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I am going to teach 10th grade students for the first time and while preparing my lecture i was just curious to know about this so as i can explain it to my students

Thanks

J. Deff
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    I'd venture that it depends on how you teach them that $f(x)=x$ is a straight line. From there on, it's just scaling and translation. – dxiv May 01 '17 at 05:35
  • Seems a good idea. But just to confirm does 2x means values of x are halved on x axis, but why? – J. Deff May 01 '17 at 05:37
  • Because, in order for the argument to the function to stay the same, $x=1$ on the old axis means $2x'=1 \iff x'=\frac{1}{2}$ on the rescaled axis. See also this answer to an unrelated question. – dxiv May 01 '17 at 05:41

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Your question is really more about pedagogy than mathematics. The reason "why" that graph is a straight line really depends on how much prior knowledge of algebra and geometry you bring to the problem. Tenth graders and professional mathematcians will have answers at different levels of sophistication.

So before you teach the class, I'd suggest you do several things.

First, think through why you think that graph is a straight line. That may not be the way to convince a tenth grader, but it's an important place to start.

Then think about what your class already knows. Have they drawn lots of graphs from tables of data, sometimes seeing straight lines, sometimes not? Have they done that for data generated by evaluating different kinds of functions? If not, you should begin there. They will observe empirically that linear functions have lines for graphs. The geometric meaning of $b$ as the $y$-intercept should be pretty clear. The slope $a$ us more subtle. Similar triangles will help. So will applications to physical situations: think about plotting distance covered as a function of time traveled at a constant rate, bank balance over time with simple interest, water filling a pool.

You can't really do this in "a lecture". Teaching that kind of "why" is a process that needs time and dialogue. You can learn some of what works from your students.

As you work through your tenth grade teaching, consider asking questions at https://matheducators.stackexchange.com/ . Perhaps this question would be better there.

Good luck.

Edit: there are linked questions here that may help:

Ethan Bolker
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  • Okay thanks, .... – J. Deff May 01 '17 at 05:54
  • i am thinking of example of example of car. How can i do this? – J. Deff May 01 '17 at 06:00
  • If you travel at $30$ miles per hour how far do you go in $h$ hours? If your trip started at mile $P$ on the turnpike where are you after $m$ minutes. Think about using good variable names, not just $x$ and $y$.In the western states highway exits are numbered with "miles from the state border" rather than just 1,2,3 ... You can probably make up good exercises from google maps. – Ethan Bolker May 01 '17 at 06:05
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If they know functions as "rules which assign numbers to other numbers" or something like that, first show them the graph of $y =x$ by taking numbers for $x$ and plotting them pointwise. Then show $2x$, $2x+1$, etc, and work up to the general formula. Once you start plotting points, the students should easily--and literally--connect the dots.

JMJ
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What is a geometric line? The concept is basically that it is straight. What does that mean. It means ... well, it means it has a constant steepness, if it got steeper or shallower it ... would be bending. What does constant steepness or slope mean? It means that for any distance you go horizontally you will go a proportional distance vertically.

Hence we can talk of slopes and rise over run and things like that. The take away is every line will have a distinct slope that is a ratio of proportional rise over run.

Okay, what does the x-y plane mean. It means we can plot x, y pairs in the plane where each point represents a pair of numbers and any shape or curve is a visual representation of some relation between values of numbers. And vice versa.

So two questions: 1) if I draw a physical line on the plane what relation between numbers does it represent. 2) What curve describes the relation $y=mx+b$.

2) is easier to answer. Every increase in X positively will result in a proportional increase in y. In other words this is a steady proportional rise for a proportional run; a slope. This is a line. The $m $ represents it's slope. (The $b $ simply represents it's relative position, where it starts. This is the yz intercept but ... that's a whole different lesson plan.)

To be thorough we should answer 1) as well. A line has a slope. So for any increase in x value there most be a proportional increase in y. If the x value increases by $x $ then the $y$ value most increase by some $mx $ where $m $ represents the slope increase of the line. So $y=m+b $ where $b $ is just some constant value representing some starting orientation for the line.

fleablood
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$\triangle f = f(x+\triangle h)-f(x)=a(x+\triangle h)+b-(ax+b)=a\triangle x$.

Therefore, $\triangle f = a\triangle x$, so for every change in $x$, the there is a proportional change in $a$, corresponding to the slope (being constant).

Saketh Malyala
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The function y=ax+b is a straight line because the rate with which the function grows is constant i.e. the slope of the line is always constant i.e. the angle that it makes with the positive direction of the x-axis is always constant. Hope this helps.

Piyush Ranjan
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