Three runners are training for a marathon. One day, they all run about ten miles, each at their own constant speed. This graph shows how long, in minutes, it takes Runner #1 to run LaTeX: dd miles.
graph. horizontal axis, distance in miles, scale = 0 to 9 by 1's. vertical axis, scale 0 to 80, by 20's. line passing through origin and 2 comma 20.
The equation that relates Runner #2's distance (in miles) with time (in minutes) is LaTeX: t=8.5dt = 8.5 d.
Runner #3's information is in the table:
distance (miles) time (minutes)
2 18
4 36
6 54
8 72
10 90
Which of the three runners has the fastest pace? Explain how you know.

When Will I Ever Use This?

An Essay for Students Who Have Ever Asked This Question in Math Class

DOUGLAS COREY, BRIGHAM YOUNG UNIVERSITY

The most-asked question in math class is some variation of “When am I ever

going to use this?” In fact, I began typing this question into a search engine, and every

one of the 10 popular completions for “when am I ever going to use . . .” dealt with

school mathematics. Now, as a student, I know this question is a compelling one. Of

course, some students ask it as a challenge to the teacher, using it to mean “Prove to me

that I need this in my life.” However, some students ask it sincerely, honestly wanting to

know how it might be used in the future.

In either case, teachers most often do one of the following. One, they respond

with some trite response about how you will need it next week on the exam or next year

in another class. Not very compelling. Two, they give the quick reply that “if you don’t

know it, you will never use it” or a similar comment that, although true, is not much

more satisfying than the first. Three, which is traditionally about the best a teacher can

do in this situation, they try to give the students what they think students want. Either

the teacher tries to tie the mathematics to some field of employment (there is a popular

poster on the Internet that helps teachers with this), or they present an application of

the mathematics to some area that might be interesting to the student or will at least

justify the mathematical topic. If you have been in a class where this happens, you

know it is rarely satisfying.

This has become a paradox that I have spent time thinking about—math is

certainly useful, yet why is it so hard to explain to or show students how it can be

useful to them? In this essay I analyze this paradox, and I give my response to the

student question, “When am I ever going to use this?”

To start off with, let’s do a thought experiment. Take a moment and think about

the last time you used multiplication outside of a school environment. It might be hard,

but try to remember. If you can’t remember the last time, try to recall at least some

time recently where you used multiplication. Think . . . Think . . . Think. Do you have

situation yet? It doesn’t have to be a time that you wrote down a multiplication

problem or worked it out on paper. You could have done it in your head or on a

calculator. If you are like me, you are having a hard time remembering a specific

situation. I had a difficult time recalling a time, yet I use multiplication all the time. It

is so ingrained in my thought process, I don’t usually notice it. I bet that is true for

most of you. I am sure that you use the idea of multiplication with its closely related

mathematical cousins of finding areas, counting things in groups or arrays, scaling

things up or down, or working with proportions. It has become such a natural a part of

our thinking that we don’t consciously think, “OK, this is a multiplication problem.

How was I taught to do this in school?”

This may be the first piece of understanding this paradox. Math is useful, but

the large majority of the use is done mentally and subconsciously. When you look at a

graph, a number, a formula, a chart, an algorithm, a quantitative situation, or anything

of the sort, you draw upon myriad mathematical connections made during the many

hours spent in math classes and doing your math homework (and in other situations) to

immediately make meaning (or begin making meaning) of what you are experiencing

(seeing, reading, and so on). Rarely could you go back and figure out when you learned

the skills, even one skill, you are using in the moment of making sense of something. It

is nearly impossible to pinpoint the moments that you learned specific skills, especially

those that create the schema we use to make sense of the world. The consequence is

often a loss of credit to the teachers who helped students learn, simply because finding

the source of the knowledge becomes nearly intractable.

Let’s get back to our thought experiment. I hope by now you have a situation in

which you used multiplication. When I first conducted this experiment, I thought of two

instances. I used multiplication to calculate the area of a raspberry patch I was putting

in my backyard so I could compare the area to some information about raspberry

patches online. I also used multiplication to quickly figure out if I had picked up enough

cans of cream of mushroom soup at the grocery store; I needed 24 cans, so I organized

them in a four-by-six array.

option a, c,d are correct.

step-by-step explanation:

from the given figure, it is given that z is equidistant from the sides of the triangle rst, then from triangle tzb and triangle szb, we have

tz=sz(given)

bz=zb(common)

therefore, by rhs rule,δtzb ≅δszb

by cpctc, sz≅tz

also, from δctz and δasz,

tz=sz(given)

∠tcz=∠saz(90°)

by rhs rule, δctz ≅ δasz, therefore by cpctc, ∠ctz≅∠asz

also,from δasz and δzsb,

zs=sz(common)

∠zbs=∠saz=90°

by rhs rule, δasz ≅δzsb, therefore, by cpctc, ∠asz≅∠zsb

hence, option a, c,d are correct.