Your student should know that multiplication is a short form, or abbreviation, for addition. For instance, 3 x 5 = 5 + 5 + 5. The statement 3 x 5 can also be interpreted as 3 + 3 + 3 + 3 + 3, but you can explain this when your student is more familiar with the concept of multiplication. In the beginning, always have your student write the addition statement using the second number in the given product.
Have your student write various multiplication statements as addition statements. For instance:
Your student should also know how to represent a multiplication statement with an array of dots. For instance, 3 x 5 can be represented as 3 rows of 5 dots:
Have your student draw arrays for the following statements (make sure the number of rows matches the first number in the statement, and the number of dots in each row matches the second number).
Your student should know how to draw arrays to represent statements that involve addition and multiplication. For instance,
3 x 5 + 2 x 5
may be represented by
Your student should see from the array that
3 x 5 + 2 x 5 = (3 + 2) x 5
(This is the distributive law.)
Have your student draw arrays for:
If your student seems confident about multiplication, you could point out that 3 x 5 can be represented as 3 rows of 5 dots or 5 rows of 3 dots: in either representation, one still has the same number of dots. Hence, in translating a multiplication statement into an addition statement, one may use either the first or the second number: e.g., 3 x 5 = 5 + 5 + 5 or 3 + 3 + 3 + 3 + 3.
From this, it is easy to see that multiplication commutes, i.e., it doesn’t matter what order you write the numbers in (3 x 5 = 5 x 3).
Finally, tell your student that knowing their times tables will help them solve problems that would otherwise require a good deal of work adding up numbers. For instance, have them solve the following problems by multiplying:
1. There are 4 pencils in a box. How many pencils are there in 5 boxes?
2. A stool has 3 legs. How many legs would 6 stools have?
3. A boat can hold 2 people. How many people can 7 boats hold?
(See section M-3 for more complicated problems of this sort.)
Show your student how to divide a number into groups or sets.
Example: Divide 10 into sets of 2.
Step 1: Draw 10 lines:
Step 2: Draw boxes around each group of 2:
Have your student say out loud, “10 divided into groups of 2 is 5,” and then write a division statement:
Using boxes (for sets or groups) have your student divide:
(1) 6 into groups of 2
(2) 12 into sets of 3
(3) 9 into groups of 3
To write a division statement based on a picture, your student should write the total number of lines divided by the number of lines in each box equals the number of boxes, e.g., dividing 12 into sets of 3 gives 4 sets:
Your student should also be aware that there is another way to interpret the picture: dividing 12 into 4 parts (or sets) gives 3 objects in each set. Hence, they could also write:
Have your student write two division statements for each of the following pictures:
Have your student write multiplication statements for the pictures above as follows: the number of boxes times the number of lines in each box equals the total number of lines.
A student will find word problems involving multiplication and division easy if they are familiar with the notion of a set (or group) of objects or things.
A box can be used to represent a set, and lines to represent the objects in each set. Make sure your student understands that the lines could stand for anything: pencils in a box, people in a boat, or legs on a dog. Ask your student to fill in the blanks in questions of the following sort:
When you are sure your student is completely familiar with the terms “set,” “group,” and “per” (which means “for every,” or “in each”), and you are certain that they understand the difference between the phrases “objects in each group,” and “objects” (or “objects all together,” or “objects in total”), have them write a description of the diagrams themselves, for example:
There are three types of word problems involving multiplication and division.
Type 1: You know the number of sets and the number of objects in each set.
Example: You have 4 sets of objects and 2 objects in each set. How many objects do you have in total?
Draw 4 boxes to represent the 4 sets:
Fill each box with 2 objects:
Count the number of objects:
8 objects (or “8 objects all together” or “8 objects in total”)
Ask your student to write a multiplication statement to represent the solution:
4 x 2 = 8
Give your student practice with questions of the following sort:
(1) 3 sets
5 objects
How many objects?
(2) 3 groups
7 objects in each group
How many objects in total?
Type 2: You know how many objects there are all together and how many objects there are in each set.
Example: You have 6 objects all together and 3 objects in each set. How many sets do you have?
Draw the total number of objects:
Put three objects at a time into a box until you’ve put all the objects in boxes:
Count the number of boxes:
2 boxes (or sets)
Ask your student to write a division statement representing the solution:
Give your student practice with questions of the following sort:
(1) 12 objects all together
4 objects in each set
How many sets?
(2) 16 objects
2 objects per set
How many sets?
Type 3: You know how many objects and how many sets there are.
Example: You have 10 objects and 5 sets. How many objects are there in each set?
Draw the total number of sets:
Put one object in each set:
Check to see if you have placed all 10 objects. If not, put one more object in each set. Continue until you have placed all the objects:
Count the number of objects in each set:
2 objects in each set
Ask your student to write a division statement to represent the solution:
When your student is able to distinguish between and solve problems of types 1, 2, and 3 readily, you can teach them how to solve more general word problems involving multiplication and division. Tell them to think of a container (like a box or pot) or a carrier (like a car or a boat) as a set, and the things contained or carried as the objects.
Example: 10 people need to cross a river. A boat can hold 2 people. How many boats are needed to take everyone across?
Think of the boats as sets (or boxes) and the people as objects placed in the sets. This is a problem of type 2 (you know the total number of objects and the number of objects in each set).
Draw 10 lines to represent 10 people:
Put boxes around every 2 lines (each box represents a boat):
Count the number of boats:
5 boats
This approach also works for things that have parts (think of the things that have parts as sets, and the parts as objects in the set).
Example: A cat has 2 eyes. How many eyes are there on 5 cats?
Use boxes to represent each cat and lines in each box to represent the eyes. This is a problem of type 1 (you know the number of sets and the number of objects in each set).
Draw 5 boxes to represent the 5 cats:
Draw 2 lines in each box (representing the eyes):
Count the number of lines to give you the answer:
10 eyes
This approach also works for things that have a value or a price (think of the thing with value as a set or box, and the price or value as the objects in the set, e.g., you can think of dollars or cents as lines that you can place inside the box representing the thing you are buying).
Example: A piece of gum costs 5 cents. You have 15 cents. How many pieces of gum can you buy?
This is a problem of type 2.
Draw 15 lines to represent 15 cents:
Put boxes around every 5 lines (each box represents a piece of gum):
Count the number of boxes:
3 boxes or pieces of gum
If you give your student enough practice with this type of problem, eventually they should see that they simply have to divide 15 by 5 to find the answer. For sample problems, see worksheet M-4A, starting on page 137.
1. Write the following multiplication statements as addition statements:
2. Draw arrays for the following statements:
3. Draw arrays for the following statements:
4. Write a multiplication statement for the following problems:
1. Using lines and boxes divide:
2. Write two division statements for each of the following pictures:
3. Draw circles to divide these arrays into:
Now go back and write two division statements for each picture.
For each picture, write two division statements and a multiplication statement.
1. Fill in the blanks:
2. State the number of sets, the number of objects in each set, and the total number of objects:
3. Draw a picture, using boxes for sets and lines for objects, to show how many objects there are in total. Then write a multiplication statement.
a) 3 sets of objects
2 objects in each set
b) 5 sets of objects
2 objects in each set
c) 4 sets
3 objects in each set
d) 4 objects in each set
3 sets
e) 3 groups of objects
3 objects per group
f) 3 objects per set
5 sets
4. Using boxes and lines, show how many sets you need. Then write a division statement.
a) 6 objects in total
3 objects in each set
b) 10 objects in total
2 objects per set
c) 3 objects per set
9 objects all together
5. Using boxes and lines, show how many objects are in each set. Then write a division statement.
a) 6 objects in total
3 objects in each set
b) 10 objects in total
2 objects per set
c) 3 objects per set
9 objects all together
6. Use a drawing to solve each of the following:
a) 6 objects in total
3 sets
How many objects in each set?
b) 3 objects in each set
15 objects
How many sets?
c) 4 objects in each set
5 sets
How many objects?
1. To solve each question, think of containers or carriers as sets, and the people or things carried or contained as objects. Use boxes and lines in your answers.
a) 3 cars
2 people in each car
How many people?
b) 20 people
5 boats
How many people in
each boat?
c) 5 plates
3 cookies on each plate
How many cookies?
d) 20 pens
4 pens in each box
How many boxes?
e) 2 buses
6 people on each bus
How many people?
f) 15 flowers
5 flowers in each pot
How many pots?
2. To solve these problems, think of things that have parts as sets, and think of the parts as objects in the set.
a) 5 chairs
4 legs per chair
How many legs?
b) 16 legs
4 legs per chair
How many chairs?
c) 3 wheels per bicycle
5 bicycles
How many wheels?
d) 2 ears on each cat
10 ears
How many cats?
e) 3 spots on each dog
15 spots
How many dogs?
f) 5 fingers on a hand
20 fingers
How many hands?
3. To solve the following problems, think of rows as sets and chairs as objects. Draw a picture to show your answer.
a) 5 rows
3 chairs in each row
How many chairs?
b) 10 chairs
2 rows
How many chairs in each row?
c) 3 chairs in each row
15 chairs
How many rows?
4. Think of things that have value, or cost money, as sets, and the cost of each thing as the number of objects in each set. (Use lines and boxes.)
a) 5 tickets
4 dollars for each ticket
How many dollars do the tickets cost?
b) 5 cents for each piece of gum
You paid 20 cents
How many pieces of gum did you buy?
c) You paid 12 dollars
You bought 4 cups
How much did each cup cost?
5. Write a multiplication or division statement to solve each of the following:
a) 4 objects in each set
16 objects in total
How many sets?
b) 25 objects in total
5 sets
How many objects per set?
c) 3 sets
4 objects per set
How many objects?
6. Write a multiplication or division statement for each of the following:
a) 5 cats
3 fleas on each cat
How many fleas?
b) 12 birds
3 birds in each tree
How many trees?
c) 4 bowls
3 fish in each bowl
How many fish?
