Area of Larger Rectangles

Overview: In our last geometry lesson, we learned that to compute the area of a rectangle, you would multiply the length of one pair of sides by the length of the other pair. However, the only examples we used were of simple multiplications that were one digit by one digit. Today, we will use rectangles to help us figure out how to multiply larger numbers. Not only will we be able to compute larger areas, but it will also help us multiply larger numbers in any situation!

Objective: Learn how to visualize the distributive property using the area of rectangles. Learn how to easily multiply by tens, hundreds, and thousands. Learn how to split a large complicated number into smaller, easier to multiply numbers, and add the result.


Suggested Lesson Development

I. Visualizing the Distributive Property

In your previous sets lesson, you learned about the distributive property of multiplication over addition. This property will come in handy today in learning to compute the area of larger rectangles. 

Review the distributive property of multiplication over addition with the students. Ask the student to first use this property to rewrite (a + b) * c. Then, have them take a simple multiplication (e.g., 3*4), rewrite one of the multiplicands as the sum of two addends, and then solve all three versions. For example:

  3 * 4 = 12
  (1 + 2) * 4 = 3 * 4 = 12
  (1 * 4) + (2 * 4) = 4 + 8 = 12

Now today, we're going to learn to visualize this property using the area of rectangles. Do you remember how the area of a rectangle is computed?

Make sure the student remembers computing the area of a rectangle. Have them draw a rectangle based on the multiplication they picked in the previous problem (e.g., 3*4). Have them count the squares inside the rectangle and see that it is the same as the multiplication of the two sides.

Can you see how the distributive property of multiplication over addition might be visualized now?

Allow the student some time to play around with their example. Encourage them to try to visualize every piece of the example they've already worked out (e.g., 3*4 = (1+2)*4 = (1*4) + (2*4)). Ask them questions about what they're doing and try to steer them in the right direction if they get lost.

Look at your visualization. What have you learned? The first thing you might have noticed is that, just like you can split up a multiplicand into the sum of two addends, you can split up one of the sides of the rectangle. Here's my version:


Do you see how just like I split 3x4 into (1+2)x4, I also split up the length 3 side into a length of 1, and a length of 2?

Now, what does the distributive property say I can do with (1+2)x4? That's right, I can rewrite it as (1x4)+(2x4). Do you see (1x4) and (2x4) in the picture? If I computed the answer to 1x4 and 2x4, what parts of the area would I have? What would I have to do with these two pieces to get the area for the original 3x4 rectangle?

Discuss this example with the students and then review their examples. Try to get them to talk about what they're learning. Hopefully, they are starting to visualize how distribution breaks down and reassembles the problems. At the very least, they need to be able to go through the mechanics of breaking down and reassembling an area problem using the distributive property.

II. Multiplying by 10's, 100's, 1000's, etc.

If I have one apple, and you give me two more, how many apples do I have? You're probably thinking that what I just asked was a very silly problem.

But what if you had one billion, and then you got two more? How many billion would you have? You can treat adding billions like you treat adding apples. If you have one apple, and get two more, you have three apples. If you have one billion, and you get two more, you have three billion! Wow! look how fast you added that!!!

You can, of course, do something similar with tens. If you have one "ten", and you get two more, you could say you have three tens. But because we use our tens so much, we have given them all special names. Two tens, for example, is also known as "twenty". Three tens is known as "thirty" and so forth.

Have the student practice thinking of 20, 30, 40, 50, 60, 70, 80, and 90 as groups of ten. You could, perhaps, play a game where you say, "four tens", and they would respond "forty!"

Hundreds, thousands, and other such numbers are used less often and so we use less imaginative names like "two hundred", or "three thousand", and so forth.

It's pretty easy to multiply by tens because our number system is based in tens. If you have 3x10, for example, you're saying you have three tens, which, as we just discussed, is just 30. 

Now, let's draw this with our area examples. Draw a rectangle that is 3x10. You should see that you have three rows of ten columns (or three columns of ten rows). Look at each row of 10. Do you see that you have three such rows? And if you have 3 tens, you know you have 30 squares.

Have the student draw some rectangles that are a single digit number by 10. If they want to go past single digits, ask them to be patient, that is coming up. For now, single digit numbers on one side, and 10 on the other. Have them practice counting the tens and saying the name for that many tens (e.g., "4 tens... 40!").

You may have noticed something. When you multiply a single digit by 10, you get that same number in the tens column! So, multiplying 3 by 10 puts 3 in the tens column, or 30. Multiplying 9 by 10 puts the 9 in the tens columns, or 90. Just like before, we're just counting 10s!

The very special number "10 tens" is what you know as one hundred (100)!

So, what if you multiplied 15 by 10?

Discuss 15x10 with the student. They will probably get to 150 without too much trouble. Point out that 150 is 15 10's. They can draw this if they want to see it. Once the have that down, have them convert 15 into (10+5) and use the distributive property of multiplication over addition to solve. Also have them try (8+7) and (9+6).

Multiplying by hundreds is similarly easy. If you have 3 hundreds, how much do you have? 300! Notice that you put the number into the hundreds column indicating how many hundreds you have. So, if you have 9 hundreds, you put the 9 in the hundreds column and get 900!

Now try having the student multiply 15x100. There won't be any problems multiplied by hundreds, so this is just to see how much they're getting.

III. Multiplying double digit numbers.

Suppose you wanted to get the area of a 12x15 rectangle. How might you do it? You maybe don't realize this, but you already know how to multiply this kind of number!!!

The trick is to use the property of distributing multiplication over addition to make this problem easier. Perhaps you don't know to multiply 12x15, but, can you figure out how to break 12 into two addends that you can multiply 15 by?

Let the student try breaking 12 into some easy to multiply numbers. Chances are, they won't actually think to multiply by 10, although they might get there accidentally if they want to multiply by 2. If they don't split it into 10+2, try to hint at it by asking them if there are any other numbers they've multiplied 15 by in this lesson. Finally, have them draw out 15X12, dividing the 12 as appropriate so they can see the two computations. As the last step, ask them how they combine the two "sub results" into the final answer

Have them practice this on a number of examples. For now, don't multiply by anything bigger than 19 so that, at most, they have the multiply by 10 and not 20 or 30 etc.


EXERCISES

For problems 1-10, use any methods you learned in the lesson to figure out the answer. However, you must draw a picture for each one.
  1. What is the area of a rectangle that is 9x9




  2. What is the area of a rectangle that is 5x10




  3. What is the area of a rectangle that is 10x10




  4. What is the area of a rectangle that is 11x11




  5. what is the area of rectangle that is 11x13




  6. What is the area of a rectangle that is 14x14





  7. What is the area of a rectangle that is 14x16




  8. What is the area of a rectangle that is 10x19





  9. What is the area of a rectangle that is 9x19





  10. What is the area of a rectangle that is 19x19




  11. Let A = {1,...,150}, B = {200,...,210}, C = {140,150,160,170,180,190,200,210}. Compute (A  B)  C.




  12. Using the same sets as in #11, rewrite it using the distributive property of intersection over union and recompute.




  13. Using the same sets as in #11, Compute (A - B) ∪ C




  14. Using the same sets as in #13, demonstrate that there is no distributive property of union over set difference.




  15. Solve x + 10 = 20




  16. Solve 23 + x = 27




  17. Solve 10 - x = 5





  18. A Tyrannosaurus attacks a herd of Hadrosaurs and kills several old and sick ones. If the pack originally had 205 members, but now has only 200, how many did the Tyrannosaur kill?




  19. Write #18 as an equation and solve.




  20. A chemist is making an explosive and needs 200 pounds of compound X to complete the project. If he gets 100 pounds from one store, and 64 from another, how many pounds does he still need to get? Write this problem as an equation and solve.




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