While Cuisenaire Rods are a great way to teach basic multiplication (using a repeated
addition model as the basis), there are some serious logistical problems with using
them.
I have one 1-quart ziplock bag containing 10 of each Cuisenaire piece (not including
ones), and one square with numbered rods as pictured at right, for each pair of
students. This amounts to more Cuisenaire rods than is contained in a classroom set.
Also, it takes two student desks of the same height placed end-to-end to fit a meter
stick.
Third is the time it takes to move desks, distribute and collect Cuisenaire rods and
meter sticks, etc., etc. If your Math class is confined to a 40-minute period, time is an
obstacle.
However, if you can manage all this, your students will love you for it. It gives an
immediate, visual, concrete presentation of what happens when we multiply (and
divide) numbers. It’s extremely important that students get this early, when they first
begin dealing with multiplication and division. And you won’t have to do it very often.
Coloring the Multiples – an Alternative to Cuisenaire Rods
For teachers who don’t have Cuisenaire rods or can’t manage the practicals for
whatever reason, here is the next best thing. You can also use these activities after
first using Cuisenaire rods and meter sticks. Or you can give students copies of the
number lines to complete as they are using Cuisenaire rods.
These activities have a 0-to-81 number line printed on legal paper. The number line
extends to 81 to allow enough space for the individual numbers to be legible. By the
time students have modeled 9 × 9, they get the point. There is a version of this
activity for multiplication only, and one for division as well.
Students color each multiple of a given number (6 is pictured at right), using
alternating and contrasting colors. Matching the colors of the Cuisenaire Rods is not
the point. They then circle each multiple. Then they work the problems at the bottom
of the page.
I have students complete the multiples in the left column after finishing the coloring
and circling. Then they cover the first column with a book or piece of paper, and do
the column in the middle of the page. For the version that includes division, students
then cover both left and center columns before doing the division problems at right.
You can decide whether you want them to also cover the number line.
The first time you do this, you should plan to spend serious time on the coloring. I
demonstrate this on the Smartboard by audibly tapping the marker in each space to
make a dot, and counting out loud: “One…two…three…four…five…SIX!” I then circle
the number 6, and color the bar to the 6 as shown. Then I begin after 6 and again
count, “One…two…three…four…five…SIX!” If I am accurate, I am then at 12, which I
circle. And so on. The counting out loud is important and you should not allow
students to skip it. You’ll need to monitor students’ progress closely, or have
students working in pairs and monitoring each other’s progress. It’s easy to miscount
once and end the first 10 multiples of 6 at 59.
Just as with all number line work, you need to stipulate that the things we are counting
are SPACES, not ticks or numbers. The first 6 spaces, ending at the tick for 6, are
colored orange. The second 6 spaces, ending at 12, are colored blue…and so on.
Cuisenaire Rods Multiplying by Coloring all Factors (PDF)
This has the activity pictured above right for all factors 2 through 9.
Cuisenaire Rods Multiplying and dividing by Coloring all Factors (PDF)
Includes division.
Cuisenaire Rods Multiplying by Coloring all Factors (Smartboard)
Cuisenaire Rods Multiplying and dividing by Coloring all Factors (Smartboard)
Here are practice problems. The following documents include a mix of problems
sorted by factor, as shown at right, for basic multiplication, or multiplication and
division.
Multiplication Problems with Missing Factors (PDF)
Multiplication Problems with Missing Factors 3 columns (PDF)
Many more problems than above
Multiplication and Division Problems with Missing Factors (PDF)
Multiplication Problems with Missing Factors (Smartboard)
Multiplication Problems with Missing Factors 3 columns (Smartboard)
Multiplication and Division Problems with Missing Factors (Smartboard)
Click here for the Decimal Computation page.
It’s possible to illustrate multiplication and division of decimal numbers using the
same Base 10 Block model as on the Introduction to Decimals page.
The “100 Grid” is the One.
Understanding place value is all-important when it comes to multiplying and dividing
with decimal numbers, just as it is when multiplying by powers of 10 or multiples of
10. The patterns inherent in these operations is the subject of Common Core Standard
NBT.1 for both Grades 4 and 5 (4.NBT.1 and 5.NBT.1): “…a digit in one place
represents ten times what it represents in the place to its right…”
Base 10 Blocks are helpful because they can be used to illustrate the pattern, or
progression, on both sides of the decimal point. The activity at right is intended to
show that the value of a number increases by 10 times for each place to the left, and
decreases by 10 times for each place to the right.
The activity pictured at right simply has students illustrate on grids the multiplication
of whole numbers by decimal numbers. Using contrasting colors, students color a grid
to show, for instance, the expression 4 × .6, or “four groups of six-tenths.” After
this is done, the connection between the multiplication and the placement of the
decimal point is straightforward:
“4 × 6 = 24.”
“OK, but twenty-four whats?“
“Twenty-four tenths.”
(just as 4 × 6 pencils = 24 pencils).
“So where does the decimal point need to be?”
Students should do a couple of different activities before they’re ready to generate a
rule or procedure to place the decimal point in the product. I think it’s a mistake to
simply state a rule at this point; more about this below.
Along with clearly illustrating the place value of the products, coloring the factors on
grids leads to another insight that should be discussed: How can the product be so
small? How can you multiply two numbers and get a product that’s less than 1? Or
less than one of the factors you started with? I thought multiplication always gave you
a bigger number!
For instance: 4 × .2 =
.8
There are many ways to approach this. One is to remind students that any number
multiplied by 1 is itself. So if you multiply a number by less than 1, you will get a
product that is less than the original number.
The grids make this obvious. Two-tenths is a small number. Four of them aren’t
really going to be much bigger.
I think students should do these activities in a particular order. There should be some
multiplication illustrated by coloring grids. This helps them visualize the relative size
of factors and product. Then students should do some problems involving the same
factors with only the location of the decimal point changed, such as in the
“Multiplication and Division Patterns” file. This helps them see the patterns involving
the placement of the decimal point relative to the numbers.
Then perhaps students can generalize what is done and write a rule. However it’s
stated, a rule should refer to the location of the decimal point being related to
multiplying or dividing the number by 10, 100, or 1000.
Here’s a way to state it for the problem given above:
4 × .6
1. Multiply the factors as if they were whole numbers.
In other words, ignore the decimal point.
4 × 6 = 24
2. Note that one of those factors is actually divided by 10.
In other words, one factor has one decimal place
(or, one digit to the right of the decimal point).
.6 = 6 ÷ 10
3. Make the same change to the product – divide it by 10.
In other words, locate the decimal point so there is
one decimal place in the product.
24 ÷ 10 = 2.4
Division with Decimal Divisors
There’s a big difference between division with decimal dividends, and divisors. These
activities address the use of decimal divisors.
In the division problems at right, the grids representing the dividend are outlined. We
then divide this dividend into parts the size of the divisor.
In other words,
“How many groups of 5 are in 10?”
10 ÷ 5 = 2
“How many groups of five-tenths are in 10?”
10 ÷ .5 = 20
Notice that this is an example of Measurement Division. For a discussion of
Measurement Division vs. Partitive Division, click here.
The question needs to be worded correctly when introducing this to students.
“How many groups of 5 fit into 10?”, or, “How many groups of five-tenths fit into 10?”
Not:
“What is 10 divided into 5 equal groups?”
How would you word this with a decimal divisor? This would be tough to interpret for
students just starting to divide with decimals.
So you color areas equal to the divisor, using alternating and contrasting colors. The
insight for the students comes with the second problem, when the quotient is larger
than the dividend. This gives rise to the same sort of questions asked above with
multiplication of decimals:
“Isn’t the quotient supposed to be a little number?”
What is made obvious with the coloring is that if we are dividing a number into groups
smaller than one, there are going to be more of those groups than there are ones in
the dividend.
The series of activities in the file “Decimal Division with Decimal Divisors” is intended
to lead students to an understanding of how the algorithm works. And just as with
decimal multiplication, the key is that the placement of a decimal point is related to the
multiplication or division of a number by 10, or 100, or 1000. Students also see that
if the same change is made to both dividend and divisor, the quotient is constant.
A suggested series of steps is given, following some practice with coloring grids to
illustrate decimal divisors and discussion of the activities in “Multiplication and
Division Patterns.” The important steps are:
Step 1: Make the divisor a whole number by multiplying it by 10. This means moving
the decimal point to the right.
Step 2: Make the same change to the dividend -- multiply it by 10. Rewrite the
decimal point in its new place. If you add a new place in the dividend,
write a 0 into it.
Division with Decimal Dividends
Activities designed to help students understand division of decimal dividends will be
forthcoming.
Algebraic Topics
This page will be added to over time.
Introduction to Exponents
One of the most common ways to get across the idea of what exponents do is through
squares and cubes. Probably every Math textbook and piece of software has images of
squares and cubes to illustrate the second and third powers, usually associated with
area and volume.
But if we agree that children learn Math by encountering it in reality – in three
dimensions – why not have them build their own models? And why stop at the third
power?
You can use Base 10 Blocks, of course. But those little 1 cm cubes are pretty hard to
stack. One-inch cubes (some cubes are actually 2 cm, or about four-fifths of an inch)
are pretty common. You could also use Unifix or some type of interlocking cubes if
need be.
Pictured at the right is the progression with 2. Students start with 2 in “one
dimension” – a “line” that is 2 cubes in length. The equation used to calculate the
value of this line is 1 × 2 = 2.
Next is the second dimension: width. The “line” has been doubled. The model is a
square with side equal to 2. The equation is 2 × 2 = 4.
Next is the third dimension: height. The square has been doubled. The model
is a cube, or the square built up so the height is equal to the length. The
equation is 2 × 2 × 2 = 8.
Next is that same cube doubled. The model is 2 cubes. I like to allow students
to name this object themselves. The equation is 2 × 2 × 2 × 2 = 16.
Notice we haven’t used the word “exponent” yet.
The same progression is shown for 3. The line is 3. It’s tripled to a square of 9. The
square is tripled to a cube of 27. The cube is tripled to a “line of cubes” of 81. And so
on.
Building these lines, squares, cubes, and so on is an eye-opener for students,
especially as they see how the equation is related to the dimensions, and to the
number of cubes in the object. Students who have already been exposed to exponents
will recognize that this is what they are demonstrating, but I’ve never had anyone lose
interest.
I like to have all students build using at least 2 different bases: in other words, they
build objects using 2 as a base up to 25, or 32. If there aren’t enough cubes, they get
far enough to project what that object would look like and what the number would be.
Then they start with 3, and build up to 34 , or 81. Again, they need to get far enough
along so they can visualize the object and calculate the number.
At least some students should work with 4 and 5 after finishing their first progression
using 2 or 3.
Space and the availability of cubes probably dictates that students will work in groups
of at least 2. Groups of 3 work well with this – there’s enough to do and students can
take turns serving as the group’s “recorder/mathematician”. Groups larger than 3
sometimes have difficulty maintaining focus.
By the way, no one uses calculators. Students almost always take the initiative in
calculating the value of objects there aren’t enough cubes for them to build.
Followup activities include an explicit introduction to exponents and some practice
with naming and calculating exponential numbers in their various expressions:
exponent, expression, and in words. Textbook activities can follow these activities
without need for much explaining on your part at that point.