x2 + y2 = z2
Then I say, "We´ve seen how there are an infinite number of solutions to the Pythagorean Equation. For example, we could take our common 3, 4, 5 triplet and multiply each value by any number, and obtain another Pythagorean Triplet."
I pause here to let the students think about this. Then I say, "Is it possible that this holds true with other exponents besides 2?" Now I show slide 2:
xn + yn = zn
I ask, "Are there any other values for n that work?" At this point I will take conjectures or questions. Then, I will give the class a few minutes to explore. After students start to make some conclusions, I will ask a few volunteers to tell us what they concluded and which values they tested.
Our next Activity consists of 6 tasks organized so that the level of difficulty continues to increase. I form homogenous student pairs before handing out the task. Then, I ask that each pair try to solve the problems consecutively from 1 to 6. I tell the class that no group should skip problems and they should do all work in their journal or notebooks.
The activity is a good formative assessment. By observing how students progress through the series, I gain insight into how well students can apply the Pythagorean Theorem.
If I think that it will help motivate a particular group of students, I will prepare to distribute Task Tickets. As pairs work, I hand out tickets for each problem that they have solved correctly. Sometimes, I will even scale the reward, giving more tickets for the more challenging problems. At the end of the activity, each team puts all their tickets in a box (making sure they have written down their names behind each ticket). I will then draw five winning tickets. The winners get to choose from a list of awards.
Going up to the board is an excellent motivator for my students. I use this strategy today, asking six volunteers to go up and write the work for each problem from our prior work (see Activity 6 Tasks). By this time I know who struggled to complete a task, but got it right at the end. These students are usually eager to go up, to show that they persevered and succeeded. So I try to select these students.
Once the answers are posted, I try to leave as much of the question answering and clearing up of confusion to the students themselves. I also encourage students who did not successfully complete all six problems to the example solutions in their journals or notebooks.
Tonight's homework assignment provides more practice on using the Pythagorean theorem on coordinate plane problems.
The first three Homework problems are direct practice. The last problem is an area problem which we haven´t seen much of in this unit, but which can be solved by applying the Pythagorean Theorem correctly.