The square root of a number is usually introduced via an activity that involves getting the side of a square with the given area. For example the side of a square with area 25 sq unit is 5 unit because 5 x 5 = 25. To introduce the existence of 
The construction shows the following equivalence:
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This post outlines a teaching sequence for introducing the concept of square roots in a GeoGebra environment. Of course you can do the same activity using grid papers, ruler and calculator. However, if the students have access to computers then I highly recommend that you use GeoGebra to do this. In my post GeoGebra and Mathematics, I argued that the more the students understand the mathematics behind GeoGebra, the more confident they could become in using this tool. The earlier the exposure to this environment, the better. The way to do this is to integrate the learning of the tool in learning mathematics.
The figure below is the result of the final activity in my proposed teaching sequence for teaching square roots of numbers and some surds or irrational numbers. The GeoGebra tool that the students is expected to learn is the tool for constructing general polygons and regular polygons (the one in the middle of the toolbar).
The teaching sequence is composed of four activities.
Activity 1 involves exploration of the two polygon tools: polygons and regular polygons. To draw a polygon using the polygon tool is the same as drawing polygons using a ruler. You draw two pints then you use the ruler/straight edge to draw a side. But with Geogebra you click the points to determine the corners of the polygon and Geogebra will draw the lines for you. In the algebra window you will see the length of the segment and the area of the polygon. Click here to explore.
GeoGebra shows further its intelligence and economy of steps in Activity 2 which involves drawing regular polygons. Using the regular polygon tool and then clicking two points in the drawing pad, GeoGebra will ask for the number of sides of the polygon. All the students need to do is to type the number of sides of their choice and presto they will have a regular polygon. Click here to explore.
Activity 3 is the main activity which involves solving the problem Draw a square which is double the area of another square. Click here to take you to the task.
Activity 4 consolidates ideas in Activity 3. Ask the students to click File then New to get a new window from the previous activity’s applet then ask them to draw the figure above – Squares and Square Roots. You can also use the figure to compare geometrically the values of 
Like the rest of the activities I post here, the learning of mathematics, in this case the square roots of numbers, is in the context of solving a problem. The activities link number, algebra, geometry and technology. Click here for the sequel of this post.
This is the second in the series of posts about integrating the teaching of GeoGebra and Mathematics in lower secondary school. The first post was about teaching the point tool and investigating coordinates of points in a Cartesian plane.
GeoGebra book:
Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra
This is my third post on constructing test items based on TIMSS Assessment Framework. My first set of examples is about assessing understanding of zeros of polynomial function and the second post is about graphs of rational functions. Of course there are other frameworks that may be used for constructing test item like the Bloom’s Taxonomy. However, in my experience, Bloom’s is not very useful in mathematics, even its revised version. The best framework so far for mathematics is that of TIMMS’s which I summarized here.
Here are three examples of trigonometry test questions using the three different cognitive levels: knowing facts, procedures and concepts, applying, and reasoning.
Knowing
If cos2(3x-3) = 5 , what is the value of 1-sin2 (3x-3) equal to?
a. 0
b. 1
c. 3
d. 4
e. 5
Applying
Given the graphs of f and g, sketch the graph of f+g.
Reasoning
Which of the following functions will have the same set zeroes as function g, given that g(x) = sin kx and f(x) = k?
a. f+g
b. f–g
c. fg
d. g/f
e. gof
