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Geometry Solution Page 146-1 / Comments

Geometry Solution Page 146-1:

Hi,

We believe there is an error in the volume calculations, and we don't see an errata for it.

We worked it out as follows:

V=1/3BH
H squared = h squared - .5 squared
H squared = .866 squared - .5 squared
H squared = .5
H = .707 or .7

On the solution page, it goes from:

H squared = .866 squared - .5 squared

to:

H = .5

We think you forgot to take the square root of H.

In that case, the Volume calculation is incorrect also. This is how we worked it out:

V= 1/3 x 1 squared x .7
V= .23 dm squared
V= 233 cm cubed

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I would also like to make this comment/suggestion to Joan. As a parent who hasn't worked with geometry in a LONG time - since 10th grade to be exact(and with a kid who gets impatient with my slowness!), I was only able to follow this solution by going thru the following additional steps.

I need to find Volume and Surface Area, so I start by looking at those formulas to see what variables I know and don't know, but first, to keep it all straight, I had to label the 3D diagram with H, h, and a. And I added a and c to the net.

V = 1/3 BH (I had to remind myself that B = l x w of pyramid because my son started out saying V = 1/3 lwh; I just need to find H)

S = A(sq) 4A(tri)
A (sq) = lw = B (knowns)
A (tri) = bh/2 (I know b is the base of triangle; I need to find h)

I realize I need to find h first (because I will need to know h to solve for H). Referencing the net, I know 'a' is half of one side which I know equals 1. So a = .5 dm. I also know that 'c' = 1 dm. or the length of one side. To find h, I can use the Pythagorean Theorem. I find when I use different variables in this equation (ie. h for b), it is helpful to show myself the original formula and then make the substitutions, and then reorder the equation to solve for the unknown:

c sq. = a sq. b sq.
c sq. = a sq. h sq.
h sq. = c sq. - a sq.
h sq. = 1 sq. - .5 sq.
h sq. = 1 - .25
h sq. = .75
h = .866 dm

Then I plug 'h' into my Surface Area formula and get that answer.

Now, on to find the Volume. Referencing the 3D drawing, I now know h = .866 dm and a = half of 1 dm or .5 dm so I need to find H. Again, I found it most helpful to start with the Pythagorean equation, substitute and reorder.

c sq. = a sq. b sq.
h sq. = a sq. H sq.
H sq. = h sq. - a sq.

After I solve for H, I plug 'H' into my Volume formula to get that answer.

I'm certainly not suggesting that you need to write all of these words on the solution page. I just did it to try to show you where I got stuck. For me, it would have been helpful if the diagram on the solution page was fully labeled and if you also included those Pythagorean Equations that I wrote above.

I can walk myself through the equations on your solution pages pretty well, but sometimes a little more (equations) would be helpful, for time's sake (even if it seems redundant to you). Granted, even in this case, we all finally figured it out, but I don't even want to tell you how long it took.

Because Geometry has involved a lot more thinking on my part than the previous levels, I wish the solution pages could help me by also laying out the logical progression we should approach a problem when going over it with our child. I understand that this isn't always possible due to space limitations and the fact that there may be more than one way to approach a problem. I also have the additional challenge with my son in that he is very linear, and he wants to follow your directions to a tee and he haggles us over any inconsistency (even imagined). I wish he could relax a little bit because life is not perfect.

Anyway, my husband and I think what you are teaching is fabulous and I pray you don't think I'm just nitpicking. We are amazed that our 11-year-old can learn and understand not only some algebra, but also geometry and trig! You are giving kids a terrific foundation from which to build upon.

As I said previously, I took Geometry in the 10th grade, which was sandwiched by two years of Algebra and a semester of Trig right into college. My husband had a lot more Math than I, including Calculus and he works with Geometry a bit in a branch of his job (technical illustration via computer). We both have always liked Math and we did well at it, but we know we're rusty at it some 20 years later. It is scary how it comes back to you though!

Anyway, we are really, really pleased with your whole program. We just felt it might be helpful for you to hear some feedback from those in the trenches (hey, sometimes it feels like a battle of the minds with kids these days).

Our many thanks,
Julia, Dave & Liam