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Where do formulas come from? How are they invented? Why do they work? How do they make sense?
All those questions have the same answer: patterns.
Patterns can be most easily studied by making a three-column table. Looking carefully at the table allows us to analyze the pattern and create a formula.
Here are a few patterns to analyze and turn into formulas. This activity is our "icebreaker" activity as the new term begins, to help you meet your classmates. Work on each pattern with a different partner, and introduce yourselves as you work quietly together.
1. This pattern is an increasing number of X shapes, each made by four toothpicks. Can you explain the formula that describes how many toothpicks it takes to make a row of X shapes?
2. Can you explain the formula that describes how many toothpicks it takes to make a row of boxes?
3. This pattern looks like a row of houses that gets longer and longer.
4. Time for some area patterns. How does the number of tiles in a square increase?
5. These rectangles have a width equal to their placement number in the pattern, and a height one greater than that. What is the pattern for their areas?
6. How about a triangle of tiles? This pattern seems harder than the previous two! But there is a trick that makes it easy. You can find the trick by comparing this pattern to the rectangle pattern.
Calories are a measure of energy. Our body uses energy even when resting for breathing, blood circulation, maintenance of body temperature and growing and repairing cells. Exercising needs more energy.
How can we find the number of calories burned when exercising?
This pattern might be complicated. It surely depends upon the type of exercise, how intensely it is done, and for how long the person exercises. It could also sensibly depend upon the person's age, weight, or sex.
As an in-class activity, let's just focus on measuring one person doing one type of exercise.
When we exercise our muscles need more oxygen. For aerobic exercise, the amount of extra oxygen needed is proportional to the amount of energy used exercising. To get that extra oxygen to our muscles, our breathing and heart rate increase.
So there are three issues to deal with as you do some experimenting with a partner.
First, how much does your breathing and heart rate increase if we do the same exercise at different intensities?
You will need to pick an exercise whose intensity you can change accurately.
As an example, you might walk through the hallways for a minute to see what distance you go. Then jog that distance twice in a minute. Then run that distance three times in a minute.
As another example, you might do liesurely jumping jacks for a minute. Then do twice as many in a minute. Then do three times as many in a minute.
Second, which better estimates exercise energy use, breathing or heart rate?
The simplest result would be if either your breathing or heart rate (or both) doubled and tripled when the exercise intenisty does.
Maybe the pattern is not that simple. Maybe either breathing or heart rate increase more or less than following the doubling and tripling of exercise intensity.
Third, can we simply use our measured breathing and heart rates? Or do we need to subtract the resting amounts to isolate the increase caused by exercise?
If neither your breathing or heart rate (or both) make a good pattern, perhaps your resting metabolism is the culprit. After all, the exercise was needing extra oxygen, so perhaps what we should look at is the extra breathing and heart rates?
Try subtracting your resting breathing and heart rates from the exercise values at all intensities. Does that help make a pattern?
The mean is the average you are most used to in math class: add up all of the numbers and then divide by how many numbers there were.
When people say "average" it should be safe to assume they are talking about the mean, unless they say otherwise. In most situations the mean is a very realistic average that provides useful information about the size of a typical value in that set of numbers.
Example of Mean
In a certain preschool classroom of 4-year-olds, the students' weights in pounds are 34, 36, 36, 36, 37, 37, 38, 39, 39, 40, 40, and 41. What is the average weight of the twelve students in the room?
The median is a different kind of average: list all of the numbers in order and pick the middle one in that list. (If the list has an even number of values then there will be no middle value, so average the two values most in the middle.)
This process "throws out" any atypical smallest or largest values. The median is appropriate and often used for company salaries, state incomes, house values, and other situations where the lowest and highest values are really should not be thought of as representative of the group.
Example of Median
In a certain preschool classroom of 4-year-olds, the students' weights in pounds are 34, 36, 36, 36, 37, 37, 38, 39, 39, 40, 40, and 41. The teachers weigh 120 and 141 lbs. What is the average weight?