Course Guide - USAID Teacher Education Project

c) Introduce the idea that the square root of certain numbers are irrational numbersand can only be expressed as approximations.4. Class Activitiesa) Distribute dot paper that is arranged in 5-dot squares. Have students draw as manydifferent size squares as possible, having them label the area of each. (The uprightsquares will have areas of 1, 4, 9, and 16. The tilted squares will have areas of 2, 5, 8,and 10.)Note which students stop at 1, 4, 9, and 16 and think they have discovered allpossibilities. Prompt them to go beyond these, letting them know there are severalmore squares for them to discover. (Do not be surprised if they draw the same 1, 4,and 9 squares in different places on their dot paper.)b) Begin by discussing the area and side lengths of the upright squares, noting thatthe area of square is found by multiplying its side lengths, and this side length iscalled the square root (or surd) of a given square.Ask for the square root of each of the upright squares and introduce the format √16 asa way of expressing "the square root of 16." This can be thought of as "4 is the sidelength of square with an area of 16."c) Next, turn students attention to the tilted squares they drew. Ask them to considerthe square with an area of 2. What is its side length? What number multiplied by itselfequals 2? If students have calculators they may try to find this number by guess-andcheck,coming up with an approximation of 1.41.However, this is not exactly the square root of 2. In fact, the most accurate way toexpress "the square root of 2" is to write it as √2. Following this line of thought, askfor the side lengths (square roots) of the other tilted squares.Mention that √2 is called an irrational number, because it cannot be written as aterminating or repeating decimal. Remind students that fractions and terminating orrepeating decimals are called "rational" numbers.You may want to note that pi is also an irrational number, although for practicalpurposes when measuring circular objects in the real world we tend to use pi'sapproximation 3.14. (Mention that computers have calculated pi to over 2577 billiondecimal places (and √2 to over a million decimal places) without finding a repeatingpattern of digits.)d) Remind students of their work with the multiplication of integers, and ask whatfactors could produce the whole number 4. Since both 2 x 2 and -2 x -2 equal 4, 4 canbe said to have two square roots, one positive (2) and the other negative (-2). In fact,every positive number has two square roots. If students ask about the square roots ofnegative numbers, briefly mention that these are called imaginary numbers.5. Assignments (to be determined by instructor)