What fraction of each of the squares is shaded? (from test for 12-13-year-olds)

[JamesWilko]

Can you help with this Math problem? It's from a scholarship paper for 12-13 year olds. I have to work out what fraction of the entire square the shaded region is.

I've worked out that it's 47/90 and have verified that by drawing it on an architect's sketching program. Thing is the method I used I know is much more complicated than it needs to be and I don't know how to explain it simply to my student.

To work out the perpendicular height of the triangle where the two larger triangles overlap I worked out where they intercept using y=mx+c, but my student isn't expected to know that at this age.

Can someone suggest the simple way of doing this that I'm missing? Thanks

 

[Dr.Peterson]

Can you help with this Math problem? It's from a scholarship paper for 12-13 year olds. I have to work out what fraction of the entire square the shaded region is.

I've worked out that it's 47/90 and have verified that by drawing it on an architect's sketching program. Thing is the method I used I know is much more complicated than it needs to be and I don't know how to explain it simply to my student.

To work out the perpendicular height of the triangle where the two larger triangles overlap I worked out where they intercept using y=mx+c, but my student isn't expected to know that at this age.

Can someone suggest the simple way of doing this that I'm missing? Thanks

For most of these, I would start by turning the picture into a grid of squares, as shown in the last example. If the dots are 1 cm apart, then your grid consists of 9 squares, each 1 cm² (square centimeter) in area; a square that is cut exactly in half diagonally leaves two triangles, each 1/2 cm². For the first, this is all you need to do; or you can notice that each of the two big triangles is exactly half of a larger square.

In the other two, you are given hints, which amount to making finer grids. Draw a line every 1/2 cm rather than every 1 cm, and do the same sort of counting. For the last, try even finer. Before you decide on the spacing of the lines, you might use some trick involving similar triangles to decide what spacing will work well (by seeing where the lines will intersect).

Apparently you are really asking only about the last; am I right? I might first use similar triangles to find where the line with positive slope intersects grid lines; then use similar triangles again to find the height and area of the unshaded triangle within the lower middle square. I can see several other possibilities. One is to use similar triangles to directly find the location of the intersection, then use that to get base and height of various components. Extending the negative-slope line to the top line, and relating the resulting unshaded triangle to a similar shaded one is particularly nice way.

Try some of these ideas and show your work; if necessary, we can then give you more hints. Or there may be some much more elegant way to do it than what I've thought of.

 

[Bob Brown MSEE]

use harmonic mean

The small shaded triangle area is
A = bh/2
where
b = 1
2h = harmonic mean [1, 3/2]