Radford City Schools

Math Notes

~ Math ~

Measuring Metric Capacity  SOL 5.8c

In a graduated cylinder, measure from the bottom of the curve (meniscus).

Capacity is the measure of the amount of liquid a container holds.

A graduated cylinder is the tool you use to measure capacity.

1000 milliliters (mL) = 1 liter (L)

An eyedropper holds about 1 milliliter (mL) of liquid.

You can buy 1 liter (L) of soda or juice.

A liter is the basic unit of metric capacity.

Measuring Metric Mass  SOL 5.8c

Mass is the measurement of the amount of matter in an object.

Mass is not how much something weighs.

Weight is how much gravity is pulling on an object.

A balance is the tool used to measure metric mass.

1000 grams (g) = 1 kilogram (kg)

1 gram (g) is about the mass if a paper clip.

1 kilogram (kg) is about the mass of a large container of peanut butter.

A gram is the basic unit of metric mass.
Measuring Metric Length  SOL 5.8c

10 millimeters (mm) = 1 centimeter (cm)

100 centimeters (cm) = 1 meter (m)

1000 meters (m) = 1 kilometer (km)

1 millimeter (mm) is about the thickness of a dime.

1 centimeter (cm) is about the size of a pencil eraser.

1 meter (m) is about the width of the classroom door.

1 kilometer (km) is less than 1 mile.  (1 mile = 1.6 kilometers)

The basic unit of metric length is meters.

Converting Metric Units  SOL 5.8c

King Henry Doesn't Usually Drink Chocolate Milk

Step 1:  Write King Henry Doesn't Usually Drink Chocolate Milk.

Step 2:  Copy the measurement with the decimal.

Step 3:  If there is no decimal, write it on the right.

Step 4:  Draw an arrow over the number.

Step 5:  Count how many spaces your decimal must travel, but don't count the space you are on.

Step 6:  Follow the arrow and move the decimal that many spaces.

Customary Length  SOL 5.8e

Rulers, yardsticks, and tape measures are tools used to measure length in standard measurement.

Rulers measure in inches and in parts of an inch.  Parts of an inch are shown in fractions:  eighths, fourths, and halves.

Fractions are always simplified to lowest terms.  2/8 is simplified to 1/4.

1 foot (ft) = 12 inches (in)

1 yard (yd) = 3 feet (ft)

1 yard (yd) = 36 inches (in)

1 mile (mi) = 5280 feet (ft)

1 mile (mi) = 1760 yards (yd)

Customary Capacity  SOL 5.8e

1 gallon (gal) = 4 quarts (qt)

1 quart (qt) = 2 pints (pt)

1 pint (pt) = 2 cups (c)

1 cup (c) = 8 fluid ounces (fl. oz)

A small cup of coffee is about 1 cup.

A tall glass of lemonade or a large bowl of soup is about 1 pint.

Oil for a car comes in a quart size container.

A large container of milk comes in a gallon-sized container.

Customary Weight  SOL 5.8e

Weight is the measure of the force of gravity on an object.

16 ounces (oz) = 1 pound (lb)

2000 pounds (lb) = 1 ton (T)

A slice of bread weighs about 1 ounce (oz).

A loaf of bread weighs about 1 pound (lb).

A large car weighs about 1 ton (T).

Place Value of Whole Numbers  SOL 5.4

Place value tells us how much each digit is worth.

To read a whole number:
  • Step 1:  Read to the first comma.
  • Step 2:  Say the name of the period.
  • Step 3:  Read to the second comma.
  • Step 4:  Say the name of the period.
  • Step 5:  Read the last three digits.

Place Value of Decimals  SOL 5.5a

Think of money to help you understand decimals.

How to read decimals:
  • Step 1: Read the whole number part.
  • Step 2:  Read the decimal point as and.
  • Step 3.  Pretend the rest of the digits form a whole number.
  • Step 4.  Say the place value of the last digit.

Remember there is no "oneths" place!

Rounding Decimals  SOL 5.1

To round decimals:
  • Step 1:  Circle the digit in the place to which you are rounding.
  • Step 2:  Underline the place to the right.
  • Step 3:  Ask yourself if the underlined number is 5 or greater.
  • Step 4:  If the answer is yes, change the circled digit to one more.
  • Step 5:  If the answer is no, leave the circled digit alone.
  • Step 6:  Drop any digits to the right of the circled digit.

Comparing and Ordering Decimals  SOL 5.1

We compare and order decimals when we look at a menu.

When we compare we use terms such as less than <, greater than >, or equal to =.

Comparing decimals is similar to comparing whole numbers.  Example:  4 < 45, 150 >105.

Equivalent decimals are decimals that name the same number.  Example:  0.60 = 0.6

To order decimals:
  • Step 1:  Line up the decimal points.
  • Step 2:  Start at the left and find the smallest number until you have put all the numbers in order.


15. 14.95 Decimals can be ordered from least to greatest or
14.95 15. greatest to least.
15.8 15.01 Make sure you check the directions!
15.01 15.8

Estimating Sums and Differences  SOL 5.4

To estimate sums or differences:
  • Step 1:  Round each addend to the largest place they have in common.
  • Step 2:  Add or subtract the rounded numbers.


  4239 4000
  2256 2000
+1975       +2000

Adding and Subtracting Decimals  SOL 5.5a

To add or subtract decimals:
  • Step 1:  Line up the decimal points.
  • Step 2:  Fill in any holes with zeros.
  • Step 3:  Add or subtract.

12.3 + 3.034

  + 3.034

Writing Equations  SOL 5.18abcd 

An equation is a number sentence that shows two quantities are equal.

A variable is a symbol or letter that stands for some number.


The python is the longest snake in the world.  At birth it is 2 feet long, and some adult pythons are 29 feet long.  How much do these pythons grow to reach adult length?   Write an equation with a variable to model the problem.

2 + f = 29


Write a problem for this equation.

3 + n = 5

Mrs. Bowen had 5 apples.  3 apples were on her desk.  Some more apples were in her lunchbox.  How many apples were in her lunchbox?

Mean, Median, Mode, and Range  SOL 5.16abcd

The mean (fair share) is the average of a set of numbers.

The median (measure of center) is the middle number in a set of numbers.

The mode is the number that appears most frequently in a set of numbers.

The range (measure of variation) is the difference in the greatest and least in a set of numbers.

To find the mean:
  • Step 1:  Add to find the sum of the numbers.
  • Step 2:  Divide the sum by the number of addends.

To find the median:
  • Step 1:  Put the numbers in order from least to greatest.
  • Step 2:  Mark off two numbers at a time, one from each end of the list until you reach the number(s) left in the middle.
  • Step 3:  If there are two numbers left in the middle, find the mean (average) of the two numbers to find the median of the list.

To find the mode:
  • Step 1:  Write the numbers in order from least to greatest.
  • Step 2:  Find the number that appears the most often.

There may be one, two or more, or no modes in a list of numbers.

To find the range:
  • Step 1:  Write the numbers in order from least to greatest.
  • Step 2:  Subtract the least number from the greatest number in the list.

Collecting, Organizing, and Displaying Data  SOL 5.15

A stem and leaf plot is a way to organize the numbers in a set of data.

To make a stem and leaf plot:
  • Step 1:  Title the plot.
  • Step 2:  Write the data in order from least to greatest.
  • Step 3:  Choose the stems.  Each digit in the tens place in your list is a stem.
  • Step 4:  Write the stems vertically from least to greatest.
  • Step 5:  The leaves are all the ones digits in your list.  Write them next to the stems that match their tens digit.

A line graph is used to show change over time.

Estimating Products  SOL 5.4

The product is the answer to a multiplication problem.

Factors are the numbers you multiply together to get another number.

To estimate products:
  • Step 1:  Round each factor to the largest place.
  • Step 2:  Count the number of zeros.
  • Step 3:  Write the same number of zeros in the product.
  • Step 4:  Multiply the remaining digits and write it in the product.


    14 10
  x26        x30

Multiplying Two Digit Numbers  SOL 5.4
To multiply by a two digit factor:
  • Step 1:  Multiply by the ones.
  • Step 2:  Write the partial product.
  • Step 3:  Write the magic zero as the place holder.
  • Step 4:  Multiply by the tens.
  • Step 5:  Write the partial product.
  • Step 6:  Add the partial products.

Estimating Decimal Products  SOL 5.5a

To estimate decimal products:
  • Step 1:  Round each factor to the nearest whole number (the ones place).
  • Step 2:  Multiply.

Multiplying with Decimals.  SOL 5.5a

To multiply numbers with decimals:
  • Step 1:  Multiply the numbers as if they were whole numbers.
  • Step 2:  Count the number of decimal places in the factors.
  • Step 3:  Beginning at the right side of the product, count over the number of places you found.  Place the decimal there.

When multiplying decimals, never line up the decimals!

Dividing Whole Numbers  SOL 5.4 

A quotient is the answer to a division problem.

The divisor is the number on the outside of the division sign.  This is the number "dividing into" another number.

The dividend is the number on the inside of the division sign.  This is the number being divided.

To divide:
  • Step 1:  Divide the tens.
  • Step 2:  Multiply the partial quotient by the divisor.
  • Step 3:  Subtract.
  • Step 4:  Compare.
  • Step 5:  Bring down the next digit.
  • Step 6:  Repeat as needed and write the remainder, if any.

Remember...Dirty Monkeys Smell Completely Bad  (Divide, Multiply, Subtract, Compare, Bring down)

Dividing with a Double Digit Divisor  SOL 5.4

To divide with a double digit divisor:
  • Step 1:  Decide in what place the quotient will begin.  Hold the unused places with an "x".
  • Step 2:  Mentally round the divisor. 
  • Step 3:  Mentally divide.
  • Step 4:  Write the digit in the quotient.
  • Step 5:  Multiply the partial quotient by the divisor.
  • Step 6:  Subtract.
  • Step 7:  Compare.
  • Step 8:  If the number is larger than the divisor, erase the partial quotient and increase it by one.
  • Step 9:  Write the remainder or repeat the process as needed.

Word Problems  SOL 5.4

Remember with any word problem, you need to read carefully.  Highlight or underline key information.

Look for key words.

Addition - total, all together, sum

Subtraction - difference, how many more or less

Multiplication - how many of the same thing repeated, times, each

Division - how many equal groups, divided evenly, same number, each

Try to think about how it would work in real life.
Order of Operations  SOL 5.7

Order of Operations is the special rules mathematicians must follow when a problem has more than one operation.

P is for parentheses.
E is for exponents.
M & D are for multiplication and division.
A & S are for addition and subtraction.

PEMDAS can stand for Please Excuse My Dear Aunt Sally.

The correct answer can only be found when using the correct order of operations.
Patterns SOL 5.17 

To find the pattern:
  • Step 1:  Look at the first two numbers in each line to see how they change.
  • Step 2:  Check to see if the pattern changes between the second and third numbers.
  • Step 3:  Continue the pattern.

An input output table is similar to a machine.  You put a number into the table (input), change the number based on the certain rule, and then get a new number (output).

Properties of Addition and Multiplication  SOL 5.18 

Commutative Property for Addition a + b = b + a

Commutative Property for Multiplication a x b = b x a

Associative Property for Addition (a + b) + c = a + (b + c)

Associative Property for Multiplication (a x b) x c = a x (b x c)

Distributive Property  a(b + c) = (a x b) + (a x c)

The Distributive Property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products.

Odd and Even SOL 5.3b

An odd number does not have 2 as a factor and it is not divisible by 2.

The factors of 17  are 1 x 17.  You cannot divide 2 into 17 evenly.

An even number does have 2 as a factor and it is divisible by 2.

The factors of 14 are 1 x 14, and 2 x 7.  You can divide 14 by 2 evenly.

Divisibility Rules  SOL 5.3a 

Divisible by 2
All even numbers are divisible by 2.

Divisible by 3
Add the digits.  If the sum is divisible by 3, so is the original number.

Divisible by 5
All numbers ending in 0 or 5 are divisible.

Divisible by 6
If a number is divisible by 2 and 3, it is divisible by 6.

Divisible by 10
All numbers ending in 0 are divisible by 10.

Factors, Prime, and Composite Numbers SOL 5.3a 

The product is the answer to a multiplication problem.

A factor is a number that is multiplied by another to give a product.

A factor can divide evenly into another number.

A prime number is a number that has only two factors, itself and 1.

7 is prime because the only numbers that will divide into it evenly is 1 and 7.
7 is prime because the only way it can be made with multiplication is 1 x 7.

A composite number is a number that has more than two factors.

8 is composite because its factors are 1 x 8 and 2 x 4.

Factor Trees  SOL 5.3 

Factor Trees are diagrams that allow you to find the prime factorization of a number.

Prime factorization shows how to made a number using only prime numbers.

Making Equivalent Fractions  SOL 5.2b 

A numerator is the top number in a fraction.

A denominator is the bottom number in a fraction.

To make an equivalent fraction, multiply the numerator and denominator by the same number.

You will get a new fraction with the same value as the original fraction.

We are not changing the value of the fraction, because we are simply multiplying by a fraction that is equivalent to 1.

When the numerator and denominator of a fraction are the same, the fraction equals 1.

Improper Fractions and Mixed Numbers  SOL 5.2b 

A improper fraction has a numerator that is greater than the denominator.

A mixed number has a whole number part and a fraction part.

To change an improper fraction to a mixed number:
  • Step 1:  Divide the denominator into the numerator.
  • Step 2:  The answer is the whole number of your new mixed number.
  • Step 3:  The remainder becomes the new numerator.
  • Step 4:  The denominator remains the same.

To change a mixed number into an improper fraction:
  • Step 1:  Multiply the denominator by the whole number.
  • Step 2:  Add the numerator.
  • Step 3:  This number is the new numerator; the denominator stays the same.

Comparing and Ordering Fractions SOL 5.2b 

To compare two fractions:
  • Step 1:  Multiply the first numerator with the second denominator.
  • Step 2:  Multiply the first denominator with the second numerator.
  • Step 3:  Use less than < or greater than > between the two fractions.

To compare a group of fractions:
  • Step 1:  Find the common denominator for the group of fractions.
  • Step 2:  Change each fraction to an equivalent fraction.
  • Step 3:  Compare the numerators.
  • Step 4:  Order the fractions from least to greatest.

Relating Fractions and Decimals SOL 5.2a 

To write a decimal as a fraction:
  • Step 1:  Say the decimal out loud.
  • Step 2:  Write the fraction you say.
  • Step 3:  Simplify.

.2 = 2/10 = 1/5

To change a fraction to a decimal:
  • Step 1:  Divide the numerator by the denominator.
  • Step 2:  Write the decimal and zeros in the dividend as needed to finish the division.
  • Step 3:  If the division problem goes on without ending, show the repeating decimal by drawing a line over the repeating digits.

Adding and Subtracting Fractions with Like Denominators SOL 5.6

To add fractions with like denominators:
  • Step 1:  Add the numerators.
  • Step 2:  The denominator remains the same.
  • Step 3:  Simplify.

To subtract fractions with like denominators:
  • Step 1:  Subtract the numerators.
  • Step 2:  The denominators remain the same.
  • Step 3:  Simplify.

Adding and Subtracting Fractions with Unlike Denominators SOL 5.6 

To add fractions with unlike denominators:
  • Step 1:  Rewrite the fractions with like denominators.
  • Step 2:  Make equivalent fractions using the new denominators.
  • Step 3:  Add the fractions.
  • Step 4:  Simplify.

To subtract fractions with unlike denominators:
  • Step 1:  Rewrite the fractions with like denominators.
  • Step 2:  Make equivalent fractions using the new denominators.
  • Step 3:  Subtract the fractions.
  • Step 4:  Simplify.

Adding Mixed Numbers with Unlike Denominators SOL 5.6 

To add mixed numbers with unlike denominators:
  • Step 1;  Find the common denominator.
  • Step 2:  Make the equivalent fractions.
  • Step 3:  Add the fractions.  Remember that if you get an improper fraction when you add, change it to a mixed number.
  • Step 4:  Add the whole numbers.
  • Step 5:  If you have a whole number and  a mixed number in the answer, add the whole numbers together.
  • Step 6:  Simplify.

Subtracting Mixed Numbers with Unlike Denominators SOL 5.6 

To subtract fractions with unlike denominators:
  • Step 1:  Find the common denominator.
  • Step 2:  Make equivalent fractions.
  • Step 3:  If you cannot subtract the fractions, you must borrow one whole from the whole number.  Write the one whole as a fraction with the same common denominator and add it to the existing fraction.
  • Step 4: Subtract the numerators. 
  • Step 5:  Subtract the whole numbers.
  • Step 6:  Simplify.

Geometry Terms  SOL 5.12a 

A line goes on and on in both directions.  A line is drawn with an arrow on each end.

An endpoint is a point at the end of a line segment or ray.

A line segment is part of a line.  It is drawn with two endpoints.

A ray goes on and on in one direction.  It is drawn with an arrow on one end and an endpoint on the other.

Parallel lines are always the same distance apart and they will never touch.

Intersecting lines are tow lines that cross each other.

Perpendicular lines are two intersecting lines that intersect for form right angles.
Angles SOL 5.12a

Angles are two rays that share an endpoint.

A right angle measures 90 degrees.

An acute angle measures less than 90 degrees.

An obtuse angle measures more than 90 degrees.

A straight angle measures exactly 180 degrees.

A circle is 360 degrees.

Measuring Angles SOL 5.11

A protractor is a tool used to measure angles.

To measure an angle, you must line up the protractor correctly by putting the circle exactly on the endpoint and the lines exactly on the rays.

Look to see if the angle is acute or obtuse.  

Read the measurement that lines up with the other ray.  Make sure the measurement matches either an acute or obtuse angle.

Circles SOL 5.9

A chord is a line segment joining any two points on a circle.

A diameter is a chord that goes through the center of a circle.

A radius is a line segment from the center of a circle to any point on the circle. It is half of a diameter.

A circumference is the perimeter of a circle.
Congruent, Noncongruent, and Similar Figures SOL 5.13b 

Congruent figures have the same size and shape.  They match exactly.

Noncongruent figures do not have the same size and shape.

Similar figures have the same shape, but may or may not have the same shape.

All congruent figures are similar.

Symmetry SOL 5.13b 

A figure has symmetry if you can fold it so that it has two parts that match exactly.

A regular polygon gas as many lines of symmetry as it has sides.

Polygons SOL 5.13 

A polygon is a closed figure made of three or more line segments.

A regular polygon has all sides of equal length.

A triangle has three sides.

A quadrilateral has four sides.

A pentagon has five sides.

A hexagon has six sides.

An octagon has eight sides.

A decagon has ten sides.

The angles of a triangle measures 180 degrees.

The angles of a quadrilateral measures 360 degrees.

A triangle is half a rectangle.

Identifying Triangles SOL 5.12b 

An isosceles triangle has two congruent sides.

An equilateral triangle has all three sides that are congruent.

A scalene triangle has no congruent sides.

A right triangle has exactly one right angle.

An acute triangle has three acute angles.

An obtuse triangle has exactly one obtuse angle.

Combining and Subdividing Shapes SOL 5.13b 

Shapes can be combined to make a new shape.

Shapes can be subdivided into two or more shapes.

Quadrilaterals SOL 5.13a 

Quadrilaterals are two-dimensional or plane figures with four sides.

A parallelogram is a four-sided figure with opposite sides the same length and parallel.

A rectangle is a parallelogram with four right angles.

A rhombus is a parallelogram with all sides the same length.

A square is a rectangle with all sides the same length.

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Elapsed Time SOL 5.10 

Elapsed time tells how much time has passed.

60 seconds = 1 minute

60 minutes = 1 hour

24 hours = 1 day

365 days = 1 year

To figure elapsed time:
  • Step 1:  Count the hours.
  • Step 2:  Count the minutes.

Another way to figure elapsed time:
  • Step 1:  Add or subtract the minutes.
  • Step 2:  Add or subtract the hours.
  • Step 3:  Simplify.  Remember there are 60 minutes in an hour.

Perimeter and Area SOL 5.8ab

Perimeter is the distance around a figure.

To find the perimeter:
  • Step 1:  Add the lengths of the sides.  P = s+s+s+s

Area is the number of square units needed to cover a figure.

To find the area:
  • Step 1:  Multiply the length times the width   A = l x w

Area of a Right Triangle SOL 5.8 

Area is the number of square units needed to cover a figure.

A triangle is half of a rectangle.

To find the area of a right triangle:
  • Step 1:  Multiply the length times the width.    A = (l x w) divided by 2
  • Step 2:  Divide by 2.

Volume  SOL 5.8 

Volume is the number of cubic units it takes to fill a solid.

Volume is measured in cubic units.

To find the volume:
  • Step 1:  Multiply the length times the width times the height.  V = l x w x h