Fourth Grade Mathematics
This review provides the sequential steps necessary to attain proficiency
in the basic skills in mathematics. The skills include the basic operations of addition, subtraction, multiplication, and division, as well as
fractions, decimals, measurement, and geometry.
The skills and supporting practice are designed for students working on or
near fourth grade level. Mastery of the skills included within the review
will provide a sound basis for fifth grade mathematics.
Nina P. Ross, Ed.D., Curriculum Development
Place Value of Whole Numbers
This chapter focuses on place-value concepts for reading and writing
numbers up to six digits. 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called digits.
Digits and place-value are used to write greater numbers in standard
form.
A digit standing alone is in the ones place. A digit placed to its left
is in the tens place. Place values from right to left are ones (1s), tens
(10s), hundreds (100s), thousands (1000s), ten thousands (10,000s), and
hundred thousands (100,000s).
- Rounding Whole Numbers
At times it is all right to give an approximate answer when the exact number
is not necessary. To tell "about" how many or how much is called
rounding. Rounding is telling which number is to the nearest ten, hundred,
thousand, ten thousand, or hundred thousand.
If the digit to the right of the place to be rounded is 5 or more, increase
the place to be rounded by 1. If the digit to the right of the
place to be rounded is less than 5, leave the place to be rounded the
same. After determining whether to increase or leave the place the same,
replace each digit to the right with a 0.
- Solving Word Problems: Rounding
In order to solve word problems using rounding, it is necessary to identify
the basic operation to be used, such as adding or subtracting. Then
the numbers to be used in the operation should be rounded. Finally, the
operation should be performed, using the rounded numbers.
Occasionally, the problem will state "Round the answer," in which case the
problem will be worked before any rounding is done.
Scratch paper and a pencil are necessary to help in solving the word problems
in this chapter.
- Adding Whole Numbers
To be successful in the basic operation of addition, it is necessary to
have mastered the addition facts involving single digit numbers. This
chapter focuses on using addition facts to add up to four 4-digit numbers,
estimation, and column addition.
Scratch paper and pencil will be helpful in working many of the problems
given.
- Subtracting Whole Numbers
To be successful in subtracting numbers, it is necessary to master subtraction
facts. The subtraction problems in this chapter include practice
in regrouping, mental math, and working with zero.
Paper and pencil will be needed for working problems.
- Solving Word Problems: Adding, Subtracting
In order to solve word problems using addition and subtraction, it is necessary
to know the addition and subtraction facts. Look for the key word
or phrase that indicates which operation to use to solve the problem. Key
phrases are how much altogether, how much more, find the difference, and
how much greater.
Write a number sentence using the given facts and the basic operation sign
for addition or subtraction. Solve the problem.
Scratch paper and a pencil will be needed.
- Multiplying: One Digit Numbers
Multiplication relates to repeated addition. Addition can be used when
the numbers are small and equivalent, but multiplication is more practical
when the numbers are greater.
Zero times any number is zero. One times any number is that number. Another
property of multiplication is that numbers can be multiplied in any
order.
It is necessary to memorize multiplication facts to be successful in solving
multiplication problems.
- Solving Word Problems: Multiplying
When trying to solve multiplication word problems, it helps to write a
number sentence using the given facts and the operation sign. The operation
to be used can be determined by the key word or phrase such as how
many. Multiplication often uses the same word clues as addition.
To be successful in solving multiplication problems, it is necessary to
memorize basic multiplication facts.
Scratch paper and pencil may be helpful for some students.
- Dividing: One Digit Numbers
Division, like addition, subtraction and multiplication, is simplified
when the student has mastered the basic facts. Division is the inverse
of multiplication. Therefore, many of the problems can be solved by
writing a multiplication sentence, using the given facts. For instance,
the multiplication fact 4 x 5 = 20 is closely related to 5 x 4 = 20,
20 divided by 5 = 4, and 20 divided by 4 = 5.
The answer to a division problem is called the quotient.
Scratch paper and pencil may be used.
- Solving Word Problems: Dividing
Knowing the division facts is helpful in solving word problems. Related
multiplication facts can be used to find the answer also. Division is
used to find out how many sets of something or how many are in each set.
When solving word problems, understand what is to be found, find the
needed information, plan which operation is necessary, find the answer,
and then check back for accuracy.
Scratch paper and pencil may be used to work the following problems.
- Roman Numerals
Roman numerals are symbols that were developed and used thousands of years
ago by the people of Rome. The numerals are still in use today for special
purposes. The Numerals are formed by combining I, V, X, L, D, and
M.
I = 1
V = 5
X = 10
L = 50
C = 100
M = 1,000
D = 500
When writing Roman numerals, remember to subtract when a lesser number is
to the left of the greater number. Add when the lesser number is to the
right of the greater number.
- Counting Money
In order to be able to count money and to make change, a student must know
the value of each coin. The monetary system is based on 100. There are
100 pennies in a dollar, 20 nickels, 10 dimes, and 4 quarters. A quarter
is one-fourth of a dollar or 25 pennies. A dime is one-tenth of a dollar
or 10 pennies. A nickel is one-twentieth of a dollar or 5 pennies.
When counting change, count the smaller coins first and end with the amount
given. Use play money for practice in counting change, adding, or
changing given amounts into equivalent coins.
Scratch paper and pencil may be helpful in determining answers.
- Solving Word Problems: Money
Although the cash registers used in many stores are computerized and the
clerks do not have to add or subtract, it is necessary for them to count
change. Students should know how to count change to be certain a clerk
has given them the correct amount.
When counting change, start with the cost of the purchase. Count the
smaller coins first. End with the amount given to the clerk. Example:
A $3.89 purchase was made. The clerk was given $5.00. The clerk should
give the following change and say, "$3.89, $3.90, $4.00, $5.00." The
change was one penny, one dime, and one dollar.
- Multiply: Multiples of 10, 100, and 1,000
To be able to successfully multiply by 10, 100, and 1,000, a student must
be able to count by 10s, 100s, and 1,000s and know the basic multiplication
facts. All factors that are multiplied by 10 have one zero as the
last digit in the product. Factors multiplied by 100 have two zeros as
the last two digits. Factors multiplied by 1,000 have three zeros as the
last three digits in the product.
Sometimes it is easier to think of 100 as 1 hundred, 200 as 2 hundreds,
etc. Then 4 times 300 becomes 4 x 3 hundreds = 12 hundreds or 1200.
Scratch paper and pencil may be used to solve the problems in Chapter 14.
- Multiply: Two Digit Factors by One Digit Factors
To multiply successfully, it is necessary to know the basic multiplication
facts. As factors increase in amount, it becomes necessary to know how to
trade ones for tens or regroup. Any multiplication fact with a product
less than 10 will not require trading but all others will.
As each problem is worked, decide whether a trade is necessary or is not
necessary.
Scratch paper and pencil may be needed to solve these problems.
- Multiply: Three Digit Factors by One Digit Factors
As with any mathematical problem, it is necessary to know basic math
facts. To multiply three digit factors by one digit factors is quite easy
to do. It goes just one step farther than multiplying two digit factors
by one digit factors.
Remember that it is necessary to trade if the product of two factors is
ten or greater. Multiply the ones and trade if necessary. Next, multiply
the tens and add any extra tens. Trade if necessary. Last, multiply the
hundreds and add any extra hundreds.
Scratch paper and pencil are needed to work the problems in vertical form.
- Multiply: Four Digit Factors by One Digit Factors
When multiplying four digit factors by one digit factors, it is possible
to have from zero to three trades. Follow the following steps:
Multiply the ones, trading if necessary.
Multiply the tens. Add any extra tens and trade if necessary.
Multiply the hundreds. Add any extra hundreds and trade if necessary.
Multiply the thousands, adding any extra thousands.
Scratch paper and pencil will be needed to work the problems.
- Solving Word Problems: Multiplication
When solving word problems, it is necessary to follow the basic steps of
problem solving, as well as remembering to use the step-by-step process of
multiplying two, three, and four digit factors by one digit:
Identify and understand the question.
Identify the given facts needed for solving the problem.
Decide on the mathematical operation to use.
Work the problem.
Check the answer for errors.
Scratch paper and pencil will be needed to solve the problems.
- Divide: One Digit Divisors to Find Two Digit Quotients
Division is the inverse relation of multiplication. Therefore, it is
often easier to think in multiplication terms. For example, instead of
thinking 80 divided by ___ = 20, think in terms of ___ x 20 = 80.
It is easier to work division problems if the number to be divided, called
the dividend, is put in the "box" and the divisor is put on the left side.
The answer, called the quotient, is put above the "box."
Scratch paper and pencil are needed to work the problems.
- Divide: One Digit Divisors to Find Three Digit Quotients
The purpose of this chapter is to give practice in dividing three digit
numbers by one digit divisors to find three digit quotients with remainders.
There are three major parts to each problem: dividing hundreds,
dividing tens, and dividing ones. Within each part there are smaller
steps that are almost identical. These multiplication steps determine how
many hundreds, tens, or ones have been used. The subtraction step determines
how many are left to be divided or the remainder if it is the last
step.
Scratch paper and pencil are needed to work the problems.
- Finding Averages
The focus of this chapter is to find averages when the sum of the numbers
to be averaged is a 3 digit number or less. The first step in finding
averages is to find the sum of the numbers to be averaged. The second
step is to divide the sum by the number of addends. The quotient is the
average of the numbers.
If there is a remainder, the quotient can be rounded by increasing it by
one if the remainder is half or more of the divisor. The directions will
usually tell whether or not to round the quotient.
Scratch paper and pencil are needed to find the average.
- Solving Word Problems: Finding Averages
When finding averages to solve word problems, it is important to find the
correct information. The addends must all be numbers relative to the
question. Unless the correct addends have been used, the answer will be
incorrect.
The second step is to divide by the number of addends. The quotient will
be the average of the numbers. Read the problem carefully to know whether
to round the quotient if there is a remainder.
Scratch paper and pencil will be needed to solve the word problems.
- Fractions: Equivalent
Fractions that name the same amount are called equivalent fractions.
Equivalent fractions have different numerators and denominators because
one of the fractions has a greater number of parts than its equivalent.
Remember that the greater number of parts are smaller parts.
Equivalent fractions can be found by multiplying the numerator and denominator
by the same number.
Scratch paper and pencil are needed to solve the problems.
- Fractions: Lowest Terms
When both the numerator and the denominator of a fraction are divided by
the same number (except zero), the value of the fraction remains the same.
A fraction is in lowest terms when no number other than 1 is exactly divisible into both its numerator and its denominator.
To change a fraction to its lowest terms, divide both the numerator and
the denominator by the largest number that divides evenly into each of
them.
Scratch paper and pencil are needed to work the problems.
- Fractions: Comparing
If fractions have the same numerator, the fraction with the smallest
denominator has the largest value.
If fractions have the same denominator, the fraction with the largest
numerator has the largest value.
To compare the values of fractions, change all the fractions to equal
fractions having the same denominator. The denominator must be a number
into which all the denominators will divide evenly.
Scratch paper and pencil are needed to solve the problems.
- Fractions: Adding With Like Denominators
Fractions cannot be added unless their denominators are alike. There are
three steps to follow when adding fractions with like denominators.
First, look at the denominators to make sure they are alike.
Second, add the numerators.
Finally, write the sum over the denominator.
In some cases, the numerator will be greater than the denominator. These
fractions are called improper fractions.
Scratch paper and a pencil are needed to solve the problems.
- Fractions: Adding With Unlike Denominators
The focus of this chapter is to add fractions with unlike denominators.
There are three steps to follow when adding fractions with unlike denominators.
First, look at the denominators to determine that they are unlike.
Next, find equivalent fractions with the same denominators.
Last, add the fractions the same way fractions with like denominators
are added.
Scratch paper and a pencil are needed to solve the problems.
- Adding Mixed Numerals
When the numerator of a fraction is greater than the denominator, the
fraction is greater than 1. Division is used to write a mixed number for
a fraction greater than 1. Use the following steps to change improper
fractions to mixed numerals.
Divide the numerator by the denominator.
Write the quotient as the whole number part.
Write the remainder over the divisor as the fraction part.
Scratch paper and pencil are needed to solve the problems.
- Fractions: Subtracting With Like Denominators
Fractions with the same denominator are called like fractions and can be
added and subtracted. Fractions with different denominators are called
unlike fractions and cannot be added or subtracted unless they are changed
to like fractions.
To subtract like fractions, subtract the numerators and put the difference
over the denominator.
Scratch paper and pencil are needed to solve the problems.
- Fractions: Subtracting With Unlike Denominators
Fractions that have different denominators are called unlike fractions.
Fractions must have the same denominator before they can be subtracted.
To subtract fractions with unlike denominators, find an equivalent fraction
with the same denominator. Then subtract the numerators and place
the difference over the denominator.
Scratch paper and pencil are needed to solve the problems.
- Units of Time
A knowledge of the units of time is important in every day activities.
Time is measured in seconds, minutes, hours, days, weeks, months, and
years. A combination of smaller units forms the larger unit:
60 seconds = 1 minute
60 minutes = 1 hour
24 hours = 1 day
7 days = 1 week
52 weeks = 1 year
12 months = 1 year
Although the number of days and weeks varies by month, it is the same each
year with the exception of every four years, called "leap year."
- Customary Units of Length
The customary units of length are inches, feet, yards, and miles. Length
can also be measured in metric units. This chapter will focus on the customary
units of length of the following:
1 foot = 12 inches
3 feet = 1 yard
1 yard = 36 inches
1 mile = 5,280 feet or 1,760 yards
Yardsticks, 12-inch rules, and tape measures are useful tools for measuring
customary units of feet, inches, and yards.
- Customary Units of Weight
Although there are metric units of weight, the customary units of weight
are ounce, pound, and ton. There are 16 ounces in a pound and 2,000
pounds in a ton.
Small items that could be measured in ounces are candy, pencils, paper
clips, and oranges.
Items thought of as being measured in pounds are meat, butter, people,
suitcases, and chickens.
Elephants, trucks, and bricks are measured by the ton.
- Customary Units of Capacity
Although there are metric units of capacity, the focus of this chapter is
on cups, pints, quarts, and gallons. These are familiar units used daily
during food preparation, eating, and snacking.
2 cups = 1 pint
2 pints = 1 quart
4 quarts = 1 gallon
- Problem Solving: Customary Units
Knowing the customary units of time, length, weight, and capacity is important
in our everyday living. The problem solving in this chapter is
representative of problems addressed by students at home and at school.
Read the problem carefully. Find and understand the question. Determine
the basic operation to be used in solving the problem. Write the facts in
a number sentence. Check for accuracy.
Scratch paper and pencil are needed to solve the problems.
- Metric Units
Although the customary units of measure are more familiar for measuring
some things, the metric system is familiar for measuring other things.
For instance, students think of soft drinks in terms of liter bottles.
The focus in this chapter is on the following metric units of measure:
1 meter (m) = 100 centimeters
1 kilometer (km) = 1,000 meters
1 liter (l) = 1,000 milliliters
1 kilogram (kg) = 1,000 grams
Temperature is measured in degrees Celsius (C).
- Perimeter
Sometimes it is necessary to know the distance around an object. This
measurement is know as the perimeter. The perimeter of an object can be
calculated by measuring the lengths of the object's sides and adding to
get the total length. The answer is expressed in the unit of measure
used for measuring, such as meters, centimeters, yards, feet, or inches.
Practice drawing figures with a centimeter ruler and then measuring to
find the perimeter.
- Area
The surface area of a region is the number of square units needed to cover
the region. Examples are the class room floor, the glass pane in a window,
a desk top, or a sheet of drawing paper. To find the area of a
square or rectangle, the length is multiplied by the width.
The surface area of a region is noted in square units.
Scratch paper and pencil are needed to solve the problems.
- Volume
The volume of a box is the number of cubic units it takes to fill the box.
Volume is found by multiplying length times width times height. The
formula or sentence showing volume is as follows: V = l x w x h
Using unit cubes or blocks is an easy way to illustrate volume in cubic
measurement.
Scratch paper and pencil are needed to solve the problems.
- Geometry: Basic Terms and Shapes
Geometry is the study of space and plane figures. The focus of this chapter
is to count vertices, faces, and sides of space and plane figures. It
identifies and names points, lines, segments, parallel and intersecting
lines, rays, angles, and right angles as well as describes congruent, similar,
and symmetric figures.
Space figures include cubes, rectangular prism, spheres, cones, cylinders,
and pyramids. Plane figures include squares, rectangles, circles, triangles,
quadrilaterals, pentagons, hexagons, and octagons.
Points, lines, segments, intersecting lines, and parallel lines are noted.
- Angles
An angle is formed by two rays that begin at the same point. When two
lines meet and intersect, angles are formed at the point of intersection.
Angles that look like square corners are called right angles.
Polygons have angles at each of their vertices. Squares have right angles
at each of their vertices.
Angles are named by the point of intersection of their sides. An example
is <A which is formed by two rays, BA and CA. The angle can also be named
<BAC with the point always the middle letter.
- Polygons
Polygons are plane figures that have all straight sides. Each pair of
sides meets at a vertex. Plane figures lie flat on a surface plane. Poly
means many. Polygons are figures with many sides. The plane figures are
named according to how many sides they have.
Triangles are polygons with three sides. Quadrilaterals have four sides.
Pentagons have five sides. Hexagons have six and octagons have eight.
Practice drawing and labeling each of the polygons named above, counting
the number of sides and angles of each figure.
- Lines: Segments, Parallel, Intersecting
Lines of geometry go on forever. They exist only in our minds. There are
many things around us that remind us of lines. Examples are long,
straight roads, telephone poles and wires, and railroad tracks.
Line segments are illustrated by a line with a dot at each end and arrows
pointing in both directions. Rays are lines with a point at one end and
an arrow at the other.
Parallel lines never intersect. Intersecting lines cross each other at
one point called the point of intersection.
- Problem Solving: Geometry
Geometric shapes are a part of the environment and are present in our
homes and classrooms. Being able to identify the shapes and to know the
part they play in the design of living is the central focus of this chapter.
Boxes, cans, balls, tents, and batteries are just a small sampling of the
multitude of uses made of the geometric shapes of cubes, cylinders,
spheres, prisms, and cones. It is important to be able to know the number
of faces, angles, edges, and other identifying features of each shape.
- Multiply: Two Digit Factors by Two Digit Factors
It is necessary to know the multiplication facts to successfully multiply
two digit factors by two digit factors. Basically, it is the same as multiplying
two separate problems and adding their products.
First, multiply by ones. Next, multiply by tens, putting the product under
the product of the ones but placing a zero in the ones place. Last,
add the two products.
The answer can be verified by putting the bottom factor on top and repeating
the process. The products must be identical if the answer is correct.
- Multiply: Three Digit Factors by Two Digit Factors
Multiplying three digit factors by two digit factors follows the same procedure
as multiplying two digit factors by two digit factors. The product
of the second set of factors will contain more digits.
First, multiply the ones digit times the factor on top and write the product.
Next, multiply by the tens digit. Write the product below the
first product. Last, add the two products.
Check your work to verify your answer.
Scratch paper and pencil will be needed to solve the problems.
- Problem Solving: Multiplication
In order to solve word problems involving multiplication it is necessary
to understand the question. Then, decide on the mathematical operation
needed to solve the problem. Choose the correct factors from the given
information and proceed with the solution.
Check your work to verify the answer.
Scratch paper and pencil will be needed to solve the problems.
- Divide: Two Digit Divisors to Find One Digit Quotients
It is important to know the basic division facts before attempting to do
problems involving two digit divisors.
First, decide where to start. Round the divisor and estimate. For dividing
ones, follow the pattern: divide, multiply, subtract, and compare.
Scratch paper and pencil will be needed to solve the problems.
- Divide: Two Digit Divisors to Find Two Digit Quotients
It is important to know the basic division facts before trying to divide
two digit divisors to find two digit quotients. The same basic procedure
is used. Decide where to start. Divide the tens by dividing, multiplying, ubtracting, and comparing. Next, divide the ones by bringing down,
dividing, multiplying, subtracting, and comparing. Verify your answer by
multiplying.
Scratch paper and pencil are necessary to solve the problems.
- Problem Solving: Division
When solving division problems, it is helpful to round the divisor to the
nearest multiple of ten before making an estimate for the quotient. When
solving word problems, determine which mathematical operation is needed to
solve the problem. Look for the information needed to write a number sentence
using the mathematical operation. Solve the problem using the steps
for working division problems. Verify your answer.
Scratch paper and pencil will needed to solve the problems.
- Decimals: Place Value
Decimals are other ways of writing fractions. Instead of having a numerator
and a denominator, decimals have a decimal point and place values.
The first place to the right of the decimal is called tenths, the second
place is hundredths, the third place thousandths, ten thousandths, hundred
thousandths, and so on.
Just as three tenths and thirty hundredths are equivalent fractions, they
have the same value as decimals. Zeros can be added to the extreme right
end of a decimal without changing its value. Zeros inserted at any other
position within the decimal will change the value of the decimal.
- Decimals: Comparing
Comparing decimals is like comparing whole numbers. First, line up the
decimal points. Starting at the left, find the first place where the digits
are different. Compare these digits and decide if they are less
than (<) or greater than (>). The numbers compare the same way the digits
compare.
2.56 The first place the digits differ is
2.52 with the 6 and 2.
Therefore, 2.56 > 2.52. Don't forget, though, that 12.52 > 2.56 because
the whole number is greater.
- Decimals: Rounding
Rounding decimals is similar to rounding whole numbers. First, decide to
which place the decimal is to be rounded. Next, look at the digit to the
right of the place to be rounded to. If the digit is 5 or greater, increase
the digit to be rounded by one and drop all places to the right of
the rounded place. If the digit to the right of the place to be rounded
is less than 5, leave the rounded place as it is but drop all places to
its right.
Example: 17.563 rounded to the nearest tenth is 17.6.
- Decimals: Adding Through Hundredths
Adding decimals is just like adding whole numbers. First, line up the
decimal points. Next add the hundredths. Trade if necessary. Then add
the tenths. Trade if necessary. Finally, add the whole numbers and place
the decimal point in line with the decimal points in the addends.
Trading with decimals is just like trading with whole numbers.
Scratch paper and pencil are needed to solve the problems.
- Decimals: Subtracting Through Hundredths
Subtracting decimals is just like subtracting whole numbers. First, line
up the decimal points. Next, subtract the hundredths and trade if necessary.
Third, subtract the tenths and trade if necessary. Subtract the
whole numbers and place the decimal point in the sum.
Subtracting money is just like subtracting decimals.
Scratch paper and pencil are needed to solve the problems.
- Problem Solving: Decimals
Solving word problems using decimals is no more difficult than solving
other word problems. Understand the question and decide which mathematical
operation is needed to solve the problem. Determine which information
is necessary and write a number sentence using the information and the
operation sign.
Follow each step in the mathematical operation being used to answer the
question. Verify the answer by checking your work.
Scratch paper and pencil are needed to solve the problems.
- Prime Numbers
A prime number is a number that cannot be divided evenly by any number
other than itself and one. One is not considered a prime number because
it has only itself as a factor.
The following are just a few of the prime numbers. Test each one to see
if it has any factors other than one and itself.
2 3 5 7 11 13 17 19 23 29 31 37 41
- Prime Factorization
Prime factorization is finding the factors of a given number that are
prime numbers. A prime number is one that has no other factors other
than one and itself. Prime factors are the factors of a number that are
prime.
Scratch paper and pencil are needed to find the factors of each number
and to circle the prime numbers within the factors.
- Greatest Common Factor
The greatest common factor of any given numbers is the greatest factor
that is common to all of the given numbers. To find the greatest common
factor of any given numbers, list all the factors of each number and
choose the greatest. If the numbers do not have a factor in common, the
greatest common factor is considered to be one.
Scratch paper and pencil are needed to find the factors and solve the
problems.
- Least Common Multiple
The least common multiple of a set of numbers is the first multiple each
of the numbers has in common. To find the least common multiple of a set
of numbers, list the multiples of the numbers until one appears that all
of the numbers have in common.
Zero is not a common multiple. Sometimes, in order to find the least common
multiple of a set of numbers, multiplying the numbers together will
indicate the least common multiple.
Scratch paper and pencil are needed to solve the problems.