Applied Algebra- relating the real world to the mathematical world.

You now have enough information about algebra to try out an applied problem. The general plan, when applying math to the real world, is to find a way to match up objects, events and properties in the real world (pieces of rope, skidoo tracks, lakes, flying, temperature) to mathematical objects, actions and relationships in the mathematical world (lines, shapes, numbers, addition, subtraction, equals), and then use what is known about the mathematical world to come to conclusions about the real world. Let's see how this works by considering a real world scenario:

Gathering Precipitation Data

Suppose Nasak and Mary are doing some research for Environment Canada at the Eureka weather station, on the amount of precipitation that falls over the course of the first week of April in Eureka, and also the first week in November. For each of these weeks, they put out a collecting bucket, and, at 12:00 pm each day collect the bucket, melt the snow in the bucket and measure the amount of water that results.

They record their data as follows
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April 1: 0.2 ml
April 2: 1.0 ml
April 3: 0 ml
April 4: 0 ml
April 5: 0.4 ml
April 6: 0 ml
April 7: 0.3 ml

November 1 0.5 ml
November 2 0.1 ml
November 3 0.1 ml
November 4 0.2 ml
November 5 0 ml
November 6 0 ml
November 7 0.3 ml

Notice that by associating a number (the number amount in ml) to the precipitation that falls in the bucket each day, Nasak and Mary are already making a connection between the natural world- where some precipitation falls in a bucket- and the world of mathematics- where there are numbers. We tend to take this connection between the natural world and the mathematical world of numbers for granted on a day to day basis, but sometimes it's worth pointing out.

Now let's consider how the algebraic skills you've practiced so far can be applied to this situation.

Exercise 1: Write instructions, only in words, for how to get the total amount of precipitation that fell in the first week of April.

Exercise 2: Write in words how you will get the total amount of precipitation that fell in the first week of November.

Exercise 3: Write in words a general description for how you would get the total amount of precipitation for the first week of any given month.

Exercise 4: Rewrite this general description using some words and some mathematical symbols (e.g. the symbols for add, + , and the symbol for equals, =).

Exercise 5: Rewrite this general description using mathematical symbols (like the ones in Exercise 4) and variable names. (If this exercise gives you a sinking feeling, or you aren't sure how to begin, remember the strategies discussed in section 1.3 and section 3.7).

If you have done the exercises, you have now written a mathematical equation that expresses the relationship between the amount of precipitation gathered each day, and the total amount of precipitation that fell for that week. The number of milliliters that you get from adding up each day's amount of precipitation will be equal to the number of milliliteres of actual precipitation that fell that week in that spot.

In section 3.9 we discussed how a mathematical statement with an equal sign can be TRUE or FALSE. In this case, we can ask ourselves- is it true that:

day1 + day2 + day3 + day4 + day5 + day6 + day7 = total_precipitation

(where day1 is the amount of precipitation that fell on the first day, day2 is the amount of precipitation that fell on the second day and so on)

It seems like the answer is yes, so this mathematical equation would be TRUE.

However, what if we wrote:

day1 + day3 + day7 = total_precipitation

In this case, the mathematical statement would be making a FALSE statement about the world, because adding the precipitation for only day 1, day 3 and day 7 wouldn't give us the total precipitation for the week.

Now lets suppose that the totals for each week have been entered into the computer. Unfortunately, a problem arises, because Mary's colleague, Frank, spills coffee on her desk. In the process, it becomes impossible to read the last piece of data in the record- the amount of precipitation measured on November 7.

The record for November now looks like:

November 1 0.5 ml
November 2 0.1 ml
November 3 0.1 ml
November 4 0.2 ml
November 5 0 ml
November 6 0 ml
November 7 (this data is covered by a coffee stain and can't be read)
Total Precipitation in November: 1.2 ml

Without looking at the information from earlier in the section, is there some way that we can use the information we still have- the total amount of precipitation and the amount that fell on the other days of the week- to find the lost piece of information? If so, how?

Exercise: Describe how you would find the missing piece of information from November 7

Exercise 8: How would you describe generally how you would find the missing piece of information for the last entry of any week?

Exercise 9: How would you rewrite the general statement from exercise 8 using mathematical symbols and variable names?

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Sample Answers for Exercises

 

Here are some possible answers for the exercises

 

Exercise 1: Write instructions, only in words, for how to get the total amount of precipitation that fell in the first week of April.

 

“Add together the amount of precipitation that fell on each day in the first week of April. This will give you the total amount of participation.”

Exercise 2: Write in words how you will get the total amount of precipitation that fell in the first week of November.
 

“Add together the amount of precipitation that fell on each day in the first week of November. This will give you the total amount of participation.”

Exercise 3: Write in words a general description for how you would get the total amount of precipitation for the first week of any given month.
 

“Add together the amount of precipitation that fell on each day of that week. This will give you the total amount of participation.”

Exercise 4: Rewrite this general description using some words and some mathematical symbols (e.g. the symbols for add, + , and the symbol for equals, =).
 

“+ the amount of precipitation that fell each day of the week. This will = the total amount of precipitation”.

Exercise 5: Rewrite this general description using mathematical symbols (like the ones in Exercise 4) and variable names. (If this exercise gives you a sinking feeling, or you aren't sure how to begin, remember the strategies discussed in section 1.3 and section 3.7).

 

day1 + day2 + day3 + day4 + day5 + day6 + day7 = total_precipitation


Exercise 8: How would you describe generally how you would find the missing piece of information for the last entry of any week?
 

“I would add up the amount of precipitation from the other days (day 1 through day 6), and I would compare this with the amount of total precipitation for the week. I would know that the difference between these two numbers would be the amount that fell on day 7.”

Exercise 9: How would you rewrite the general statement from exercise 8 using mathematical symbols and variable names?
 

day1 + day2 + day3 + day4 + day5 + day6 = the total amount other than day 7

 

total_precipitation – (day1 + day2 + day3 + day4 + day5 + day6) = the amount that fell on day 7

 

 

 

 

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Copyright Jen Schellinck, 2006