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Vocabulary
Acute Angle: An angle of less than 90 degrees.
Angle: The amount of space where two lines meet.
Area: The number of square units that covers a certain space.
Average: A value that lies within a range of values.
Circumference: The distance around a circle.
Congruent Shapes: Identical geometric shapes, usually facing in different directions.
Cubic Unit: A unit with six equal sides, like a child's block.
Customary System: Measures length in inches and feet capacity in cups and pints, weight in
ounces and pounds and temperature in Fahrenheit.
Data: Gathered information (datum-singular).
Decimal: A number that includes a period called a decimal point. The digits to the right of
the decimal point are a value less than one.
Denominator: The bottom number in a fraction.
Diameter: The length of a line that divides a circle in half.
Digit: A numeral.
Dividend: The number to be divided in a division problem.
Divisor: The number used to divide another number.
Equation: A number sentence in which the value on the left of the equal sign must equal the
value on the right.
Equilateral Triangle: A triangle with three equal sides.
Equivalent Fractions: Fractions that name the same amount, such as ½ and 5/10.
Estimating: Using an approximate number instead of an exact one.
Expanded Notation: Writing out the value of each digit in a number.
Fraction: A number that names part of something.
Geometry: The study of lines and angles, the shapes they create and how they relate to one
another.
Greatest Common Factor (GCF): The largest number that will divide evenly into a set of
numbers.
Improper Fraction: A fraction that has a larger numerator than its denominator.
Integers: Numbers above or below zero: -2, -1,0, + 1, +2, and so on.
Intersecting lines: At least two straight lines that cross each other's paths.
Isosceles Triangle: A triangle with two equal sides.
Least Common Multiple (LCM): The lowest possible multiple any pair of numbers have
in common.
Line: A series of continuous points in a straight path, extending in either direction.
Line Segment: A straight line extending from one exact point to another.
Mean: The average of a group of numbers.
Median: The number in the middle when numbers are listed in order.
Metric System: Measures length in meters, capacity in liters, mass in grams and temperature
in Celsius.
Mixed Number: A whole number and a fraction, such as 1 ¾ .
Numerator: The top number in a fraction.
Obtuse Angle: An angle of more than 90 degrees.
Opposite Integers: Two integers the same distance from 0 but in different directions, such as
-2 and +2.
Ordered Pairs: Another term used to describe pairs of integers used to locate points on a
graph.
Parallel lines: Lines that never get closer together or farther apart at any point.
Percent: A kind of ratio that compares a number with 100.
Perimeter: The distance around a shape formed by straight lines, such as a, square or
triangle.
Perpendicular Lines: Two lines that intersect each other at a 90-degree angle.
Place Value: The position of a digit in a number.
Probability: The ratio of favorable outcomes to possible outcomes in an experiment.
Proportion: A statement that two ratios are equal.
Quadrilateral: A shape with four sides and four angles.
Quotient: The answer in a division problem.
Radius: The length of a line from the center of a circle to the outside edge.
Range: The difference between the highest and lowest number in a group of numbers.
Ratio: A comparison of two quantities.
Ray: A straight line extending in one direction from one specific point.
Reciprocals: Two fractions that, when multiplied together, make 1, such as 2/7 and 7/2.
Right Angle: An angle of 90 degrees.
Rounding: Expressing a number to the nearest whole number, ten, thousand or other value.
Scalene Triangle: A triangle with no equal sides.
Similar Shapes: The same geometric shape in differing sizes.
Straight Angles: An angle of 180 degrees.
Symmetrical Shapes: Shapes that, when divided in halt are identical.
Vertex: The point at which two lines intersect.
Volume: The number of cubic units that fills a space.
X Axis: The horizontal number line in a plotting graph.
X Coordinate / Y Coordinate: Show where a point is on a plotting graph.
Y Axis: The vertical number line in a plotting graph.
Place Value
Every digit in a number belongs in a certain place. Each place has a different value.
This is a place value chart:
We can use a place value chart to identify the
value of a digit in a number. For example, if we
put a 5 in the thousands place, the value is “five
thousand” or “5,000”.
Example: in the number 12,345.678 the value of
the 7 is .07 or “seven hundredths” because it is in
the hundredths place.
Practice:
Use the number 87,981,067.3845 to answer the following questions
1) What digit is in the tens place? _____
2) What digit is in the millions place? ____
3) The “units” place is also called the “ones” place. What digit is in the ones place?
_____
4) What is the value of the 3? ________________________________
5) Create the number with the greatest value using only the digits 1, 9, 4, 0, 5 and 2.
Write the number that is 100 less and 100 more than the given number.
Example:
123 223 323
6)
________
1,341 ________
7)
________ 12,987 ________
8)
________
922 ________
9)
________
555 ________
10) Write a number that has a 5 in the hundreds place and a 2 in the hundredths
place:
Rounding
To get an approximate answer instead of an exact answer we can round numbers.
Numbers can be rounded to various places, depending on how precise an answer needs
to be. The ability to round numbers can be applied when doing mental math or quick
calculations.
When rounding, the digit to the right of the requested place tells us if we should round up
or down. If the digit to the right is 5 or more, we round up. If the digit is 4 or less we
round down. Remember the phrase “5 or more, let it soar. 4 or less, let it rest”
Example: to round 485 to the nearest tens, we look at the digit to the right of the tens
place, and the 5 tells us to round up to 490 (5 or more, let it soar).
Estimate the following sums by rounding first, then adding. You can round to
whichever place is reasonable for each problem.
1)
1,082 + 98 = ____________ + ____________ = _______________
2)
2,612 + 4,411 = ____________ + ____________ =______________
3)
822 + 922 = ____________ + ____________ = ______________
4)
10,309 + 17,984 = ____________ + ____________ = _____________
Round each number to the nearest whole number (ones place).
Example: 76.3  76 (76.3 is approximately 76)
5)
12.5  ________
6)
907.99  ________
7)
11.11  ________
8)
12.01  ________
9)
5,532.45  ________
10)
12,040.87  ________
Round each number to the nearest hundred.
Example: 11,584  11,600 (the 8 tells the 5 to go up to a 6)
11)
34,765  ________
12)
109,023  ________
13)
97  ________
14)
1,395,721  ________
Patterns
Patterns are numbers, designs, or objects that repeat in the same way. To find a pattern,
look for the change happening to each number or shape, and then repeat for the
following shapes.
Find the pattern that is happening to the following design. Using that pattern, what
would the next image be?
1)
Next design:
The pattern below is “add two, then times two”. Using that pattern, what would the
next 3 numbers be?
2)
1, 3, 6, 8, 16, 18, 36, _____, _____, _____
Identify the patterns shown below. Using that pattern, what would the next 2
numbers be in each series?
3)
3, 5, 9, 17, 33
Pattern: ___________________
Next two numbers: __________, __________
4)
100, 97, 94, 91, 88, 85
Pattern: ___________________
Next two numbers: __________, __________
5)
4, 6, 10, 18, 34
Pattern: ___________________
Next two numbers: __________, __________
Order of Operations
Order of Operations helps us to solve problems the same way so that we get the same answer. To
remember the correct order, use the phrase “Please Excuse My Dear Aunt Sally… Let’s Rock!”
P
E
MD*
AS*
*LR
parenthesis ()
exponents
multiplication or division
addition or subtraction
left to right (applies to multiplication/division
and addition/subtraction)
Example
6 x 8 – (4 + 2) + 32
6 x 8 – 6 + 32
6x8–6+9
48 – 6 + 9
42 + 9
51
Use order of operations to simplify:
1) 15 – 4 x 2 + 3
2) (3 + 9) – 7 + (10  2)
3) 7 + 8 x (5 + 3) – 1
4) 3  (5 – 2) + 36
5) (15 + 9 )  ( 8 – 2)
6) 9 + 3 x 7 – 5
Adding and Subtracting Decimals
To add or subtract decimals, the most important thing to remember is to line up the
decimal points! We line up the decimal points to line up the correct place values
(add the tens to the tens, the hundredths to the hundredths, etc.). Then, add or subtract
as you normally would. The decimal point drops straight down into the answer.
Example:
14.2 – 2.97  14.20  Decimals are lined up
- 2.97
11.23  Decimal lined up in answer
Add the following decimal numbers
1) 4.50 + 1.3 =
2) 122.8 + 19.7 =
3) 56 + 8.76 =
4) 8.99 + 9.394 =
(hint: 56 = 56.0)
Subtract the following decimal numbers
5) 9.8 – 4.2 =
7)
12.8 – 3.9 =
6) 21.44 – 3.87
8) 65.09 – 32.76 =
Application:
9) Alison had $12.50. She earned $8.75 babysitting. How much money does she
have total?
10) Doug had a piece of rope 8.1 feet long. He cut off 1.2 feet of rope to use for a
project. What is the length of his rope now?
Multiplication of Whole Numbers
Remember to regroup if you need to.
Multiply 56 x 14
2
56
x 14
224
+ 560 ← write a 0 to fill ones place
784
Absolutely NO LATICE work.
Complete.
1.
64
x94
2.
46
x74
3.
22
x18
4.
30
x28
5.
89
x12
6.
7.
84
x71
8.
55
x80
9.
45
x92
10.
75
x20
74
x70
Show your work here;
Complete.
49.
2 × (8 × 2 × 6) × 3
Complete.
53. We decided that before the bell sounded
we should count the number of students
present in the cafeteria. There were
sixteen boys. There were twice as many
girls. How many students were there in
all?
50.
4 × (7 × 9 × 1)
12.
Jose' family lives on a 17-acre farm. Carlos'
family's farm is four times as big. How many
acres are in Carlos' family farm?
Division of Whole Numbers
Divide until each digit in the dividend
27
33 893
Has been brought down.
Write remainders as fraction.
= 27 2/33
-66
233
- 231
2
Divide. Write remainders as simplified fractions. Show your work.
1. 2,178  18
2. 156  26
3. 4,833  27
4. 312  78
5. 1,328  83
6. 940  47
7. 176  88
8. 2,135  61
9. 2,623  43
10. 837  93
11. 770  77
12. 2,860  55
Fractions - Identify Shaded Portion
Write a fraction to show how much of the shape is shaded.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Fractions - Draw Shaded Portion
Draw a picture to show the fractional representation.
1. two-tenths
2. 1
3. one-third
6
4. 3
5. 1
6. 4
8
2
5
7. three-fourths
8. six-sevenths
9. five-ninths
10. one-sixth
11. 1
12. 1
4
7
Fractions –Finding Equivalents, Add & Subtract
Complete & Express all results in simplest form, compare denominators first.
1.
4

7 14
2.
3 9

4
3.
21

5 10
4.
1

6 42
5.
6 1
 =
7 7
6.
5 2

=
6 6
7.
3
1
4 2 =
8
8
8.
2
1
7 2 =
3
3
9.
6 2
 =
8 8
10.
3 1
 =
6 6
11.
1
3
3 5 =
4
4
12.
2 1
 =
3 9
15.
2
1
9 6 =
3
6
16.
19.
31 42

=
4
8
20.
13.
4 1

=
5 10
14.
17.
5 5
6  =
6 6
18.
15  3
2
5
=
6.
Ordering Decimals/Plotting on Number line
5
1
3 2 =
9
9
17
5
1
7 =
16
8
4
3
12  10 =
7
7
Use the number line to help order the following 10 numbers from smallest to
largest. First, place each point on the number line and label it. After all the
points have been plotted on the number line, list the numbers in order from
smallest to largest.
5
3
4
6
7
Multiplying Fractions/Mixed Numbers
8.
7.
The local pizza parlor is giving away a pizza to anyone who can answer all of these
multiplication problems. Marcos wants to win. Can you give him a helping hand to find the
correct answers? Write the answers in the simplest form.
1.
1
2
x
5
3
2.
1
3
x
4
7
3.
1
3
x
2
8
4.
3
3
x
7
4
Remember:
To multiply fractions, multiply the
numerators, then multiply the
denominators.
1
2
x
=?
2
3
1
2
2
x
=
2
3
6
Reduce
5.
5
1
x
9
3
6.
12
x
3
7.
42
x
1
2
8.
33
x
26
9.
12
x
23
10 4 1
x
31
5
7
5
5
2
3
4
7
4
2
2
to simplest terms.
6
2
=?
6
9.
Division of Fractions
Complete each division problem below. Then write the corresponding letter on the line in front of each
problem. The letters will spell out a tongue twister. Try to say it quickly six times.
___S__
3
1
÷
=
4
2
_____
1
2
÷
= _____
2
3
_____
4
3
÷ = _____
5
5
_____
2
5
÷
= _____
3
6
_____
4
1
÷
= _____
7
2
_____
2
2
÷
= _____
4
3
_____
8
3
÷ = _____
10 5
_____
3
1
÷ = _____
4
3
_____
6
1
÷ = _____
8
3
_____
11
2
4
2
÷
= _____
5
5
11.
10.
12.
Range/Mean/Median/Mode
Find the range.
1. 5, 7, 8, 8, 15,23
Find the mean.
7. 5, 7, 8, 8, 15, 23
2. 42, 48, 53, 54, 57,59,60,61,61
8. 42, 48, 53,54, 57, 59, 60, 61, 61
3.22,23,26,31,38,41,45,62
9. 22, 23, 26, 31, 38,41, 45, 62
Remember:
Range is the difference between the
greatest and the least number in a
set of data.
21,15,27,12, 20
Find the median.
4.5,7,8,8,15,23
5. 42,48,53,54, 57, 59, 60,61,61
Remember:
Mean is the average of a set of data.
Add the numbers, and then divide the
sum of the numbers by the number of
addends.
21 + 15 + 27 + 12 + 20 = 95
95 ÷ 5 = 19
The mean is 19.
Find the mode.
10.5, 7, 8, 8, 15, 23
6.22, 23, 26, 31, 38, 41,45,62
11.42, 48, 53, 54, 57, 59, 60, 61, 61
Remember:
Median is the middle number in a
set of data. Arrange the numbers
from least to greatest. The middle
number is the median.
12,15,20,21,27
The median is 20.
12. 22, 23, 26, 31, 38, 41, 45, 62
Remember:
Mode is the number that appears
the most often in a set of data.
Some sets have no mode.
21,15,27,12,20
There is no mode.
23,18,6,15,6
The mode is 6.
Number Lines/Mean/Median./Mode/Range
Probability/Coordinate Points
Fill in the circle next to the
correct answer.
Use this number line for numbers 1 through 4.
A
B
C
D
6. What is the probability of drawing a
white marble at random from the bag?
A. 1/2
B. 1/3
C. 2/11
D. 2/9
7. What is the probability of drawing a black marble at
random from the bag?
A. 1/2
B. 9/11
C. 3/6
D. 3/4
1. Which point is located at -2?
A. point A
C. point C
B. point B
D. point D
For numbers 8 through 11, use the following data:
13.
14. Which point is located at 3?
A. point A
C. point C
B. point B
D. point D
3. Which point is located at 1?
A. point A
C. point C
B. point B
D. point D
4. Which point is located at -3.5?
A. point A
C. point C
B. point B
D. point D
8. What is the mean of the data set?
A. 19
B. 30
C. 31
9. What is the range of the data set?
A. 25
B. 10
C. 15
10. What is the mode of the data set?
A. 29
C. both 29 and 35
B. 35
D. there is no mode
11. What is the median of the data set?
A. 29
B. 30
C. 32
5. Draw a number line and number it from -3 to
+3, with 0 right in the middle.
Write an S on the value of -1 and a W
on the VALUE OF 2.
15.
For numbers 6 and 7, use this bag of marbles.
12. Plot point A at (-2, 1) and point B at
(0, -2) on this graph.
D. 32
D. 35
D.
Graphing Methods
Tables and different kinds of graphs have different purposes. Some are more helpful for
certain kinds of information. The table and three graphs below all show basically the same
information-the amount of money Mike and Margaret made in their lawn-mowing business
over a 4-month period.
5.
6.
Combined Income per Month
Margaret
Combined Income
per Month
75
70
65
60
55
June
July
Aug
Sept
7.
Combined Income per Month
Directions: Study the graphs and table.
Then circle the one that answers each
question below.
75
70
65
60
55
June
July
Aug
Sept
1. Which one shows the fraction of the total income that Mike and Margaret made in
August?
table
line graph
bar graph
circle graph
2. Which one compares Mike's earnings with Margaret's?
table
line graph
bar graph
circle graph
3. Which one has the most exact numbers?
table
line graph
bar graph
circle graph
4. Which one has no numbers?
table
line graph
bar graph
circle graph
8. Which two best show how Mike and Margaret's income changed from month to
month?
table
line graph
bar graph
circle graph
Function Tables
Complete each of the following function tables using the given rule:
1.
2.
3.
-
Rule = +27
Input
Rule = 15
Output
Input
Rule = +4 - 3
Output
Input
1
25
4
11
19
15
16
15
23
23
13
34
4.
5.
Rule = x2 + 3
Input
6.
Rule = ÷2 + 1
Output
Input
2
4
4
16
9
Output
Input
Rule = x3 - 12
Output
Output
19
40
8
38
7.
Input
Rule = x3 - 5
13
15
Output
1
9.
8.
Rule = ÷3 - 2
Input
Output
Rule = x5 + 1
Input
Output
12
12
16
8
15
21
3
3
5
39
41
51
Measurement
Fill in the answer
Convert each measure to inches.
1.
3.
5.
7.
3 ft = _______ in
8 yd = _______ in
2 ft = _______ in
5 yd = _______ in
2.
4.
6.
8.
7 in 8 yd = _______ in
8 yd 1 in = _______ in
5 in 2 ft = _______ in
3 ft 2 in = _______ in
Convert each measure to feet.
9. 2 ft 8 yd = _______ ft
10. 3 mi = _______ ft
11. 108 in 8 yd = _______ ft
12. 4 yd = _______ ft
13. 1 mi 10 yd = _______ ft
14. 9 yd = _______ ft
Convert each measure to feet and inches
15. 39 in 3 yd = _______ ft _______ in
16. 102 in = _______ ft _______ in
17. 13 in 1 yd = _______ ft _______ in
18. 87 in = _______ ft _______ in
19. 3 mi 102 in = _______ ft _______ in
20. 85 in = _______ ft _______ in
Geometry Basics
These are some geometry words that you should be familiar with:
ANGLES
Acute angle – an angle that measures greater than 0 but less than 90
Obtuse angle – and angle that measures greater than 90 but less than 180
Straight angle – basically a straight line: measures exactly 180
Right angle – measures exactly 90
LINES
Ray –has one end point, and goes on forever in the other direction
Line – goes on forever in both directions
Line segment – has two endpoints
Practice: Identify the following angles. On the line below each angle, label each one as either “acute”,
“obtuse”, “straight” or “right”.
1)
2)
3)
__________
____________
4)
5)
________
6)
__________
_____________
________
Identify the following images. On the line below each image, label each one as either a “ray”, “line”, or
“line segment”.
7)
8)
9)
________________
_______________
__________
Lines of Symmetry
A line of symmetry is a straight line drawn through a shape that creates mirror images
on both sides. If you fold the shape in half on the line of symmetry, both sides will match
up perfectly. Shapes can have no lines of symmetry, one line of symmetry, or many lines
of symmetry.
Line of symmetry
In this pentagon, there is one line of symmetry.
This line cuts the shape in half so each side is a
mirror image of the other side
Draw as many lines of symmetry as you can in the following shapes. Below each shape, write how
many lines of symmetry you were able to find. If there are no lines of symmetry, write “none”
1)
2)
__________
3)
___________
4)
_________
5)
__________
6)
______________
____________
Graphing Coordinate Points
To graph coordinate points, remember the phrase “run before you jump”. Each coordinate
point is made up of an X coordinate and a Y coordinate. X is horizontal (left to right) and Y
is vertical (up and down). Run left or right to the X coordinate, then jump or fall to the Y
coordinate.
Example: plot the point (-2, 3)
This means run left to –2, then jump up 3
Practice:
On the coordinate plane below, plot each of the following points. As you plot each point, connect it to the
previous point by using a ruler (connect the dots). When you are done connecting all points you should see a
picture of a tired dog. The “start” and “stop” begin and end new lines.
DOG TIRED
Start (4,4) , (5,4) , (6,2) , (0,2) , (1,4) , (4,4) , (4,10) , (1,10) , (4,18) , (21,18) ,
(24,10) , (21,10) , (21,9) , (16,9) , (14,8) , (12,10) , (10,10) , (9,9) ,
(9,7) , (6,5) , (6,3) , (7,2) , (9,2) , (11,4) , (11,6) , (12,7) , (11,8) Stop
Start (14,8) , (15,6) , (15,4) , (14,3) , (11,3) , (10,2) , (17,2) , (17,3) , (16,4) Stop
Start (20,6) , (20,5) , (22,3) , (20,3) , (19,4) , (18,3) , (18,2) , (24,2) , (25,3) ,
(25,6) , (23,8) , (21,9) Stop
Start (11,4) , (12,3) Stop
Start (9,7) , (10,7) Stop
y
10
8
6
4
2
0
1
2
3
4 5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
x