![]() What do you think $1 + 3 + 5 + \ldots + 99$ will be? Don't worry about simplifying, I'd prefer you to leave the answer as $^2$. I'm currently teaching a remedial algebra class in college, and used the following as a bonus question on the first exam: $\bullet$ First, that the sum of the first $n$ odd, positive integers is $n^2$. And second, they often require relatively little prior knowledge, and offer the chance to think visually. First, the vast majority of students are never exposed to these ideas (save binomial coefficients, and these, if introduced, are just strange symbols used to expand $(a+b)^n$, in my experience). I'll expand on my comment, now that I have some time.įor high school students, I really like discrete-math type ideas, particularly combinatorics. (This repeats content from my answer to a related question.) ![]() (U.S.) 8th-graders than a two-column Euclidean proof. I have found this tactile demonstration more convincing to Naively, it could wellīe that the bisectors do not meet at a point. Proposition 4, Book IV of Euclid viscerally. What I find so pleasing is that when you perform this physically, theĪngle bisectors meet at a point $x$ (the incenter), and one grasps (Figure from How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra.) The natural solution is to mountain-crease (red below) the angle bisectors,Īnd valley-crease (green dashed) a "perpendicular" from the incenter $x$: The perimeter, but rather by folding the paper ![]() ![]() Not by punching the scissors through and cutting Suppose you desire to cut out a triangle from the middle of a piece of paper, Here is one example that I find aesthetically pleasing, and which I haveįound effective in 8 th-grade classrooms. ![]()
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