Lesson
Plan for Friday, 2/22/2008
Objectives:
(1) Learn from mistakes in §10.8 homework.
(2) Learn what is meant by prorating (not in textbook).
(3) Learn how to compute arc length of any
fractional part of a circle (§10.9).
Introduction
Greet students
Demonstrate new Smokey ringtones
(now fully connected and operational!)
HW review (§10.8)
Scan
Audit
Honor reminder
#8a: y
cannot be found without considering the unmarked segment length (mark it as x).
Backward
chaining: find x by solving equation
62 = 4(4 + x) by TSPT.
Help
students understand why secant “whole chunk” is 4 + x, not 4x or 8 + x, etc.
If
students cannot find x = 5, spend 5
minutes re-teaching Algebra I.
Assuming
x = 5 is found, write second equation
3(3 + y) = 4(4 + x) by SSPT.
Solve
to get y = 9.
Enrichment:
show students why it was never actually necessary to solve for x.
(Since
62 = 4(4 + x) by TSPT, we
know 4 + x = 9. Therefore, second
equation
becomes
3(3 + y) = 4(9) = 36 by substituting
the value for 4 + x.)
#8b: Re-teach portion of Chapter 9 where we
learned c2 = a2 + b2 iff triangle is right.
Rule
for students to remember: Look at the longest side!
If c is “too small” (that is, if c2 < a2 + b2),
then 
(acute).
If c is “too big” (that is, if c2 > a2 + b2),
then 
(obtuse).
From
part (a) we know the sides are 6, 12, 13.
Remind
students of the 5, 12, 13 family. Without doing any arithmetic, what
do we know about a 6, 12, 13 situation?
The
longest side is 13, which we call c
(the “hypotenuse candidate”).
But
look! This c is too small. (Since 132
= 52 + 122, we know 132 < 62 +
122.)
Answer:
acute.
#9. Let AE = x. Then by CCPT, x(7 – x) = 2 · 3.
Simplify
to get x2 – 7x + 6 = 0.
Factor
to get (x – 1)(x – 6) = 0.
Since
either x – 1 = 0 or x – 6 = 0, x = 1 or x = 6.
Even if
they grumble, force the students to check both answers.
If x = 1, then BE = 6, so that CCPT gives 1
· 6 = 3 · 2 (OK).
If x = 6, then BE = 1, so that CCPT gives 6
· 1 = 3 · 2 (OK).
Enrichment:
Does the location of the circle prove anything?
(If the
location of the center is meaningful, then we must reject x = 1.
However,
the dot is provided simply to show that we have
a circle,
not
to indicate relative placement of the center.)
#10. Chastise any
student who wrote PQ = 1 by inspection.
Answer
is 1, but students must be able to write equation: 32 = PQ(PQ + 8).
If they
need to let PQ = x, that is OK.
Equation
is quadratic and produces solutions PQ = 1 or PQ = –9 (reject).
If some
students can’t solve the quadratic to get PQ = 1 or PQ = –9, spend a
chunk
of time re-teaching Algebra I. Be pleasant about it, but encourage
them
to get up to speed. Otherwise, Algebra II will be a miserable time.
#11. Make sure every
student can write equation 6(10) = x(x + 7).
Explain
that on next Tuesday’s test, equation by itself is worth half credit.
Simplify
to get x2 + 7x – 60 = 0, solve to get x = 5 or x = –12.
For
students who can’t factor, reassure them that quadratic formula (with
correct
plug-ins) is usually going to get them full credit, even if they
can’t
do the arithmetic. Exception would be if simplification is required.
Since x is a length, reject negative root to
get x = 5.
Check:
5 · 12 = 6 · 10 as needed (OK).
#13. Let d = diameter = AB.
Then by
TSPT, 62 = 4(4 + d). Tell
students that this alone is worth half credit.
There
is no need to show the work of solving this equation (same as #8a).
Since d = 5, r = 2.5.
#15. Make sure everyone
can at least understand why (HE)2 =
100(8100).
Chastise
any student who needs a calculator to solve this.
#16. By CCPT, (x – 2)(x + 7) = 4(2x – 1).
Simplify
to get x2 – 3x – 10 = 0.
Solve
to get x = –2 or x = 5.
Chastise
any student who rejects x = –2
without explaining why.
Since x is not a length, we cannot automatically
reject a negative answer!
But
since one of the lengths is x – 2, x = –2 is unacceptable.
Check:
If x = 5, then CCPT gives 3 · 12 = 9
· 4 (OK).
#17. This problem is
hard based on what we know at this point.
Solve
by assuming EF = 4 and BC = 3. (Warning: These need to be proved.)
By CCPT
with small circle, 3x = 2y. Solve for y to get y = 1.5x (useful later).
By CCPT
with large circle, 7(x + 4) = 5(y + 3).
Substitute
from above to get 7(x + 4) = 5(1.5x + 3).
7x + 28 = 7.5x + 15
13
= 0.5x
26
= x = DE (answer 1)
39
= 1.5x = y = DC (answer 2)
Tell
students to give themselves half a gold star for getting this far.
To earn
a full gold star, and a couple of extra credit points, they must prove
a
little lemma.
Lemma:
Whenever two concentric circles are cut by a common secant line,
the
two segments within the larger circle but external to the smaller
circle
(i.e.,the two segments that
surround the chord of the smaller circle)
must
be congruent.
Take
5-6 minutes of class time to prove this if students seem interested.
The
proof is not hard but requires splitting into two cases.
Case I
(secant passing through common center) is easy.
Case II
(secant not passing through common center) is somewhat harder.
Offer
hint of using Thm. 74 (radius 
chord) twice.
Stretch break
Turn to p. 499 (§10.9)
Ask students if they know what the word prorate means.
Example: If 21 out of 25 students (84%) agree to
go on a trip to New Orleans that
everyone was
originally supposed to go on, and a donor to the school had agreed
to contribute
$8,000 to subsidize the trip, how much should each student receive
to defray travel
expenses?
(Let students think for a while, then use Smokey
to choose someone to answer.)
Answer: The $8,000 should be prorated since fewer students are traveling. Originally
everyone’s share
would have been $320. By prorating this at a factor of 21/25
(or if you prefer,
84%), we get $268.80.
Today’s vocabulary word: prorating means multiplying by a suitable factor.
Example #2: Mr. Hansen buys a can of asparagus, a
pack of chewing gum, and
a squirt gun for a
total of $5.42 plus sales tax of 21 cents. Because the squirt gun
is for educational
purposes, the supermarket agrees to provide a prorated sales
tax exemption, even
though food items and non-food items have different sales
tax rates. (This is
sometimes done if the amounts are small or if the difficulty of
reconstructing the
itemized subtotals makes prorating a more sensible option.)
How much money does
Mr. Hansen get back if the retail price of the squirt gun
is $2.95?
Answer: Since the squirt gun represents 2.95/5.42
of the total retail price, the suitable
prorating factor is
2.95/5.42, or about 54%. Mr. Hansen gets a whopping 11 cents
as a refund.
(Multiply 21 cents times the prorating factor, and round to the
nearest penny.)
Example #3: Joe Schmo
and his wife, Flo, earn a total of $72,350 in
calendar year
2007. When filing
their state income tax form, they are asked to allocate their
married-couple
exemption of $2,600 according to income. If Joe earned $31,000
and Flo earned $41,350, how should the exemption be split
between them?
Answer: Joe’s share of the income is 31000/72350,
giving a prorating factor of about
42.85%. Flo’s share of the income is 41350/72350, giving a
prorating factor of
about 57.15%. (Tell
students to do the division on a calculator to check the
arithmetic. Show
students how Smokey can be used as a calculator if they don’t
have one.)
Final
answers: Multiply 2600(0.4285) to get $1114 for Joe’s exemption.
Multiply
2600(0.5715) to get $1486 for Flo’s exemption.
Note: On income tax forms, amounts are
always rounded to the nearest dollar.
Example #4: Compute the arc length of a
62.9-degree arc in a circle of radius 11 cm.
Answer: The prorating factor is 62.9/360, because
that is the portion of the circle’s
circumference
represented by the arc.
We
know that the total circumference is 
.
(Remind students
that they can either multiply 
times diameter or use 
.)
Therefore, multiply
times the prorating
factor to get approximately 12.08 cm.
At this point,
someone will always ask if answer can be left unsimplified
on test.
Reassure students
that they need to simplify only if the problem demands it.
However,
simplification is always required in HW, since presumably there are
calculators
available at home.
Ask students to look at #5abcd and give each
answer in conceptual terms (unsimplified).
They will be doing these again in more detail for
HW.
Wish them a happy weekend.