A Math Forum Project

ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary

Graph Zooming - posted February 19, 2002

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Welcome to ZOOM!

Your challenge is to explore the concept of scale in mathematics and apply it to your everyday experiences. Scale is an important concept to mathematicians and people in many other fields, because it allows one to gain many different perspectives on the same thing. You may notice important things in one scale that you might not see in another.

To Do and Notice

Use the graph zooming applet to experiment with different scales of a graph.

Click to Start the Graph Zoomer

Notice what you see and don't see depending on the view you choose. Use your experiences with the applet to answer the following questions:

1. What is the point of intersection of the red and green lines? (Please write as an ordered pair.)

   Zoom in so you cannot see the red line. You have just changed the scale of the graph. Notice that the minimum and maximum x and y values of the graph have changed.

2. Describe what the Zoom In button does. Why does it make the red line disappear? (Hint: Consider what points are on the red line.)

3. Suppose you could add a fourth Zoom button that would show the intersection of the green line on the y-axis (the "y-intercept"), but not show the red line at all. What would your new button do? (Hint: What would the new scale of the graph be?)

4. Bonus question: Change the equation for the blue line. Experiment with different equations until you can see your new line when the graph is in the Zoomed In position. What is the equation of your line?

Comments

Teacher Support Page

I'm happy to say that everyone who tried the bonus got it right! But that makes me wonder if it was too easy...

I had a great time mentoring this week. I think it was a combination of having students who cared about the math and having a well-worded problem with an applet that worked.

I was really excited that so many of you revised your solutions this week (about 33%). The main problem you had was in being clear enough in your explanations, though most of you seemed to understand what you wanted to say. In many cases, when you revised, you made things very clear.

I guess the main problem was that people talked about what zooming in was without talking about why the red line didn't show up. Don't forget, these solutions are read by people who don't know how to solve these problems, so your clear explanations are very important!

The second problem that people had was in clearly explaining what the new zoom button would show. We felt that to make it clear, you should state what was in the center, plus how far out on the x- and y-axes you could see. Some people mistakenly thought that the current zoom-in button would suffice -- but that doesn't show where the green line intersects the y-axis.

Thanks to all of you who participated this week. I enjoyed the interactions. Please take a look at the problems that were highlighted to see some details that you may have missed in your solutions.

- Jody

Highlighted solutions:

1. What is the point of intersection of the red and green lines?
(Please write as an ordered pair)
(20, 80)

Zoom in so you cannot see the red line. You have just changed the
scale
of the graph! Notice that the minimum and maximum x and y values of
the
graph have changed.

2. Describe what the Zoom In button does. Why does it make the red
line
disappear? (Hint: Consider what points are on the red line).
The zoom in button is used to scale down the graph. The numbers on
the x and y axis become smaller, because you're looking more closely
at the scale. The red line disappears because it intercepts the x-
oaxis at -60, and the y-axis at 60. Therefore, if you zoom out so
that all you can see of the graph is points going up to 11 and
negative 11, then obviously you would not be able to see the red
line.

3. Suppose you could add a fourth <b>Zoom</b> button that would show
the intersection of the green line on the y-axis (the "y-intercept"),
but not show the red line at all. What would your new button do?
(Hint:
What would the new scale of the graph be?)
The fourth zoom button would serve to zoom in just to show the green
line and where it intercepts the x and y axis, but not to the point
where it would show the red line. So, if you wanted to make a new
graph to only show where the green line intercepts the x and y axis
and the green line itself, you would have several options. The graph
would have points going up to 20 and going down 20. Like, the y-axis
would have points going up to 20, and points going down to -20, so
there are 40 points in between.

Bonus question

4. Now change the equation for the blue line. Experiment with
different
equations until you can see your new line when the graph is in the
Zoomed In position. What is the equation of your line?
y = 9/7 x + 10

1. What is the point of intersection of the red and green lines?
(Please write as an ordered pair)

(20, 80)

Zoom in so you cannot see the red line. You have just changed the
scale
of the graph! Notice that the minimum and maximum x and y values of
the
graph have changed.

2. Describe what the Zoom In button does. Why does it make the red
line
disappear? (Hint: Consider what points are on the red line).
The zoom in button shows the graph more close up. Therefore the
points only go up to 11 and -11 from the origin which is (0,0). The
red line disappears because when the graph was 'zoomed out' the red
line crosses the graph at 60 and -60. But when the graph was 'zoomed
in' the graph only showed up to points 10 and -10. Thus, there is no
possible way that we could've seen the red line.

3. Suppose you could add a fourth <b>Zoom</b> button that would show
the intersection of the green line on the y-axis (the "y-intercept"),
but not show the red line at all. What would your new button do?
(Hint:
What would the new scale of the graph be?)
The y-axis should reach up to 22 and -22. The x-axis should reach up
to 22 and a little bit more and -22 and a little bit more. There can
also be more options. This button should make the graph zoom in so we
will only be able to see the green line and its y-intercept. Since
the y-intercept is 20. The scale increases and decreases by 2s.

Bonus question-
4.Now change the equation for the blue line. Experiment with
different
equations until you can see your new line when the graph is in the
Zoomed In position. What is the equation of your line?

y = 9/7x + 10

14 students received credit this week.

Alex Acosta, age 15 - Westside High School, Houston, TX
Lee Bobbitt, age 15 - Westside High School, Houston, TX
Andy G., age 13 - Georgetown Day School, Washington, DC
Julius L., age 13 - Taipei American School, Taipei, Taiwan
Katie L., age 13 - Taipei American School, Taipei, Taiwan
Tina L., age 13 - Taipei American School, Taipei, Taiwan
Kai-Ling S., age 13 - Taipei American School, Taipei, Taiwan
Mary S., age 13 - Taipei American School, Taipei, Taiwan
Max S., age 14 - Taipei American School, Taipei, Taiwan
Will S., age 15 - McLean High School, McLean, VA
Michelle Shadwell, age 17 - Westside High School, Houston, TX
Brian T., age 14 - McLean High School, McLean, VA
Denise W., age 13 - Taipei American School, Taipei, Taiwan
Grace Hsiang Wen Y., age 14 - Taipei American School, Taipei, Taiwan

View most of the solutions submitted by the students above