NCTM’s Tips for Teachers
These words of wisdom come from NCTM members and publications.
Feel free to supplement these questions with ones that you have by e-mailing Tips@nctm.org. Each month, we’ll ask for questions and tips on a different topic and post the best on this page.
Common Questions and Their Answers Part 1
Why is ?
Sine and cosine represent ratios of lengths of sides of a right triangle for a particular angle . Students often remember this relationship using the mnemonic SOHCAHTOA. Using these labels, we can set up a right triangle with sides O and A, and hypotenuse H. Next, use the Pythagorean Theorem to write O2+A2 = H2. All that remains to be done is division through by H2 and substitution in for sine and cosine.
This is a quick justification that many of your students will understand and from which they will gain a stronger understanding of Trigonometry.
For more discussion of this problem, and two others (why is the line of symmetry of a parabola and how do we know that gives us the volume of a sphere?) see Wendy B. Sanchez’s article, Helping Students Make Sense of Mathematics, in the January ’07 issue of Mathematics Teacher.
MODEL- To develop an understanding of invert-and-multiply with your students, try using a number line to illustrate problems. For instance, would look like this:
*Please note that this image is not meant to suggest that , but rather there are segments of length that fit in the space of .
This can also be used to develop an understanding of why the answer is larger when divided by a number smaller than 1.
MULTIPLICATION- Another way of thinking of this problem is to consider division as a multiplication problem. The above problem, , is the same as asking “two-thirds times what gives me five-halves?” or . To solve, we multiply both sides by 3/2, , which is equivalent to the invert-and-multiply algorithm.
DIVIDE STRAIGHT ACROSS- An intuitive student may question, “when we multiply fractions we just multiple straight across the top and straight across the bottom, why can’t we do that for division?” The answer is, “YOU CAN.” Consider a slightly different problem from above, where we will divide straight across and then simplify by multiplying by two “special ones,” 3/3 and 7/7. Starting with , divide straight across to obtain . Now we will cancel the denominator in the top by multiplying by 3/3 and the denominator in the bottom by multiplying by 7/7, so , which leads us to . Thus, , which is our invert-and-multiply algorithm.
Dividing fractions this way may help to illustrate where the invert-and-multiply algorithm comes from.
Distributive Property of Multiplication?
When teaching children the multiplication algorithm, it can be helpful to give them pictures. One way of illustrating the distributive property is to break the problem down into tens and multiply. So looks like:
As a picture, this can be represented as
Which is .
The student can now multiply length times width and add to arrive at the answer of 117.
To extend this to multi-digit multiplication, one can change the picture a little. For instance, looks like:
Illustrated with an area model, this is:
April’s Tips for Teachers topic will be determined through your responses to this update and conversations with you and your colleagues at the Annual Meeting in Atlanta. Send your questions and ideas to Tips@nctm.org.