3rd Grade Comprehensive CRCT Math Study Guide
Content Weights: Number Operations-55%, Measurement-23%, Geometry-10%, Algebra-7%, Data Analysis & Probability-5%
Number and Operations: 55% = 33 questions
Number and Operations: Students will use decimal fractions and common fractions to represent parts of a
whole. They will also understand the four arithmetic operations for whole numbers and use them in basic
calculations, and apply them in problem solving situations.
M3N1 Students will further develop their understanding of whole numbers and decimals and ways of
representing them.
a. Identify place values from tenths through ten thousands.
b. Understand the relative sizes of digits in place value notation (10 times, 100 times, 1/10 of a single
digit whole number) and always to represent them including word name, standard form and
expanded form.
Place Value through the ten thousands.
Thousands
tens
ones
2
7
Ones
hundreds
tens
ones
5
6
9
,
Each group of 3 digits separated by a comma in a number is called a period.
vocabulary
digit
place value
standard form
expanded form
word form
definition
Any one of the ten number
symbols
The value of a digit determined
by its place in a number
The usual, or common, way of
writing a number, using digits
A way of writing a number as
the sum of the values of the
digits
A way of writing a number
using words
Whole numbers
Ten
thousands
thousands
7
Standard form
Expanded form
Word form
example
1
,
0,1,2,3,4,5,6,7,8, or 9
The value of the 2 in the number 27,569 is 20,000
27,569
20,000 + 7,000 + 500 + 60 + 9
Twenty-seven thousand, five hundred sixty-nine
(and is only used when there is a decimal point)
hundreds
tens
3
8
Decimals
ones
Tenths Hundredths
1
1
10
100
or
or
1.0
0.1
0.01
2
.
2
1
71, 382.21
70,000+1,000+300+80+2+ .2+ .01
seventy-one thousand, three hundred eighty-two and twenty-one
hundredths
1
M3N2 Students will further develop their skills of addition and subtraction and apply them in problem
solving.
a. Use the properties of addition and subtraction to compute and verify the results of computation.
b. Use mental math and estimation strategies to add and subtract.
c. Solve problems requiring addition and subtraction.
d. Model addition and subtraction by counting back change using the fewest number of coins.
Properties of Addition
Commutative property: When two numbers are added, the sum is the same regardless of the order of the
addends. For example 4 + 2 = 2 + 4
Associative Property: When three or more numbers are added, the sum is the same regardless of the grouping
of the addends. For example (2 + 3) + 4 = 2 + (3 + 4)
Identity Property: The sum of any number and zero is the original number. For example 5 + 0 = 5.
There is an inverse relationship between addition and subtraction.
Example: Since 3 + 7 = 10 then the following are also true:


10 - 3 = 7
10 - 7 = 3
Similar relationships exist for subtraction.
Example: Since 10 - 3 = 7 then the following are also true:


3 + 7 = 10
7 + 3 = 10
An equation is balanced or the same on either side of the equals (=) sign. If exactly the same thing is done to
both sides of the equation, it will still be balanced or equal.
In the example above we start with the equation 3 + 7 = 10:



Subtract the same number from both sides
3 + 7 - 3 = 10 - 3
On the left side the 3 and -3 produce 0 which leaves
7 = 10 - 3
Turning the equation around to be in more normal form
10 - 3 = 7
2
M3N3 Students will further develop their understanding of multiplication of whole numbers and develop
the ability to apply it in problem solving.
a. Describe the relationship between addition and multiplication, i.e., multiplication is defined as
repeated addition.
For example, 3 x 7 = 21, instead you may consider it as 3 x 7 = 7 + 7 + 7 = 14 + 7 = 21. This particular method
is a bit challenging in terms of addition for a child, but it lucidly shows the relationship between multiplication
and addition. Accordingly, you may also calculate 9 x 7 = 10 x 7 - 7 = 63 or 6 x 7 = 3 x 7 + 3 x 7 = 21 + 21 =
42. It is also a creative method to learn the multiplication table, which explains you the exact working of math.
b. Know the multiplication facts with understanding and fluency to 10x10.
×
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2
3
4
5
6
7
8
9
10
11
12
2
2
4
6
8
10
12
14
16
18
20
22
24
3
3
6
9
12
15
18
21
24
27
30
33
36
4
4
8
12
16
20
24
28
32
36
40
44
48
5
5
10
15
20
25
30
35
40
45
50
55
60
6
6
12
18
24
30
36
42
48
54
60
66
72
7
7
14
21
28
35
42
49
56
63
70
77
84
8
8
16
24
32
40
48
56
64
72
80
88
96
9
9
18
27
36
45
54
63
72
81
90
99
108
10
10
20
30
40
50
60
70
80
90
100
110
120
11
11
22
33
44
55
66
77
88
99
110
121
132
12
12
24
36
48
60
72
84
96
108
120
132
144
3
Multiplication by a single digit
23
x 4
23 is 2 tens and 3 ones.
3 ones multiplied by 4 gives 12 ones and
2 tens multiplied by 4 gives 8 tens (that is 80).
80 and 12 are added to give the final product 92.
The 3 ones are first multiplied by 4 giving the product 12, which is 1 ten and 2 ones. 2 is written in the ones
column and the 1 is recorded in the tens column. Now the 2 tens are multiplied by 4 to give 8 tens. The 1 ten
recorded before is added on, so the product has 9 tens.
M3N3
c. Use arrays and area models to develop understanding of the distributive property and to
determine partial products for multiplication of 2- or 3- digit number.
Multiplication can be defined in terms of repeated addition. For example, 3 × 6 can be viewed as 6 + 6 + 6. More
generally, for any positive integer n, n × b can be represented as n × b = b + b + … + b
where the sum on the right consists of n addends.
A rectangular array provides a visual model for multiplication. For example, the product 3 × 6 can be represented as
By displaying 18 dots as 3 rows with 6 dots in each row, this array provides a visual representation of 3 × 6 as 6 + 6 + 6.
An equivalent area model can be made in which the dots of the array are replaced by unit squares.
4
Besides representing 3 × 6 as an array of 18 unit squares, this model also shows that the area of a rectangle with a height
of 3 units and a base of 6 units is 3 × 6 square units, or 18 square units.
Given a pair of numbers a and b called factors, multiplication assigns them a value a × b = c, called their product.
The Commutative Property of Multiplication states that changing the order in which two numbers are multiplied does
not change the product. That is, for all numbers a and b, a × b = b × a.
The array model can be used to make this plausible. For example, because 3 × 6 = 6 × 3, an array with 3 rows and 6 dots
in each row has the same number of dots as an array with 6 rows and 3 dots in each row.
Another important property of multiplication is the Identity Property of Multiplication. It states that the product of any
number and 1 is that number. That is, for all numbers a, a × 1 = 1 × a = a.
The Zero Property of Multiplication states that when a number is multiplied by zero, the product is zero. That is, for all
numbers a, a × 0 = 0 × a = 0.
M3N3
d. Understand the effect on the product when multiplying by multiples of 10.
8x1=8
8x2=16
8x10=80
8x2=160
8x100=800
8x200=1600
8x1000=8,000
8x2000=16,00
e. Apply the identity and commutative and associative properties of multiplication and verify the
results.
Multiplication Property
definition
Example
When you change the order of the
5x4 = 4x5
Commutative Property
factors, the product stays the same.
When you group factors in different
(6 x 7) x 9 = 6 x (7 x 9)
Associative Property
ways, the product stays the same. The
parentheses tell you which numbers to
multiply first.
When two addends are multiplied by a
Distributive Property
factor, the product is the same when
(3 + 5) x 2 = (3 x 2) + (5 x 2)
each addend is multiplied by the factor
and those products are added.
5
Property of One
Zero Property
When you multiply any number by 1,
the product is equal to that number.
When you multiply any number by 0,
the product is 0.
Factor–the number used in a multiplication problem:
Product-the answer in a multiplication problem:
Addend-the number to be added in an addition problem:
6 x 5 = 30
6 x 5 = 30
(3+5) x 2
12 x 1 = 12
178 x 0 = 0
6 and 5 are factors
30 is the product
3 and 5 are addends
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3N4 Students will understand the meaning of division and develop the ability to apply it in problem
solving.
a. Understand the relationship between division and multiplication and between division and
subtraction.
b. Recognize that division may be two situations: the first is determining how many equal parts of a
given size or amount may be taken away from the whole as in repeated subtraction, and the
second is determining the size of the parts when the whole is separated into a given number of
equal parts as in a sharing model
c. Recognize problem-solving situations in which division may be applied and write corresponding
mathematical expressions.
When you divide to find the number of objects in each group, the division is called fair sharing or partitioning. For
example:
A farmer is filling baskets of apples. The farmer has 24 apples and 4 baskets. If she divides them equally, how many
apples will she put in each basket?
When you divide to find the number of groups, the division is called measuring or repeated subtraction. It is easy to see
that you can keep subtracting 4 from 24 until you reach zero. Each 4 you subtract is a group or basket.
A farmer has 24 apples. She wants to sell them at 4 apples for $1.00. How many baskets of 4 can she fill?
Manipulatives and visual aids are important when teaching multiplication and division. Students have used arrays to
illustrate the multiplication process. Arrays can also illustrate division.
6
Since division is the inverse, or opposite, of multiplication, you can use arrays to help students understand how
multiplication and division are related. If in multiplication we find the product of two factors, in division we find the missing
factor if the other factor and the product are known.
In the multiplication model below, you multiply to find the number of counters in all. In the division model you divide to find
the number of counters in each group. The same three numbers are used. The model shows that division “undoes”
multiplication and multiplication “undoes” division. So when multiplying or dividing, students can use a fact from the
inverse operation. For example, since you know that 4 x 5 = 20, you also know the related division fact 20 ÷ 4 = 5 or 20 ÷
5 = 4. Students can also check their work by using the inverse operation.
Notice that the numbers in multiplication and division sentences have special names. In multiplication the numbers you
multiply are called factors; the answer is called the product. In division the number being divided is the dividend, the
number that divides it is the divisor, and the answer is the quotient. Discuss the relationship of these numbers as you
explain how multiplication and division are related.
There are other models your students can use to explore the relationship between multiplication and division. Expose your
students to the different models and let each student choose which model is most helpful to him or her. Here is an
example using counters to multiply and divide.
7
factor
4
number of
groups
dividend
12
total number
of counters
x
factor
3
counters in
each group
÷
divisor
4
number of
groups
=
product
12
total number of
counters
=
quotient
3
counters in
each group
Here is an example using a number line.
factor
4
x
factor
5
=
product
20
dividend
20
÷
divisor
5
=
quotient
4
Another strategy your students may find helpful is using a related multiplication fact to divide. The lesson Relating
Multiplication and Division focuses on this strategy. Here is an example.
18 ÷ 6 = ?
Think: 6 x ? = 18 Six times what number is 18?
6 x 3 = 18,
so 18 ÷ 6 = 3.
When students understand the concept of division, they can proceed to explore the rules for dividing with 0 and 1. Lead
students to discover the rules themselves by having them use counters to model the division. A few examples follow.
8
Divide 4 counters into 4 groups.
4÷4=1
Divide 2 counters into 2 groups.
2÷2=1
When any number except 0 is divided by itself, the quotient is 1.
Put 3 counters in 1 group.
Put 5 counters in 1 group.
3÷1=3
5÷1=5
When any number is divided by 1, the quotient is that number.
Divide 0 counters into 2 groups.
0÷2=0
Divide 0 counters into 4 groups.
0÷4=0
When 0 is divided by any number except 0, the quotient is 0.
Divide 6 counters into 0 groups.
Divide 1 counter into 0 groups.
You cannot divide a number by 0.
9
Encourage students to think about the relationship between multiplication and division when they solve problems. For
example, they can use a related multiplication fact to find the unit cost of an item—for example the cost of one baseball
cap priced at 3 for $18.
$18 ÷ 3 = $6
Think: 3 x ? = $18
3 x $6 = $18
So $18 ÷ 3 = ?
The cost is $6 for one baseball cap.
Dividing by Repeated Subtractions
The result of division is to separate a group of objects into several equal smaller groups. The starting group
is called the dividend. The number of groups that are separated out is called the divisor. The number of objects
in each smaller group is called the quotient.
ex: 24 divided by 6 (24 / 6 = 4)
24 is the Dividend
6 is the divisor
and the quotient is 4
Which means 24 can be divided into 4 equal parts of 6 each.
The results of division can be obtained by repeated subtraction. If we are separating 24 objects into 4 equal
groups of six, we would take (or subtract) six objects at a time from the large group and place them in 4 equal
groups. In mathematical terms this would be: 24-6-6-6-6.
M3N4
d.
e.
f.
g.
Explain the meaning of a remainder in division in different circumstances.
Divide a 2- and 3- digit number by a 1- digit divisor.
Solve problems requiring division.
Use mental math strategies to divide.
If you divide 13 bananas evenly between
Joe and Sally, how much does each one
get?
13 ÷ 2 = ?
We say that Joe and Sally both get 6 bananas and one is left over. The leftover banana is
called the remainder. Or, if we don't want leftovers or remainders, both would get 6 1/2
bananas.
13 ÷ 2 = 6, remainder 1.
10
14 bananas divided between 3 people
gives 4 bananas to each and 2 bananas
that cannot be divided.
14 ÷ 3 = 4,
remainder 2
8 scissors divided between 5 people
gives 1 scissors to each and 3 scissors
that cannot be divided.
8 ÷ 5= 1,
remainder 3
Division Steps
There are five steps of dividing to remember.
1.
Divide.
2.
Multiply.
3.
Subtract.
4.
Compare.
5.
Long Division
4 -- Quotient
Divisor -- 3 |12 -- Dividend
Bring down the next number.
11
1
6|84
1
1. Divide 84 ÷ 6 is the problem. Look at the first number only in the dividend so the
problem is 8 ÷ 6. The
answer is 1. Write it above the 8.
2.
Multiply. 1 x 6 = 6. Write the number 6 under the 8.
6|84
6
3.
1
4.
6|84
-6
2
5.
Subtract. 8 - 6. The answer is 2. Write it down under the 6
Compare what is left over after subtracting with the divisor. It must be less
than the divisor, if not, then go back to step 1 and choose a larger number to
multiple.
Bring down the 4. Then go through steps 1 through 4 again. 24 ÷ 6 = 4.
Multiple. Subtract. Compare. There is not another number to bring down.
The final answer is 14. 84 ÷ 6 = 14.
14
6|84
-6
24
-24
0
M3N5 Students will understand the meaning of decimal fractions and common fractions in simple cases
and apply them in problem-solving situations.
a. Identify fractions that are decimal fractions and/or common fractions.
b. Understand that a common fraction (i.e., 3/10) can be written as a decimal (i.e. 0.3)
8
8=
One decimal digit; one 0 in the denominator.
10
8
.08 =
Two decimal digits; two 0's in the denominator.
100
8
.008 =
1000 Three decimal digits; three 0's in the denominator
12
c. Understand the fraction a/b represents an equal sized parts of a whole that is divided into b equal
sized parts.
d. Know and use decimal fractions and common fractions to represent the size of parts created by
equal divisions of a whole.
To show half, draw 2 equal boxes to represent the total number of equal parts and shade 1 part out of the
2 equal parts.
Similarly, to show one-third, draw 3 equal boxes to represent the total number of equal parts and shade 1
part out of the 3 equal parts.
To show three-quarters, draw 4 equal boxes to represent the total number of equal parts and shade 3
parts out of the 4 equal parts.
e. Understand the concept of addition and subtraction of decimal fractions and common fractions
with like denominators.
f. Model addition and subtraction of decimal fractions and common fractions with like
denominators.
g. Use mental math and estimation strategies to add and subtract decimal fractions with like
denominators.
h. Solve problems involving decimal fractions and common fractions with like denominators.
ADDING OR SUBTRACTING FRACTIONS WITH THE SAME DENOMINATORS
Adding or subtracting fractions with the same denominator is easy. All you have to do is to add or subtract
the numerators.
And it is always a good idea to make your result "nice" by converting it to a mixed number and simplifying
if possible.
13
Example:
Find
1
2
+
5
5
Both fractions have the same denominator of 5, so we can simply add the numerators:
1
2
3
+
=
5
5
5
1
5
+
2
5
=
3
5
Example:
Find
7
3
8
8
Both fractions have the same denominator of 8, so we can simply subtract the numerators:
7 3 4
1
- =
=
8 8 8
2
NOTE that the result was simplified (the numerator and the denominator divided by 4).
7
8
-
3
8
=
4
8
=
1
2
14
Measurement: 23% =14 questions
Students will understand and measure time and length. They will also model and calculate perimeter and
area of simple geometric figures.
M3M1 Students will further develop their understanding of the concept of time by determining elapsed
time of a full, half and quarter-hour.
Start
End
Elapsed Time?
0:00
Time
Time
Start
End
Elapsed Time?
1:30
Time
Time
Start
End
Elapsed Time?
1:00
Time
Time
Start
End
Elapsed Time?
4:30
Time
Time
15
Start
End
Elapsed Time?
1:00
Time
Time
Start
End
Elapsed Time?
2:15
Time
Time
Start
End
Elapsed Time?
2:15
Time
Time
Start
End
Elapsed Time?
2:30
Time
Time
16
Start
End
Elapsed Time?
2:00
Time
Time
Start
End
Elapsed Time?
4:00
Time
Time
Start
End
Elapsed Time?
0:00
Time
Time
Start
End
Elapsed Time?
2:00
Time
Time
M3M2 Students will measure length choosing appropriate units and tools.
a. Use the units, kilometer (km) and mile (mi.) to discuss the measure of long distances.
b. Measure to the nearest ¼ inch, ½ inch and millimeter (mm) in addition to the previously
learned inch, foot, yard, centimeter, and meter.
17
c. Estimate length and represent it using appropriate units.
d. Compare one unit to another within a single system of measurement.
Units of Length
When measuring length in the customary system the common units used are inches, feet, yards, and miles. Units of length measure height, width,
length, depth, and distance.
People use "feet" to measure their height. Construction worker and architects use feet to measure walls, floors, and ceilings. It is also used to determine
one point to another.
The end of your thumb to your first joint is about 1 inch long.
An egg carton is about a foot long.
A standard doorway is about a yard in width. A man six-feet tall is 2 yards high.
Distances between cities are measured in miles
From smallest to greatest length measurements are compared as:
Inches
Feet
Yards
Miles
Reading a Ruler
When reading a ruler, you want to find out how far the item is from zero. When reading a ruler, you must locate the zero marking, this may vary
depending on the ruler. On some rulers, the zeros start at the end, but on others it starts about 1/4 of an inch from the end. If you do not check your ruler
beforehand you will not be getting an accurate measurement.
Point A is 1 1/4 inches or 1 1/4 inches away from zero.
Point B is 3 3/4 inches or 3 3/4 inches away from zero.
Point C is 4 1/8 inches or 4 1/8 inches away from zero.
18
Customary
Metric
inch - in
foot - ft
yard - yd
mile - mi
1 ft = 12 in
1 yd = 36 in
1 yd = 3 ft
1 mi=63,360 in
1 mi = 5280 ft
1 mi=1760 yd
centimeter cm
decimeter dm
meter - m
kilometer - km
1 cm=10 mm
1 dm=100
mm
1 dm=10 cm
1 m= 1000
mm
1 m=100 cm
1 m=10dm
1 km=1,000,000
mm
1 km=100,000
cm
1 km=10,000 dm
1 km=1000 m
A paperclip is
approximately
1 centimeter
wide.
A crayon is
approximately
1 decimeter
long.
The distance
from the floor
to the door
knob is
approximately
1 meter.
The length of 6
city blocks is
approximately 1
kilometer long.
Length
The distance
between the
knuckles on
your index
finger is
approximately
1 inch.
ounce - oz
Your
notebook is
approximately
1 foot tall.
A baseball bat
is
approximately
1 yard long.
Long distances
are measured in
miles. The
distance a
vehicle travels
is measured in
miles.
pound - lb
ton - T
1 lb = 16 oz
1 T = 32,000 oz
1 T = 2000 lb
gram
kilogram
1 kilogram = 1000 grams
Weight
A slice a
bread
weighs
about 1
ounce.
cups - c
A loaf of bread weighs
about 1 pound.
A car weighs about 1
ton.
pint - pt
quarts - qt
gallon - gal
1 pt = 2 c
1 qt = 4 c
1 qt = 2 pt
1 gal = 16 c
1 gal = 8 pt
1 gal = 4 qt
Oil comes in a
quart-sized
container.
A large
container of
milk comes in
a 1 gallon
container.
A packet of sugar
weighs about 1
gram.
A car weighs
about 1000
kilograms.
A book weighs about 1
kilogram.
milliliter - ml
liter - l
1 liter =
1000 milliliters
Capacity
A small cup of
coffee holds
approximately
1 cup.
A tall glass of
lemonade or a
bowl of soup
holds
approximately
1 pint.
A bottle of cola is one liter. A
very large bottle holds 3 liters.
Ten drops from a medicine
dropper is approximately 1
milliliter.
When converting any unit of measurements if you want:


To change to a larger unit, divide.
To change to a smaller unit, multiply.
19
M3M3 Students will understand and measure the perimeter of geometric figures.
a. Understand the meaning of linear unit and measurement in perimeter.
b. Understand the concept of perimeter as being the length of the boundary of a geometric figure.
c. Determine the perimeter of a geometric figure by measuring and summing the lengths of the sides.
M3M4 Students will understand and measure the area of simple geometric figures (squares and
rectangles).
a. Understand the meaning of the square unit and measurement in area.
b. Model (by tiling) the area of a simple geometric figure using square units (square inch, square
foot, etc.)
c. Determine the area of squares and rectangles by counting, addition, and multiplication with
models.
\Surface
Area and Perimeter of a Rectangle
The area of other squares can be found by counting squares or by multiplying the length of the sides.
20
The perimeter of a square is the total length of the four sides.
Area of a Square
If l is the side-length of a square, the area of the square is l2 or l × l.
What is the area of a square having side-length 3.4?
The area is the square of the side-length, which is 3.4 × 3.4 = 11.56.
Area of a Rectangle
The area of a rectangle is the product of its width and length.
What is the area of a rectangle having a length of 6 and a width of 2.2?
The area is the product of these two side-lengths, which is 6 × 2.2 = 13
21
Geometry: 10% =6 questions
Students will further develop their understanding of characteristics of previously studied geometric figures.
M3G1 Students will further develop their understanding of geometric figures by drawing them. They
will also state and explain their properties.
M3G1a Draw and classify previously learned fundamental geometric figures and scalene, isosceles, and
equilateral triangles.
There are three special names given to triangles that tell how many sides (or angles) are equal.
Equilateral Triangle
Three equal sides
Three equal angles, always
60°
Isosceles Triangle
Two equal sides
Two equal angles
Scalene Triangle
No equal sides
No equal angles
Triangles can also have names that tell you what type of angle is inside:
Acute Triangle
All angles are less than 90°
Right Triangle
Has a right angle (90°)
Obtuse Triangle
Has an angle more than 90°
22
M3G1b. Identify and compare the properties of fundamental geometric figures
c. Examine and compare angles of fundamental geometric figures
The Rectangle
means "right angle"
and
show equal sides
A rectangle is a four-sided shape where every angle is a right angle (90°). Also opposite sides are
parallel and of equal length.
The Rhombus
A rhombus is a four-sided shape where all sides have equal length. Also opposite sides are parallel and
opposite angles are equal. Another interesting thing is that the diagonals (dashed lines in second figure) of
a rhombus bisect each other at right angles
The Square
means "right angle"
show equal sides
A square has equal sides and every angle is a right angle (90°). Also opposite sides are parallel. A square
also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).
The Parallelogram
Opposite sides are parallel and equal in length, and opposite angles are
equal (angles "a" are the same, and angles "b" are the same)
23
The Trapezoid
A trapezoid has one pair of opposite sides parallel.
Acute Angles
An acute angle is one which is less than 90°
Right Angles
A right angle is an internal angle which is equal to 90°
Obtuse Angles
An obtuse angle is one which is more than 90° but less than 180°
Straight Angle
A straight angle is 180 degrees
24
M3G1d. Identify the center, diameter, and radius of a circle
A circle is easy to make:
Draw a curve that is "radius" away
from a central point.
And so:
All points are the same distance
from the center.
Radius and Diameter
The Radius is the distance from the center to the edge.
The Diameter starts at one side of the circle, goes through the
center and ends on the other side.
So the Diameter is twice the Radius:
Diameter = 2 × Radius
M3M3c. Determine the perimeter of a geometric figure by measuring and summing/adding the lengths
of the sides.
Perimeter
The distance around a two-dimensional shape.
Example: the perimeter of this rectangle is
a+b+a+b = 2(a+b)
The perimeter of a circle is called the
circumference.
See: Circumference
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Algebra: 7% = 4 questions
Students will understand how to express relationships as mathematical expressions.
M3G1.Students will use mathematical expressions to represent relationships between quantities and
interpret given expressions.
a. Describe and extend numeric and geometric patterns.
b. Describe and explain a quantitative relationship represented by a formula (such as
perimeter of a geometric figure).
c. Use a symbol, such as  and ∆, to represent an unknown in a number sentence.
60 x
= 120
=
2
60 x 2 = 120
Evaluate 5n when n=12
Step 1
Step 2
Step 3
Write the expression
Replace n with 12
Solve
5n
5 x 12
5 x 12 = 60
Data Analysis and Probability: 5% = 3 questions
Students will gather, organize, and display data and interpret graphs.
M3D1.Students will gather, organize and display data and interpret graphs.
a. Solve problems by organizing and displaying data in charts, tables and graphs.
b. Construct and interpret line plot graphs, pictographs, Venn diagrams, and bar graphs using scale
increments of 1,2,5 and 10.
Graph
Line plot
Line graph
Bar graph
Pictograph
Circle graph
Venn diagram
defined
A diagram that organizes data using a number line.
A graph that uses lines to show changes in data over time.
A graph in which information is shown by means of rectangular bars.
A graph in which information is shown by means of pictures or symbols.
A graph that represents data as part of a circle.
Two or more overlapping circles that show the relationship between sets(similarity/differences).
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Line Plot
Suppose thirty people live in an apartment building. These are the
following ages:
58, 30, 37, 36, 34, 49, 35, 40, 47, 47, 39, 54, 47, 48, 54, 50, 35, 40, 38, 47,
48, 34, 40, 46, 49, 47, 35, 48, 47, 46
Your first step should be: placing the values in numerical order.
30, 34, 34, 35, 35, 35, 36, 37, 38, 39, 40, 40, 40, 46, 46, 47, 47, 47, 47, 47,
47, 48, 48, 48, 49, 49, 50, 54, 54, 58
Now create your graph.
This graph shows all the ages of the people who live in the apartment building. It
shows the youngest person is 30, and the oldest is 58. Most people in the building are
over 46 years of age. The most common age is 47.
Line plots allow several features of the data to become more obvious.
For example, outliers, clusters, and gaps are apparent.



Outliers are data points whose values are significantly larger or smaller than
other values, such as the ages of 30, and 58.
Clusters are isolated groups of points, such as the ages of 46 through 50.
Gaps are large spaces between points, such as 41 and 45.
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Line Graphs
A line graph is a way to summarize how two pieces of information are related and how they vary
depending on one another. The numbers along a side of the line graph are called the scale.
Example 1:
The graph above shows how John's weight varied from the beginning of 1991 to the beginning of 1995.
The weight scale runs vertically, while the time scale is on the horizontal axis. Following the gridlines up
from the beginning of the years, we see that John's weight was 68 kg in 1991, 70 kg in 1992, 74 kg in
1993, 74 kg in 1994, and 73 kg in 1995. Examining the graph also tells us that John's weight increased
during 1991 and 1995, stayed the same during 1991, and fell during 1994.
Pictograph
10 students chose cat as their favorite pet
4 students chose dog as their favorite pet
6 students chose hamster as their favorite pet
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Bar Graph
The bar chart below shows the weight in kilograms of some fruit sold one day by a local market. We can
see that 52 kg of apples were sold, 40 kg of oranges were sold, and 8 kg of star fruit were sold.
Circle Graph
The circle graph below shows the ingredients used to make a sausage and mushroom pizza. The fraction of each
ingredient by weight shown in the pie chart below is now given as a percent. Again, we see that half of the
pizza's weight, 50%, comes from the crust. Note that the sum of the percent sizes of each slice is equal to 100%.
Graphically, the same information is given, but the data labels are different. Always be aware of how any chart
or graph is labeled.
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Venn Diagram
An example of a three-part Venn diagram follows:
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References (websites)
Math Worksheet Center
AA Math
Math Is Fun
Houghton Mifflin Math
Math Slice
Learning NC
Jennifer Nord
Revise Soft
Homeschool Math
Model Method
Jamit
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