Download 7th form math review

7TH FORM
MATH REVIEW
GRAPHING IN THE COORDINATE PLANE
A coordinate plane is a grid formed by the intersection of two number lines.
ACTIVITY
Graph each point in the same coordinate plane:
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A(4, -5)
B (3, 4)
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C(5, 0)
D(0, -3)
EQUATIONS TABLES AND GRAPHS
Any order pair that makes an equation true is a solution of the equation.
An equation is a linear equation if all of its solutions lie on a line.
ACTIVITY
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In 2000, about four babies were born worldwide every second. Make a table and write an equation to
represent the total number of babies born.
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For a certain repair, an auto shop charges a $20 fee for materials plus $40 per hour for labor. Graph
the linear equation y = 40x + 20, where y represents the total cost and x represents the hours of labor.
TRANSLATIONS
A transformation is a change in the position, shape, or size of a figure.
A translation is a transformation that moves each point of a figure the same distance and in the same
direction.
The figure you get after a transformation is an image.
ACTIVITY
Graph each point and its image after the given translation:
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T(1,3), left 2 units
V(-4,4), down 6 units
S(4,0), right 1 unit, down 3 units
X(0,-2), right 7 units
Match each rule with the correct translation:
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(x, y) à (x – 6, y + 2)
(x, y) à (x + 3, y)
(x, y) à (x – 1, y – 5)
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P(4, -1) à P’(3, -6)
Q(3, 0) à Q’(-3, 2)
R(-2, 4) à R’(1, 4)
REFLECTIONS AND SYMMETRY
A reflection is a transformation that flips a figure over a line. This line is the line of reflection.
ACTIVITY
Graph the given point in its image after each reflection over the given axis. Name the coordinates of the
reflected point:
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H(-3,2), x – axis
G(2, 4), y – axis
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B(-3, -4), y – axis
D(0, - 2), x – axis
RATIOS AND RATES
A ratio is a comparison of two quantities by division.
You can write a ratio three ways:
a:b
a to b
!
"
A rate is a ratio that compares quantities measured in different units, such as miles to gallons or feet to
seconds.
A unit rate is the rate for one unit of given quantity.
ACTIVITY
Write each ratio in simplest form:
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50m: 30m
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22 inches to 3 feet
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6 feet to 6 yards
Find each unit rate:
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36 gals in 12 minutes
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$42 for 3 books
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676 miles in 13 hours
PYTHAGOREAN THEOREM
In a right triangle, the two shortest sides are legs. The longest side, which is opposite the right angle, is the
hypotenuse. The Pythagorean Theorem is an equation that shows the relationship between the legs and the
hypotenuse.
ACTIVITY
a) Find the length of the hypotenuse of a right triangle with legs of 12 cm and 16cm.
b) A television screen is 16 in. high and 22 in. wide. What is its diagonal dimension to the nearest integer?
SOLVING PROPORTIONS
A proportion is an equation stating that two ratios are equal.
The cross-product of two ratios are two products found by multiplying the denominator of each ratio by the
numerator of the other ratio. In a proportion, a cross products are equal.
ACTIVITY
Solve each proportion:
#
"
$
#'
a) = b)
c)
((
)
,
(.#
=
**
=
.*
+
#
2
)
1$
e) = f) 5 is to 8 as 15 is to w
g) 10 is to 7 as m is to 10.5
$./
h) 30 is to 16 as j is to 8
d)
*
(.
=
0
1
FRACTIONS, DECIMALS AND PERCENTS.
A percent is a ratio that compares a number to 100.
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To write a number as a percent you must move the decimal point two times to the right (in other words,
you multiply by 100)
To write a percent as a number you must move the decimal two times to the left (in other words you
must divide by 100)
To write a percent as a fraction in simplest form:
o If the percent is a whole number just have it like #/100 and simplify is possible.
o If the percent is a decimal, first you must write is as a decimal (move the decimal point two
times to the left). Then write the decimal part as the numerator, and the denominator will be the
last place value you have. For example: 0.25% -> 0.0025 -> 25/10,000
Write each decimal as a percent:
a) 0.39 __________
c) 0.08 __________
b) 0.5 __________
d) 0.056 __________
Write each fraction as a percent:
a) 3/4 __________
c) 1/5 __________
b) 5/8 __________
d) 7/10 __________
Write each percent as a decimal:
a) 45% __________
b) 150% __________
c) 90% __________
d) 0.2% __________
A cereal supplies 1.25% of the RDA calcium. Write 1.25% as a fraction in simplest form.
PERCENTS AND PROPORTIONS
ACTIVITY
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What percent is 21 of 50?
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45 is what percent of 65?
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83 is 70% of what number?
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What is 45% of 72?
Beth and her mom picked a bushel of red and yellow delicious apples, totaling 52 apples. They made tarts with
8 yellow apples. After making tarts, 25% of the remaining apples were yellow. What percent of the original
apples were red? Round to the nearest tenth of a percent.
6.1 Solving two-step equations p. 261
To solve two-step equations remember that the first step undoes the addition or subtraction; the second
undoes the multiplication or division.
ACTIVITY
Solve each equation:
a) 4r + 6 = 14
b) 9y – 11 = 7
c) m/4 + 6 = 3
e) 4.6b + 26.8 = 50.72
d) v/-7 + 8 = 19
f) -2.06d + 18 = -10.84
Hugo received $100 for his birthday. He then saved $20 per week until he had a total of $460 to buy a printer.
Use an equation to show how many weeks it took him to save money.
You spent $10.50 at the fair. If it costs $4.50 for admission and you rode 8 rides which all cost the same, how
much does one ride ticket cost?
SIMPLIFYING ALGEBRAIC EXPRESSIONS
A term is a number, a variable, or the product of a number and one or more variables.
Like terms are terms that have exactly the same variable factors.
ACTIVITY
Combine like terms:
a) 9j + 34j
c) 5t – 12t + 17t
b) 23s – 12s
d) 6q + 14q – 8q
Simplify each expression:
a) 4a + 7 + 2a
d) 4.3(5.6 + c)
b) 8(k – 9)
e) 4(m + 6) – 3
c) (w + 3)7
f) -9 + 8(x + 6)
Tyrone bought 15.3 gal of gasoline prices at g dollars per gallon, 2qt of oil priced at q dollars per quart, and a
wiper blade priced at $3.79. Write an expression that represents the total cost of these items.
Karen buys 4 boxes of cereal and 3 bags of almonds at the grocery store. Her brother, David, buys 2 boxes of
cereal. Define and use a variable to represent the total cost.
SOLVING MULTI-STEP EQUATIONS
Steps:
1.
2.
3.
4.
Solve the distributive property.
Combine like-terms.
Undo addition or subtraction.
Undo multiplication or division.
ACTIVITY
Solve each equation:
a) 2(2.5b – 9) + 6b = -7
c) 0.7w + 16 + 4w = 27.28
b) 24 = -6(m + 1) + 18
d) 20 = -4(f + 6) + 14
You want to join the tennis team. You go to the sporting goods store with $100. If the tennis racket you want
costs $80 and the tennis balls cost $4 per can, how many cans can of tennis balls can you buy?
Johnny wants to ship a package to his friend. A shipping company charges $2.49 for the first pound and $1.24
for each additional pound. If it cost Johnny $11.17 to ship the package, how much did his package weigh?
You buy 4 Cortland apples and some Gala apples. Each variety of apples cost $1.20/lb. You can spend $7.20.
How many pounds of Gala apples can you buy?
SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES
To solve an equation with variables on both sides, bring all the variable terms to one side of the equation.
Remember to use the distributive property to simplify an equation before you can bring the variable terms to
one side.
ACTIVITY
Solve each equation:
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2 + 14z = -8 + 9z
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7m = 9(m + 4)
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-8 – 5y = 12 – 9y
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8(4 – a) = 2a
At Video Shack, movie rentals cost $3.99 each. The cost of renting three movies and one video game is $0.11
less than the cost of renting five video games. How much does renting a video game cost?
A croquet ball weighs 460 gams. Together a golf ball and a croquet ball weigh the same as 11 golf balls. How
much does one golf ball weigh?
SOLVING INEQUALITIES (ADDING/SUBTRACTING)
An inequality is a mathematical sentence that contains <, >, ≤, ≥,or ≠.
ACTIVITY
Solve each inequality. Graph your solution:
a) m + 6 > 2
c) y – 3 ≥ - 4
b) q + 4 < 9
d) w – 6 ≤ - 9
The amount of snow on the ground increased by 8 inches between 7PM and 10PM. By 10PM, there was less
than 2ft of snow. How much snow was there by 7PM?
The school record for points scored in a basketball season by one player is 462. Maria has 235 points so far
this season. How many more points does she need to break the record?
SOLVING INEQUALITIES (MULTIPLYING/DIVIDING)
When you multiply or divide each side of an inequality by a positive number, the relationship between the two
sides does not change.
When you multiply or divide each side of an inequality by a negative number, reverse the direction of the
inequality sign.
ACTIVITY
Solve each inequality. Graph the solution:
a) y/-4 > 3
c) y/2 > 0
b) -5b < 15
d) 2b > 8
Write an inequality for each problem. Solve the inequality.
a) Dom wants to buy 5 baseballs. He has $20. What is the most each baseball can cost?
b) A typing service charges $5.00 per page. Mrs. Garza does not want to spend more than $50 for the typing.
What is the maximum number of pages she can have typed?
c) The tables at a restaurant can each sit 8 people. A dinner at the restaurant will be attended by 125 people.
How many tables does the restaurant need in order for every person at the dinner to have a seat?
MEAN, MEDIAN, MODE, RANGE.
A measure of central tendency is a single value that summarizes how a set of data is centered. Mean,
median, and mode are measures of central tendency.
The mean is the sum of the data values divided by the number of data items.
The median is the middle value when the data values are arranged in numerical order. For an even number of
data values, the median is the mean of the two middle numbers.
The mode is the item with the greatest frequency. A data set may have no mode, one mode, or more than one
mode.
The range of a set of data is the difference between the greatest and least values in the set. Range is a
measure of how spread out the data in a set are.
If one data item is much higher or lower than the other data items, it is an outlier.
ACTIVITY
Find the mean, median, mode, and range of each data set.
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0, 0, 0, 0, 1, 1, 2, 2, 3
7, 8, 8, 9, 9, 9, 9, 10, 10
0, 1, 1, 2, 2, 3, 3, 3, 4, 6
VENN DIAGRAMS
A Venn Diagram is a diagram that uses regions, usually circles, to show how sets of numbers or objects are
related.
ACTIVITY
A softball team has 18 players. Fourteen players bat right-handed. Two players can bat left-or-right-handed.
Four players bat only left-handed. Draw a Venn diagram for this situation.
Draw a Venn diagram for each situation:
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18 students play a sport
15 students are in the band
11 do both activities
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75 people are downhill skiers
51 people are cross-country skiers
26 ski both downhill and cross-country
SURFACE AREA OF SOLIDS
Find the Surface Area of the following solids.
A)
B)
LINEAR EQUATIONS
Graph the following linear equations.
a) y = 5x + 2
b) y = - 2x + 4