(A new question of the week) We’ll look at a very complicated logarithmic equation, which leads to quartic equations and some very interesting graphs. We won’t find a fully satisfying solution method, but we’ll have some fun trying – and reveal the fallibility of at least one Math Doctor!
(A new question of the week) Here we have a different kind of question than usual: A conjecture about distances between points, with a request for confirmation. Normally we like to just give hints to help a student figure something out; this was a request for a theorem that ought to exist, and trying to …
(A new question of the week) A couple recent questions involved factoring numbers, in interesting ways. One involves the volume and perimeter of a block of cubes, and the other involves finding numbers with a given HCF (Highest Common Factor) and sum. Both illustrate thinking through a non-routine problem about factors.
(A new question of the week) We’ve looked in the past at place values, but here we’ll see some tricks for doing multiplication and division with both decimals and large numbers by moving the decimal point around. The first question is primarily a matter of arithmetic, then the second extends it to the algebraic concept …
(A new question of the week) May was a particularly good month for interesting questions! Here is one requiring us to find one value of a function, based on an unusual property: If \(a+b=2^x\), then \(f(a)+f(b)=x^2\). The problem turned out to be not as hard as it looked, yet the function itself is quite interesting …
There are a number of standard techniques for graphing functions, such as transforming simple functions, or finding asymptotes and holes for rational functions, and using calculus to find slopes. What if you have a rational function of a trig function, and can’t yet use calculus to figure it out? We’ll look at how we can …
(A new question of the week) Today we’ll look at a classic algebra word problem: Finding how long it takes to fill a cistern through two pipes, with a drain open. But usually these problems are given with specific numbers, as a simple exercise in algebra. What if it’s all variables? the discussion provides some …
Last week we looked at problems about counting the squares of all sizes in a checkerboard. Some solutions required finding the sum of consecutive squares, \(1^2+2^2+3^2+\dots+n^2\), for which we used a formula whose derivation I deferred to this week. Here we’ll see a couple proofs that require knowing the formula ahead of time, and a …
Last week we looked at ways to count paths along the edges of a rectangular grid. Now we’ll look at a companion problem: counting the number of squares (or rectangles) of all sizes in a square (or rectangular) grid. This, too, is a very common question, and I’ll be picking just a few of many …
Here is a recent question about arithmetic sequences and series (specifically, reversing the process to find the number of terms given the sum), that nicely illustrates a common type of interaction with a student: gathering information about both problem and student, then guiding them to use what they know, or giving new information as needed. …