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. Rearrange into One Rectangle. . Warm-up: Make a Rectangle. A warm-up is provided at the start of class.
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.
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.
“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.
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. Don’t change problems.
Part 1 You work toward solutions and I try to be quiet! Active Problem Solving
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.
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?
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.
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.
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 step…once you take that first step you may see the problem from different perspectives.
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.
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.
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?
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
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.
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.)
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.
Part 2 Activities promote discovery and problem solving. The calculator table provides perspectives. IntroDuce Topics with Activities
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?
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 =
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
Average Velocity • Suppose we have completed the first 100 miles at V1= 300 miles per hour and can travel any positive velocity, V2, 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.
Average Velocity • In the equations below, D drops out
Average Velocity Limit • The average velocity approaches 600, twice V1 = 300 mph
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.
Limit problem • At what speed will we be within 10 miles per hour of the limit?
Average Velocity with V₁ = 300 • Equation for average velocity, x = V₂:
II. Motivating Related Rates (p. 6) Ripley’s 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.
Pythagorean theorem • Wall: 35 inches • Base: y, unknown inches • Hypotenuse: yard stick 36 inches • Substitute mentally and quickly give an estimate. No calculator needed!!!
Property of Consecutive Squares Between consecutive squares we have
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.
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.
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.
III. Comparative Ages (p. 7) Start with a new-born child and its mother’s age. Choose a mother’s 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.)
Comparative Ages (p. 7) What type of functions are y₁, y₂, and y₃? 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?
Anticipate formal limit statements 6. At what child’s age will the quotient be ≈1.3? At what child’s age will the quotient be ≈1.1? Are either likely in a normal lifetime? 7. For x > ____, |y₃ – 1|< 0.25 For x greater than ___, the quotient of ages will be between 1.25 and 1.
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.
V. Build Rectangles from Tiles (p.9) • Tiles arranged in a rectangle.
V. Build Rectangles from Tiles (p.9) • Tiles arranged in a rectangle.
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? Why Tiles? (page 8, bottom) • Distinguish x from x² • 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