Each of these

rows contains

the number

2, so tha

t's a

common f

ac

tor.

All the lengths are

H

ALF the size of

the 10-8-6 ramp.

number

s tell

y

ou wha

t the

lengths of sides

on y

our sca

led

ramp will be.

10

6

the pythagorean theorem

Any good jump has some similar scaled cousins

When you scale the ramp—making it bigger or smaller—none

of the angles change, so it stays a good ramp with a right angle.

To find whole number lengths for the smaller-scaled sizes, look

for a common factor in your current lengths. That’s easiest to do

by using a set of factor trees.

8

Use factor trees to find whole-number miniatures

Factor Tree Factor Tree Factor Tree

10 = 2 x 5

10 8 6

2 5 2 4 2 3

T

he other

5

3

4

What’s the next largest ramp you can build that is similar to the

10-8-6 design?

you are here 4 121

sharpen solution

What’s the next largest ramp you can build that is similar to the

10-8-6 design?

The lengths 15, 12, and 9 have the same ratios as

10-8-6 and as 5-4-3.

5 x 3 = 15

4 x 3 = 12

3 x 3 = 9

10 x 1.5 = 15

8 x 1.5 = 12

6 x 1.5 = 9

It doesn’t ma

tter whether y

ou

used the or

iginal ramp or the

mini ver

sion to do y

our sca

ling—

i

t w

orks out jus

t the same!

15

12

9

Q:

What if I wanted a ramp even smaller? Can I just keep

doing more factor trees?

A:The Kwik-klik units don’t come in half sizes—there isn’t a 1.5

length, so you’d quickly run out of parts, but assuming you weren’t

just talking about building it with the kit parts, you still only need

to do your factor trees until one of the numbers on the bottom is a

prime number—that means it can’t be divided by anything except

itself and one. Then to make a really small ramp you’d multiply

those factors by a fraction.

Q:

How can we skate on a 2D ramp? Isn’t this gonna be

more like a rail that you can slide on?

A:What we’re actually representing is the side of the ramp.

There would be two of these side triangles the same, with a panel

connected to the sloping beam on each triangle. This is a 3D

problem which has a 2D solution.

If you’re interested in exploring 3D problems further, come and

catch up with Sam in Head First 3D Geometry.

122 Chapter 3

W

ha

t the jumps

ac

tua

ll

y look like

in 3D

the pythagorean theorem

Kwik-Klik tips for easy right angles

Select lengths so that:

Longest side

2

= shortest side

2

+ middle side

2

Longest side

Middle side

Shortest

side

So how do we know which sets of ratios

can give us a right triangle? The dude at the

store gave me this slip of paper with some

odd stuff on it—tips for building right angles

or something.... I didn’t really pay it any

attention—what do you think it means?

you are here 4 123

study hall conversation

Kwik-Klik tips for easy right angles

Select lengths so that:

Longest side

2

= shortest side

2

+ middle side

2

Longest side

Middle side

Shortest

side

Well, it looks like

complete gibberish to me.

No wonder she didn’t pay

attention to it.

Frank: But it must mean something. I mean, nobody goes to

the trouble of writing something down unless it’s useful.

Jim: True. But how would you use it? And why would

squaring the side lengths have anything to do with the angle of

the triangle?

Joe: Oh…those twos are for squaring! Yeah—no way that

would work.

Frank: OK, don’t freak out, but if you just try it, like for the

3-4-5 ramp design…it works out perfect.

Jim: What? Are you sure you got your numbers right?

Frank: Yeah—I’m sure. Longest side is 5, and 5 squared is 25.

Shortest side is 3, and 3 squared is 9, and the middle side is 4,

and 4 squared is 16. So—add up the middle and shortest sides

squared—9 plus 16—and you get.…

Joe: 25. The same as the square of the longest side. That has

got to be a coincidence.

Frank: There’s only one way to find out—let’s check the

others.…

Frank

Jim

Joe

124 Chapter 3

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