Do not mess with this applet or look at the hints until you have thought about this problem for AT LEAST 15 minutes.

Problem 2

Do not mess with this applet until you have spent AT LEAST 15 minutes trying to solve this puzzle on your own. Remember! Arranging the pieces is not the only thing to do in the problem. You need to explain why you are sure that the pieces fit exactly

Problem 3

I am going to use the idea that if 3, 4, and 5 are the sides of a right triangle, then any non-zero scalar multiple of 3, 4, and 5 are also the sides of a right triangle.

There are, of course, two possible answers (WHY?).

If two sides are

then notice that

and

so the missing side of the triangle must be

Again, this is just one solution. Can you work with the numbers 3 and 4 as the sides of the right triangle arranged in another way to find another solution?

Problem 4 and 5

In each applet below, drag the GREEN point. Click on the check box to get a VISUAL HINT as to how to find the point on the line that will help you find the dimensions of the square or rectangle.

Problem 6

Please do not mess with this applet until you have spent AT LEAST 15 minutes on your own working on this problem. Remember! You should find out what is special about these two points (which the three hints will help you to see)

Problem 7

This problem is related to prior problem, and illustrates an important property concerning the DISTANCES between three points (taken two at a time). Look back at Problem 6. How are the lengths PR, PQ, and QR related? Make a general statement about the distances between any three points. Apply this generalization to this problem.

Problem 8

(a) This is a vector idea. The distance from -7 to 17 is calculated by subtracting: 17 - ( -7 ) = 24. If I want to find a point that is two-thirds of the way from -7 to 17, then I need to take two-thirds of the distance between the two points, and add that on to -7. Hence, -7 + 16 = 9. Therefor, 9 is the point two-thirds of the way from -7 to 17.

(b) This is asking you to generalize what you did in part a. Where did the 24 come from? How did I get that number? What did I do with this number?

What number is two-thirds of the way from m to n? Think about how you found 24 and do the same thing. Think about what you did to 24 after you found it and do the same thing.

EXTENSION: What would be different if you went from 17 to -7?

Problem 9

You should spend a little time on this problem by yourself (at least 15 minutes) before you study this pencast.

Again, do not mess with this applet until you have spent AT LEAST 15 minutes pondering this problem, or have reached some sort of satisfactory (in your mind) answer.

Problem 11

This is another problem that will test your perseverance. I know many of you just gave up on the 9 by 16 rectangle puzzle. Others just googled it.

If it can be done, do it. If you think that it can not, create a credible argument as to why it can not be done.

Problem 12

Apply what you learned in Problem 8.

Page 14

Problem 1

If the vector given is [ 24, 7 ], then the vector that is three-fifths as long is found by multiplying the components by three-fifths (this is related to Pythagorean Triples)

Problem 2

I will do the first part (three-fifths of the way from A to B) and I will leave the other part (three-fifths of the way from B to A).

I am going to use the same idea from problems 6 and 7 on Page 13 by going three-fifths of the way along the vector that points from A(-5,2) to B(19,9). The vector that points from A to B has the components [24,7]. I just found the vector that is three-fifths as long as this vector in Problem 1. Therefor, if I took this shorter vector, and placed its tail at (-5,2), the tip of the vector would be at (9.4, 6.2).

Problem 3

Again, kids. Do not mess with this applet until you have thought about this problem and arrived at a reasonable solution (in your mind) for AT LEAST 15 minutes.

Problem 4

And AGAIN! Do not mess with this applet until you have thought about this problem and arrived at a reasonable solution (in your mind) for AT LEAST 15 minutes.

Note: There is an error in HINT 2. It should read S+T=? and T+U=?

Problem 5

Give a good 15 minutes on this problem, or reach a reasonable solution in your mind, before you look at this pencast.

Give some time to play around with this problem. Once you think you have a reasonable answer in your mind, compare it to the answer given by the applet. Double click on the applet so it opens in a new window. Look at the formulas used for the coordinates for each of the points. Do they make sense to you? Why are they what they are? Why 45/60 and 60/60? Why the (Minutes-25) ?

Problem 7

Do I need to even write it? Do not mess with this applet until you have thought about this problem and arrived at a reasonable solution (in your mind) for AT LEAST 15 minutes.

Problem 8

When will you mess with this applet? Only when you have thought about this problem and arrived at a reasonable solution (in your mind) for AT LEAST 15 minutes.

Problem 9

You can certainly solve this by using parametric equations, writing the distance formula, and looking for the minimum on this graph. Our, you can try using our NEW trick for finding the closest point.

Spend some time trying to get an a reasonable answer BEFORE you play with the applet. Then, click on the ANIMATION BUTTON in the lower left corner of the applet. Click on the RESET icon to erase the traces. You can also stop the animation and drag the sliders individually.

Looking at the two parametric equations given in the problem. HINT: The path that point P travels on is described by the parametric equations

From the parametric equations, what are the slopes of the two lines? Are they perpendicular (they certainly look like it in the applet)? How do you know?

Problem 12

Think about it! 23 and

Problem 13

This is a question of SLOPES! Solve each of the following proportions for k

## Table of Contents

## Page 13

## Problem 1

Do not mess with this applet or look at the hints until you have thought about this problem for AT LEAST 15 minutes.

## Problem 2

Do not mess with this applet until you have spent AT LEAST 15 minutes trying to solve this puzzle on your own. Remember! Arranging the pieces is not the only thing to do in the problem. You need to explain why you are sure that the pieces fit exactly

## Problem 3

I am going to use the idea that if 3, 4, and 5 are the sides of a right triangle, then any non-zero scalar multiple of 3, 4, and 5 are also the sides of a right triangle.

There are, of course, two possible answers (WHY?).

If two sides are

then notice that

and

so the missing side of the triangle must be

Again, this is just one solution. Can you work with the numbers 3 and 4 as the sides of the right triangle arranged in another way to find another solution?

## Problem 4 and 5

In each applet below, drag the GREEN point. Click on the check box to get a VISUAL HINT as to how to find the point on the line that will help you find the dimensions of the square or rectangle.

## Problem 6

Please do not mess with this applet until you have spent AT LEAST 15 minutes on your own working on this problem. Remember! You should find out what is special about these two points (which the three hints will help you to see)

## Problem 7

This problem is related to prior problem, and illustrates an important property concerning the DISTANCES between three points (taken two at a time). Look back at Problem 6. How are the lengths PR, PQ, and QR related? Make a general statement about the distances between any three points. Apply this generalization to this problem.

## Problem 8

(a) This is a vector idea. The distance from -7 to 17 is calculated by subtracting: 17 - ( -7 ) = 24. If I want to find a point that is two-thirds of the way from -7 to 17, then I need to take two-thirds of the distance between the two points, and add that on to -7. Hence, -7 + 16 = 9. Therefor, 9 is the point two-thirds of the way from -7 to 17.

(b) This is asking you to generalize what you did in part a. Where did the 24 come from? How did I get that number? What did I do with this number?

What number is two-thirds of the way from

mton? Think about how you found 24 and do the same thing. Think about what you did to 24 after you found it and do the same thing.EXTENSION: What would be different if you went from 17 to -7?

## Problem 9

You should spend a little time on this problem by yourself (at least 15 minutes) before you study this pencast.

brought to you by Livescribe

## Problem 10

Again, do not mess with this applet until you have spent AT LEAST 15 minutes pondering this problem, or have reached some sort of satisfactory (in your mind) answer.

## Problem 11

This is another problem that will test your perseverance. I know many of you just gave up on the 9 by 16 rectangle puzzle. Others just googled it.

If it can be done, do it. If you think that it can not, create a credible argument as to why it can not be done.

## Problem 12

Apply what you learned in Problem 8.

## Page 14

## Problem 1

If the vector given is [ 24, 7 ], then the vector that is three-fifths as long is found by multiplying the components by three-fifths (this is related to Pythagorean Triples)

## Problem 2

I will do the first part (three-fifths of the way from A to B) and I will leave the other part (three-fifths of the way from B to A).

I am going to use the same idea from problems 6 and 7 on Page 13 by going three-fifths of the way along the vector that points from A(-5,2) to B(19,9). The vector that points from A to B has the components [24,7]. I just found the vector that is three-fifths as long as this vector in Problem 1. Therefor, if I took this shorter vector, and placed its tail at (-5,2), the tip of the vector would be at (9.4, 6.2).

## Problem 3

Again, kids. Do not mess with this applet until you have thought about this problem and arrived at a reasonable solution (in your mind) for AT LEAST 15 minutes.

## Problem 4

And AGAIN! Do not mess with this applet until you have thought about this problem and arrived at a reasonable solution (in your mind) for AT LEAST 15 minutes.

Note: There is an error in HINT 2. It should read S+T=? and T+U=?

## Problem 5

Give a good 15 minutes on this problem, or reach a reasonable solution in your mind, before you look at this pencast.

brought to you by Livescribe

## Problem 6

Give some time to play around with this problem. Once you think you have a reasonable answer in your mind, compare it to the answer given by the applet. Double click on the applet so it opens in a new window. Look at the formulas used for the coordinates for each of the points. Do they make sense to you? Why are they what they are? Why 45/60 and 60/60? Why the (Minutes-25) ?

## Problem 7

Do I need to even write it? Do not mess with this applet until you have thought about this problem and arrived at a reasonable solution (in your mind) for AT LEAST 15 minutes.

## Problem 8

When will you mess with this applet? Only when you have thought about this problem and arrived at a reasonable solution (in your mind) for AT LEAST 15 minutes.

## Problem 9

You can certainly solve this by using parametric equations, writing the distance formula, and looking for the minimum on this graph. Our, you can try using our NEW trick for finding the closest point.

## Problem 10

brought to you by Livescribe

## Problem 11

Spend some time trying to get an a reasonable answer BEFORE you play with the applet. Then, click on the ANIMATION BUTTON in the lower left corner of the applet. Click on the RESET icon to erase the traces. You can also stop the animation and drag the sliders individually.

Looking at the two parametric equations given in the problem.

HINT:The path that point P travels on is described by the parametric equationsFrom the parametric equations, what are the slopes of the two lines? Are they perpendicular (they certainly look like it in the applet)? How do you know?

## Problem 12

Think about it! 23 and

## Problem 13

This is a question of SLOPES! Solve each of the following proportions for

k(a)

(b)

## Problem 14

Again, a question of slopes.