1. A problem from the Common Core-inspired PARCC exam, which a commenter on Dan Meyer's blog thought I would particularly like, and felt was comparable to the sample problems on the high school exam in Finland that I blogged about earlier:
2. The Finnish problems I blogged about earlier:
1 c. Simplify the expression (a2-b2)/(a - b) + (a2-b2)/(a + b) with a not equal to b or –b.
5. A circle is tangent to the line 3x-4y = 0 at the point (8, 6), and it touches the positive x-axis. Define the circle center and radius.
6. Let a1…an be real numbers. What value of the variable x make the sum (x + a1)2 + (x + a2)2 + ….+ (x + an)n as small as possible?
9. The plane 9 + x + 2y + 3z = 6 intersects the positive coordinate axis at the points A, B and C.
a) Determine the volume of a tetrahedron whose vertices are at the origin O and the points A, B and C.
b) Determine the area of the triangle ABC.
13. Let us consider positive integers n and k for which n + (n +1) + (n + 2) +… + (n + k) = 1007
a) Prove that these numbers n and k satisfy the equation (k +1)(2n + k) = 2014.
III. Extra Credit
1. Consider what is involved in solving the PARCC problem (recognizing that it's a quadratic; seeing what the a, b, and c coefficients of the standard form of the quadratic equation correspond to here, and seeing that 4ac must be positive). Compare these recognitions, apparent with minimal symbol manipulations, with what's involved in solving the Finnish problems.
2. Some problems can appear mathematically abstract without involving much math. Discuss.
3. About Finnish problem 13, the commenter on Dan Meyer's blog writes:
Problem 13 is a variation on the proof for the sum of an arithmetic sequence. Algebra 1 in the CCSS, btw. We don’t require a proof, but if we were going to require one, that would be one of the easiest to pick. And reform mathematics programs will typically explore that proof (geometrically, and then with algebraic symbols to formalise). Traditional American textbooks just give the formula as something to memorize because, get this, accurate procedures without understanding are considered sufficient.Discuss.