3RD-4TH GRADE MATH LESSONS
By Max Millard
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  1. Count, read and write whole numbers up to 10,000
  2. Understand the special properties of 0 and 1 in multiplication and division
  3. Round off numbers up to 10,000 to the nearest ten, hundred and thousand
  4. Use expanded notation to represent numbers
  5. Find the sum or difference of two whole numbers between 0 and 10,000
  6. Memorize the multiplication table for numbers between 1 and 12
  7. Understand the special properties of 0 and 1 in multiplication and division
  8. Determine the unit cost when given the total cost and number of units
  9. Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context
  10. Add and subtract simple fractions
  11. Solve problems using addition, subtraction, multiplication, and division of money amounts in decimal notation. Multiply and divide money amounts in decimal notation by using whole-number multipliers and divisors
  12. Materials for 3rd-4th grade math lessons

1. Count, read and write whole numbers up to 10,000. The students should understand the place values of whole numbers.

a. Practice the four places for these numbers: ones, tens, hundreds, thousands. Write some 4-digit numbers on the blackboard, point to digits at random, and ask the class which place the digits are. Draw pictures that represent the places: ones are shown as little lines, tens are little boxes about the thickness of 10 lines, hundreds are much bigger boxes, and thousands are much bigger still. Represent a number such as 2,356 by writing the number on the board, then drawing the appropriate lines or boxes underneath each digit.

b. Write some 4-digit numbers on the board, then point to a number, call out one of the digits and ask the class which place it represents.

c. Phrase some questions differently. You can also ask: how many ones does it have? How many hundreds? This will help cement the idea that this means the same as places.


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2. Compare and order whole numbers up to 10,000.

a. Write a series of random numbers on the board, some with one digit, others with two digits, others with three or four digits. Have the students practice putting them in numerical order, starting with the smallest.

b. Write down some numbers that all have the same number of digits, and practicing putting them in numerical order. Start with 2-digit numbers in the 20s. Do the same with 3-digit numbers in the 200s. This will help them practice checking other digits besides the first one that appears.

c. Write down some 4-digit numbers that have the same digits but in different order, such as 2332, 2323, 3223. Remind the students that when comparing two numbers for which one is larger, they need to start with the digit on the left, and if they are the same digit, check the next one, or second left.

Project: have the students each write a 2-digit number on a 3x5 card without showing anyone else, then exchange cards with a classmate and ask everyone to line up in the order of their numbers. Do the same with 3-digit and 4-digit numbers. This will teach them to discuss numbers among themselves and cooperate. You might want to divide the class into two groups and see which group can do it fastest.


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3. Round off numbers up to 10,000 to the nearest ten, hundred and thousand.

a. Start by giving some example of numbers that are very close to the number that you're rounding it off to. For example, what is 19 rounded off to the nearest ten? What is 399 rounded off to the nearest hundred?

b. Give some harder examples, such as 27 and 33 rounded off to the nearest ten, 264 rounded off to the nearest hundred, 5244 rounded off to the nearest thousand.

c. When is it important to round something off? For example, when you pay your taxes, you round everything off to the nearest dollar. When you pay sales tax, you have to round it up to the next dollar.

d. Teach the rule about how to round off something that is exactly in between, such as 5, 50, 450. The rule is that you round it upward, not downward, so 5 rounded off to the nearest ten is 10, 450 rounded off to the nearest hundred is 500, not 400. Explain that this rule is an exception, and that 500 is not the nearest hundred, but that 400 is just as near.

e. Discuss other examples in life in which something that's a tie is decided one way or the other. In baseball, the tie goes to the runner. In card games such as 21, the tie goes to the dealer. In soccer, a tie is a tie. In horse racing, a race can be so close that it's called a photo finish, but it's not really a tie: one horse always wins, even if it's by a nose.


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4. Use expanded notation to represent numbers (e.g., 3,206 = 3,000 + 200 + 6.

a. Write some 4-digit numbers on the board and ask the class to break them down into their components of ones, tens, hundreds and thousands. There are usually four components, but there may be fewer.

b. Give some examples of 4-digit numbers with only two or three components so that the class won't always expect exactly four.

c. Write down the components separately and in no particular order, and ask the class to make them into a 4-digit number. For example, 3 + 5000 + 200 + 40.


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5. Find the sum or difference of two whole numbers between 0 and 10,000.

a. Practice by showing a pair of dice, and make sure the students know how many dots each one has by just looking at them. It is important that they not count the dots, but have the numbers memorized by sight. Practice with the individual dice first, throwing them down asking the class to tell their sum, which is from 2 to 12.

b. Show some printed pages that represent dice, with dots in the same position as real dice, but big enough for the whole class to see. Let the class see how quickly they can name the sum and difference between the two dice. This should be easy for most students, but it's a good warmup for harder exercises to follow.

c. Hold up a pair of 3x5 cards, each containing a two-digit number and practice calculating the sum and the difference. Start with easier numbers in the 20s and 30s, and progress to harder ones in the 80s and 90s.

d. Do the same exercise with numbers up to thousands.

Project: divide the class into groups of four or five students each, and play a simplified form of Yahtsee. You need 4 dice per game, plus a small box to throw the dice in, a writing pad and pencil to keep score, and a cup for shaking the dice before throwing them. A yogurt cup works fine. Regular Yahtsee is played with 5 dice, but the 4-dice version involves more math practice. Each player throw all 4 dice at first, then can keep any of them before throwing a second and third time. You earn points by getting pairs, three of a kind, or four of a kind. Your score is the total number of dots that are in a pair, threesome or foursome. The only other way to get points is by getting a straight (four numbers in a row), which scores 20 points. The highest score for any turn is 24, for 4 sixes. The second highest score is 22, for 2 fives and 2 sixes. Players have to decide whether they want to try for a straight and risk scoring 0, or trying for combinations of high numbers. A whole game consists of 10 scores per player, and the scores must be added up after each turn. This game teaches not only addition, but multiplication and statistics as players try to guess the odds of saving certain dice and throwing away others.


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6. Memorize the multiplication table for numbers between 1 and 12.

a. Make a big magic square consisting of a 10x10 grid, with the numbers from 1 to 12 on the top and the left, and their products in the little squares. It should be big enough for all the students to see from their seats. Explain the square and ask the students to fill in all the products that everyone knows already. They should all know the products of 1, 10 and 11 except for 11x11. Review the way to multiply by 10 and 11. The rest of the squares will remain blank, and they should be filled in as soon as the students learn to memorize the products.

b. Teach the children to use their fingers to multiply by 9. This is a neat trick that they will enjoy practicing. Hold up both hands with fingers spread apart, and starting on the far left (this will be on the teacher's far right if standing in front of the class). Number the fingers from 1 to 10. To multiply any number between 1 and 10, bend that finger down. The fingers on the left of the bent finger represent tens, and the fingers on the right represent ones. For example, if you bend down the 4th finger, you will have 3 finger on the left and 6 on the right. Practice and drill on this. All kids must do the finger bending with their own hands, too. In this way, they learn that they always have their 9 times tables with them.

c. The 5s are the next easiest. Practice reciting the 5s and show students how to count 5s on their fingers (hands-on). There's a song you can teach them called "Five Times Five." It goes like this:

5 times 5 is 25, 5 times 6 is 30
5 times 7 is 35, 5 times 8 is 40
5 times 9 is 45, 5 times 10 is 50
5 times 11 is 55, 5 times 12 is 60.

d. Teach the 2s by counting by 2 as a group. Practice doubling numbers. Use visual aids to show what happens when you double the number of coins or other objects.

e. Here's a song to practice doubling. You can play a tape of the song and let the kids sing along.

INCHWORM

Chorus:
2 and 2 are 4
4 and 4 are 8
8 and 8 are 16
16 and 16 are 32

Inchworm, inchworm, measuring the marigolds You and your arithmetic will probably go far Inchworm, inchworm, measuring the marigolds Seems to me you'd stop and see how beautiful they are

f. Teach the 3s. Students use hands to count out "0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36. Drill on these each day.

g. Hand out facts that rhyme: 6x4=24, 6x8=48. Sing them, rap them, say them, etc.

h. Teach "5, 6, 7, 8": 56=7x8

i. To multiply by 4, double the number and then double it again. Often this can be done in your head. Practice on paper first.

j. Squaring the numbers. The students should already know 2x2, 3x3, 4x4, 5x5. Here are some mnemonics to help them learn the others:

6x6: someone drank six six-packs because he was "thirsty," which sounds like "36" if someone is slurring their words.

7x7: seven is a lucky number. The luckiest year in San Francisco history of was 1849, when the gold rush began. If the students forget 7x7, ask: "What's 7? A lucky number. What was really lucky?" They may think of the San Francisco 49ers.

8x8: They might remember the Beatles song "When I'm 64," which has the line, "Will you still need me, will you still feed me, when I'm 64?" They "ate" and "ate" when they were 64.

k. There are other mnemonics you can use to help the students memorize the products, such as assigning an aspect or a picture to represent each number from 1 to 12, and making up stories about them. For example, 3 is a magic number and 7 is a lucky number. When you combine them, you get 21, the luckiest hand in blackjack.

l. Besides filling in the magic square gradually with the products that the whole class learns, give a filled-in magic square to everyone so that they practice reading it and looking up products they don't know.


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7. Understand the special properties of 0 and 1 in multiplication and division.

a. Explain how to divide by one, divide by 0, divide by 2. Practice with some easy numbers. Show with coins or pictures of dice. If you hold up two dice showing the same number, the class will see what happens when the multiply by 2 and divide by 2.


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8. Determine the unit cost when given the total cost and number of units.

a. Practice with pennies and pieces of fruits or other objects. Show that a certain number of items costs a certain number of pennies, then match each object with the pennies. Combine the pennies for 2 and 3 objects, so the students can practice the math of dividing and multiplying these small numbers.

b. Do the exercise with dimes, or use Monopoly money to practice with dollars.

c. Change the number of items and the amount of money, and ask the class to calculate the unit cost. Let them come up and handle the physical objects if they want.


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9. Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context (e.g., of a pizza is the same amount as 2/4 of another pizza that is the same size; show that 3/8 is larger than .

a. Use the book "How Many Ways Can You Cut a Pie?" By Jane Moncure, which covers this topic.

Project: Using a compass, Velcro and some manila folders, make several "pies" that are the same size. The students can help with this. Keep one pie whole. Cut the others into slices of 1/2, 1/3, 1/4, 1/6, 1/8. Put Velcro on the back of each slice and on the whole circle. Then practice manipulating the pieces to show that 1/3 = 2/6, 1/2 = 2/4 = 4/8, etc.


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10. Add and subtract simple fractions (e.g., determine that 1/8 + 3/8 = 1/2.

a. Use the same models as the previous example. Tape them to the blackboard and write equations about them. Have the students practice looking at the drawings and solving the equations. Show them how to do them on paper as well, by finding the lowest common denominator.

b. Put away the drawings and erase the equations from the blackboard, and ask the students about the fractions orally, so that they have practice with word problems. "How much is one-third plus one-sixth? How much is one-eighth plus one-fourth?"


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11. Solve problems using addition, subtraction, multiplication, and division of money amounts in decimal notation. Multiply and divide money amounts in decimal notation by using whole-number multipliers and divisors.

a. Write a lot of addition problems on the board, starting with the smallest ones in which you don't need to carry the 1. For example:

	2.55
+	7.32
	-------

b. Then write problems in which the first digit (whole number) needs to be carried, such as:

	5.25
+	5.42
	-------

c. Then write problems where the second digit (right after the decimal point) needs to be carried, such as:

	5.48
+	2.61
	-------

d. Then make the 3rd digit the one that needs to be carried, such as:

	5.55
+	1.48
	-------

e. Do the same for subtraction, starting with the easiest and gradually progressing to the hardest.

f. Multiplication: start with whole dollar amounts, then include tens, such as $2.10, $2.20, $2.30. First practice multiplying these numbers by 2 and 3, with totals that are less than $10.

g. Do the same with division, starting with whole dollars, then using dimes and quarters too.


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Materials for 3rd-4th grade math lessons:

book: How Many Ways Can You Cut a Pie? By Jane Moncure
compass, manila folders, Velcro (for making cardboard pie)
plastic coins
Monopoly money

many dice
recording of "Inchworm"
worksheets for addition, subtraction, multiplication


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