By Soo Jin Lee and Jaehong Shin

**Objective**: Help
students compare values among rational or irrational numbers

**Grade**: 9

**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.

Compare the size of following three numbers,

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,

**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

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