ATTACKING THE MULTIPLICATION FACTS: A BULLETIN BOARD IDEA
Teaching the basic facts for any of the four fundamental operations of
arithmetic is a task most teachers approach in a very organized manner...beginning
with the easier facts and gradually adding more and more of the complex
ones until all one-hundred have been conquered. However, students often
see this as a seemingly endless task that is attacked with no rhyme or reason.
The following activity is designed to help the teacher promote a feeling
within the classroom that says, "Here is our task, to learn one-hundred
multiplication facts. Let's set up a game plan to get this accomplished!"
At the time this activity is commenced, students should have a solid understanding
of the concept of multiplication based on the manipulation of concrete materials
using array patterns. A main thrust of this activity is to help bridge the
gap from the concrete stage (building arrays) to the symbolic stage (memorization
of the facts). Mark off an eleven by eleven grid on a bulletin board using
square decimeters. Use a strip of paper to cover up the first two columns
and the bottom two rows of the grid as shown below. Place a dot in the square
in the lower left-hand corner.
Because students already know facts like 2 fours make 8, the twos are
a good starting point. Prepare a grid to be placed on the overhead. Provide
students graph paper with squares the
same size as those on the grid. Have them cut out as many different array
patterns as they can in which one side is two units long and no more than
eighteen total squares are included. Several
examples are pictured here.
Some interesting discussions can take place as to whether the number
of arrays that fit the conditions will be even or odd (odd because of the
2 by 2 array) and what the total number will be(17) As students begin to
cut out such array patterns the questions will usually arise as to whether
the common shaded arrays above (4 by 2 and 2 by 4 and 1 by 2 and 2 by 1)
are the same or different. Discuss the pros and cons of this issue. Decide
(temporarily) to consider such pairs as being different...one showing the
problem 2 x 4 = 8 and the other that 4 x 2 = 8. To help
distinguish between them, have students place a dot in the lower left hand
corner of each array.
Have students continue cutting out array patterns until all seventeen
have been found.
Have a student hold up on of the arrays. "What are the dimensions of
your array pattern?" (4 x 2)
"How many squares in a four by two array pattern?" (8) Say, "Four
twos make eight. Four times two equals eight." Have the student place
the array pattern on the overhead grid with the dot on the dot, fold back
the corner opposite the dot, and write the numeral 8 in that square as you
say, "Four twos make eight. Four times two equals eight."
Have another student hold up a different array pattern. "What are the
dimensions of your array
pattern?" (2 x 5) "How many squares in all?" (10) "Five
twos make ten. Two times five equals ten." Have the student hold this
array pattern on the grid with the dot on the dot. Fold back the corner
opposite the dot and record the numeral 10 as you repeat,
"Five twos make ten. Two times five equals ten." Continue until
all possible arrays have been utilized, then fill in the facts on the permanent
board. The finished product appears below on the left.
Say, "We have started today to fill in the multiplication tables.
Remember the addition tables you had to learn?" Show a picture of the
addition table. Return to the bulletin board and say, "We sure have
a lot of facts left to learn, don't we? Was there a big idea that helped
make it easier to learn the addition tables?" By studying the addition
tables, lead the class to a verbalization of the commutative property of
addition ... that 3 + 4 = 4 + 3, 8 + 6 = 6 + 8, etc. Show how this property
cut the number of addition facts we had to learn almost in half.
Ask, "Do you think this idea also works with multiplication?"
Show that it does work using
array patterns. Have students demonstrate on the grid how the same array
pattern could be used to show both 2 x 4 and 4 x 2. Repeat for several other
array patterns. Say, "This idea will also cut the number of multiplication
facts we have to learn almost in half." Place a large sheet of colored
paper over the lower part of the table as shown below.
Leave this partially completed bulletin board up for students to use to practice on the twos. Emphasize the fact that they really know facts like 2 x 7 because they know 7 + 7. Be sure to cover the permanent bulletin board when fact mastery work is in progress. Note: All facts should be taught in context, i.e., presented in verbal problems made up by teacher and students alike.
Many teachers next work on the fives. This is probably because they can
be easily attained by skip-counting. Have students cut out array patterns
in which one side is five units long using no more than 45 total squares.
Remind the class that we now think of the two arrays below as being the
same. Show with several examples how one can be rotated to look like the
other.
Ask, "How many different arrays can we come up with?" (9) Do
not put dots in the corners from this point on. Have a student hold up one
of the arrays. "What are the dimensions of your array pattern?"
(5 by 6) "How many squares in all?" (30) "What multiplication
problem does it show?" (6 times 5 = 30) Place the array pattern on
the bulletin board, fold back the appropriate corner, and say, "Six
fives make 30 - six times five equals 30." If the student places the
array on the grid so the square under the turned down corner is black, have
them rotate the array the other way.
Continue until all of the fives have been placed on the grid. Various games should be used to make the twos and fives automatic.
Next, the fours should be attacked. The fours are attainable by using
prior knowledge of the twos and doubling twice. For example, 4 x 7 is twice
as much as 2 x 7. This should be shown concretely as pictured below. Since
2 sevens make 14, 4 sevens should be twice as much or 28.
This basic strategy can be applied any time one of the factors is even.
Although they have already covered 4 x 5, a helpful strategy is to use the
perfect squares as below.
For some reason, students seem to find the perfect squares easier to remember. Have students cut out arrays for the fours and add them to the multiplication table.
The threes are also usually introduced using skip counting. However, since skip counting by threes is not as easy for students as is skip counting by twos and fives, some of the thinking strategies below might be utilized. All such situations are free rides, i.e., using what you do know to get at what you do not know.
This strategy, which has several sub-categories, can be used to get any
unknown fact from a
known fact using the free ride strategy. As at the right, you might use
prior knowledge of 2 x 3 to make sense out of 3 x 3. Provide a 2 by 3 array
and add a third column of 3. "If 2 threes are 6, what would 3 threes
be?
In this example, knowledge of 3 x 5 can help make sense of 3 x 6. "If
5 threes make 15, what would 6 threes be?"
This same strategy can be used in reverse by starting with the known
fact, in this case 4 x3 and eliminating a column. "If 4 threes make
12, what would 3 threes be?"
Here, some students might get at 7 x 3 by thinking, "If 5 threes
are 15 and 2 threes are 6, then 7 threes should be 15 + 6 or 21." As
in the examples above, one should start the thinking with manipulatives
and move on to doing the same thinking mentally.
The threes should now be added to the table and included in the various fact games and activities. Each time a new set of facts is added to the chart, be sure to stop and point out how the number of facts left to be learned has decreased. (We are now down to 15.)
At this point, the ones may be added to the table. Although the ones
are very easy, it is not a good idea to include them until the concept of
multiplication has been established. Array patterns including the ones should
be used to place these products on the table. Now only ten additional
facts remain.
Because of some interesting patterns, the remaining nines should be covered
at this time. Write the nines that have already been covered on the chalkboard
as shown below. Allow students to study this information looking for patterns.
Some of the patterns usually quickly discovered include:
1. The sum of the digits of each product is nine.
2. As the ones digit of each product is decreasing by one, the tens
digit is increasing by one.
3. The tens digit of each product is one less than the number by
which nine
is being multiplied.
Discovery of and discussion about these patterns should lead students to
the finding the products of the remaining nines. The discussion should center
around why these patterns exist. Some of the explanations develop from knowing
how to multiply by ten (10 x 3 = 30 so 9 x 3 should be in the twenties.
10 x 3 = 30 so 6 x 3 should be 3 less or 27.). Although multiplication by
ten has not been discussed, it should follow from place value work (3 tens
make 30).
Discussion should also lead to a method for determining a fact such as 7 x 9 without having to list all of the combinations. If 7 tens is seventy, then 7 nines should be in the sixties. If the answer is sixty-something and the sum of the digits must be nine, then the product must be sixty-three. Array patterns should be used to add the new nines to the table. Now only six facts remain!
Next, the remaining "doubles" should be added to the table
(6 x 6, 7 x 7 and 8 x 8). These facts can be covered using the "splitting
a product into known parts" and "twice as much as a known fact"
strategies discussed earlier.
7 x 7 = ?
Since 5 x 7 = 35,
and 2 x 7 = 14
7 x 7 should = 49
6 x 6 = ?
Since 5 x 6 = 30,
6 x 6 should be 6 more
or 36.
8 x 8 = ?
Since 4 x 8 = 32,
8 x 8 should be
twice that or 64.
Any of this type of reasoning is a free ride, using what you do know to
solve a problem you don't know. Students should be given the opportunity
to make such math connections on their own. Adding the doubles to the
table leaves only three facts to be covered!
To fill in the remaining three facts (6 x 7, 6 x 8, and 7 x 8) in addition
to "splitting" and "twice as much" strategies used earlier,
students can also make use of the doubles. These, of course are all free
rides.
6 x 7 = ?
Since 6 x 6 = 36,
so 7 x 6 should 6 more
or 42
7 x 8 = ?
Since 5 x 8 = 40,
and 2 x 8 = 16,
so 7 x 8 should be 56
6 x 8 = ?
3 x 8 = 24,
so 6 x 8 should be
twice that or 48
Finally, multiplication with zero as a factor can be covered. Although
these facts may actually be dealt with earlier, it is better not to put
them on the chart until all other facts have been covered. Now the strips
covering the first two columns and rows can be removed to reveal a completed
multiplication table.
Again, as suggested in teaching addition and subtraction strategies,
a good activity is to provide a troublesome fact and have different students
verbalize their free ride for finding its product.
Highlighting troublesome facts as shown below is a good bulletin board activity.
A GAME PLAN FOR LEARNING THE MULTIPLICATION FACTS
The following steps are one possibility for planning instruction for
learning the multiplication facts. Children are encouraged to develop mature
thinking strategies so they can successfully find quick responses to fact
problems. The bulletin board idea on the preceding pages lays out this plan
in more detail.
Step 1:
Introduce multiplication using the array model.
Step 2:
Use prior knowledge of addition to attack the twos. These and all other
facts
should be taught in context rather than as isolated bits of information.
Step 3:
Use the model to develop skip counting to solve appropriate problems
involving the fives.
Step 4:
Provide practice on each new set of facts introduced setting them in
real situations.
Step 5:
Attack the fours by doubling twice.
Step 6:
Use prior knowledge of twos, fours, and fives to attack the threes.
Step 7:
Use patterns to develop the nines and ones.
Step 8:
Add new facts to drill games and activities.
Step 9:
Use drill and appropriate strategies for the doubles.
Step 10:
Use the array model to develop the strategies that involve splitting
the
product into parts. Develop one strategy at a time. Use these strategies
to develop the remaining facts.
Step 11:
Consider facts with zero a factor.
Step 12:
Test over all facts. Identify specific strategies for troublesome facts
with individual students.
Step 13:
Drill on all the facts to develop speed and accuracy.
Dr. Harry Bohan
33 Elkins Lake
Huntsville, Texas 77340