Essential Math
Two Numbers Have the Same Sum. Why Can Their Products Be So Different?
The pairs 1 and 11, 3 and 9, and 6 and 6 all add to 12. Their products are 11, 27, and 36. A fixed sum does not…
Essential Math
Articles
Essential Math
What a Quadratic Graph Can Tell You Before You Solve It
The graphs of \(y=x^{2}-6x+5\), \(y=x^{2}-6x+9\), and \(y=x^{2}-6x+13\) have the same shape. Moving the constant term shifts the parabola vertically. Those shifts change whether the curve crosses, touches, or…
Essential Math
Why Successive Percentage Changes Do Not Simply Add
A price falls from $100 by 20%, then the reduced price falls by another 10%. The final price is $72, giving a total decrease of 28%. Adding the…
Essential Math
Two Timers Ring Every 12 and 18 Minutes. When Do They Ring Together?
Two timers ring together at noon. One rings every 12 minutes and the other every 18 minutes. Their next simultaneous ring is at 12:36 p.m. The answer is…
Essential Math
The Same Line Can Look Steep or Flat: Read the Axes First
A line rising from \((0,0)\) to \((4,2)\) has slope \(\frac{1}{2}\). Draw it on two charts with different vertical scales and it can look almost flat on one and…
Essential Math
Why Doubling Every Length Makes the Area Four Times Larger
A rectangular photo measures 4 inches by 6 inches. Enlarge both dimensions by a factor of two and it becomes 8 inches by 12 inches. The length and…
Essential Math
Dividing by a Fraction Can Make a Number Bigger
You have three quarters of a yard of ribbon. Each decoration needs one eighth of a yard. Dividing tells you how many decorations you can make: \(\frac{3}{4}\div\frac{1}{8}=6\). The…
Essential Math
James Grime: FOOL FOR MATHS*
James Grime is a mathematician with a personal passion for maths communication and the promotion of mathematics in schools and to the general public. He can be…
Essential Math
Models and Delving Deeper Into Division
Models and Delving Deeper Into Division Last month, we unpacked the long-division procedure taught in American K-6 classrooms, tied it into the standard “long multiplication” algorithm, discussed…
Essential Math
What’s Hiding Within Long Division?
Look familiar? For most people, if they ever learned to do long division properly, they had to “master” something similar to the above procedure/algorithm. What was not…
Essential Math
James Tanton: An Australian Mathematician in Uncle Sam’s Court
My guest this month is James S. Tanton, a mathematician, math educator, and author. He was awarded the Kidder Faculty Prize for his teaching at the The…
Essential Math
Thirteen Ways of Looking at Multiplication (with apologies to Wallace Stevens)
Well, no, I’ve lied: not thirteen, but merely four ways of looking at (and doing) multiplication are contained herein. And I hope to convince you that they are…