These activities on this page start with the easiest
and progress to the more complex
step by step:
Understand Simple Multiplication
Multiply With Counters
For this activity, you will need simple counters. Counting bears, pennies, paper clips or other small objects will work.
We are going to start with multiplication by three's. Take the objects, and make sets of 3.
Put two sets of three in front. Ask, "If there are two sets of three objects, how many are there?" (6)
Give five or six other problems with sets of three.
Multiplication as Addition
Show that multiplication is simply addition.
Write on paper "3 X 3". Next to it, write "3 + 3 + 3.
Show the three sets of three objects. Ask what the answer is.
Now write "3 X 4" and also write "3 + 3 + 3 + 3."
Ask the student to display the correct counters to represent this problem.
Multiplication Match Game
This is a simple matching game that reinforces the basic concept of multiplication. You will need:
How to Play
- Simple objects as counters (pennies, marbles, counting bears, stickers, crayons, etc.)
- 10 flashcards with simple multiplication facts (3 X 4, 5 X 2, etc) These can be homemade or purchased.
- 10 homemade flashcards on index cards to match the multiplication equations:
- For 3 x 4, write 3 + 3 + 3 + 3
- For 5 X 2, write 5 + 5
- For 6 X 3, write 6 + 6 + 6
The child has to pair the addition card and the multiplication card that match.
Then, he or she needs to take the counting objects and demonstrate the concept by making sets that match.
- For instance, with the cards 5 X 2 and 5 + 5, the student would create two sets that each have five objects.
Our next section below describes interesting ways to teach individual multiplication facts. However, in the section after that, we continue to explore techniques to further increase the students' comprehension of multiplication.
Memorize the Multiplication Facts
It is important to use manipulatives to explain the concept and use
of multiplication. However, learning the basic multiplication facts by memory is also essential.
How do we do that? Drill, baby, drill.
Do we have to make this a boring drill?
No! There are many ways to learn multiplication facts (as well as addition, subtraction, and division.) Some students like to use one type of memory system, while others prefer variety.
Important Teaching Principle
The most effective teaching of multiplication or any other math operation requires these three things:
- 1. Concept Development (manipulatives and strategies in the section above)
- 2. Drill to memorize the facts (this section with strategies listed below)
- 3. Further comprehension by using those facts and recognizing patterns and relationships (the section below)
While learning the multiplication facts, we recommend daily practice to memorize the facts. We list a number of strategies below for doing this (flashcards, worksheets, wrap-ups, computer games, etc.)
You may want to use variety and try to use two different strategies every day for memorizing the multiplication facts. On the other hand, some families have found one method that works best for their child and stick with that.
At the same time
they are memorizing the multiplication facts, you also need to proceed to multiplication activities in the next section below to increase their comprehension of those facts.
So, what strategies can you use to help your child memorize the multiplication facts. Here are some of the best:
Look What I Know on the Multiplication Chart
To a student, it may seem like the number of math facts is over-whelming to memorize. It is not.
With no effort, they learn multiplication facts with 0, 1, 2, and 4. (Multiplication by 4 is learned with Double Double - above.)
Use the Multiplication Chart, or make your own, to demonstrate the facts that they have learned.
- Color in the 0, 1, 2, and 4 lines.
- Color both the horizontal rows and vertical columns when a number is learned.
- 3's, 5's, and 10's are also easy to learn, and will be colored as they learn them.
- It is encouraging to see large parts of the chart colored in and realize they are more than half the way there.
Flashcards are an old fashioned way of learning math facts. They also happen to be the cheapest and one of the most effective.
Downside to flashcards: Requires a partner to quiz the student on their facts.
You may purchase or make multiplication fact cards. If you purchase them, the best cards have the multiplication fact on one side, and the reverse division fact on the other.
This is also an old-fashioned, cheap, and effective way to practice multiplication facts. It also can be done independently without a partner. For some students, worksheets increase speed
more effectively than other strategies.
Hundreds of Worksheets - Only $5.99
Over 900 pages of math worksheets - addition, subtraction, multiplication and division are available from www.A-to-Z-worksheets.com. This will give your K-3rd grader plenty of practice, for a very reasonable price.
On-line games are the "modern flashcards." They are fun, multi-colored, and no partner is needed. The downside may be that some students already spend too much time in front of the screen. Of course, the other side of coin is that they might as well be practicing their math skills if they're going to be on-line anyway.
A do-it yourself version of flashcards. These are commercially available wraps have strings that the student uses to wrap around the plastic board. If they get the correct answer, the pattern on the back matches.
Advantage: No partner needed.
Disadvantage: Only one set of numbers is practiced at a time. With flashcards they can be mixed up.
It would be impossible to describe all the computer games to teach math. You can buy educational systems just for children. Software is available for your family computer, and of course on-line games can be played.
The math whiz is the one computer math game that I particularly like. It is portable, so can be used in the car or other locations. It has different levels of play. My one regret is that it doesn't display the student's score for a longer period of time. Otherwise, it could be used as a quick on-the-spot quiz.
Below, we have additional information on strategies for learning the multiplication facts for each number.
The student has learned the basic concept of mutliplication and is working on mastering individual multiplication facts.
This section provides multiplication activities that help children master the meaning of multiplying.
Multiplicand and Multiplier
The multiplicand is the number of be multiplied. If there are three nickles and someone wanted to know the value, they math problems would be 5 X 3. The multiplicand is the five.
The multiplier is the number of times we add the multiplicand. Three nickels is 5 + 5 + 5, or 5X3.
Since we all know that 5 X 3 and 3 X 5 both equal 15, it doesn't seem to matter much, does it? In higher math it does. However, for the time being, we want the students to understand that the two problems are recipricals, and have the same answer.
These rods were great for teaching addition and subtraction. Cuisenaire rods are even better at teaching multiplication.
The advantage of the cuisenaire rods over simple counting objects is the number needed. It is fairly quickly for a student to put 5 seven rods in a chain to show 35. It would take longer to make 5 sets of seven using individual objects.
Students who are familiar with the different Cuisenaire rods and their different colors will progess faster. Students who have not used them before should be given additional time to become acquainted with these colorful and versatile tools.
How does it work? Let's say you want to ask a student what 7 X 5 is equal to.
- They need to take 5 of the black seven rods.
- Lay the 5 rods lengthwise in "choo choo train" style.
- Hand the student several orange ten rods and ask them to find how long the rods are.
- The answer is equal to 3 orange ten rods and 1 yellow five rod. That is 35.
Ask students to demonstrate the answer to various multiplication problems, even with the math facts they already have memorized. The cuisenaire rods provide a visualization of the math concept that deepens their understanding, far greater than simply memorizing a "math fact."
Also, give the student the opportunity to "see" that 5 seven rods is as long 7 five rods.
The math balance reinforces multiplication concepts already learned, and demonstrates that weight can be multiplied, as well as individual objects.
One of the multiplication activities that can be done with a math balance is to find the reciprical. It also reinforces the difference between the multiplicand and the multiplier.
Find the Reciprical
Give the student the math fact 5 X 3.
On the left side of the balance the child should put three weights on the five.
On the right side of the balance he or she should put five weights on the three.
Find the Answer
Here's another of the multiplication activities done with the math balance. Give the student a flashcard with a multiplication problem. We'll use the example 7 X 3.
- The student puts 3 "seven" weights on the left side of the balance.
- Then, he adds a "ten" weight to the right side of the balance. The scale tilts to the left so he knows it is heavier than ten.
- He adds another "ten" weight to the right side. It still tilts slightly to the left, so the answer is more than 20.
- When a third "ten" weight is added to the right, the balance tilts to the right. 30 is too heavy, so one ten is taken off.
- It will balance when "one" is added to the two "tens." The answer is 21.
The same activity can be done with any other multiplication facts.
The Hundreds Board provides another avenue for exploring the meaning of multiplication.
- A Hundreds Board with the numbers 1 to 100
- Tokens to put on the board (the transparent are best - pennies can also be used.)
Skip counting in another one of the multiplication activites that can be done on the Hundred's Board for any number. We will use the number "6" as our example.
- The student counts from 1 to 6, and leaves a token on the 6.
- She then counts to 6 again, starting on the next space (which is the 7 space.) She will be on 12 when she counts to 6.
- She leaves a token on 12.
- Then she counts to 6 again, leaving a token on 18.
- Continue until she puts the tenth token on 60.
Now five the student a multiplication problem:
- If you ask 6 X 3, she will count to the third token, and find the answer is 18.
- If you ask 6 X 8, she will count to the eighth token and find the answer is 48.
If desired, you can make your own Hundreds Chart out of paper for each of the multiplication families. Instead of tokens, color the spaces with a light color (like yellow) so it can be seen.
Strategies for Different Multiplication Families
Multiplication by Two - "Double That"
A fun verbal math game is "Double That." It is one of the addition games
, but is also used at the beginning of multiplication.
- Give the student a number between one and ten.
- They have to give you the double. (Multiplying by 2)
- If you desire, they can give you a number between one and ten, and you give them an answer (half the time give the wrong answer.)
- They have three seconds to decide if you told them the correct answer.
With no effort at all, they have learned to multiply by two!
Multiplication by 4 - "Double Double"
This is the same game as "Double That" (above), but teaches multiplication by 4 instead of 2.
- Say a number, they give you the double of the number.
- You say "double double" and they have to double it again.
Multiply by 3's
Students need to memorize the multiplication by 3's. Use the tools for learning math facts listed above to help learn them quickly.
Multiply by 5's
Many children already know how to count by 5's. This will aid them in learning the 5 family.
Other strategies include:
- Use nickles: How much are 4 nickles worth, how much are 8 nickles worth, etc.
- Use Cuisenaire Rods: They quickly see that two fives are equal to a ten.
Multiply by 6's
Once students can multiply a number by 3, they can double that answer to get 6.
Use a combination of manipulatives such as Cuisenaire rods and hundreds board to demonstrate that their strategy of multiplying by 3 and then doubling works.
Multiply by 8's
If a child can multiply by 4, he can double the answer to find multiplication by 8.
Or, one can also play "Double, Double, Double." Let them check their answer using math manipulatives.
Multiply by 7's
This one has to be memorized. Students can use the doubling strategy for 2, 4, 6, and 8.
But 3's and 7's must be memorized.
Multiply by 9's
Many students are familiar with the 9 Strategy. We will use the problem 9 X 6 as an example.
- When you multiply by 9; the ten's place is one less than the multiplier.
- Since 9 X 6 is less than 10 X 6, the answer is less than 60. It will be in the 50's.
- The two digits of the sum will add up to equal nine.
- 5 + ? = 9
- Since 5 and 4 equals 9; the answer is 54.
My Finger Calculator - Multiply by 9
Here's another of the multiplication activities for learning to multiply by 9:
- Put all ten fingers of your two hands in front of you.
- Bend the finger of the multiplier.
For instance, if the problem is 9 X 4, bend the 4th finger.
- The digit in the "tens" place will be the number of fingers to the left of the bent finger.
In the example above, there are three fingers in front of the fourth finger. The answer will be in the 30's.
- The digit in the "ones" place will be the number of fingers to the right of the bent finger.
For 9 X 4, there are six fingers after the bent finger.
- The answer is 36.
Double Digit Multiplication
Now that your student has learned the basic facts and principals of multiplying, he or she is ready for multiplication activities that teach multiplying with double digit numbers.
Two Digit Multiplicand and 1 Digit Multiplier
Let's start with the easiest problems, that don't require any regrouping.
You will want to use either base ten blocks, cuisenaire rods, or dimes and pennies. Let's start with a simple problem.
23 X 2
Here is how we demonstrate this problem:
- Show the student the problem on paper, with the multiplier (2) on the bottom.
- Place 2 ten rods and 3 one rods on the table.
- Ask, "If I multiplied the one rods by 2, how many would I have?" (Answer: 6)
- Put 4 one rods on the table, and write "6" on the paper.
- Ask, "If I multiply the ten rods by 2, how many would I have?" (Answer: 4)
- Place 4 ten rods on the table, and write "4" in the tens place on the paper.
The second step in this process is to repeat the problem, but have the student write the answer on the paper. Continue to place the greater number of rods on the table after they have multiplied the numbers.
When the student is comfortable with the second step, progress to the third step. At this point, the rods representing the original number are in front of the student, but do not place the larger number that resulted when it was multiplied.
Finally, have them work the problems without the rods as a visual cue.
Two Digit Multiplicand and 1 Digit Multiplier
Now it is time to demonstrate multiplication activities with regrouping.
Let's use the same multiplicand with a larger multiplier: 23 X 4.
Again, we'll use either base ten blocks, the ten and one rods from Cuisenaire Rods, or dimes and pennies.
Let's demonstrate the problem:
- Write the problem 23 X 4 on a piece of paper.
- Place 2 Ten blocks and 3 One blocks on the table.
- Ask, "If I multiplied 3 one's by the number 4, how many one's would I have." (Answer: 12)
- Give the student 1 Ten block and 2 One blocks. Have them agree with you that this is 12.
- On the paper, put the 2 below the one's column, and the 1 at the top of the ten's column.
- Now ask, "If I multiplied 2 Ten's by 4, how many Ten's would I have?" (Answer 8 Ten's - or 80.)
- Give the student 8 Ten blocks. Have them agree that this is their answer.
- Show them that they also have the 1 Ten block from when they multiplied the one's column.
- Have them agree that you have 9 Ten's, or 90.
- Write 9 in the tens column.
- Show them they have the answer "92" both on paper, and with the blocks.
- Show them how they could do the math without the blocks.
Repeat the math problem with other equations. Here are some other problems:
- 36 X 2
- 17 X 5
- 43 X 2
- 28 X 3
When the student is comfortable, begin to withdraw the tools.
- Start with the multiplicand in front of them, when they multiply one column, take away the original manipulatives and give them the final number.
- As they progress, have the multiplicand in front of them at the start, but do not give them the product.
- Then have them do the problems without the manipulatives.
Two Digit Multiplicand and 1 Digit Multiplier
with 3 Digit Answer
The next step is larger answers to the two digit problems above. Here are some example problems:
- 23 X 5
- 35 X 6
- 38 X 4
- 17 X 9
- 53 X 2
With these problems, the answer is greater than 100. You will need the following:
- If using Base Ten Blocks, use the 100 flat.
- If using Cuisenaire Rods, place 10 ten rods side by side and trace them. Cut 10 of these out of light cardboard. The student should recognize that 100 of the 1 rods would fit on this template.
- If using money, use a $1 bill. The student needs to recognize that $1 is the same as 10 dimes or 100 pennies.
Three Digit Multiplication
Once a student can multiply with two digit multiplicands, it is a simple step to start multiplying with place value into the thousands.
- Manipulatives to the hundreds place
Base Ten Blocks: Hundreds Flats, Ten Rods, One blocks
- Cuinsenaire Rods: Draw hundred template (directions above), 10 Rods, 1 Rods
- Money: One dollar bills (for hundreds); dimes (for tens); pennies (for ones)
- Paper and pencil
- Problems to Solve
Three Digit Multiplicand and 1 Digit Multiplier
Let's follow the same pattern we used with two digit multiplicands.
Our problem to solve will be 132 X 3:
- Write the problem on the paper.
- Place one hundred flat, three ten rods, and two one blocks on the table.
- Starting with the one column, ask, "If we multiplied this 2 by 3, how many ones would we have?" (6)
- Take the 2 one blocks away and place 6 one blocks on the table.
- Write the answer "6" on the paper.
- Ask, "If we multiplied the 3 tens by 3, how many tens would we have?" (9)
- Place 9 ten rods on the table.
- Write the "9" on the paper below the tens column.
- Ask, "If we multiply the 1 one hundred by 3, how many hundreds would we have?" (3)
- Place 3 hundred flats on the table.
- Write 3 in the hundreds column on the paper.
- Ask the student if he or she agrees that the number "396" that you have written is the same as the 300 hundreds, 9 tens, and 6 ones in front of them.
Just as we did with the double digit multiplication activities, we will now add regrouping.
Let's use 147 X 5 as our problem:
- Write the problem on the paper.
- Place 1 hundred flat, 4 Ten rods, and 7 one blocks on the table.
- Starting with the one column, ask, "If we multiplied this seven by 5, how many ones would we have?" (35)
- Take the 7 one blocks away and place 3 ten rods and 5 ones on the table.
- Ask if they agree that the 3 tens and 5 ones is the same as "35."
- On the paper, write the 5 as the answer in the ones column and show how the 3 is carried to the top of the tens column.
- Ask, "If we multiplied the 4 tens by 5, how many tens would we have?" (20)
- Remove the original 4 ten rods (but not the 3 ten rods given as part of the answer to 7 X 5).
- Place 2 hundred flats on the table. Ask if they agree that this is 20 tens.
- Now ask, "How many tens are there?" (3)
- Write the "3" on the paper below the tens column. Make sure they realize those 3 tens came from multiplying the ones column and were "saved" on top of the tens column on your paper.
- Ask, "If we multiply the 1 one hundred by 5, how many hundreds would we have?" (5)
- Remove the original hundreds flat (but not the 2 hundreds flats from multiplying the tens column. Add the 5 hundreds flats.
- Ask, "How many hundreds are there now?" (7)
- Write 7 in the hundreds column on the paper.
- Ask the student if he or she agrees that the number "725" that you have written is the same as the 700 hundreds, 2 tens, and 5 ones in front of them.
Students continue to gain mastery over these multiplication activities through the following progression, as they did with double digit multiplication:
- Put the manipulatives in front of them, change the manipulatives with each step to reflect the product, write the answer on the paper.
- Put the manipulatives in front of them as a visual aid, do not exchange with the multiplied number, write the answer on the paper.
- Do the problem on paper without the manipulatives.
Double Digit Multiplier
We are ready to hit the big time! Students have learned to do multiplication problems with two and three digit multiplicands, and one digit multiplier. Now we are ready to add multipliers over 10.
The Long Way - The Easy Way
Here's a trick that I have found the best to help students grasp multiplication activities with larger multipliers. I call it "The Long Way - The Easy Way."
The steps are easy to remember:
- Start with a problem they can figure out on their own.
- Show them "the long way" to figure out the problem.
- Then show them "the easy way."
I like to start with the problem
25 X 20.
(Then we will move to 25 X 21)
You already know the answer
Give the student the problem 25 X 20. At first, he or she may not know how to solve it.
Let's make it easier. What is 25 X 2. (Hint: 2 quarters.)
If 25 X 2 is 50, what is 25 X 20?
Many students will be able to solve this. However, if they can't solve it mentally, figure it out by counting or drawing quarters.
Now the short way
Demonstrate with paper and pencil the way to find the answer.
Before moving on, give practice with more problems with multipliers ending in "0".
Multiply by 21
Let's use the problem 25 X 21 again.
You know the answer
How much would we have if we had 21 quarters? Let them calculate the answer using real quarters or drawings.
Let's find it the long way
They discovered that 25 X 21 is 525 by working with manipulatives. Now, show them with paper and pencil:
Let's find it the short way
- 25 X 1 = 25
- 25 X 20 = 500
- The answer is 525.
Now show them the multiplication process by multiplying in columns.
I have found that multiplying by "21" is a great tool for introducing advanced multiplication.
- Students can easily double.
- They can easily multiply by 20.
- They can easily multiply by 1.
Since they can easily grasp the concept of multiplying by "21", it is easier to follow the logic of why we multiply in columns.
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