Date: September 18, 2002

Grade Level: 5, 6

Subject(s):

• Mathematics/Process Skills
• Language Arts/Literature/Children's Literature

Duration: 45 minutes

Description: The lesson plan centers on Tang's selection of children's literature, The Grapes of Math . The riddles within the book are meant to stretch students' thinking skills and to force them to find efficient ways to solve mathematical problems.

Goals: Pennsylvania Academic Standards for Mathematics :

• 2.1.3. Numbers, Number Systems, and Number Relationships [D. Use drawings, diagrams, or models to show the concept of fractions as part of a whole. J. Estimate, approximate, round or use exact numbers as appropriate.]
• 2.1.5. Numbers, Number Systems, and Number Relationships [D. Use models to represent fractions and decimals.]
• 2.2.3. Computation and Estimation [E. Use estimation skills to arrive at conclusions.]
• 2.2.5. Computation and Estimation [D. Demonstrate the ability to round numbers. E. Determine through estimations the reasonableness of answers to problems involving addition, subtraction, multiplication and division of whole numbers. I. Select a method for computation and explain why it is appropriate.]
• 2.2.8. Computation and Estimation [F. Identify the difference between the exact value and approximation and determine which is appropriate for a given situation.]
• 2.4.5. Mathematical Reasoning and Connections [B. Use models, number facts, properties and relationships to check and verify predictions and explain reasoning].
• 2.5.3. Mathematical Problem Solving and Communication [A. Use appropriate problem-solving strategies (e.g., guess and check, working backwards).]
• 2.5.5. Mathematical Problem Solving and Communication [A. Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense and explain how the problem was solved. B. Use appropriate mathematical terms, vocabulary, language symbols and graphs to clearly and logically explain solutions to problems. E. Select, use, and justify the methods, materials and strategies used to solve problems. F. Use appropriate problem-solving strategies (e.g., solving a simpler problem, drawing a picture or diagram).]
• 2.5.8. Mathematical Problem Solving and Communication [C. Justify strategies and defend approaches used and conclusions reached.]
  
• 2.6.3. Statistics and Data Analysis [B. Formulate and answer questions based on data shown on graphs.]

1. Students will be able to work cooperatively in teams of four or five to develop strategies for solving the "mind-bending" riddles.
2. Students will be able to address their peers in other groups to share their strategies for solving the riddles.
3. Students will be able to provide feedback to their peers about the strengths and weaknesses of the strategies.
4. Students will be able to create a picture and devise a strategy to solve a problem similar to the riddles solved in the main part of the lesson.

• big book of The Grapes of Math by Greg Tang
• photocopies of each riddle presented in the book (with the hints removed from the page)
• overhead transparency of each riddle (with the hints present on the page)
• large sheets of oak tag for the students to complete their own riddle
• an easel to hold the big book and for the presentation of the students' riddles
• pencils
• paper
• markers or crayons

Vocabulary:

1. riddle - puzzling question posed as a problem to be solved
2. pattern - repeated characteristic in an object
3. strategy - the art of devising or employing plans or stratagems toward a goal
4. reason - to justify or support

Begin the lesson by telling the students that they will be working on riddles and creating problem-solving strategies to solve the riddles. Ask for volunteers to tell a riddle to the class (in a perfect world, the riddle will have some relation to mathematics). As a class, solve the riddle. Present the following words to the students: riddle, pattern, strategy, and reason. Tap the students' knowledge about the words to see what they already know about the terms. Define the words (the teacher can use an overhead projector or a PowerPoint presentation).

Present the first riddle (minus the hint) in the book (via PowerPoint or the overhead projector). Show the students that counting each object on the page takes far too long and is not efficient. Devise an efficient strategy to solve the riddle. In the case of this riddle, "Fish School," show the students that they have to look askew (i.e. count the fish in the first diagonal, (4), and count how many rows (4); thus, there are 16 fish). Model a problem-solving strategy used to solve the riddle, "Fish School." Divide the students into cooperative teams of four or five students. Give each team 3 riddles to solve. Be sure to tell them that they are not to count each object individually on the page; rather, they should devise a more efficient strategy to solve each riddle. Students will have 15-20 minutes to complete the task.

As students work in their cooperative teams, circulate around the room providing feedback and reinforcement as needed. Be careful to praise all students to foster an enjoyment of the activity. After 15 or 20 minutes of work has passed, assemble the students in a circle on the floor around the blackboard and the overhead projector. Show the students each riddle via a transparency (hint omitted). Ask for a volunteer from each group to stand up in front of the class and communicate to the groups the strategies that they used to solve each riddle. Allow for feedback from all students. Perhaps students from another group may find an equally valid way to solve the problem. (Be careful with praise and do not merely correct answers; explain how the answers could be tailored to make them correct). After all groups have completed the riddles and have shown the class how to solve each one, the groups should return to their desks to create their own riddles. Their riddles should include a hint on how to solve the problem. Although the riddles in the text may be used as a guide, the riddle may NOT be an exact replica of one in the text. The class will end with the discussion of each riddle. Each group will present its riddle, and students' work will be hung on the class bulletin board for all to see.

Assessment: Note how the students worked cooperatively in teams to solve the riddles. Note how the students addressed their peers to share their strategies for solving the riddles. Note the students' accuracy in providing feedback to their peers about the strengths and weaknesses of the strategies used. Observe students' abilities to create a picture and devise a strategy to solve a riddle on their own.

Useful Internet Resource:
* Pennsylvania Academic Standards for Mathematics
http://www.pde.state.pa.us/k12/lib/k12/MATH.pdf

Special Comments:

Special Needs Adaptations:

• For hearing impaired students, the riddles would be printed out on sheets of paper and distributed to each group member. The hearing impaired child(ren) would be able to read and visually comprehend the task because s/he would be able to see the patterns and directions. Accommodations -- Instead of merely having the students present their opinion orally to the class, the instructor may wish to have the students write journal entries about their conclusions. The journals could be passed from student to student so that all students could read what their peers had written about their findings.

• For a mathematically promising student: Because the lesson involves a fair amount of abstract thinking, the mathematically promising student will find this lesson challenging and interesting. Accommodations -- Instead of assigning only three riddles to the student, the instructor may wish to assign the student more riddles (e.g., five or six). The student could then be asked to write about her/his conclusions for each riddle.

• Scenario One - In this scenario, the classroom has one computer with the capabilities to project what is on the computer screen on to a large multi-media screen. In this setting, the instructor should use PowerPoint software or similar programs to create slides to discuss the relevant vocabulary used in the lesson. Additionally, the instructor should use the technology to present, discuss, and model the solving of the first riddle with the class.

• Scenario Two - In this scenario, there are six computers in the room. In this setting, the instructor should have each group of students visit their assigned computer and write one or two paragraphs containing directions on how the group solved the riddle. Once the directions are written and validated by the instructor, they should be printed and then copied/printed for every student in the class.

• Scenario Three - In this scenario, a lab exists where each student has her/his own computer. In this setting, the instructor should ask each student to type a learning-log (i.e. a few paragraphs) describing what s/he learned, what s/he still wants to know, and what s/he thinks of the lesson. The student should then e-mail her/his learning-log to the instructor. The instructor will then reply to the student with the answers to any questions and comments on the learning log.