CRCT Math Study Guide
Listed below are some of the concepts that you will be expected to know for this year’s CRCT.
Review the concepts, and practice in the areas that are difficult for you.
Numbers and Operations
Be able to equate a number’s word name, standard form, and expanded form.
Example: fifty-seven thousand, three-hundred and twenty-nine = 57,329 =
50,000 + 7,000 + 300 + 20 + 9
Understand place value through one million. Understand value of a number.
Example: What is the value of the underlined number? 17,532 Value is 7,000
Be able to order numbers. Least to greatest means smallest to largest and greatest to least means largest to smallest.
Examples: When asked to order 2 numbers, place one above the other. Start with the greatest place value, and compare the digits moving left to right. When you come to a point where one digit is larger than the other, the entire number is larger.
1,763,025
1,763,123
Moving left to right, when you reach the hundreds, 1 is greater than 0, so the bottom number is larger or greater.
Multiplication: Know your multiplication facts. Be able to multiply 1 digit by 2 digit numbers, as well as 2 digit by 2 digit numbers.
Understand multiples. 4, 8, 12, 16 __, __, __ (The next 3 multiples of 4 are 20, 24, and 28.)
Understand factors. Any numbers that when multiplied equate to a given number.
Example: What are all the factors of 24? Answer: 1, 2, 3, 4, 6, 8, 12, 24
1x24= 24 2x12=24 3x8=24 4x6=24
Rounding: Be able to round to the nearest 10, 100, and 1,000
Tips: First, identify the place value that you are being asked to round to and underline it. Look at the digit to the right. If it is 5 or more, round the underlined digit up by 1. Then change everything to the right of the underlined digit to zeros.
If the digit to the right is less than 5, leave the underlined digit alone and change everything to the right to zeros.
Example: Round the following number to the nearest hundred. 2,768
2,768 Change the 7 to an 8, and everything to the right of the hundreds place to
zeros. 2,768 rounds to 2,800.
Round the following number to the nearest ten. 863
863 Leave the 6 alone because the digit to the right is less than 5.
Change everything to the right of the tens place to zeros. 863 rounds to 860.
Estimation: Understand that rounding to the nearest ten will give you a closer estimation than rounding to the nearest hundred or thousand.
5,6245,624 5,6205,624 5,600
+ 7,2087,208 7,210 7,208 7,200
12,832 (actual) 12,830 12,800
Multiplication Properties: Be able to recognize the following properties.
Commutative: Moving the numbers when multiplying does not change the product.
3 x 5 x 8 = 5 x 3 x 8 = 8 x 5 x 3
Associative: Changing the grouping of numbers in a multiplication problem does not change the product.
(3 x 5) x 8 = 3 x (5 x 8)
Distributive: When multiplying 2 numbers, you can take one factor and change it into the sum of its parts, multiply both parts by the other factor, add them together, and get the same answer as multiplying both factors.
8 x 9 = 72 Distributive Property: (8 x 5) + (8 x 4) = 72 the 9 is broken into 5+4
or (8 x 2) + (8 x 7) = 72 the 9 is broken into 2+7
or (8 x 3) + (8 x 6) = 72 the 9 is broken into 3+6
Other Properties:
Identity: Any number multiplied by 1 is that number. 5 x 1 = 5 648 x 1 = 648
Zero Property: Any number multiplied by 0 is 0. 5 x 0 = 0 648 x 0 = 0
15 x 12 x 0 x 6 = 0 (anytime you see a 0 in any position in this type of problem, the answer will be 0)
Order of Operations: When working an equation or number sentence, follow these rules.
- If there are parentheses, work that part of the problem first.
- If there are no parentheses, work the multiplication and/or division first from left to right, followed by the addition and/or subtraction.
3 + (8 x 2) 8 x 2 = 16 then 3 + 16 = 19
8 x (12 – 3) 12 – 3 = 9 then 8 x 9 = 72
25 – 4 x 2 4 x 2 = 8 then 25 – 8 = 17
36 – 3 x 2 3 3 x 2 = 6 then 6 3 = 2 then 36 – 2 = 34
Decimals: Recognize place value of decimals
.4 the 4 is in the tenths place (pronounced four-tenths)
.24 the 2 is in the tenths place, the 4 is in the hundredths place (twenty-four hundredths)
Be able to convert decimals to fractions, and fractions to decimals
.2 = 2 .07 = 7 .15 = 15 7 = .7 3 = .03 5 = .5
10 100 100 10 100 10
Be able to recognize equivalent decimals.
.2 = .20 .6 = .60 Adding a zero to the back of a decimal number
does not change its value.
Add and subtract decimal numbers. (Watch for problems that require regrouping. In this example you will need to place a zero after the top number to regroup correctly. Otherwise you may end up with the wrong answer of .27 in the example below)
.4 place a zero after the 4 as a place holder .40
- .27 - .27
.13
(When adding and subtracting decimal numbers, remember the decimal
point drops straight down when placed in the answer.)
Division Vocabulary: Be able to identify the dividend, divisor, and quotient
Dividend – the number being divided (the largest number)
Divisor – the number you are dividing by
Quotient – the answer in a division problem
70
8 560 560 is the dividend, 8 is the divisor, and 70 is the quotient
Fractions:
Be able to recognize fractions in pie models and fraction bars.
Know that the top number is the numerator and the bottom number is the denominator.
Be able to order fractions from smallest to greatest and vice versa.
Be able to add and subtract fractions with like denominators
Be able to add and subtract mixed numbers with like denominators
3 + 1 = 4 6 5
5 5 5 8 ( remember that a fraction
with the same numerator
- 2 3 and denominator = 1)
8 6 = 1
6
4 2
8
Inverse Operations: Opposite operations. Addition and subtraction are inverse operations. Multiplication and division are inverse operations.
Algebra: Recognize that means less than, and means more than.
25 4016 12
Recognize that symbols are used to represent unknown numbers.
15 + = 20 ( 15 plus what number is = 20) 5 is the unknown number.
What is the value of the expression 25 - if = 6 25 – 6 = 19
Geometry:
Be able to identify parallel and perpendicular lines. Be able to recognize both types of lines on a rectangular prism.
Be able to identify acute, obtuse, and right angles.
Be able to identify the number of faces, edges, and vertices of a cube and a rectangular prism.
Be able to classify triangles as to angles. Acute triangle, obtuse triangle, and right triangle.
Be able to classify triangles as to length of sides. Equilateral, isosceles, scalene.
Be able to identify parallel and perpendicular planes(faces) on a cube or rectangular prism.
Know how many sides these figures have:
Pentagon = 5
Hexagon = 6
Octagon = 8
Be able to recognize the following quadrilaterals:
squarerectangleparallelogramtrapezoidrhombus
Identify types of prisms based on the shape of the face. Rectangular, triangular, pentagonal, hexagonal, octagonal.
Understand rotations. 0ne-fourth(90 degrees), one-half (180 degrees), three-fourths (270 degrees), and full rotation (360 degrees). Remember that a rotation can be clockwise or counterclockwise.
Congruent: Exactly the same.Similar: Same shape but different size
These figures are congruent:
These figures are similar:
Graphs: Be able to interpret information in bar, line, and pictographs. Remember to always look for the key in a pictograph. The key will let you know the value of the symbol being used.
A line graph is best used to show changes over time.
A double bar graph would be used to show 2 separate groups of data. Examples could be rainfall in two different cities for each month of the year, or food or sport choices for 2 groups of people. (boys and girls)
The “middle number” of a set of data or a graph cannot be determined by simply looking at the bars or numbers. Remember, you have to order the data first in order to determine the “middle number.”
Example: What is the middle number in the following set of data?
12, 5, 15, 8, 22, 11, 30 If you chose 8 you are wrong!
The data must be ordered first, from least to greatest.
5, 8, 11, 12, 15, 22, 30Now you can see that 12 is the “middle number”
Measurement:
Know the basic units used to measure weight.
Standard units – inch, feet, yards, miles
12 inches = 1 foot; 36 inches = 1 yard; 3 feet= 1yard, 1 mile= 5,280 feet)
Metric units – millimeters, centimeters, meters, kilometers
10 millimeters= 1 centimeters 1,000 millimeters= 1 meter
100 centimeters= 1 meter 1,000 meters= 1 kilometer
Abbreviations:
Inches (in)millimeter (mm)
Feet (ft)centimeter (cm)
Yards (yd)meter (m)
Miles (mi)kilometer (km)
Word Problems: Look for clue or key words to help you choose the operation.
Addition: how many in all, or how many altogether
Subtraction: how many more, or how many less, how much is left
Division: separating items into equal groups. ( 50 pictures and 5 pages. How many pictures can you put on each page?) or (80 pieces of candy and 10 friends. How many pieces can each friend have?)
Multiplication: 1 box holds 12 cookies. How many cookies are in 25 boxes?
Each row in the theater holds 27 people. There are 23 rows in the theater. How many people can the theater hold?
One-half of: divide by 2Twice as many: multiply by 2
One dozen = 12Pair = 2
2 –step word problems – Some word problems will require more than one operation.
Example: Jane bought 3 chairs for $75 each, and a sofa for $250. How much did she spend?
$75 x 3 = $225 $225
+ $250
Example: James bought a CD for $12.75, a T-shirt for $3.99, and a candy bar for
75 cents. He gave the clerk $20. How much change did James receive?
$12.75$20.00
3.99 - 17.49
.75
$2.51
$17.49
Example: John drove 53 miles. Sally drove 3 times as far. How many miles did
they drive altogether?
53 x 3 = 159159
+ 53
212
Word Problem Vocabulary
Addition: sum
“How many in all”
“How many total”
“How many altogether”
“How much did he/she spend”
Subtraction: difference
“How many more”
“How many less”
“How much longer, shorter, taller, older, heavier”
“How much change did he/she receive”
Division: quotient
“Equal groups”
“Sharing”
“Three like items weigh 9 pounds, how much does one item weigh”
“Each bus/car holds a certain number of people. How many buses/cars are needed”
“How many left over” (remainder)
Rounding:
“About how many”
“Estimate”
“Nearest ten, hundred, thousand”
Multiplication: product
Look for the following phrases:
“One item costs a certain amount. How much does 7 items cost”
Arrays: “5 rows of seats, 12 seats in each row.
Ordering numbers:
“least to greatest means smallest to largest”
“greatest to least means largest to smallest”
*remember to “stack” numbers when comparing
Algebra:
Variables are symbols representing unknown numbers
›means more than‹ means less than
Other terminology:
Increase means adddecrease means subtract
Pair means 2one-half of something= divide by 2
Dozen= 12one-third of a number= divide by 3
Days in a week= 7round trip= multiply one way by 2
Months in a year= 121centimeter = the tip of your index finger
Hours in a day= 24 1 millimeter= width of a pin
Days in a year= 3651 meter = about the width of 1 ½ doors
Minutes in an hour= 601 kilometer= a little over ½ mile
Parallel means equal distance apart
Intersecting means lines that cross
Perpendicular means lines that form right angles
Acute angle- smaller than a right angle
Right angle- forms a square corner
Obtuse angle- larger than a right angle
Acute triangleRight triangle obtuse triangle
Plane figures:
Equilateral triangle – a triangle with 3 equal sides 2cm 2cm
2cm
Isosceles triangle- a triangle with 2 equal sides 4cm4cm
8cm
Scalene triangle – a triangle in which all 3 sides have different lengths
12 cm
8cm
10 cm
Pentagon- 5 sides and 5 angleshexagon- 6 sides and 6 angles
Parallelogram – opposite sides are parallel
Trapezoid- a quadrilateral with one set of parallel sides
Solids- a cube and a rectangular prism have the same number of sides (faces) 6, edges 12, and vertices (corners)8.
Vertices= cornersfaces= sidesedges= the place where two faces (sides) meet
Cube rectangular prismtriangular prism
More terminology:
Factors: numbers being multiplied
Product: answer in multiplication
Quotient: answer in division
Dividend: number being divided
Divisor: number being divided by
Remainder: number left over
Numerator: top number in a fraction
Denominator: bottom number in a fraction
Radius: a line segment that contains one endpoint at the center of a circle and the other endpoint on the circle
Diameter: a line segment that contains the center of a circle and has both endpoints on a circle
Center: the point in the middle of the circle
Radiusdiameter
Elapsed time: the amount of time that has passed
One hour= 60 minutes
½ hour = 30 minutes
¼ hour (quarter hour) = 15 minutes
¼ after (quarter after)= 15 minutes after
¼ till (quarter till) = 15 minutes until or before
½ past = 30 minutes
a.m. = morningp.m. = afternoon and night
noon= 12:00 p.m.midnight= 12:00a.m.
clockwise= going forwardcounterclockwise= going backwards
Area and perimeter:
Area= the number of square units needed to cover a flat surface
Area= length x width or A= l x w
12 in
8inA= 12x8= 96
A=96 square inches
Perimeter= the distance around a figure
Perimeter= length+ width + length+ width
9 in
8 in 8 in
9 inP= l+ w+ l+ w=
P= 9 + 8 + 9 + 8 = 34 in