Tagged with ' Middle School Math'

I like to think of number sense as the linchpin to learning mathematics well. It plays a critical role in students’ confidence and risk taking. It helps them determine the reasonableness of their solutions and tinker with strategies and approaches when their solutions to problems are off. Number sense enables students to be flexible and efficient with both their computation and reasoning.

As the proverb claims, Slopeometer was born out of necessity. It happened when I was teaching calculus but has since proven incredibly useful in a variety of ways--from a middle-school setting on up. The inspiration came when I kept tracing along the graph of a function while I asked my AP calculus students at each point to “imagine the slope of the tangent line” as it gradually changed--from negative to zero to positive--representing the values of the “derivative” of the function. When, after some puzzled looks from my students, I said “imagine a plumb bob, always pointing to the correct value as it changes slope…” the lightbulb moment happened. Over the weekend, after a trip to a building supply store, I was able to cut a semicircle out of a piece of plexiglass and mark it with appropriate numbers and fractions, then loosely fasten a gravity-activated indicator arm that did the pointing... and presto!--the Slopeometer. My students found it much easier to picture the process, often called “curve-sketching,” and apply it to their own work. My students found it so helpful. Slopeometer has since proven useful in many areas: from “best fit” regression lines, to estimating tangent ratios, to slope-fields for differential equations--but the device has shown to be especially useful when first introducing slope.

We often see children who struggle to perform even the most basic operations, even after repeated instruction. Without an understanding of base ten and place value, children are very limited in their ability to do ordinary arithmetic; that is, to add, subtract, multiply and divide. If, for example, they can only think of 48 crayons as 48 individual loose crayons, and 25 more crayons as 25 individual loose crayons, then the only strategies available for finding the total number of crayons are “counting all” (1, 2, . . ., 48, 49, 50, . . . 73) and “counting on” (48 - 49, 50, . . . , 73).