ATMOPAV 2012

22
Encouraging Perseverance in Problem Solving: Strategies and Resources Robert Lochel Hatboro-Horsham School District

description

Presentation at ATMOPAV 2012, "Encouraging Perseverance in Problem Solving" by Bob Lochel

Transcript of ATMOPAV 2012

Page 1: ATMOPAV 2012

Encouraging Perseverance in Problem Solving: Strategies and Resources

Robert Lochel

Hatboro-Horsham School District

Page 2: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

FROM THE COMMON CORE “STANDARDS FOR MATHEMATICAL PRACTICE” STANDARD 1: Make sense of problems and

persevere in solving them. Mathematically proficient students start by

explaining to themselves the meaning of a problem and looking for entry points to its solution.

They monitor and evaluate their progress and change course if necessary.

Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?”

They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Page 3: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

DO THESE PROBLEMS ENCOURAGE PERSISTANCE? Landscapers plan to spread a layer of stone on a path. The

number s of bags of stone needed depends on the depth d (in inches) of the layer. They need 10 bags to spread a layer of stone that is 2 inches deep. Write a direct variation equation that relates d and s. Then find the number of bags needed to spread a layer that is 3 inches deep. From Larson Algebra 1

Main Street and Market Street intersect to form an 80-degree angle. Use a protractor and a ruler to sketch the intersection.

From algebra.com: “One good use for rational equations is the shared work problem. This solution would be of great help in scheduling employees. For example, If Bob can mow a lawn in 3 hours and Joe can do it in 5 hours, how long would it take them together?”

Page 4: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 1: PROVIDE AUTHENTIC TASKS AND ENCOURAGE DISCUSSION Landscapers plan to spread a layer of stone

on a path. The number s of bags of stone needed depends on the depth d (in inches) of the layer. They need 10 bags to spread a layer of stone that is 2 inches deep. Write a direct variation equation that relates d and s. Then find the number of bags needed to spread a layer that is 3 inches deep. Direct Variation, Linear Functions – Algebra 1 Table Tennis Problem: see links below

Page 5: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 1: PROVIDE AUTHENTIC TASKS AND ENCOURAGE DISCUSSION Main Street and Market Street intersect to

form an 80-degree angle. Use a protractor and a ruler to sketch the intersection.

6th Grade – Angle Measurement, Acute, Obtuse angles.

“Sharp Turn” problem

Page 6: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 1: PROVIDE AUTHENTIC TASKS AND ENCOURAGE DISCUSSION “One good use for rational equations is the

shared work problem. This solution would be of great help in scheduling employees. For example, If Bob can mow a lawn in 3 hours and Joe can do it in 5 hours, how long would it take them together?”

Rational Functions and Equations A.K.A.: the most fun unit of the year!

“Looking for an Assistant” video. Link below.

Page 7: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 2: DON’T SAVE PROJECTS AND ACTIVITIES FOR THE END OF UNITS Conic Sections Drawing Project – Algebra 2

My! How the technology has changed! DOS drawing program – black, white AND

magenta! Math Toolkit grapher – use Print Screen! NOW: www.desmos.com

Lecture on Conics

Test (bombed it!)

Drawing Project (if we fit it in after all the snow

days…)

Introduce

Drawing Project

Let’s explore

our conics!

Do we need

such a big test?

Page 8: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 2: DON’T SAVE PROJECTS AND ACTIVITIES FOR THE END OF UNITS

Page 9: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 2: DON’T SAVE PROJECTS AND ACTIVITIES FOR THE END OF UNITS

Page 10: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 2: DON’T SAVE PROJECTS AND ACTIVITIES FOR THE END OF UNITS

Page 11: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 3: EMBRACE ALTERNATE MEANS OF “TURNING IN” PROBLEMS “Elvis on the Beach” Optimization – Old Dogs

and New Tasks”, Kaplan and Otten, Mathematics Teacher, May 2012 8th grade Algebra I class

What is your immediate “gut” reaction of this problem’s difficulty? “I feel like this will be in the middle of hard and

easy. It won’t be the hardest, but it won’t be just a quick problem.”

“I think this problem will be really easy.” “I think if I kept trying, it wouldn’t be that hard.” “I think it’s easy, peazy, lemon squeezy.”

Page 12: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 3: EMBRACE ALTERNATE MEANS OF “TURNING IN” PROBLEMS Students work through ideas together on day

1.

Page 13: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 4: ALLOW STUDENTS TO SELF-SELECT PROBLEMS OF INTEREST www.101qs.com, by Dan Meyer. Excellent

source of lesson starters. Teacher-contributed pictures and videos

Academic Geometry class (10th grade) Choose a picture which poses an “interesting”

geometric question. List information and formulas you will need to

solve the problem. Carry out the steps needed to solve the problem. Summarize your work using presentation software

(PPT or Prezi)

Page 14: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 4: ALLOW STUDENTS TO SELF-SELECT PROBLEMS OF INTEREST

Page 15: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 4: ALLOW STUDENTS TO SELF-SELECT PROBLEMS OF INTEREST

Page 16: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 4: ALLOW STUDENTS TO SELF-SELECT PROBLEMS OF INTEREST

Page 17: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 5: ENCOURAGE THOUGHTFUL REFLECTION AND SHARING Contest problems often provide interesting

challenges: A 3x3 square is partitioned into 9 unit squares. 

Each unit square is painted either white or black with each color being equally likely, chosen independently and at random.  The square is then rotated 90 degrees clockwise about its center, and every white square in a position formerly occupied by a black square is painted black.  The colors of all other squares are left unchanged.  What is the probability that the grid is now entirely black? (2012 AMC-12 Exam)

Worked with Prob/Stat classes to discuss problem-solving strategies.

Page 18: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 5: ENCOURAGE THOUGHTFUL REFLECTION AND SHARING What is your immediate, gut reaction of the

difficulty of this problem? “This may take some effort, but I might be able to

do it.” “I have no idea what I just read.” “This is hard and confusing, but I think I can do it

eventually.” “This sucks, and I don’t want to do it.” “Huh?”

Page 19: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 5: ENCOURAGE THOUGHTFUL REFLECTION AND SHARING Suggest some opening steps.

“Draw a picture” “Underline and interpret difficult words”

With your team, sketch examples of squares which work, and don’t work:

Page 20: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 5: ENCOURAGE THOUGHTFUL REFLECTION AND SHARING After working as a team, what has been

established? “Center has to be shaded. At least 4 others have

to be shaded.” “More than 1 way to accomplish task.” Need to “re-read and figure out what has been

said.”

Page 21: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

STRATEGY 5: ENCOURAGE THOUGHTFUL REFLECTION AND SHARING Has your assessment of this problem’s

difficulty changed? “No, because this usually happens to me on tests

and quizzes. I glance at it and freak but read it over a few times and understand.”

“Yes. I have become more confident in this type of problem.”

“My opinion {this sucks} has changed. While it is a hard problem, I can do it now.”

“Very slightly. When I found a way that worked it looked easy, but as I went on it got harder.”

Page 22: ATMOPAV 2012

R. Lochel. ATMOPAV 2012

IN SUMMARY….. STRATEGY 1: PROVIDE AUTHENTIC TASKS AND

ENCOURAGE DISCUSSION STRATEGY 2: DON’T SAVE PROJECTS AND

ACTIVITIES FOR THE END OF UNITS STRATEGY 3: EMBRACE ALTERNATE MEANS OF

“TURNING IN” PROBLEMS STRATEGY 4: ALLOW STUDENTS TO SELF-

SELECT PROBLEMS OF INTEREST STRATEGY 5: ENCOURAGE THOUGHTFUL

REFLECTION AND SHARING