What Follows “Showing Up?” Problem Solving Alice Kaseberg AMATYC 2013 Warm-up: Make a Rectangle...
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What Follows “Showing Up?” Problem Solving Alice Kaseberg AMATYC 2013 Warm-up: Make a Rectangle Do not share problem solutions. Let everyone enjoy success, now or later.
What Follows “Showing Up?” Problem Solving Alice Kaseberg AMATYC 2013 Warm-up: Make a Rectangle Do not share problem solutions. Let everyone enjoy success,
What Follows Showing Up? Problem Solving Alice Kaseberg AMATYC
2013 Warm-up: Make a Rectangle Do not share problem solutions. Let
everyone enjoy success, now or later.
Slide 2
Rearrange into One Rectangle.
Slide 3
Warm-up: Make a Rectangle A warm-up is provided at the start of
class. The warm-up provides a transition from events outside the
classroom to the activities, thinking, and learning inside the
classroom. The warm-up may review key ideas or lead into some part
of the lesson.
Slide 4
Showing Up is commonly given as the first step in Student
Success Trying Hard or Persistence is often given as the next step.
Promote problem solving as a checklist for a student to
self-measure persistence.
Slide 5
Current Problem Solving Terms: Brain Fitness and Brain
Aerobics
Slide 6
Understand the Problem Understanding is the first step in
problem solving. We start today by spending 15 minutes individually
studying a problem. No talking (no phones, no Internet). Take a
minute to choose a problem and then 15 minutes to wrap your head
around the problem. Start now. Dont change problems.
Slide 7
ACTIVE PROBLEM SOLVING Part 1 You work toward solutions and I
try to be quiet!
Slide 8
15 minutes are UP! STOP One problem-solving strategy is to take
a mental break from the problem. Now, as a mental break, we review
some problem solving steps and related strategies.
Slide 9
Problem Solving Steps Understand the Problem Make a Plan Carry
Out the Plan Check and Extend Without describing the problem, what
general strategies do you use to understand or plan a
solution?
Slide 10
Strategies for Understand Read the problem several times. Take
notes on the problem; Identify conditions and record assumptions.
Paraphrase the problem. Write (research as needed) definitions of
key words; write operations suggested by key words; write
implications suggested by non-mathematical words.
Slide 11
Strategies for Make a Plan Try a simpler problem; identify and
record how one might simplify the problem. Decide on an appropriate
picture for the problem. Define variables, as appropriate. Consider
use of a calculator table or spreadsheet, plan column headings.
Predict likely patterns and make estimates of reasonable answers.
Recall a similar problem.
Slide 12
Strategies for Make a Plan And on difficult problems: What can
I do while I figure out what to do? A journey of a thousand miles
begins with a single steponce you take that first step you may see
the problem from different perspectives.
Slide 13
Strategies for Carry out the Plan Be tidy enough in your
writing so as to be able to come back later and not need to repeat
the effort. Make systematic lists. Set up graph paper or graphing
calculator. Set up calculator tables or spreadsheets. Draw careful
pictures, to scale. Stand up and stretch periodically. After
getting stuck, take an appropriate break.
Slide 14
Interpreting Take a Break Starting homework right after class,
early in the evening, and early in the weekend means being able to
work for an hour, do something else and come back later with a
fresh view. Research has proven appropriate breaks to be an
effective learning strategy.
Slide 15
Strategies for Check and Extend Does the answer agree with the
estimate? Does the answer make sense? Have I explained changes in
assumptions? Have I explained the thinking or reasoning that
allowed me to solve the problem? How would the results change if I
used different conditions or assumptions?
Slide 16
A. Martin Gardner: Digit Challenge Find a ten-digit number such
that the first number tells how many zeros in the number; the
second number tells how many ones in the number; the third digit
tells how many twos in the number; and so forth, until the last
digit tells how many nines in the number. __ __ __ __ __ 0s 1s 2s
3s 4s 5s 6s 7s 8s 9s
Slide 17
B. Dudeney: Paper Folding Fold the paper [Map 1] on the lines
so the square sections are in serial order. The square numbered 1
should be face up on top of the stack.
Slide 18
The numbers on the following sections need not be face up nor
all facing in the same direction. Map 2 (when you finish Map 1
because both were given by Martin Gardner in the same
article.)
Slide 19
Party Puzzle (George Polya) How many children have you, and how
old are they? asked the guest, a mathematics teacher. I have three
girls, said Mr. Smith. The product of their ages is 72 and the sum
of their ages is the street number. The guest went to look at the
entrance, came back and said, The problem is indeterminate. Yes,
that is so, said Mr. Smith, but I still hope that the oldest girl
will someday win the Stanford competition. Tell the ages of the
girls, stating your reasons.
Slide 20
INTRODUCE TOPICS WITH ACTIVITIES Part 2 Activities promote
discovery and problem solving. The calculator table provides
perspectives.
Slide 21
I. Average Velocity (page 5) An airplane travels 100 miles at
500 miles per hour and a second 100 miles at 300 miles per hour.
What is the average velocity for the whole trip?
Slide 22
Average Velocity An airplane travels 100 miles at 500 miles per
hour and a second 100 miles at 300 miles per hour. For the first
100 miles, t = D/r = 1/5 hr. For the second 100, t = 1/3 hr. Total
time is r = D/t =
Slide 23
Average Velocity An airplane travels 100 miles at 500 miles per
hour and a second 100 miles at 300 miles per hour. For the first
100 miles, t = D/r = 1/5 hr. For the second 100, t = 1/3 hr. Total
time is 1/5 + 1/3 = 8/15 hr. r = D/t = 200/(8/15) = 375 miles per
hour
Slide 24
Average Velocity Suppose we have completed the first 100 miles
at V 1 = 300 miles per hour and can travel any positive velocity, V
2, for the second 100 miles. Is there a fastest average velocity
for the total trip? If so, what is that velocity? If not, show your
reasoning.
Slide 25
Average Velocity In the equations below, D drops out
Slide 26
Average Velocity on TI-84
Slide 27
Average Velocity Limit The average velocity approaches 600,
twice V 1 = 300 mph
Slide 28
Mention Calculus Reduce fear of higher courses. Motivate
students to take additional courses. Give pride in doing a calculus
problem in Intermediate (or College Algebra) Prepare students for
vocabulary such as limit, related rates, etc.
Slide 29
Limit problem At what speed will we be within 10 miles per hour
of the limit?
Slide 30
Average Velocity with V = 300 Equation for average velocity, x
= V :
Slide 31
THE NEXT THREE ACTIVITIES: A FAST OVERVIEW
Slide 32
II. Motivating Related Rates (p. 6) Ripleys Believe It or Not
1. A yardstick (36 inches) is vertically flat against a wall with
end on floor. How far from the wall should I pull the bottom end in
order for the top of the yardstick to drop by 1 inch? Make a
guess.
Slide 33
Pythagorean theorem Wall: 35 inches Base: y, unknown inches
Hypotenuse: yard stick 36 inches Substitute mentally and quickly
give an estimate. No calculator needed!!!
Slide 34
Property of Consecutive Squares Between consecutive squares we
have
Slide 35
Property of Consecutive Squares Between consecutive squares we
have Thus the difference between the squares is 71 and the base is
the square root of 71, approximately 8.5 inches.
Slide 36
Developing Calculus: 2. For related rates the activity might:
Continue dropping the top of the yardstick one inch at a time.
Calculate the total distance from the wall and also the change in
the distance from the wall. Describe the changes with a function in
terms of x, the number of inches the top has dropped. Explore with
a calculator table.
Slide 37
Developing Calculus: 3. For maximization the activity might
Calculate the area of each triangle formed by the yardstick, the
wall and the floor. Describe the area with a function in terms of
x, the number of inches the top has dropped. Explore with a
calculator table.
Slide 38
III. Comparative Ages (p. 7) III. Comparative Ages (p. 7) Start
with a new-born child and its mothers age. Choose a mothers age
with which you are familiar. Complete the rest of the table
individually and work with a partner or group on the questions
following the table. (Gives a variety of data with similar
results.)
Slide 39
Comparative Ages (p. 7) ChildMotherDifferenceQuotient: y /x
Ratio: y to x 0(any mother) 1 2 3 4 5 70 x y =y =y =
Slide 40
Comparative Ages (p. 7) What type of functions are y , y , and
y ? a. Give another example where 20 to 0 makes sense. b. At what
age will the child be when the quotient is 2? c. What is the
quotient when the child is 70? d. What number does the quotient
column seem to be approaching? Could it be zero? e. Will the ratio
of ages ever be exactly 1 to 1?
Slide 41
Anticipate formal limit statements 6. At what childs age will
the quotient be 1.3? At what childs age will the quotient be 1.1?
Are either likely in a normal lifetime? 7. For x > ____, |y
1|< 0.25 For x greater than ___, the quotient of ages will be
between 1.25 and 1.
Slide 42
IV. Ellipse: Cut an ellipse (potato) along its axes (page 8)
Assume axes are 2a and 2b in length. Rearrange the ellipse to keep
the same circumference but obtain a larger area.
Slide 43
V. Build Rectangles from Tiles (p.9)
Slide 44
Slide 45
Tiles arranged in a rectangle.
Slide 46
V. Build Rectangles from Tiles (p.9) Tiles arranged in a
rectangle.
Slide 47
V. Build Rectangles from Tiles (p.9)
Slide 48
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Slide 50
Why Tiles? (page 8, bottom) Distinguish x from x Distinguish an
area of x square units from a length of x units Clarify adding like
terms Introduce variables into area and perimeter Give visual form
to multiplying and factoring polynomials Statistics and Eyeglasses
There are lots of glasses and contact lenses in your classroom.
Great source of data for introductory statistics. In fact, more
than 150 million Americans use corrective eyewear, spending $15
billion annually on eyewear. What is the meaning of the numbers on
eyeglasses? How do they vary for children, women, men?
Slide 51
Polynomials: Multiply and Factor The tables on page 10 make a
visual connection between the area model for multiplying and
factoring polynomials and the purely symbolic forms. a general
form, unlike FOIL. a tool for completing the square.
Slide 52
Multiplying Polynomials (p. 10)
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Slide 58
Background: Multiplication Table (Look for equal diagonal
products.)
Slide 59
Factoring Polynomials (p. 10)
Slide 60
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Slide 71
Factoring by Table (p. 11) The left column and top row contain
the factors. The inner rows and columns contain the products.
Locate ax and c on the backward inner diagonal. Find equal diagonal
product. Factor and place resulting like terms on the forward
diagonal.
Slide 72
Factor given (x y) on left.
Slide 73
VI. Exponents on Ten (p. 12) Ten with Exponents (5 dec.
places)Equivalent Value (5 dec. places) 10 ^ 0 10 ^ ___________ 2
10 ^ 0.47712 3 10 ^ 0.5 10 ^ 0.698975 10^ 1 10 ^ ___________13 10 ^
1.1760915 10 ^ ___________30 10 ^ 1.531.62278 10 ^ 1.6989750 10 ^
2
Slide 74
SIMPLIFY OR EXTEND THE ORIGINAL PROBLEMS Part 3 Pages
14-15
Slide 75
Ten Digit Challenge (p. 14) This can be done for simpler cases,
so you might try them first. Two and three numbers are impossible
but help to understand the ten digit problem. Find a two-digit
number so the first digit tells how many zeros and the second digit
tells how many ones. Note: Two digits would require only zeros and
ones: 00, 01, 10, 11. Why does each fail? __ __ 0s 1s
Slide 76
Ten Digit Challenge Possible three-digit numbers 000222111
001010100012021 002020200102120 110101011201210 112121211 220202022
221212122
Slide 77
Ten Digit Challenge (p. 14) There is a solution for four digits
as well as certain other numbers of digits up to ten digits.
Slide 78
More Math Maps (p. 14)
Slide 79
Book Signatures (math maps, p.15) Pages for a book are
traditionally printed in multiples of four (called signatures) on a
single sheet of paper which is folded, sewn, and trimmed before
being bound with a cover. 1. How might the individual pages in a
32- page signature be printed on a single piece of paper to place
the pages in the correct orientation when folded?
Slide 80
Four Money Values (p. 14) Find four money values which add to
$6.75 and which multiply to $6.75.
Slide 81
Part 4: More P-S Problems Sixteen Sheep (Henry Dudeney, p.
15)
Slide 82
More P-S: Logistics (p. 16) Ruffles Motel (true story) Sailing
(Par 5) Car Loading (Par 7) These lead to the missionary and
cannibal problem or jealous husband problem (See Wikipedia)
Slide 83
Order of Operations (p. 16) Use the digits in the year 2013 and
any operation to obtain the values 1 to 20.
Slide 84
Order of Operations (p. 16) Use the digits in the year 2013 and
any operation to obtain the values 1 to 20.
Slide 85
P-SBreaking Mind Set (p. 16) Social Security Number Grandmother
Masked Man Split $3
Slide 86
Biography and Bibliography (p.17) Jack Lochhead (also in a
letter to my grandfather, and other engineers): Steel Cable. Art
Whimbey: Code Breaking (p. 18) Plus Martin Gardner, Harry Dudeney,
George Polya, Ren Descartes, and W.W. Sawyer: Human Blood Groups
(p.19) And online sources (p. 20)
Slide 87
Part 6: Selected Answers Further hint on Party Puzzle (p. 21)
Breaking Mind Set Building rectangles from Algebra Tile Sets Tiles,
Fourfold Approach (p. 22-24) The email version will contain answers
to the Course/Topic Openers.
Slide 88
THANK YOU [email protected] This has been my pleasure.
Email me for electronic version of handout.