Page 288 - ABCTE Study Guide_Neat

Problem

Two trees with heights of 20 m and 30 m respectively have ropes running from the top of each tree to the
bottom of the other tree. The trees are 40 meters apart. We’ll assume that the ropes are pulled tight
enough that we can ignore any bending or drooping. How high above the ground do the ropes intersect?

Let’s solve this problem by representing it in a visual way, in this case, a diagram:

You can see that we have a much simpler problem on our hands after drawing the diagram. A, B, C, D,
E, and Fare vertices of the triangles in the diagram. Now also notice that:

b = the base of triangle EFA

h = the height of triangle EFA and the height above the ground at which the ropes intersect

If we had not drawn this diagram, it would have been very hard to solve this problem, since we need the
triangles and their properties to solve for h. Also, this diagram allows us to see that triangle BCA is similar
to triangle EFC, and triangle DCA is similar to triangle EFA. Solving for h shows that the ropes intersect
twelve meters above the ground.

Students frequently complain that mathematics is too difficult for them, because it is too abstract and
unapproachable. Explaining mathematical reasoning and problem solving by using a variety of
methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and concrete models can
help students understand the problem better by making it more concrete and approachable.

Let’s try another one.

Problem

Given a pickle jar filled with marbles, about how many marbles does the jar contain?

Problems like this one require the student to make and use estimations. In this case, an estimation is all
that is required, although, in more complex problems, estimates may help the student arrive at the final
answer.

How would a student do this? A good estimation can be found by counting how many marbles are on the
base of the jar and multiplying that by the number of marbles that make up the height of the marbles in the
jar.

Now to make sure that we understand when and how to use these methods, let's solve a problem on our
own: