"How much concrete would you need for a new driveway?"Areas of Objects
Objective: When done with this lesson, you will be able to calculate the area of rectangles, triangles, parallelograms, and trapezoids given the coordinates of vertices.
Approximate completion time: 3 hours
"How much concrete would you need for a new driveway?"
"How much might that cost?"
"They'll need some grass for that house. How much will they need?"
"Won't they need some 'topsoil' for before they plant grass?"
To answer any of these questions, one would need to know a few things about area.
INVESTIGATE:
Let's see what you know already about area. We'll start with an easy shape.
Take a look at this rectangle that was plotted on a nice coordinate system. (Do you remember what this type of coordinate system is called?)
Take a guess at how much area you think this rectangle has. When you think you know what the area is, click on the orgin of the coordinate grid to see if you were thinking correctly.Hopefully you were counting the number of squares in your answer. There are many different ways to do this. Some count all of them like in the animation you got when you clicked the origin. Others do some fancy multiplying. However you do it really doesn't matter. Yet there is something very, very, extremely important.....units.
How you describe the area of a shape is totally dependent on the units you use. This rectangle could be 20 square feet or 20 square inches or 20 square miles or 20 square yards. If you don't answer in terms of units, then I guess it could be anything--maybe even square grandmothers!
Now for another example. Notice that in this graph, the distance between two adjacent grid points is 1 foot.
Notice that there will be 12 small squares inside the rectangle. So we say the area is "12 square feet" or "12 sq ft" or "12 ft2".
Hopefully you've seen this idea before. Yet please note the detail of units.Notice that the area can also be found by multiplying the lengths of the sides: 3 feet by 4 feet = 3ft · 4ft = 12 ft2. (Do you remember that x times x is x 2 ? That's why it is ft2 for this case.)
In this example note that the distance between two adjacent grid points is 1 mile.
This one is a bit tougher. Some might try to add up the "parts" of square miles that are there. Yet to be exact, you'll need some clear thinking. Here's a hint: do you see a triangle missing that would have made this a rectangle? As it is, this shape is a trapezoid and not quite a rectangle. Before you read on, try to figure out what you think is the area of this trapezoid.
If you think you have an idea how to find the area or if you're looking for a hint, click on the (3 mile, 2 mile) coordinate. You'll get a new window with an animation that shows some thinking behind a good strategy some of us call "framing".
So how exactly would the area of the trapezoid be found with this "framing" technique? Take a look at this work. The correct answer is 15 square miles for the area of this trapezoid.
Here are 3 shapes for you to practice. Try to calculate the area of each of these shapes. Don't forget your units! If you'd like to check the number part of your answer, then use this nice tool or this one to build the shape and check its area (without the units--which is pretty silly if you think about it). You can move your mouse over the last two to get some help with the "framing" idea.
PRACTICE:
Now it's time for you to dig in a little deeper to practice and make some conclusions of your own.
Download your practice problems here. (Be patient, this one takes a bit more memory.)
