How to teach areas of triangle, parallelogram, rectangle

[EngOZ]

My otherwise bright 9 year old daughter seems to have trouble comprehending how to find the areas of triangles, parallelograms and rectangles.

 

We've started using MUS Delta, so I'm not sure if these topics were covered in previous levels.

 

Please share some ideas and resources so I can effectively teach dd. 

[wendyroo]

I would use graph paper.  

 

Start with rectangles; draw some (on the grid lines) and have her count how many squares are inside the rectangles.  Do it over and over until she is completely comfortable with the idea that multiplying the lengths of the sides is simply a short cut to figure out how many squares are inside.  Really emphasize that area is simply the number of square units inside a shape.  Practice that concept (just with rectangles) in more abstract problems in MUS.

 

Next move on to parallelograms...they require a bit of cutting.  Back on the graph paper draw some parallelograms and have her physically cut them apart and rearrange them into rectangles so she can really see how to find their area.

 

Lastly, triangles.  Start by drawing and cutting out two identical right triangles.  Show her how to put them together to make a rectangle, find the area and then split it to find the area of the triangle.  Do the same with an equilateral triangle which will be a bit more complicated to make into a rectangle, but will reinforce the concept.  Once she is familiar with the process, challenge her to prove that the formula works by carrying out the rectangle-making process with several "non-standard" triangles.

 

Wendy

[EKS]

Have you had her do multiplication problems with the blocks?  In the MUS sequence, finding the area of a rectangle is a natural outgrowth of this process and it is assumed that the student is rock solid on it.   I honestly would consider backing up to Gamma and running though it quickly (MUS is really easy to accelerate).  After you do that you may find that Mr Demme's explanations of area are much more comprehensible to her.

[daijobu]

I like PP's ideas.  You can also draw an equilateral triangle with it's base and height marked: b and h or maybe use actual numbers.  Then draw another triangle with the same base and the same height, but move the apex of the triangle a little to the the right.  Draw another triangle, same base and height, but with the apex a little more to the right.  Like this or this.
 
To reinforce, whenever you are solving an area problem, always begin by writing A = 1/2 b * h.  Always.   I also like to make a little subscripted image of a tiny triangle right after the A, to further indicate this is the formula for the area of a triangle.

 

Then under that (lining up the equal signs) replace b and h with the actual numbers.  So it looks something like this:

 

b=3

h=4 

A = 1/2 b * h

   = 1/2 3 * 4

   = 6

 

Parallelograms have a nicely similar formula to the area of a triangle.  In fact, one can think of a parallelogram as a generic rectangle, with parallel sides but without the right angles.  You can similarly draw parallelograms with equal areas, same bases and same heights.  Like this.

[fdrinca]

First, definitely pull out the graph paper to figure out areas of rectangles. Maybe reference that it's an array?

 

Moving on, I'd have her start by figuring out herself how to find the area of a triangle. 

 

Start with a square. What's the area? Divide the square in half diagonally - two triangles. What would be the area of the triangles? Half of the square, right? 

 

Let her discover this, though. That power of discovery can be huge.

 

Do the same thing with parallelograms. Have her physically reconfigure the shape so that it becomes something she understands - a rectangle. Then recall the area of a rectangle. Make sure to pay attention to what "height" is for a parallelogram. 

 

I think we spent a week with graph paper and construction paper when we went over shapes, area, perimeter, and volume. They could motivate theories for the area of other regular shapes (pentagons, hexagons, etc) by the time we were done.

[MrSmith]

I think Wendy nailed the concrete aspects. For a more abstract treatment consider the following.

 

On graph paper, draw any size rectangle. Work with it until there is no more doubt that A=bh.

 

Then split the rectangle diagonally into two identical triangles. Draw many sizes of rectangle until again no doubt that there are always two triangles inside, and the triangles are the same size. Thus area of triangle is A=(bh)/2. (One half base times height is due to commutative property).

 

Then introduce the 'related rectangle' for a triangle: For any triangle, you can put a rectangle around it such that one side of the triangle forms one side of the rectangle. The other side of the rectangle is given by the height of the triangle. Work this until convinced that the Area of the triangle really is still A=(bh)/2 (break up the rectangle into two or three pieces).

 

The parallogram is just two triangle stuck together, so A=2*(bh/2).

[Kathleen.]

Paper and pencil examples above are considered abstract. A good concrete way would be to use a geoboard with some rubber bands and chex or cheese crackers or anything that fits. Make a shape woth the rubber band and fill it in with what ever cracker you have on hand. Once she sees how you found the area you can then move the rubber band around to find the area of other shapes. If you don't have a geoboard draw a rectangle on the paper and fill it in with crackers or unit blocks - something she can see to make it concrete.

[wapiti]

You got some great ideas!  With the caveat the MUS sequence is not typical and I'm not familiar, I just wanted to note that in other programs, areas of these shapes will be revisited every year through high school geometry and emphasized at the prealgebra level.  For 4th grade, I'd probably be happy with understanding area of a rectangle with seeing that a triangle is half a rectangle being a bonus.

Edited May 8, 2016 by wapiti