By Max Millard
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  1. Estimate, round and manipulate very large numbers
  2. Interpret percents as part of a hundred...
  3. Understand and compute positive integer powers of nonnegative numbers
  4. Determine the prime factors of all numbers through 50
  5. Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers
  6. Add, subtract, multiply and divide with decimals. Add with negative integers
  7. Understand the concept of multiplication and division of fractions
  8. Apply the multiplication and division of fractions to solving problems
  9. Materials for 5th-6th grade math lessons

1. Estimate, round and manipulate very large numbers.

a. Talk about estimating, its meaning and usefulness. About how many students are there in the class? What's the population of San Francisco? How old are the children? If someone says he is 10, explain that 10 is an estimate, and that he's really 10 years plus something. Sales clerks estimate how old someone is before offering a senior discount. Newspapers estimate the population of cities because it's impossible to know the exact number. Bus drivers estimate how old a passenger is deciding whether to accept a youth fare. One reason it's important to learn to estimate is to prevent you from making mistakes on the calculator. You might use a calculator to get exact amounts, but pressing one wrong key can get you a result that's way off. By estimating first, you'll know the approximate amount, and will notice any mistakes.

b. Use a small postal scale to weigh some items that weigh about one ounce each -- for example, cassette tapes. See what the total is, and count the number of items. Make an estimate about how much each tape weighs.

c. Weigh an item and estimate how much it will cost to send it, at the rate of 22 an ounce. Use the figure 25 for the estimate. Then calculate the actual cost, which is 37 for the first ounce and 22 for each extra ounce. See if the two figures are close.

d. Write down some numbers on the blackboard that are in the millions, and let kids estimate their total. For example:


Practice rounding off each number to the nearest million, then adding the estimates. Make two columns, one for the actual number and the second for the estimate. Add up both columns and compare the results.


2. Interpret percents as part of a hundred. Find decimal and percent equivalents for common fractions and explain why they represent the same value. Compute a given percent of the whole number.

a. Stress that 100% means all of something. If you give a 100% effort, it means you are trying your best. If you have read 100% of the Harry Potter books, that means 5 books. If you have seen 100 of the Harry Potter movies, that's 3 movies. What are some other things that are 100%?

b. If there are some basketball fans in the class, they might know about percentages of free throws. If Shaquille O'Neal takes 100 free throws and misses 40 of them, what is his percentage? What if he takes 20 shots and misses 7? Name some other basketball stars that the class might be familiar with, or even bring some stat sheets from real basketball games and check the free throws.

c. Another good source is baseball stats, listed in any daily newspaper, showing the number times at bat and the number of hits, followed by the batting average. Look for the box that shows the major league leaders in batting. Give the students the at-bats and number of hits, and see if they can calculate the batting average, which is a percentage.

d. Show a big square of 100 small squares, or ask the class to draw one. Fold it in half and show that it now has 50 squares. Fold it in half again and show that there are 25 squares. Show 1/10 of the sheet, and ask the class how many little squares are shown. Practice converting 50/100, 25/100, 10/100 and other numbers to fractions by finding the high common denominator.

e. To compute a given percentage of a whole number, first show that a percent is the same as a decimal.

100% = 1
50% = .5
75% = .75


3. Understand and compute positive integer powers of nonnegative numbers; compute examples as repeated multiplication.

a. Explain about the powers of 2. Show 2 to the 2nd power, 3rd and 4th power.

b. Play the song "Inchworm," which goes:

2 and 2 are 4
4 and 4 are 8
8 and 8 are 16
16 and 16 are 32

Explain that doubling the total is the same as calculating the next power of 2.

c. Multiply 3 x 3, then times another 3. Bring in a Rubik's Cube, which measures 3 x 3 x 3. Show that cubing a number is the same as calculating the 3rd power of a number.

d. Use a ruler that has inches on one side and centimeters on the other to measure the Rubik's cube both ways. Show that either way, the area of one face is the square of the side, and the volume is the cube of the side.


4. Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show the multiples of a factor (e.g., 24 = 2 x 2 x 2 x 3).

a. First, define a prime number and give some examples. A prime number cannot be divided by any number except itself and 1. The prime numbers up to 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Why is 2 the only one that is even?

b. Except for the primes themselves, every number up to 50 is the product of a prime number and one other number. These two numbers are called multiplicands or multipliers. Give the class some numbers less than 50 and see who can tell you the multipliers. If no one can, approach the problem systematically by first trying to divide by 2, then by 3, then by the other prime numbers until you get the answer.

c. Starting with 4, make a list of the multipliers that make up every number up to 50.

d. Take the list and break it down further, to show that every number except the primes is a product of a series of primes. 8 = 2 x 2 x 2. 12 = 2 x 2 x 3. Go through the list until you do this for every nonprime number up to 50.


5. Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.

a. Draw a number line, which should have a 0 in the middle, negative numbers on the left and positive numbers on the right. Or you could represent it with two rulers touching each other at 0, and tell the class that one of the rulers shows positive numbers and the other shows negative numbers. Point to different places on the two rulers and ask the class what numbers they are. Start with whole numbers.

b. Mark some places on the number line that are halfway between two numbers, so the class will not give a whole number as the answer. Practice having the class identify both positive and negative numbers that end with .

c. Draw a line on the blackboard that is bigger in scale than a ruler, so that it goes only as far as 3 or so. Then make some marks between the numbers to represent fractions and decimals. First write the values they represent, and practice saying these values as a group.

d. Erase the values you have written and ask students to go up to the blackboard and make a mark for numbers that you name. They can include whole numbers, mixed numbers and negative numbers.


6. Add, subtract, multiply and divide with decimals. Add with negative integers. Subtract positive integers from negative integers. Verify the reasonableness of the results.

a. Practice the 5s times table, and demonstrate that multiplying a number by .5 gets the same answer, except that the decimal is in a different place.

b. When adding or subtracting decimals, make estimates first.

c. Write down some rows of several numbers containing decimals and practice putting the decimal in the right place. Mix them up so that some numbers have one or two decimal places, and others have three or four. Get them in the habit of adding zeroes so that they all have the same number of decimal places.

d. Do the same with subtraction. Start with simple numbers that don't involve carrying the 1. Then make them more complicated by having a different number of decimal places and needing to carry the 1. See what the class is capable of doing. You might want to hand out some worksheets that start off easy and become harder, to see what the students are capable of doing.


-	2423.50


-	2953.5

e. Write down some multiplication problems involving decimals, starting with whole numbers that are multiplied by a decimal, then two decimals multiplied together. Go slowly, so that no students are left behind. If some students find the exercise too easy, let them work on a math worksheet.

f. Practice division with decimals. Show that division is the same as making a fraction. A negative divided by a negative equals a positive. If two numbers in the fraction both have the same number of decimal places, you can erase the decimal places and rewrite the fraction with positive numbers. For example:

.5 ÷ .5 = .5/.5

.5/.5 = 5/5 = 1

-100 ÷ -50 = 100/50 = 1

g. Write a list of positive and negative numbers and add them together. If the list is long, show how to do this by using a calculator. Done by hand, one way is to add all the positive numbers first, then add all the negative numbers and subtract that number from the first total.

h. When using a long list, practice making estimates as you go along, writing the approximate running total on the right side of the program.


7. Understand the concept of multiplication and division of fractions.

a. Practice first with fractions that represent whole numbers, such as:

3/3 x 2/2

6/3 x 4/2

Do this by multiplying the top numbers first, then the bottom numbers. Check the answers by converting the fractions to whole numbers and doing the multiplication that way too.

b. Do more exercises with simple fractions, such as x , x . Show how to multiply the tops and bottoms, then reduce the fraction by finding the largest common denominator.

c. Discuss what is meant by the largest common denominator. Write some fractions on the board and ask the class to identify the largest common denominator and reduce the fraction.

c. Use plastic coins to represent fractions. A quarter equals , a half dollar = , a dime equals 1/10. Show that a half dollar equals two quarters, and divide the quarters in half to get one quarter. Show that there are 10 nickels in a half dollar, and that 1/10 of equals 1/20, which represents 5.

d. Write down the equations of multiplication and division of fractions that correspond to the plastic coins, and ask students to show you with the coins how to prove the equations.


8. Apply the multiplication and division of fractions to solving problems.

a. Devise some word problems of real-life situations in which it's necessary to multiply and divide fractions. For example:

Your mother says that you and your brother can each have half of the cookies. you give half of your cookies to your sister. She has 3 cookies. How many does your brother have?

A store is having a sale and everything is "half off." You want to buy some candy bars that normally cost 3 for $1. What is the sale price?

You hear that the price of eggs used to be $1.60 per dozen, but now the price has risen by one-half. How much does each egg cost now? Solution: $1.60 x 1/2 = .80. $1.60 + .80 = $2.40. $2.40 ÷ 12 = .20.


Materials for 5th-6th grade math lessons:

postal scale
large calculator
Rubik's cube
stat sheets from baseball games
math worksheets
magic square sheet