Canons and Fugues

The idea of a canon is that one single theme is played against itself. This is done by having "copies" of the theme played by the various participating voices. But there are means' ways to do this. The most straightforward of all canons is the round, such as "Three Blind Mice", "Row, Row, Row Your Boat", or " Frere Jacques". Here, the theme enters in the first voice and, after a fixed time-delay, a "copy" of it enters, in precisely the same key. After the same fixed time-delay in the second v...

 Roman Arithmetic

...basic Roman arithmetic is largely rather simple, even for those of us spoiled by Arabic notation. Addition is no sweat, because complex Roman numbers already use what math pros call additive notation, with numerals set beside one another to create a larger number. VI is just V plus I, after all. To add large numbers, simply pile all the letters together, arrange them in descending order, and there’s your sum. CLXVI plus CLXVI? CCLLXXVVII, or CCCXXXII. And one of the advantages of the Rom...

 Mathematics is Hard Work, Not Genius

What I fight against most in some sense, [when talking to the public,] is the kind of message, for example as put out by the film Good Will Hunting, that there is something you're born with and either you have it or you don't. That's really not the experience of mathematicians. We all find it difficult, it's not that we're any different from someone who struggles with maths problems in third grade. It's really the same process. We're just prepared to handle that struggle on a much larger scal...

 Math Exercise: Multiple Approaches to Problem-Solving

For example, if the problem was to fi nd the answer to 8 × 6, students may suggest three options: memorizing the multiplication table for 6, knowing that 8 × 5 = 40 and adding another 8 to equal 48, or adding a column of six 8s. Allowing students to personally choose among approaches all confi rmed as correct and to support their choice will increase their comfort levels. Th is process also builds math logic, intuition, and reasoning skills that extend into other academic subjects and real-...

 Math Games

Buzz. An example of a low-stress, win-win game is Prime Number Buzz. Students stand in a circle or at their desks and go around the room in order, saying either the next sequential number if it is a composite or “buzz” if it is a prime. If they are incorrect, they sit down, but they keep listening and when they catch another student’s error, they stand up and rejoin the game. (The same game format works for Multiples Buzz, using multiples of, for example, 3, 4, and so on.) Telephone. T...

 Math Exercise: Comparisons

Select two boxes or cans of food that weigh 8 ounces and 16 ounces, respectively. Have students hold each as you tell them (or they read) the weights of the containers. Give students a box or can with the weight covered and have them compare the weight of the new package to the weight of the 8- and 16-ounce samples. Th ey can then estimate whether the new item’s weight is closer to 8 or 16 ounces. As students become more successful, they may want to predict a more specifi c weight. Ask them...

 Use a Big Opening for Class

In your planning, consider what you most want students to know and then work backward to develop an opening that promotes sustained interest toward that goal. If possible, represent the unit in several different ways that appeal to different learning strengths and levels of achievable challenge so you can continually engage all students. Here are some fascinating facts you can use as “big openings” with your students to help them with number sense, specifi cally with understanding large ...

 Mathematical Cue Words

Addition: add, plus, sum, total, altogether, increased by, grew, gained, total of, combined, more than (as in, “3 more than 7 is 10”), put together, in all Subtraction: minus, take away, diff erence, less than, from, remove, subtract, gives away, sells, loses, fewer than, decreased by, diff erence between Multiplication: product, times, doubled (tripled, etc.), some problems give information about one and ask for total amounts (also, when dealing with multiplication of fractions, of us...

 Delay Method of Errorless Math Practice

Prepare a list of the calculations from the flash cards on a sheet of paper. These can be on a template, with multiplication facts at the appropriate level pulled and copied for the student. On these forms, include three columns next to each multiplication question, labeled “correct repeat,” “correct wait,” and “correct response.” Start with review and confi dence building. For example, show the question 3 × 4 = __ on the card and without any delay say the answer. Th e student re...

 Understanding Factors Through Divisibility

This is appropriate for 4th or 5th graders and involves finding the smallest number divisible by 2, 3, 4, 5, 6, 9, and 10. Directions: Teach the students how to tell if a number is divisible by 2, 3, 4, 5, 6, 9, & 10. Give groups ofu00a0two tou00a0four students the divisibility rules on a sheet of paper, and challenge them to find the smallest number divisible by all of the numbers listed above. Eventually the students, with or without help from the teacher, will come to reali...