The arrangement of elements in the periodic table contains lots of clues about the characteristics of those elements, their families, and their relation to each other. The table is like a multi-facetted gemstone and the more you study it, the more you discover. It’s hard to think of a more meaningfully arranged two-dimensional table (although in some ways it’s three-dimensional). For any given element in the table, its location – column and row – tells you a lot about it, as do its neighbors – above, below, left side, right side.
A good place to start when introducing students to the periodic table is with the number that appears above each symbol. That number is three things simultaneously. I like to use the acronym A.P.E: Atomic number, number of Protons, number of Electrons. Take selenium (Se), element 34. The fact that 34 is its atomic number tells us that selenium has 34 protons in its nucleus, and 34 electrons orbiting that nucleus. This is a pretty good starting point in visualizing the structure of selenium – what we’d see if we had a really powerful microscope, one that could also snap a photo and freeze the electrons in motion.
The element to the left of selenium has one less proton and one less electron. And the element to the right of selenium has one more proton and one more electron. Simple as that.
The periodic table can get complicated, and there are many exceptions to its rules. So it’s important to start with those facts and trends which can be trusted to give consistently accurate and revealing information.
Believe it or not you can cook chestnuts in a microwave oven. And they taste pretty good provided you get the cooking time just right. Microwaves are powerful, so there’s little room for error. Too few seconds and they’re virtually raw; too many seconds and they’re rubbery or hard. The table below shows the cook times I arrived at through trial and error for chestnuts in batches of two, three, four, and five. These results were derived by using medium-to-large fresh chestnuts in a 1000-watt oven that has a turntable. (I also carved an “X” into at least one side of each chestnut with a sharp knife prior to cooking.) Chestnuts were stored at room temperature.
The freshness matters a lot because freshness determines the moisture content, and this impacts cooking times.
Now to the Algebra part. Suppose you wanted to use the data above to create a “recipe” for microwaving chestnuts – a formula that you or someone else could use for cooking batches of six, seven, eight chestnuts or more (theoretically).
Think of the left column as your x-values and the right column as your y-values. Notice how for each additional chestnut the number of seconds added increases each time, it’s not constant. So we know this would not be a linear function. Let’s assume that it is an exponential function – that it follows the form y = abˣ. How could you use this data to create an exponential function, and what would the resulting “recipe” be?
One of the benefits of tutoring students from dozens of different schools with different classroom teachers is that I hear, secondhand, a wide variety of tips, advice and devices that teachers pass along to their students. The most clever of these, two or three, I have kept and added to my own toolbox to increase my effectiveness as a tutor. A memorable tip or mnemonic device is a big help when guiding students through a new or tricky task, provided that they remember it correctly and know when it applies.
“Keep Change Flip” is one of the more common tips used in middle school math. It guides students through the process of dividing fractions. It’s a shorthand way of reminding them to take the following three steps: 1) Keep the first number as is, 2) Change the division sign to multiplication, 3) Flip the second fraction upside down (…then multiply the fractions). Multiplying fractions is always taught before dividing fractions, so these steps revert division problems back to multiplication, which presumably the student has mastered.
Keep Change Flip sounds very similar to another math mnemonic…
“Keep Change Change” is a memory aid used to teach students how to subtract signed numbers. This one, too, lets students convert a trickier operation back to something they already know. Specifically, it converts subtraction problems to addition problems. (This is around the time when many students are taught that subtracting is “adding the opposite.”) As an example, take a problem like -7 – (-5). The Keep would have you keep the -7 as is. The first Change refers to the operation symbol, a subtraction sign which becomes an addition sign. The second Change refers to the sign of the second number: if it’s negative make it positive; if positive make it negative. Following these steps would remake the problem into: -7 + 5.