
(Parenthesis first)
4 + 6(3) ÷ 3
(Multiply)
4 + 18 ÷ 3
(Divide)
4+6
(Add)
10
o Solve 36 – ([3 x 22] + 24)
(Exponents within the parenthesis)
36 – ([3 x 4] + 24)
(Multiplication within the parenthesis)
36 – ([12] + 24)
(Addition within the parenthesis)
36 – (36)
(Subtract)
0
What are operations? Procedures used to combine numbers, expressions, or polynomials into a single
result





Addition
Subtraction
Multiplication
Division
Exponents
What is substitution? Substitution is when variables in a given expression or equation are replaced with
designated values in order to evaluate the expression or equation.
 Examples:
o If x = 3 solve x2 + 5
(Substitute 3 in for x)
2
3 +5
(Solve the exponent)
9+5
(Add)
Study Guide – 7Math-Q2-V12008
14
o
If x = 4 solve 3x2 – 2x + 10
3• 42 – 2 • 4 + 10
3 • 16 – 2 • 4 + 10
48 – 8 + 10
40 + 10
50
(Substitute 4 in for x)
(Solve for the exponent)
(Multiply)
(Subtract)
(Add)
What are variables? A symbol used to represent a number or group of numbers in an expression or an
equation.
Practice Questions:
Correct responses are marked with an asterisk (*)
1) 32 – 3 + 7 = __________
a) 19
b) 13 *
c) 12
d) 10
2) 30 ÷ 5 x (2 + 8) ÷ 2 + 5 = __________
a)
60
7
b) 12.5
c) 35 *
d) 67
3) What does the expression 5b + 2 mean?
a) Multiply b times 5
b) Add 5 to the square of b
c) Multiply 2 by the quantity 5 times b
d) Add 2 to the quantity 5 times b *
4) Evaluate the expression x – 3y, if x = 4 and y = 2.
a) 4
b) 6
c) 10 *
d) 8
5) Evaluate the expression -3a2 + ab2c – 2a, if a = 1, b = 2, and c = 3.
a) 7 *
b) 6
c) 5
d) 4
Study Guide – 7Math-Q2-V12008
TWO-STEP EQUATIONS
What is a verbal expression? A way to write a mathematical expression using key words such as product,
more than, and decreased by.
What is an algebraic expression? A mathematical phrase that is written using one or more
variables and constants, but which does not contain a relation symbol (<, >, ≤, ≥, =, ≠)
 Example: 3y + 6
What is a variable? A symbol used to represent a number or group of numbers in an expression or an
equation.
What is a constant? A quantity that does not change its value in a given expression or equation
 Example: In the expression 3s + 4, 3 and 4 are constants
How do you translate a two-step verbal expression into an algebraic expression? All you need to
do is translate each of the key words into their appropriate symbols.
 Examples:
o Write an algebraic expression for the statement “the product of a number and seven, increased
by 9.”
The number is the unknown so pick any variable to represent the unknown number.
The word product means multiplication.
The word increased by means add.
So, the product of a number and seven, increased by 9 becomes 7x + 9.
o Write an algebraic expression for the statement “four less than half a number.”
The number is the unknown so pick any variable to represent the unknown number.
The words half of a number indicate multiplication of the number by ½.
The words four less than indicate subtraction of 4.
So, four less than half a number becomes ½x – 4.
What is an equation? A mathematical sentence stating that two expressions are equal.
How do I solve two-step equations? The goal of all equations is to isolate the variable, or get it alone.



The first step in achieving that goal is to undo the addition or subtraction by using the inverse
operation.
The last step is to undo the multiplication or division by using the inverse operation.
o What are inverse operations? An operation that is the opposite of, or undoes, another
operation
 addition and subtraction are inverse operations
 multiplication and division are inverse operations.
Examples:
o Solve for x: 3x + 2 = 17
-2 -2
3x 15

3
3
x=5
o
Solve for y:
y
43
5
Study Guide – 7Math-Q2-V12008
+4 +4
5•
y
=7•5
5
y = 35
Practice Questions:
Correct responses are marked with an asterisk (*)
1) Which of the following is an example of an equation?
a) 4x + 2
b) 4x + 2 = 18 *
c) 4x + 2  18
d) 4x + 2 ‹ 18
2) Translate the following into an algebraic expression: six added to the quotient of eight and a number, n.
a) 6 + 8n
b) 8 + 6n
8
6 *
n
n
d)  6
8
c)
3) The cost of renting a snowmobile includes a $5 rental fee plus $20 per hour. Which expression represents
the total cost of renting a snowmobile?
a) 20h + 5 *
b) 20(h + 5)
c) 20h – 5
d) 5h + 20
4) Solve the following two-step equation for x: 2x - 7 = 15
a) 7
b) 11 *
c) 4
d) 44
5) Solve the following two-step equation for y:
y
68
4
a) 56
b) 2
c) 3.5
d) 8 *
Study Guide – 7Math-Q2-V12008
ONE-STEP INEQUALITIES
What is an inequality? An inequality is a math sentence comparing two expressions that are not equal,
using one of the following symbols: <, >, ≤, ≥, ≠.
How do I read each of these symbols?





<
>
≤
≥
≠
-
less than
greater than
less than or equal to
greater than or equal to
not equal to
How do I solve a one-step inequality? The goal of solving a one-step inequality is to isolate the variable.


To achieve the goal, undo the given operation by using the inverse operation.
o addition and subtraction are inverse operations
o multiplication and division are inverse operations
Examples:
O Solve for x: x + 5 > 7
-5 -5
x >2
O Solve for x: x – 4 < 8
o
Solve for x:
+4 +4
x < 12
3x ≥ 18
18
3x
≥
3
3
x≥6
o
Solve for x:
x
5
2
x
2•
≤5•2
2
x ≤ 10
How do I graph an inequality? To graph an inequality first you draw a number line. Then you use an open
or closed circle to plot the solution on the line. Finally, shade the line so that the values make the inequality
true.
 Use an open circle (◦) when you use the greater than (>) and less than (<) symbols.
 Use a closed circle (•) when you use the greater than or equal to (≥) and less than or equal to (≤)
symbols.
 Examples:
o Graph: x ≥ 3
o
Graph: x < 1
Study Guide – 7Math-Q2-V12008
Practice Questions:
Correct responses are marked with an asterisk (*)
1) Which one of the choices below represents x is greater than or equal to 4?
a) x > 4
b) x < 4
c) x ≥ 4 *
d) x ≤ 4
2) What is the solution to the following inequality? x + 5 > 7
a) x > 12
b) x > 2 *
c) x = 2
d) x < 2
3) What is the solution to the following inequality?
x
 6
2
a) x ≤ -12 *
b) x ≤ 12
c) x ≤ -3
d) x ≤ 3
4) What is the inequality represented by the graph below?
a)
b)
c)
d)
x
x
x
x
<
>
≥
≤
2
2
2
2*
5) What is the inequality represented by the graph below?
a)
b)
c)
d)
x
x
x
x
≥
>
>
<
-1
1
-1 *
-1
Study Guide – 7Math-Q2-V12008
PROBABILITY
What is a compound event? A compound event is an event made up of two or more single events. There
are two types of compound events, independent and dependent.
What is an independent event? An independent event is when the outcome of the first event does not
affect the outcome of the events that follow.
How do I find the probability of an independent event?

For example: What is the probability of Kathi drawing a 4 from a set of cards numbered 1 – 10 and
rolling a 2 on a number cube.
1 1 1
 
10 6 60
What is a dependent event? A dependent event is when the outcome of the first event affects the outcome
of the events that follow.
How do I find the probability of a dependent event?

For example: What is the probability of choosing a red marble and then another red marble from a bag
with 5 red, 4 blue, and 1 yellow without replacing the first marble?
5 4 20
 
10 9 90
What is experimental probability? Experimental probability is the ratio of the number of times an event
occurs to the total number of trials.
How do I find experimental probability?
Study Guide – 7Math-Q2-V12008

For example: Tanya made saves on 15 out of 25 shots. What is the experimental probability that she
will make a save on the next shot?
What is theoretical probability? Theoretical Probability is the ratio of the number of equally likely
outcomes in an event to the total number of possible outcomes.
How do I find theoretical probability?

For example: What is theoretical probability of drawing one of the 12 E’s from a bag of 100 Scrabble
tiles?
Practice Questions:
Correct responses are marked with an asterisk (*)
1) A bag contains 5 red marbles and 5 purple marbles. What is the probability of drawing a red marble and
then a purple marble, without replacing the first marble before drawing the second marble?
10
19
25
b)
*
90
25
c)
100
5
d)
10
a)
2) There are 8 blue marbles and 7 red marbles in a bag. Julie pulls two marbles. What is the probability that
she first pulls a blue marble and then a red marble, without replacing the first marble before drawing the
second marble ?
56
*
210
56
b)
225
15
c)
29
15
d)
225
a)
Study Guide – 7Math-Q2-V12008
3) Jamie has 3 raffle tickets. One hundred tickets were sold. Her name was not drawn for the first prize. What
is the probability that her name will be drawn for the second prize?
1
2
1
b)
9
1
c)
6
1
d)
*
36
a)
4) What is the probability of drawing a red marble from a bag of 6 red and 4 blue marbles, replacing it, and
then drawing a blue marble?
10
100
10
b)
24
24
c)
*
100
6
d)
10
a)
5) Mr. Rose draws names to see who will give the first book report. There are 10 boys and 14 girls in his class.
What is the theoretical probability that he will draw a girl’s name?
10
24
14
b)
*
24
10
c)
100
14
d)
100
a)
6) Macy makes 15 of 20 free throws at basketball practice. What is the experimental probability that she will
miss her next free throw?
3
4
15
b)
20
5
c)
*
20
5
d)
10
a)
Study Guide – 7Math-Q2-V12008
MEASURES OF CENTRAL TENDENCY AND RANGE
What is central tendency? Central tendency is used to represent the middle of a data set.
How do we measure central tendency? Measures of central tendency are mean, median, and mode.
What is mean? The mean is the sum of the values divided by the number of values in the data set.
What is median? The median is the middle value when the numbers are listed in numerical order. If there
are two numbers in the middle, take the mean of the two numbers.
What is mode? The mode is the value that occurs most often. Sometimes there is no mode or more than
one mode.

Example: Find the mean, median, and mode of 8, 3, 5, 4, 1, and 3.
o
Step 1: List in order:
1, 3, 3, 4, 5, 8
o
Step 2: Find the mean.
o
Step 3: Find the median.
o
Step 4: Find the mode.
1 + 3 + 3 + 4 + 5 + 8 = 24
24 ÷ 6 = 4
(There are 6 numbers in the data set.)
mean = 4
3+4=7
(The middle two numbers are 3 & 4.)
7÷2 = 3.5
median = 3.5
There are two number 3s.
mode = 3
What is range? Range is the difference between the greatest and least number.
 Example: Find the range of 8, 3, 5, 4, 1, and 3
8-1=7
range = 7
Practice Questions:
Correct responses are marked with an asterisk (*)
1) What is the mean of the following data? 62, 63, 65, 71, 75, 69, 72, 75
a) 70
b) 69 *
c) 75
d) 65
2) Which of the following is the median for the set of data? 5, 7, 6, 8, 5, 7, 5, 10
a) 7
b) 6.5 *
c) 5
d) 6
3) What is the mode of the following data? 7, 9, 8, 7, 2, 6, 7, 11, 9, 7, 5, 7, 3
a) 2
b) 5
c) 7 *
d) 11
Study Guide – 7Math-Q2-V12008
4) What is
a)
b)
c)
d)
the mode of the following data? 8, 12, 2, 16, 7
14
2
0
No mode *
5) Find the range for the following data set: 3, 8, 12, 4, 1, 3, 6, 3, 6, 6, 3
a) 9
b) 7
c) 11 *
d) 6
Study Guide – 7Math-Q2-V12008
READ & INTERPRET DATA REPRESENTED GRAPHICALLY
AND CONVERT RAW DATA TO DOUBLE BAR AND LINE
GRAPHS
What is a bar graph? A bar graph can be used to display and compare data. You can use a double bar
graph when comparing two related sets of data.
How do I interpret a bar graph? Compare the bars first, then if needed find the corresponding number on
the scale.

Example: Use the bar graph to answer each question.
a) Which language has the most native speakers?
The bar for Mandarin is the longest, so Mandarin has the most native speakers.
b) About how many more people speak Mandarin than speak Hindi?
About 500 million more people speak Mandarin than speak Hindi.
How do I make a double bar graph?
The table shows the life
expectancies of people in three
Central American countries.
Step 1: Choose a scale and
interval for the vertical
axis.
Step 2: Draw a pair of bars for
each country’s data. Use
different colors to show
males and females.
Step 3: Label the axes and give
the graph a title.
Step 4: Make a key to show what
each bar represents.
What is a line graph? You use a line graph to show how data changes over a period of time. You can use
a double line graph to show change over time for two sets of related data.
How do you interpret a line graph? Find the data points and the corresponding number on the scale.
Study Guide – 7Math-Q2-V12008

Example: Use the graph to answer the following questions.
a) At what age did the kitten weigh 6 lbs?
The kitten weighed 6 lbs at the age of 8 months
b) Between what two months did the most growth occur?
Between the 2nd and 4th months.
How do I make a double line graph?
The table shows the normal daily
temperatures in degrees Fahrenheit
in two Alaskan cities.
Step 1: Choose a scale and
interval for the vertical
axis.
Step 2: Plot a point for each
temperature in Nome and
connect the points.
Step 3: Then, using a different color,
plot a point for each
temperature in Anchorage
and connect the points.
Step 4: Label the axes, give the graph
a title, and make a key to show
what each line represents.
What is circle graph? A circle graph, also called a pie chart, shows how a set of data is divided into parts.
The entire circle contains 100% of the data. Each sector, or slice, of the circle represents one part of the entire
data set.
How do I interpret a circle graph? Look at each sector to determine what percent of the graph each piece
represents.

Example: Use the circle graph to answer the following questions.
a) Which group of CDs sold the most?
Rock
b) What percent of the CD sales are country?
27%
Study Guide – 7Math-Q2-V12008
What is a histogram? A histogram is a bar graph that shows the frequency of data within equal intervals.
There is no space between the bars in a histogram.
How do I interpret a histogram? Compare the bars first, then if needed find the corresponding number on
the scale.

Example: Use the histogram above to answer the following questions.
a) Which interval represents the greatest number of CDs?
11 – 15
b) Estimate how many people own 1 – 10 CDs?
Approximately 55
What is a misleading graph? A data display that distorts information in order to persuade.
How do I determine if a graph is misleading?

Example: The above graph is misleading because while there is only a $.10 per minute difference,
Company B’s cell phone makes it look like it would be twice the cost of Company A’s cell service.

Example: Both bar graphs show the percent of people in California, Maryland, Michigan, and
Washington who use seat belts. Which graph could be misleading? Why?
Study Guide – 7Math-Q2-V12008
Graph B could be misleading. Because the vertical axis on graph B is broken, it appears that the percent of
people in California who wear seat belts is twice as great as the percent in Michigan. In fact, it is only 5%
greater. People might conclude from graph B that the percent of people in California who wear seat belts is
much greater than the percents in the other states.
Practice Questions:
Correct responses are marked with an asterisk (*)
1)
Why is the graph to the right misleading?
a) The distances are incorrect.
b) The bars are mislabeled.
c) The bars are too tall.
d) The intervals on the vertical axis are not equal. *
2) The double-bar graph shows the win-loss records for the Carolina Panthers football
team in the years 1998-2004. In which year did the panthers win mores games than they lost?
a) 2003 *
b) 2001
c) 1999
d) 1998
Study Guide – 7Math-Q2-V12008
3) The histogram shows the ages of all members in a fan club. How many fan club members are between the
ages of 30-39?
a) 5
b) 22 *
c) 8
d) 17
4.) Which 2 age groups make up 3 times as many members as those who are between 60 and 69?
a) 50–59 and 70+
b) 30–39 and 18–29
c) 18–29 and 50–59 *
d) 40–49 and 50–59
5.) The double-line graph shows the number of tornadoes in the United States during part of 2002 and 2003.
Which time period showed the greatest decrease in the number of tornadoes?
a) Aug. to Sept. 2002
b) Aug. to Sept. 2003
c) Oct. to Nov. 2002
d) Nov. to Dec. 2003*
Study Guide – 7Math-Q2-V12008