Which is Larger?

By Soo Jin Lee and Jaehong Shin

Objective: Help students compare values among rational or irrational numbers

Procedure/Acitvities:
The purpose of following activities is to help students compare among rational or irrational numbers by visualization of the each numbers' value using GSP. Using GS, students can draw out a figure containing all values to be compared and understand intuitively relations among those numbers in terms of their size.

Exploration 1

First, for construction , draw out a right-angled triangle which has two sides other than hypotenuses with size 1, 2 each.

By Pythagorean Theorem

Then, for construction , draw out a right-angled triangle which has two sides other than hypotenuses with size 1, 4 each.

By Pythagorean Theorem

Let's connect above two constructed triangles to make a side with size as follows.

By Pythagorean Theorem

Now, students can compare the sum of and with the value visually since one side of a triangle can not be more than the sum of the other two sides. For exploration, click!!

Discussion: Make students individually compare with using GSP in a similar method and present their result of exploration.

Extension: Make students to challenge above problem without the aid of GSP and suggest an algebraical solution.

Exploration 2
There are four positive numbers satisfying following inequality.

a/b < c/d

Compare the size of following three numbers,

a/b, c/d, (a+c)/(b+d)

In a similar way to exploration 1, draw out a triangle which has two sides other than hypotenuses with size a, b and a triangle with two sides c, d each such that satisfy an assumption.

Students should have no problems in seeing that a/b < c/d since they can compare between the size of each triangle's tangent visually.

Now, to construct (a+c)/(b+d) connect two triangles properly as follows.

For exploration, click!!

As students can see from above drawing by comparing their tangents one aother, the size of (a+c)/(b+d) is the middle of the size of a/b and c/d. that is,

a/b < (a+c)/(b+d) < c/d.

Extension : Make students to be aware that they can not solve comparison problems by drawing method and lead to an algebraical method.

A > B _ A- B > 0
A < B _ A- B < 0
A = B _ A-B = 0

Ex) (a+c)/(b+d) ­ a/b -> (bc-ad)/b*(b+d) > 0 (Because bc-ad>0 by assumption)

Therefore, (a+c)/(b+d) > a/b