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4th Grade Comprehensive CRCT Study Guide
Number & Operations:
Please review the standards on the study guide to review some of the VERY IMPORTANT CRCT skills.
Number and Operations: Students will further develop their understanding of whole numbers and master four basic
operations with whole numbers by solving problems. They will also understand rounding and when to appropriately
use it. Students will further develop their understanding of addition and subtraction of decimal fractions and
common fractions with like denominators.
M4N1 Students will further develop their understanding of how whole numbers and decimals are represented in the base-ten
numeration system.
M4N1a. Identify place value names and places from hundredths through one million.
b. Equate a number’s word name, its standard form, and its expanded form.
Place Value through the hundred millions p.16,17
Millions
Thousands
Ones
hundreds
tens
ones
hundreds
tens
ones
4
0
1
8
2
7
,
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
1
,
hundreds
5
tens
6
ones
9
example
0,1,2,3,4,5,6,7,8, or 9
The value of the 4 in the number 401,827,569 is 400,000,000
401,827,569
400,000,000 + 1,000,000 + 800,000 + 20,000 + 7,000 + 500 + 60 + 9
Four hundred one million, eight hundred twenty-seven thousand, five
hundred sixty-nine
hundreds
tens
ones
3
8
1.0
2
Decimals
Tenths
1
10
or
0.1
2
.
Hundredths
1
100
or
0.01
1
71, 382.21
70,000+1,000+300+80+2+ .2+ .01
seventy-one thousand, three hundred eighty-two and twenty-one hundredths
M4N2 Students will understand and apply the concept of rounding numbers.
M4N2a.Round numbers to the nearest ten, hundred, or thousand.
b. Describe situations in which rounding numbers would be appropriate and determine whether to round to the nearest ten,
hundred, or thousand.
Round Numbers (to find about how many (estimate) or how much by expressing a number to the nearest ten, hundred,
thousand, and so on) 38-39, 43, 68-69, 79
Steps to round 5,843 to the nearest thousand:
Step 1
Step 2
Find the place you want to
Look at the digit to its right.
Step 3
*If the circled digit is 5 or greater (5, 6, 7, 8, or 9)
1
round to. Underline the
digit in that place.
Circle that digit.
5,
5,843
round up.
*If the circled digit is less than 5 (4, 3, 2, 1, or 0),
round down.
43
8
*8 is greater than 5 so round up.
5,843 rounds to 6,000
M4N2c. Determine to which whole number or tenth a given decimal is closest using tools such as a number line and/or
charts.
To understand decimal numbers you need to understand how ten forms the basis of our number system.
Let's first explore decimals on the number line. Draw little lines between 0 and 1 so that it is divided into ten tiny
parts - tenths!
Now let's zoom in between 0 and 1. See how the number line between 0 and 1 is divided into ten increments or
parts. Each of these ten parts is a tenth.
The first digit after the decimal point tells how many tenth parts or tenths are in the number.
4
So 0.4 means four tenths - the same as
2
. And 1.2 is the same as 1
10
- 1 whole 2 tenths.
10
M4N2d. Round a decimal to the nearest whole number or tenth.
(see rounding rules in previous rounding section and rounding to nearest whole number in next section)
1.48 rounded to nearest 0.1:
|\
| \ replace 8 with 0
V \
1.40 8 is greater than 5, so add 0.1:
|
|
V
1.50
This means that 1.48 is between 1.4 and 1.5, but is closer to
1.5. If the number had been 1.45, exactly between the two, you would
still round up to 1.5. If it were 1.43, you would have rounded down,
leaving it at 1.40, because 3 is less than 5.
The only real difference is that when you put zeroes to the right of a
2
decimal, you can ignore them completely because they don't affect the
number. So you can write the answer as 1.5 rather than 1.50.
M4N2e. Represents the results of computation as a rounded number when appropriate and estimate a sum or difference by
rounding numbers.
Estimate a sum or difference by rounding numbers. P.64-68
estimate- a number close to an exact amount; to find out about how many
sum- the answer to an addition problem: 5 + 5 = 10, the sum is 10
difference- the answer in a subtraction problem: 15 – 10 = 5, the difference is 5
Round to the nearest dollar
$4.69 rounds to $5.00
$1.50 rounds to $2.00
$7.00
Round to the nearest thousand
6,762 rounds to 7,000
-2,501 rounds to -3,000
4,000
Round to the nearest ten thousand
73,466
rounds to 70,000
-19,387
rounds to -20,000
50,000
M4N3 Students will solve problems involving multiplication of 2-3 digit numbers by 1 or 2 digit numbers.
Addition Method
Let's look at how to solve 4 x 3.
1.
1
Understand that the mathematical sentence above is just another way to say 3 groups of four.
2.
2
Knowing that each of the four groups have three objects, add 4 three times. 4 + 4 + 4 = 12
Long Multiplication
Let's look at how to solve 187 x 54.
1.
1
Put the numbers on top of each other and put a line under them like how the picture on the right shows.
3
2.
2
Multiply the number on the bottom right with the number above it and then with the number next to it, so
on and so forth. If your result has two digits, put the first digit above the number next to the left of the top first
right number. Add the number above it (if any) to the result.
o
7 x 4 = 28.
Place 2 above 8 as shown.
o
8 x 4 = 32.
32 + 2 = 34.
Place 3 above 1 as shown.
o
1 x 4 = 4.
4 + 3 = 7.
o
4
Place 0 at where the orange box is. This is just a placeholder for this method to show that you are moving
on to multiply the tens place value.
7 x 5 = 35.
Place 3 below 2 as shown to avoid confusion.
o
8 x 5 = 40.
40 + 3 = 43.
Place 4 below 3 as shown.
o
1 x 5 = 5.
5 + 4 = 9.
o
Do simple addition and you're done!
M4N4 Students will further develop their understanding of division of whole numbers and divide in problem solving
situations without calculators.
M4N4a. Know the division facts with understanding and fluency
b. Solve problems involving division by 1 or 2 digit numbers (including those that generate a remainder).
5
Division Steps
There are five steps of dividing to remember.
1.
Divide.
Long Division
2.
Multiply.
3.
Subtract.
4.
Compare.
4
Divisor -- 3 |12
-- Quotient
-- Dividend
5. Bring down the next number.
1.
1
6|84
1
6|84
The
2.
Divide 84 ÷ 6 is the problem. Look at the first number only in the dividend so the problem is 8 ÷ 6.
answer is 1. Write it above the 8.
Multiply. 1 x 6 = 6. Write the number 6 under the 8.
6
3.
1
6|84
-6
2
Subtract. 8 - 6. The answer is 2. Write it down under the 6
4. Compare what is left over after subtracting with the divisor. It must be less
divisor, if not, then go back to step 1 and choose a larger number to
multiple.
than the
5. 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
M4n4c. Understand the relationship between dividend, divisor, quotient and remainder
Long Division with Remainders
435 ÷ 25
4 ÷ 25 = 0 remainder 4
The first number of the dividend is divided by
the divisor.
The whole number result is placed at the top.
Any remainders are ignored at this point.
6
25 × 0 = 0
The answer from the first operation is
multiplied by the divisor. The result is placed
under the number divided into.
4–0=4
Now we take away the bottom number from
the top number.
Bring down the next number of the dividend.
43 ÷ 25 = 1 remainder 18
Divide this number by the divisor.
The whole number result is placed at the top.
Any remainders are ignored at this point.
25 × 1 = 25
The answer from the above operation is
multiplied by the divisor. The result is placed
under the last number divided into.
43 – 25 = 18
Now we take away the bottom number from
the top number.
7
Bring down the next number of the dividend.
185 ÷ 25 = 7 remainder 10
Divide this number by the divisor.
The whole number result is placed at the top.
Any remainders are ignored at this point.
25 × 7 = 175
The answer from the above operation is
multiplied by the divisor. The result is placed
under the number divided into.
185 – 175 = 10
Now we take away the bottom number from
the top number.
There is still 10 left over but no more
numbers to bring down.
8
With a long division with remainders the answer
is expressed as 17 remainder 10 as shown in
the diagram
In the problem 435/25:
the dividend is the number being divided, 435
the divisor is the number that the dividend is being divided by, 24
the quotient is the answer to 435/25 which is 17
the remainder is the number left over which is smaller than the divisor, which is 10.
To check a division problem, you can multiply the divisor by the quotient to get the dividend.
If there is a remainder, multiply the divisor by the quotient, then add the remainder to get the dividend
25 x 17 = 425 + 10 = 435 (the dividend)
M4N4d Understand and explain the effect on the quotient of multiplying or dividing both the divisor and dividend by
the same number (2050/50 yields the same answer as 205/5).
205/5 = 41
2,050/50 = 41
20,500/500 = 41
205,000/5,000 = 41
M4N5 Students will further develop their understanding of the meaning of decimals and use them in computations.
a. Understand decimals are a part of the base-ten system
b. Understand the relative size of numbers and order two digit decimals
First, compare the whole numbers to the left of the decimal point.
If they are not the same, the smaller decimal number is the one with the smaller whole number.
For instance, compare 52.432 with 45.989
52 is bigger than 45, so the bigger decimal number is 52.432
We write 52.432 > 45.989 or 45.989 < 52.989
On the other hand, if they are the same, compare the whole number to the right of the decimal point.
The smaller decimal number is the one with the smaller whole number on the right of the decimal point.
for instance, compare 60.802 with 60.504
9
The whole numbers to the left of the decimal point are equal, so compare the whole numbers to the right of the
decimal point.
504 is smaller than 802, so the smaller decimal number is 60.504.
We write 60.504 < 60.802 or 60.802 > 60.504
Sometimes, they may not have the same number of decimal places to the right of the decimal point.
Just add zero(s) in this case!
For instance, compare 10.598 with 10.61
add a 0 after 61 to get 10.610
610 is bigger than 598, so 10.598 < 10.61
Other examples of comparing decimals:
4.7 > 4.4
3.23 < 3.25
7.34 < 7.304
Other times, it may not be obvious which one of the whole numbers to the right of the decimal point is bigger or
smaller.
In this case, compare each digit to the right of the decimal point starting with the tenths place
If the digits in the tenths place are equal, compare the digits in the hundredths place, and so forth...
for instance, compare 0.043 with 0.00985
10
Compare 1.2045 with 1.2050
The digits in the tenths place, which are 2 and 2 are equal, so we cannot conclude.
The digits in the hundredths place, which are 0 and 0 are equal, so we cannot conclude
The digits in the thousandths place are 4 and 5.
4 < 5, so 1.2045 < 1.2050
c.
Add and subtract both one and two digit decimals.
Detailed Example of Addition
Add the following numbers 1.19 and .16
The answer to this is: 1.19 + .16 = 1.35.
 First line the numbers up in a column, lining up the decimal points.
 Add down the columns, starting at the right. Notice that 9 + 6 = 15, so we need to carry a
1 to the tenths column.
 Continue to add down the columns, moving from right to left.
Detailed Example of Subtraction
Subtract 1.387 from 12.17.
The answer to this is: 12.17 - 1.387 = 10.783
 First line the numbers up in a column, lining up the decimal points. Since the number
1.387 has three digits after the decimal point, we add a zero on the end of 12.17 so it
also has three digits showing past the decimal point.
 Subtract down the columns, starting at the right. Notice that in the thousandths column 7
> 0. We must "borrow" from the hundredths column.
 When we move to the hundredths column, we notice that 8 > 6. We must "borrow" from
the tenths column.
11
 Continue to subtract down the columns, moving from right to left. Again, we need to
borrow from the ones place to be able to subtract the tenths.
d. Model Multiplication and division of decimals.
3 x 0.2:
In the problem 3 x 0.2, the 3 refers to the number of groups, and the 0.2 indicates the size of each group. That
is, 3 x 0.2 means 0.2 + 0.2 + 0.2, or 3 groups of 0.2.
0.2 can be represented by cutting a unit square into tenths and shading two of the tenths. Thus, the gray region
below represents two groups of size 0.1, or a total of one group of size 0.2:
3 x 0.2 can be represented as adding 0.2 together 3 times, in other words, as the sum of 3 groups of 0.2:
Another way to represent 3 x 0.2 is to show the 3 groups of 0.2 on the same whole:
Both representations show that 3 x 0.2 = 0.6 because, with 3 groups of 0.2, six-tenths of a whole (0.6) is
shaded.
Model 0.24 ÷ 0.6 on the hundreds block.
Follow these steps:
12

You need to shade a rectangle with
an area of 0.24. So, shade 24 small
squares, in a decimal model.
There are many rectangles with an
area of 0.24. You need to shade
one that has a length of 0.6.
The missing factor is 0.4.


?
The area of a 0.4 by 0.6 rectangle is
0.24. Therefore, 0.24  0.6 = 0.4
0.6
0.24
Model 0.2 ÷ 0.4 on the
Follow these steps:



Since you are using
decimal models, first
write 0.2 as 0.20.
Shade a rectangle with
an area of 0.20 and a
length of 0.4.
The missing factor is
0.5.
hundreds block.
?
The area of a 0.4 by 0.5
rectangle is 0.20. Therefore,
0.20 0.4 = 0.5
e.
0.4
Multiply and divide both one and two digit decimals by whole numbers.
0.20
Multiplication is repeated addition.
This can be represented on the number line as follows:
Note the following:
This suggests that:
13
To multiply a decimal number by a whole number:


ignore the decimal point and multiply the digits
place the decimal point in the answer so that it has the same number of decimal places as the number being
multiplied
Example 11
Calculate 45.27 × 6.
Solution:
Note:
There are two decimal places in the decimal number being multiplied. So, we place the decimal point two places from the right-hand end of the answer.
The procedure for the division of decimals is very similar to the division of whole numbers.
How to divide a four digit decimal number by a two digit number (e.g. 0.4131 ÷ 17).









Place the divisor (17) before the division bracket and place the dividend (0.4131) under it.
17)0.4131
Proceed with the division as you normally would except put the decimal point in the answer or quotient
exactly above where it occurs in the dividend. For example:
0.0243
17)0.4131
M4N6 Students will further develop their understanding of meaning of decimal fractions and common fractions and
use them in computation.
a. Understand representations of equivalent common fractions and/or decimals fractions.
Equivalent Fractions
Equivalent Fractions have the same value, even though they may look different.
These fractions are really the same:
1
2
=
2
4
=
4
8
Why are they the same? Because when you multiply or divide both the top and bottom by the same number, the fraction
keeps it's value. The rule to remember is:
What you do to the top of the fraction
you must also do to the bottom of the fraction !
So, here is why those fractions are really the same:
×2
×2
14
1
2
=
4
=
2
4
×2
8
×2
And visually it looks like this:
/2
/4
1
/8
2
4
=
=
Here are some more equivalent fractions, this time by dividing:
÷3
18
÷6
6
=
36
1
=
12
÷3
2
÷6
If we keep dividing until we can't go any further, then we have simplified the fraction (made it
as simple as possible).
Decimals are a type of fractional number. The decimal 0.5 represents the fraction 5/10. The decimal 0.25
represents the fraction 25/100. Decimal fractions always have a denominator based on a power of 10.
We know that 5/10 is equivalent to 1/2 since 1/2 times 5/5 is 5/10. Therefore, the decimal 0.5 is equivalent
to 1/2 or 2/4, etc.
Some common Equivalent Decimals and Fractions:







0.1 and 1/10
0.2 and 1/5
0.5 and 1/2
0.25 and 1/4
0.50 and 1/2
0.75 and 3/4
1.0 and 1/1 or 2/2 or 1
b.
Add and subtract fractions and mixed numbers with like denominators (Denominators should not exceed
twelve).
Steps
1. Add numerators
15
2. Do NOT Add the denominators; the denominator remains the same.
3. Add whole numbers
4. Examine your answer. If an improper fraction exist, convert into a mixed numeral. If not, go to step 6.
5. Add the whole number in step 4 to the whole number in step 3 and keep the new fraction.
6. Reduce, if necessary
Example: 13 4/8 + 3 6/8 =
1. Add numerators --- 4 + 6 = 10
2. The denominator remains the same. 8
3. Add whole numbers 13 + 3 = 16
4. Examine your answer. 13 4/8 + 3 6/8 = 16 10/8 The fraction 10/8 is an improper fraction and must be converted to a
mixed number. Thus 10/8 = 1 2/8.
5. Add the whole number in step 4 to the whole number in step 3 and keep the new fraction. 16 + 1 2/8 = 17 2/8
6. Reduce 17 2/8 = 17 1/4
7. In summary, 13 4/8 + 3 6/8 = 16 10/8 = 17 2/8 = 17 ¼
c. Use mixed numbers and improper fractions interchangeably.
Changing an Improper Fraction to a Mixed Fraction
Example
Convert
22
7
into a mixed number
Solution
We divide
3
7 | 22
16
21
1
The result is 3R1. Now to write this as a mixed number, the whole number part is 3 the numerator is 1 and the denominator
is the original denominator 7
22
1
= 3
7
7
MIXED NUMBER TO IMPROPER FRACTION
Write the mixed fraction
3¾
as an improper fraction
Solution
Write
3
3
3x4
=
4
4
12 + 3
=
3
+
4
15
=
4
4
M4N7 Students will explain and use properties of the four arithmetic operations to solve and check problems.
a. Describe situations in which the four operations may be used and the relationship among them.
b. Compute using the order of operations, including parentheses.
c. Compute using the commutative, associative, and distributive properties.
d. Use mental math and estimation strategies to compute.
Order of Operations p110-111
An expression is a number or group of numbers with operation symbols:
4 + (5 x 4) – 1 x 2
You must follow the rules for the order of operations to find the value of this expression.
Order of Operations
First, do operations inside the parentheses ().
Always do operations inside parentheses first, it can be
addition, subtraction, multiplication or division.
Then do multiplication and division in order from left to
right.
Finally, do addition and subtraction in order from left to
right.
Multiplication Properties p. 84, 100, 176
Multiplication Property
definition
Commutative Property
When you change the order of the factors, the
product stays the same.
Associative Property
When you group factors in different ways, the
product stays the same. The parentheses tell
you which numbers to multiply first.
Distributive Property
When two addends are multiplied by a factor,
the product is the same when each addend is
4 + (5 x 4) -1 x 2
4 + 20 - 1 x 2
4 + 20 - 2
24 – 2 = 22
Example
5x4 = 4x5
(6 x 7) x 9 = 6 x (7 x 9)
(3 + 5) x 2 = (3 x 2) + (5 x 2)
17
Property of One
Zero Property
multiplied by the factor and those products are
added.
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
References
www.steck-vaughn.hmhco
www.mathsteacher.com
www.Itcconline.net
www.personal.umich.edu
www.mathisfun.com
www.aamath.com
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www.wikihow.com
www.mca.k12.nf.ca
www.bellaonline.com
Ms. Glosser Math Goodies
www.mcek12tn.net
www.math.com
www.tutorvista.com
www.mathisfun.com
www.coolmath.net
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