Operations Using Rectangles

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In this lesson, we build on concepts explored in the last lesson, and in the lessons linked in red directly above (from Course I), to develop more mental strategies for computations with whole numbers, decimals, and fractions.

We have discovered that some models work for representing a variety of computations. For example, we can represent a product by the dimensions of a rectangle (lots happens in this animation - watch it more than once!):

• a) 223 x 132 = 29,436
• b) 22.3 x 13.2 = 294.36
• c) 2.23 x 1.32 = 2.9436

Inherent in all of these kinds of models as well, is the distributive property. Here's an example relating to these particular numbers:

By adding up the partial products, which are easy to compute, we can find the solution:

2 + .64 + .23 + .0736 = 2.9436

Mental math is where it's at!

By exploring these kinds of models, some important generalizations about computations can be made. Dividing each of the dimensions by 10 divided the product by 100 (a to b, b to c). Dividing the dimensions by 100 divided the product by 10,000 (a to c). In class, we also explore the result of dividing only one dimension by 10 or 100; for example 223 x 13.2. What would the resulting product be? If you thought 2,943.6, you're right. The point is if we know the answer to 223 x 132, then we also know the answer to lots of other related problems, because we know the effect of dividing the factors by powers of ten.

More problems to discuss with your child:

• 2.3 x .9
• 350 x 460
• 142 x 21
• 4.3 x 2.7
• 3 3/4 x 2 1/3
• 1 5/6 x 7/10

Using generic rectangles, we can explore some other ways to think about multiplication and division computations. In class, we use paper rectangles and scissors to experiment with ways of maneuvering, cutting, rearranging, copying and so on to make a problem simpler to solve mentaly:

 

 

This method is known as the equal products method.

Here's another example...

Or, we can use the distributive property again...

Here are some more problems to try some of these methods on. The methods you use might depend on a personal preference or on the particular problem; some lend themselves more easily to one solution strategy than another...

• 18 x 45
• 250 x 28
• 22 x 16
• 2.2 x 1.6
• 2.4 x 1.5
• 12 x 99
• 1/3 x 2/5
• 3/4 x 1/2

Did the last two seem harder to think about? Try making one of the dimensions a whole number and see what happens. The idea behind these kinds of models is to help students develop mental strategies that can liberate them from calculators and memorized procedures. Students can use variations and combinations of these straategies to create their own personal methods.

We can use similar types of concepts with division. Here's some ways of looking at 126 ÷ 20:

 

 

This is the equal quotients method.

Here's another idea...

We can also make use of the distributive property...

The rectangle area concept can also be used to develop some methods for adding and subtracting fractions. We can picture the second dimension of a rectangle as a fraction with the area as a numerator, and the dimension as a denominator:

We can make use of this idea in the following way. Watch as the model develops into a geometric form of fraction addition...

 This method can also used with subtraction

of fractions, just treat the right-hand rectangle

as if it had negative area.

Try these problems using the adjacent rectangle method:

• 11/8 + 7/6
• 5/4 + 7/3
• 3/2 - 1/3
• 9/4 - 8/3
• 2/3 - 3/4
• x/3 + 1/2
• 2x/5 - 3/4

If you feel like a challenge, try to generalize this method for a/b + c/d and a/b - c/d.

This lesson, along with the previous one, contains many of important ideas about operations in the decimal and fraction number systems. If students have a firm grasp of these concepts, they should have no problem handling most daily problem-solving computations.