About the Diagnostic Quiz, Questions 1 and 3
(more extensive notes on the quiz: later this week, on website)
John Barnden
School of Computer Science, University of Birmingham
Term 1, 2014–15
Reminder: Simple Algebraic Manipulation
Question 1, Diagnostic Quiz
Part (a) QUESTION: The temperature F in Fahrenheit is obtained from the
temperature C in Celsius (Centigrade) as follows: F =
for conversion in the opposite direction.
ANSWER: C = 59 (F − 32).
(0) We’re given:
F =
9C
5
9C
5
+ 32. Provide a formula
THE WORKING:
+ 32
(1) Move the 32 over, changing its sign (i.e. subtract 32 from each side),
to get: F − 32 = 9C
5
(2) Move the 5 over, flipping it from dividing to multiplying (i.e. multiply each side by 5),
to get: 5(F − 32) = 9C
(3) Move the 9 over, flipping it from multiplying to dividing (i.e. divide each side by 9),
to get: 59 (F − 32) = C
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Diagnostic Question 1, Part (a) contd
OR: Steps (2) and (3) can be combined into one step:
(1) gave:
F − 32 =
9C
5
(2and3) Move the
5
(F
9
9
5
over, flipping it upside down (i.e. multiply each side by 95 ), to get:
− 32) = C
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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1, 2014–15
this week, on3website)
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SYSTEMATIC ERRORS on Diagnostic Question 1, Part (a)
Many of you did (1) fine (moving the 32 over), to get
F − 32 =
9C
5
but then in trying to do (2) you got the incorrect equation
5F − 32 = 9C
That is, you didn’t multiply the whole of the left hand side by 5.
You then made a further, similar error in dealing with the 9, ending up with
5
F
9
− 32 = C
This equation says you first take five ninths of F and then subtract 32.
THE CORRECT ANSWER SAYS you first subtract 32 from F and then take five ninths
of the result. A crucial difference!!
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Diagnostic Question 1, Part (b)—Mental!
We have
C = 59 (F − 32)
F = 115
The exact value for C is (5 × 83)/9. You could use a calculator, to get 46.111 . . ., and
then round down to 46,
but it’s easy to do it mentally or by hand:
83 = 81 + 2
so we have 5/9 of 81, i.e. 45,
plus 10/9, which is clearly 1 to the nearest integer (whole number).
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Diagnostic Question 1, Part (c)
Part (c) QUESTION: There’s a temperature scale called Réaumur, with abbreviation
Re. It’s like Celsius except that water boils at 80 degrees Re. What’s the formula
for converting from Fahrenheit to Réaumur?
ANSWER: R = 94 (F − 32).
THE WORKING:
(0) The question implies that water freezes at 0◦ Re, because it freezes at 0◦ C.
So we’re just
re-scaling the 0–100 range to 0–80.
So each Celsius degree is one hundredth of 80 Re degrees,
i.e. it is 80/100 of an Re degree.
So C degrees on the Celsius scale is (80/100)C degrees on the Re scale, i.e.:
R = (80/100)C , so more simply
R = 4C /5.
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Diagnostic Question 1, Part (c) contd
(1) From (a) we have C = 59 (F − 32).
(2) So from (0) and (1) we get R =
4 5
(F
5 9
− 32).
(3) Cancel the 5s to get R = 49 (F − 32).
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Diagnostic Question 1, Part (d)—Mental Again!
We have
R = 49 (F − 32)
F = 115
The exact value for R is (4 × 83)/9. You could use a calculator, to get 36.888 . . ., and
then round up to 37 if you wanted.
Again, easy to do it by hand or mentally:
4/9 of 81 is 36, and
4/9 of 2 is 8/9, which again has 1 as the nearest integer.
Or, if you know that 1/9 = 0.1111..., then 8/9 = 0.8888....
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Diagnostic Question 1—A DETAILED LESSON
About moving things from one side of an equation to another:
The intended effect is to apply the same mathematical operation to the WHOLE of each
side—e.g. subtracting 32, multiplying by 5, dividing by 9.
Special case: dividing by a fraction, such as 9/5, is the same as multiplying by the
reciprocal of the fraction, 5/9 in that case.
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Diagnostic Question 1—GENERAL LESSONS
◮
Set your work out NEATLY and CLEARLY: lined-up well, written fairly large, etc.
Unless you are really good at something, or the answer is very brief, do NOT now
use the question paper in future: use a clean extra sheet of paper and allow yourself
lots of space.
◮
CHECK YOUR WORKING carefully as far as is possible in the time available.
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Diagnostic Question 1—GENERAL LESSONS contd
CHECK YOUR RESULTS carefully in sensible ways.
E.g., check that already-known effects come out right.
In Question 1(a), if you already know what value C should corresponding to a given
value F , you can see if your formula gives C from F .
In particular:
◮
Could use the water’s freezing point (0◦ C, 32◦ F) or boiling point (100◦ C, 212◦ F).
Would show that, say, C = 95 F − 32 cannot be the answer to (a).
◮
Could take any Celsius temperature C , work out the corresponding F from the
formula I gave, and then check that your formula converts back properly from F to
C.
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Diagnostic Question 1—GENERAL LESSONS contd
CHECKING YOUR RESULTS, contd.
Another example of a check:
check that the values are of the right sort of size,
whether in an absolute sense or relative to other values.
In particular:
◮
In Question 1 it’s pretty obvious that the Réaumur value for a temperature above
0◦ C must be less than the Celsius value. So your answer for (d) should be less than
your answer to (b). Not all of you achieved this!
◮
On the other hand, if we’re scaling 0–100 C as 0–80 Re, the Réaumur value can’t be
much less than the Celsius value, even if we don’t know the exact conversion
formula. So an answer to (d) such as 12 looks suspiciously low against a value of 46
for (b).
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Diagnostic Question 3—Dividing by Zero
Basically, can’t do it! Division by zero is not defined.
Alternative ways of saying this, in the case of 1/0 say:
1/0 has no value
1/0 is not defined
1/0 is undefined.
WHY IS THIS?
Because y /x should be a number z that when multiplied by x gives you y , i.e.:
y = xz
But there is no such z when, say, y = 1, x = 0.
Case of 0/0: see below.
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
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Dividing by Zero, contd
It is NOT true that y/0 equals “infinity” (∞)
BUT this is SOMETIMES a HEURISTICALLY USEFUL way of putting things.
The main idea here is that for a FIXED positive number c,
c/x “approaches infinity” or “tends to infinity” as positive x tends to 0:
c/x → ∞
as
x →0
(with x > 0)
OR: “the limit of c/x as x tends to zero is infinity”:
lim (c/x) = ∞
x→0
(with x > 0)
But this is merely a way of talking and writing something.
What’s ACTUALLY going on is that
you can make c/x as large as you like by making x small enough
Caution, if you do think of 1/0, 2/0 as infinite:
Barnden (SoCS)
2/0 is no bigger than 1/0 !
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
Term later
1, 2014–15
this week, on
14website)
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0/0
See above: we want a number z such that 0 = 0z.
◮
The trouble is now that any z will work.
So we still don’t have a possibility of a well-defined value
(or even of a small number of well-defined values, as with the case of -2 and 2 both
being square roots of 4).
So we still say the division is undefined.
◮
0/x = 0 for all non-zero x, so you might think it would be an easy extension to take
0/0 to be 0.
But y /x could tend to anything depending on circumstances (or fail to tend to
anything) when both x and y tend to 0. Taking 0/0 to be 0 would not fit in general.
Barnden (SoCS)
About the Diagnostic Quiz, Questions 1 and 3 (more extensive notes on the quiz:
Term later
1, 2014–15
this week, on
15website)
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