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Math Lecture Notes
Arithmetic Facts

Formulas link to here link to index

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.

1. Can you explain the formula that describes how many toothpicks it takes to make a row of boxes?

a toothpick boxes pattern

Here are a few more patterns to analyze and turn into formulas. This activity will also be our "icebreaker" activity to meet your classmates as the new term begins. Work on each pattern with a different partner, and if you solve it quickly introduce yourselves quietly.

2. This pattern is an increasing number of X shapes, each made by four toothpicks.

a toothpick x pattern

3. This pattern looks like a row of houses that gets longer and longer.

a toothpick house pattern

4. Time for some area patterns. How does the number of tiles in a square increase?

square area pattern

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?

rectangle area pattern

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.

triangle area pattern

7. What is the most number of pieces you can make with straight cuts on a pizza? You will have to cut messy and not have every cut go through the center! One cut must make 2 pieces. Two cuts cannot make more than four pieces. The picture below shows a way three cuts can make seven pieces. Four cuts can make eleven pieces! And so on. (Turning this pattern into a formula again seems very hard, until you notice a trick by comparing it to the triangle area pattern.)

pizza pieces pattern

8. Back to toothpick patterns. This pattern involves how many toothpicks are in a triangle that grows downward. Each step in the pattern adds another row to the bottom of the previous triangle. Once again, there is a trick about comparing this to the triangle area pattern.

toothpick triangles pattern

Calories link to here link to index

Calories in a Gram

Not all calories are created equal! One gram of fat has more than twice the calories of a gram of carbohydrate or protein:

Calories Per Gram

Carbohydrates have 4 calories per gram
Proteins have 4 calories per gram
Fats have 9 calories per gram

Calories in a Day

The USDA has recommendations for what someone on a 2,000 calorie per day diet you should be eating. (Labels on food packages refer to this information.)

2,000 Calories per day Diet

• 300 grams of carbohydrate per day (1,200 calories)
• at least 53 grams of protein per day (212 calories or more)
• less than 65 grams of fat per day (585 calories or less)

Of these amounts, at least 25 of the carbohydrate grams should be dietary fiber, and no more than 20 of the fat grams should be saturated fat.

What about a person who is more active or larger and eats more? The recommendations for a 2,500 calorie per day diet are:

2,500 Calories per day Diet

• 375 grams of carbohydrate per day (1,500 calories)
• at least 70 grams of protein per day (280 calories or more)
• less than 80 grams of fat per day (720 calories or less)

Of these amounts, at least 30 of the carbohydrate grams should be dietary fiber, and no more than 25 of the fat grams should be saturated fat.

Burning Calories

Activity
 
Calories Per Pound
Per Minute
ballroom dancing0.023
grocery shopping0.028
walking0.037
weight lifting0.039
bicycling0.045
aerobic dancing0.061
basketball0.063
swimming0.070
running0.090

How can we find the number of calories burned when exercising?

This question is easy to answer because burning calories only depends upon the weight of the person exercising and how long they exercise. Unlike with the formulas we already learned for BMR and DCI, height and age do not matter for calculating calories burned.

The chart on the right that estimates, for different activities, how many calories are burned per minute per pound of the person exercising.

If you are curious about other activities, more charts such as this one are easy to find in books and on the internet. But their information should always be taken with some skepticism. Two different people can participate in the same activity with very different levels of physical exertion. For example, this chart does not differentiate between bicycling at a lazy pace to the corner store or rushing over the Donald Street mountain pass as part of a cycling race.

Fat vs. Carbohydrates Burned

Another important aspect to the issue of how many calories are burned by different activities is what kind of calories are burned. A person wanting to lose weight desires to burn as many fat calories as possible, not only carbohydrate calories.

The practical lesson is simply "never mind and exercise a lot." But this is a math class, so we will make the issue complicated.

The trickiness comes from two effects working against each other.

Your metabolism burns a smaller percentage of fat calories when doing more intense exercise. This happens because carbohydrate calories are needed to keep busy muscles healthy.

For example, compare walking and running. When walking, about three-quarters of the calories burned will be fat calories and one-quarter will be carbohydrate calories. When running, about half of the calories burned will be fat calories and half carbohydrate calories.

But a more intense exercise burns so many more calories total! The total number of fat calories burned still comes out ahead. (You can check that for walking versus running.)

Body Fat link to here link to index

The simplest measure for if someone's weight is healthy is their percent body fat. The American Council on Exercise has categorized how much body fat people should have to be healthy, based upon the person's usual activity levels.

WomenMenAbdominal Muscle DefinitionImpression
Not Enough Fat body fat image for women 1
13% or less
body fat image for men 1
5% or less
crazy "scary"
Athletes body fat image for women 2
14% to 20%
body fat image for men 2
6% to 13%
well-defined "hard"
Fit body fat image for women 3
21% to 24%
body fat image for men 3
14% to 17%
slight "lean"
Average body fat image for women 4
25% to 31%
body fat image for men 4
18% to 24%
neither muscular nor flabby "love handles"
Obese body fat image for women 5
32% or more
body fat image for men 5
25% or more
more flab than muscle "more than love handles"

For every 20 years older than 18 years old, a person should expect to have 2% more body fat for their activity level than shown in the table.

Unfortunately, percent body fat is very difficult to measure accurately. A measurement with even ±4% accuracy requires a trip to the doctor and all sorts of caliper measurements. A more accurate measurement requires a fancy room for underwater weighing or air displacement measurements.

The easy and practical way to think about body fat is to consider the saying, "Nothing tastes as good as feeling fit." If that saying is genuinely true for you, and your waist feels more lean than love-handlish, you almost certainly have a very healthy percent body fat.

Three Kinds of Averages link to here link to index

Mean

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?

Median

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?

Mode

The mode of a set of numbers is the value that repeats most often. If there is a tie and more than one value is "most often" then all values that tie are modes.

The mode is almost never used.