gr8_squares+and+roots+p2

= Roots and Squares p2 =

//It is expected that students will://
 * **Prescribed Learning Outcomes**

C1 Develop and apply the Pythagorean theorem to solve problems. [CN, PS, R, T, V] ||
 * **Planning for Assessment** || **Assessment Strategies** ||
 * Have students cut squares from grid paper with the following side lengths, and use the squares to model and explain the Pythagorean theorem: 9, 12, 15; 5, 12, 13; and 8, 15, 17. Ask students to multiply or divide the sides of any of these triangles by a common factor and test to see if the newly created triangle still works with the Pythagorean theorem. Have them explain the results using what they know of perfect squares.

Have students create several different right triangles (in succession) on a geoboard (physical or virtual) initially with a horizontal-vertical orientation. Assuming the vertical or horizontal distance between two pegs to be 1 unit, have students calculate the length of the hypotenuse for each triangle they create.

Have students create a triangle the geoboard which does not have a right angle. Have students calculate the length of any non-vertical or non-horizontal sides by creating right triangles around the original triangle (each of which have two sides of whole number units) and applying the principle of the Pythagorean theorem to calculate the needed sides of the original triangle. (See geoboard figure)

Have students create a square on a geoboard whose area is NOT a square number and calculate the length of each side. || Verify that students can
 * model and explain the Pythagorean theorem correctly.
 * determine the measure of the third side of a right triangle, given the measures of the other two sides, to solve a problem.

The geoboard figure ([|geoboard figure.pdf]) shows triangle ABC, a scalene triangle. The length of side AB can be calculated by counting the units (3). Side AC can be calculated by considering a right triangle using AC as the hypotenuse. Thus, creating triangle ACD, the Pythagorean theorem can be used to calculate the length of AC. Similarly, by creating another right triangle BCE such that BC is the hypotenuse, BC can be calculated.l

Students should be aware when solving equations as a result of applying the Pythagorean theorem that there are two solutions to this (quadratic) equation both a positive and a negative root. Since the square root function only gives the principle (positive) root the fact that there are two answers is indicated by ± in front of the square root sign. However, only the positive roots is logical as a measure of length. Still, they should be aware that given 3^2 + 4^2 = c^2, 9 + 19 = c^2, and 25 = c^2,that c=±sqr(25), therefore c = +5 or c = -5

Note: It is important for students to get into the habit now of writing both answers to the equation and choosing the solution that makes sense to answer the question that was being asked (in this case a length), to avoid mistakes latter when quadratic equations are being applied in other situations and both roots are required to answer the question being asked. ||