New on my Web site:→ Animated slides on the Lab Gear model for signed number arithmetic.Note that for each operation, the model is based on what students already know. For addition, you put down the first number, then the second number, and finally count. For subtraction, you put down the first number, take away the… Continue reading Animated Demonstrations
Last winter, I attended an interesting meeting of mathematicians and math educators in Banff, Canada. Our charge was to compile a list of integer sequences that would offer suitable problems for students (and teachers) at each level from Kindergarten to 12th grade. It was a sequel to 2014's Unsolved K-12 meeting, and once again was… Continue reading Integer Sequences
One way to discover or apply the formula for the area of a triangle is to explore area on the geoboard. The initial activities should be along the lines suggested in my Geometry Labs Lab 8.4. (Free download.)After that, one can zero in on triangles, by asking a question like: "Find triangles of area 15."… Continue reading Geoboard Triangles
A few days ago, I saw a raging debate on Twitter about hint-giving in math class. It was triggered by a short talk by Michael Pershan, a teacher in NYC. Michael argues that high school teachers need to share good hints with each other, and he proposes some guidelines as to what makes a good… Continue reading About hints
In my last post, I described a problem I encountered more than twenty years ago, and my recent attempt at solving it. The problem: Partition the numbers from 1 to 2n into pairs, so that the sum of the numbers in each pair is a perfect square. For what numbers is this possible? I decided… Continue reading Getting Help
I’ve Got a Problem!
Many, many years ago, I saw this problem somewhere:Arrange the whole numbers from 1 to 18 into nine pairs, so that the sum of the numbers in each pair is a perfect square.I liked the problem, and included it in a book I co-authored (Algebra: Themes, Tools, Concepts, following lesson 5.5). In the Teacher's Edition,… Continue reading I’ve Got a Problem!
One good thing about the Common Core middle school standards is the emphasis on proportional relationships, and the fact that they are approached in a multidimensional way. In addition to "set up a proportion and solve it", which is probably the most common way to teach this, the standards propose multiple representations and a variety… Continue reading Proportional Relationships
A correspondent writes:We emphasize the idea that students should approach problems in multiple ways. This has caused me to wonder about patterns. For example:students might conclude that 3^0=1 because of the pattern 3^4=81, 3^3=27, 3^2=9, 3^1=1orwhen the second difference is constant, students will conclude that the function is quadraticorwhen a function is concave up, the… Continue reading Patterns
I had a great time at the Julia Robinson Math Festival the weekend before last. Hundreds of kids attended, most of them girls, it seemed to me. The setup: many, many tables; at each table, one or two adult guides, and a math problem that combines access and depth. Students choose a table, and work… Continue reading Egyptian Fractions
No Three on a Line
In a recent post, I mentioned K-12 Unsolved, the project I'm involved in that aims to publicize 13 unsolved math problems, in the hope that an appropriate version of each problem will find its way into K-12 classrooms. One problem we looked at was posed by Henry Dudeney in 1917. Here is the problem: Consider… Continue reading No Three on a Line