Challenge Problem: Introduction - WordPress.com€¦ · 0 = Incorrect Total Score *Working backwards Work is presented in a logical order. Explanations are clear and easy to understand.

Day 1

Math Content For Challenge Problems students explain in writing their method for solving brain teasers. In other words, students are not only to solve the problem and show their work, they must also �gure out a clear and succinct way to document their thought process. Since this concept is new and students have not written Challenge Problems before, extra time and care needs to be taken to orient students to what a quality response looks like (as well as those that fall short of expectations). Similar to other lessons in this collection that introduce terms using examples and non-examples, this lessons has students evaluating sample responses. In doing so the students become intimately involved in comparing the guidelines and grading rubric with students’ work. They narrow in on mistakes, omissions, and unclear wording and make suggestions for improvement. In my experience having students walk in the teacher’s shoes for a day is one of the most successful ways introduce students to a new type of activity and motivate them to work to their highest potential. Though students will be evaluating sample responses rather than writing their own report this time around, it is still important for students to solve the problem. This gives them enough familiarity with the solution to grade other’s work.

Introducing the Checkerboard Problem “How many of you have played checkers before?” “Does anyone remember what a checkerboard looks like?” Have students get in pairs, pass out Checkerboard Problem, and direct their attention to Overhead #1. “The question is, how many squares are there on a checkerboard? I want to point out that not all squares on a checkerboard are obvious.” Demonstrate by outlining squares such as the ones below.

Challenge Problem: Introduction

▣ USE A PROBLEM-SOLVING MODEL THAT INCORPORATES UNDERSTANDING THE PROBLEM, MAKING A PLAN, CARRYING OUT THE PLAN, AND EVALUATING THE SOLUTION FOR REASONABLENESS

▣ SELECT OR DEVELOP AN APPROPRIATE PROBLEM-SOLVING STRATEGY FROM A VARIETY OF DIFFERENT TYPES

▣ SELECT TOOLS SUCH AS REAL OBJECTS, MANIPULATIVES, PAPER/PENCIL, AND TECHNOLOGY OR TECHNIQUES SUCH AS MENTAL MATH, ESTIMATION, AND NUMBER SENSE TO SOLVE PROBLEMS

Prerequisites

none

Preparation

Guidelines for Challenge Problems for each student Worksheet #1 for each student

Overhead #1 as well as a copy of all four student responses on overheads Worksheet #2 for each pair as well as a copy of all four student responses

Encouraging Mathematical Reasoning: www.MathLessonBank.com

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“What methods can you and your partner use to make sure you’ve found them all?” Discuss suggestions and methods for staying organized. “Before you begin I want you and your partner to take an educated guess at the answer. Record all your work in an organized way so I can see what you were thinking” Circulate as students work.

Sharing & Discussing the Checkerboard Problem Discuss the di>erent methods pairs used for solving the Checkerboard Problem. The following questions may aide the discussion:

• What was your strategy?

• How did you know if your answer included all the squares? • How did your guess compare to the actual answer? • Was the problem too easy, just right, or too di(cult? Why?

• If you had it to solve over again, would you do anything di)erently?

Introducing Challenge Problems “The Checkerboard problem is your +rst Challenge Problem. For every Challenge Problem, you’ll write an answer keys that includes a complete description for working the problem. Imagine a student who is struggling in math is reading your Challenge Problem. Your goal is to take them step-by-step through solving the problem. Don’t assume they know anything and make sure you explain clearly. I’ve prepared some guidelines so you know what I expect.” Pass out Guidelines for Challenge Problems to be kept in students’ Reference Folder (if you are doing this in your classroom). Ask students to read it silently to themselves and then chose a volunteer to read it aloud to the class. Answer any questions students have. “For this +rst Challenge Problem, we are going to do something di)erent. Each pair is going to get four student responses to grade. Make sure you use the guidelines and grading rubric to determine how to score each Challenge Problem. Let’s work through the +rst one together.” Pass out the Worksheet #2 and Student #1’s paper. Work through grading it together. Include in your discussion how to improve the Challenge Problem to bring the score up to 100%. You may �nd it helpful to make an overhead out of Student #1’s paper so you may make improvements directly on the work itself. Overall Student #1 work is organized and clear. Most of the discussion should center around the quality of the explanation and what a struggling math student might think if they were reading it. When brainstorming improvements it is still worth mentioning improvements including

• Part 1 is missing a question mark • Part 2 is not a complete sentence. • Part 3 is not a complete sentence and is missing units.

• Part 4 “I used the short cut 8 x 8” might be improved to say “I used the short cut 8 x 8 = 64 to �nd the total number of small squares”.

• At the top of page 3, “3’s were harder but I counted 6 x 6.” could be improved to say “The squares that are size 3 by 3 were harder to add up. I counted 6 across the top and 6 down the side. If you take 6 x 6 = 36 you get the total number of 3 by 3 squares.”

• The table just beneath that would be clearer if the columns were labeled.

Facilitating, Discussing & Sharing Pass out the remaining Challenge Problems for pairs to grade. Circulate while students work. After most pairs have �nished, discuss scores, reasoning, and suggestions for improvement. See if the class can come to a consensus. Make sure students are basing their reasoning on the grading rubric. If the class comes to a consensus on a score di>erent from your own, explain to the class your reasoning and ask for suggestions of statements to add to the grading rubric that would make your grading clearer. At the end you may want to ask:

• If you were to cut and paste from all four response to create the best paper, which parts would you choose from each paper?

Day 2

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- LESSON PAGES

Encouraging Mathematical Reasoning: www.MathLessonBank.com

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Challenge Problem: Checkerboard

How many squares are on a standard checkerboard?

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- OVERHEAD #1

Encouraging Mathematical Reasoning

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Name: Date:

Challenge Problem: Checkerboard Squares

How many squares are on a standard checkerboard?

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- WORKSHEET #1


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� Guidelines for Challenge Problems

Imagine a student who is struggling in math is reading your Challenge Problem. Your goal is to take them step-by-step through solving the problem. Don’t assume he/she knows anything and make sure you explain clearly.

� Strategy is appropriate and eKcient for solving the problem.

� All six parts are answered thoroughly.

1. Understanding the Problem In your own words, restate the information you were given as well as the question asked. Are there any constraints or special requirements? Is this problem similar to any you’ve solved in the past? How can that help you? 2. Questions and Strategy What questions come to mind? List them. What is your strategy? Could any of the following help? *Drawing a picture *Formulas you know *Guess and check method 3. Guess Take an educated guess at the answer. Be sure to include units. 4. Work In a logical order, explain how you solved the problem. Remember your audience is a student who is struggling in math. Don’t make assumptions, and include any drawings, charts, equations, etc. that will help make your explanation clear. 5. Solution State your answer with units. Does your answer seem reasonable to you? How do you know? How does your answer compare to your educated guess in part 3. 6. Re0ection Was this problem too easy, just right, or too diKcult? Why? If you had it to do over again, would you do anything di>erently?

Score Struggling Student Says

4 “I completely understand this Challenge Problem.”

Challenge Problem has all 4 qualities.

3 “I understand most but not all of this Challenge Problem.”

Challenge Problem has 3 of the 4 qualities.

2 “I understand about half of this Challenge Problem.”


1 “I don’t understand most of this Challenge Problem.”


I Missing any of the 6 parts.

4 ½ = 100% 2 ½ = 80%

4 = 95% 2 = 75%

3 ½ = 90% 1 ½ = 70%

3 = 85% 1 = 65%

I = No grade.

Please �nish &

turn back in.

*A strategy for similar past problem *Creating a chart or graph *Breaking the problem down into smaller steps

Qualities

Parts

Grading

Rubric Score Correct Answer ½ = Correct 0 = Incorrect

+ =

Total Score

*Working backwards

� Work is presented in a logical order.

� Explanations are clear and easy to understand.


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- GUIDELINES

Name: Date:

Evaluating Checkerboard Challenge Problems

What speci8c corrections are needed to bring the score to 4 ½ ?





Student #1


+ =

Total Score





Student #2


+ =

Total Score



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- WORKSHEET #2






Student #3


+ =

Total Score





Student #4


+ =

Total Score



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- WORKSHEET #2

Student #1


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- STUDENT RESPONSE #1

Student #1

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Student #1

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Student #2

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- STUDENT RESONSE #3


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Student #3

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Student #4

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Challenge Problem: Introduction - WordPress.com€¦ · 0 = Incorrect Total Score *Working backwards Work is presented in a logical order. Explanations are clear and easy to understand.

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