You will soon be assigned five problems to test whether you have internalized the
material in Lesson 5 of our algebra course. The Keystone Illustration below is a
prototype of the problems you'll be doing. Work out the problem on your own.
Afterwards, study the detailed solutions we've provided. In particular, notice that several different ways are presented that
could be used to solve the problem.
Instructions for the Keystone Problem
next
© 2007 Herbert I. Gross
As a teacher/trainer, it is important for you to understand and be able to respond
in different ways to the different ways individual students learn. The more ways
you are ready to explain a problem, the better the chances are that the students
will come to understand.
next
© 2007 Herbert I. Gross
next
A ball projected vertically upward at a speed of 160 feet per second, in the absence of air resistance, reaches a height of h feet at the end t seconds
according to the rule: h = 400 – 16(t – 5)2
Keystone Illustration for Lesson 5
next
Answer: 336 feet© 2007 Herbert I. Gross
(a) How high up is the ball at the end of 7 seconds?
next
Solution for Part a:
next
© 2007 Herbert I. Gross
h = 400 – 16 ( t – 5 )27
To solve part (a) we replace t by 7 in the formula.
Using our PEMDAS agreement, we do what's inside the parentheses first, (7 – 5).
(2)2
We next replace (2)2 by 4 to obtain
(4)
nextnextnext
Solution for Part a:
next
© 2007 Herbert I. Gross
h = 400 – 16 (4)
And since we multiply 16 by 4 before we subtract, the equation becomes…
Finally subtracting 64 from 400, the answer is 336.
nextnextnext
64336
h is measured in feet so the answer to part (a) is 336 feet.
feet
next
A ball projected vertically upward at a speed of 160 feet per second, in the absence of air resistance, reaches a height of h feet at the end t seconds
according to the rule: h = 400 – 16(t – 5)2
Keystone Illustration for Lesson 5
next
Answer: 336 feet© 2007 Herbert I. Gross
(b) How high up is the ball at the end of 3 seconds?
next
Solution for Part b:
next
© 2007 Herbert I. Gross
h = 400 – 16 ( t – 5 )23The procedure for solving part (b) is
exactly the same as the procedure for solving part (a). Namely, we replace
t by 3 in the formula.
Using our PEMDAS agreement, we do what's inside the parentheses first, (3 – 5).
(-2)2
nextnextnext
Solution for Part b:
next
© 2007 Herbert I. Gross
h = 400 – 16 (-2)2
(-2)2 means -2 × -2; which by our rule for multiplying two negative numbers is 4, so
we next replace (-2)2 by 4 to obtain…
We multiply 16 by 4, before we subtract.
nextnextnext
h is measured in feet so the answer to part (b) is 336 feet.
(+4)
Subtracting 64 from 400, the answer is 336.
64336 feet
next
• The fact that the product of two negative numbers is positive tells us that the square
of any signed number is non-negative. More specifically, a signed number is either
positive, negative or 0.
next
© 2007 Herbert I. Gross
Note
next
© 2007 Herbert I. Gross
Note
If we multiply a positive number by itself the product will be positive.
If we multiply a negative number by itself the product will be positive.
If we multiply 0 by itself the product will be 0.
• With respect to this example, if we replace t by 7, t – 5 = 2, and if we replace t
by 3, t – 5 = -2. While +2 ≠ -2, (+2)2 = (-2)2.More generally if two numbers have the
same magnitude their squares are equal.
next
© 2007 Herbert I. Gross
Note
• Based on the above note, it is incorrect to
talk about the square root of a number. Every positive number has two square roots. In other
words, for example, if (t – 5)2 = 4, t – 5 can be either 2 or -2.
When we talk about the square root of a number we usually mean the positive square
root of the number. However as we shall see in our next note, if we neglect the negative square root, we miss part of the answer to the present
example.
next
© 2007 Herbert I. Gross
Note
• Because addition has nicer properties than subtraction, a good approach might be
to rewrite h = 400 – 16 (t – 5)2 in theequivalent form h = 400 + -16 (t – 5)2 and then translate the formula into a verbal “recipe”.
next
© 2007 Herbert I. Gross
Note
Step 1Step 2Step 3Step 4Step 5Step 6
Start with tSubtract 5
Square the resultMultiply by -16
Add 400The answer is h.
tt – 5
(t – 5)2
-16(t – 5)2
400 + -16(t – 5)2
h = 400 + -16(t – 5)2
next
• In the present example we found that when t = 3
or t = 7, h = 336. Let's now undo the above recipe and show why this occurred when t = 3 and t = 7. Recall that when we “undo” a recipe we start with
the last step and replace each operation by the one that “undoes" it. In this case we see that…
next
© 2007 Herbert I. Gross
Note
Step 1Step 2Step 3Step 4Step 5Step 6
Start with 336Subtract 400Divide by -16
Take the (2) square root(s).Add 5
The answer is t.
336336 – 400
-64 ÷ -16√4
2 + 5 or -2 + 5t = 7 or 3
next
-64 +4
+2 or -2 7 or 3
next
© 2007 Herbert I. Gross
and this would lead to our missing that t = 3 was also an answer.
• Notice that without the knowledge that a positive number has two square roots,
step 4 would have read, “Take the square root of 4”, and the answer would have
been only +2.
next
Step 4 Take the (2) square root(s). √4+2 or -2+2 Take the square root.
The reason that two different
values of t produce the same value for h is that the ball is at a given height
twice, once on the way up, and once on the way down.
next
© 2007 Herbert I. Gross
144 feet
256 feet
336 feet
384 feet
400 feet
0 feet
1s
2s
3s
4s
5s
6s
7s
8s
9s
10s0s
5s
nextnextnext
next
© 2007 Herbert I. Gross
Note
(t – 5)2 has to be non-negative; and the only time it can be 0 is if t – 5 = 0; that is,
if t = 5.
When t = 5, 16(t – 5)2 =0.
• The fact that the square of a signed number
can never be negative gives us additional information that is contained in the formula. Namely…
nextnext
next
© 2007 Herbert I. Gross
Note
Since 16 (t – 5)2 can never be negative, whenever t ≠ 0, we are subtracting a
positive number from 400. In other words if t represents any number other than 0, h,
which equals 400 – 16(t – 5)2, is less that 400 feet.
• Therefore, the conclusion is that the ball reaches its greatest height (400 feet)
when the time is 5 seconds.
next
next
© 2007 Herbert I. Gross
Note
So if we think of 5 seconds as being our reference point, 3 seconds would be
represented by -2, and 7 seconds would be represented by +2, if t = 5.
In the above context -2 is just as meaningful as +2.
• Notice that 3 seconds is 2 seconds before the ball reaches its greatest height and that 7 seconds is 2 seconds after the
ball reaches its maximum height.
nextnext
The use of “profit and loss", “increase and decrease”, “below zero and above zero” give us good ways to visualize signed
numbers. At the same time, however, they eliminate the need for us to deal with
positive and negative numbers per se. That is we can talk about a $7 loss rather than a
transaction of -$7, etc.
next
© 2007 Herbert I. Gross
Summary
Moreover, even if we elect to use the terms “profit” and “loss” it would be difficult to
give a physical reason as to why the product of two negative numbers is
positive.
next
© 2007 Herbert I. Gross
For example we can interpret 3 × -2 = -6 by saying that, if we have a $2 loss three
times, the net result is a $6 loss. However in looking at -3 × -2, it makes little sense to talk about a $2 loss “negative three” times
or a $3 loss “negative two” times.
next