Part-Part-Whole Relationships

Goals   (IMPORTANT - You will be basing your whole plan on what you choose here.  Choose well) : To develop flexible part-part-whole thinking about numbers 1 through 10; to develop understanding of zero.

NCTM Standard (IMPORTANT) : Number and Operations. Understand Numbers, ways of representing numbers, relationships among numbers, and number systems.

• Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers.

Chronological Plan:

1. Before: Story about Halloween. Every Halloween, I offer trick-and-treaters a bowl of mini-Snickers and a bowl of mini-Mars. They can take a total of six candies. What would you choose? Think about it. Pick a couple of volunteers, and diagram their choices on the board. If no one chooses all Snickers or all Mars, make that your own preference, and draw that one too. Present the question, "How many other different ways of chosing do we have? You will try to figure this out for six candies, and other number of candies as well."

2. During: Break up the class in heterogeneous groups of two. Explain the manipulatives and the cards at each station. One student will try different ways of making up the assigned number, while the other will record the findings and provide suggestions. Recorder and Trier will rotate roles for each.

3. After: (IMPORTANT) You will led the class into a discussion about the findings about four numbers: 1, 3, 6, 10. All groups should have one of these numbers. Make sure they resolve any differences. Ask them if they see any patterns in the results. My main goal during this part of the lesson is to have them discuss among them that :

• the same number can be broken in many ways,
• what does breaking 7 as 7 and 0 mean (discuss 0 as a possibility),
• have some patterns in the organization of the partitions emerge (hopefully to start generalizing to other numbers.)

(Notice how the "After" or discussion points above reflect the "Goals" section of the Lesson Plan)

4. Closure. Collect their worksheets while they write in their journal what they learned about the different ways two numbers make another number.

Key Questions: (VERY IMPORTANT - BOTH SCENARIOS AND QUESTIONS)

Scenario 1: A student is not thinking of zero as one of the possibilities for part-part-whole (as in 6 is 6 and 0).

Possible question:  What if you want to have all Snickers and no Mars? (Thinking zero).

Scenario 2: A student is missing some possible part-part-whole possibilities (only giving 6 as 4 and 2, and 5 and 1).

Possible questions: Can you order your findings in some way? (no whites, one white, etc.)  Now that youÔø‡Ôø‡Ôø‡ve got them ordered, do you think you missed any arrangements?

(Notice again how the Scenarios and Key Questions deal with the elements identified in the "Goals" section of the Lesson Plan)

Assessment Plan: (IMPORTANT - BUT ONLY FOR LPs #2 and #3) Three assessment tools will be used for this plan. The first will be the teacher roaming around each group, and using a checklist with the students' names and three columns where I will note:

• Whether the student can break-up the number at least in one way (one mark in column one)
• Whether the student can break-up the number in multiple ways (two marks in column one)
• Whether the student found all the possible ways to break-up the given number (three marks in column one)
• Whether the student organizes his number pairs in a pattern, or is otherwise aware of pattern in the different ways to break-up numbers (Column two)
• Whether the student uses 0 as one of the possibilities, or is open to discussion zero pairs as a possibility (Column three).

(Compare again how with the elements identified in the "Goals" section of the Lesson Plan)

The second  assessment tool will be the use of the worksheets to triangulate the in-class data, as well as the mathematics communication skills (representation of choices). Finally the journal will serve both as another triangulation tool with the previous two, as well as a way to assess self-confidence in doing mathematics, and sense-making. Appropriate homework could also be given.

Outcome Differentiation Plan: (IMPORTANT -Notice how the following relate to the goals identified above.  They are not just more of the same, but extensions along the main ideas of my overall goals) Students that encounter this activity not challenging enough can be given the following extensions (to be done during the class discussion period): use three (or more) types of candy and try to answer the same questions as above. Use geometrical arrays to break up a number into two; then for each possible part-part-whole combination, there are many possible geometric configurations. How many? (Start with two rows of 3 1"x1" squares representing six. How many ways of breaking that into two threes are there).
For students having trouble finding more than one PPW combination, provide them with a worksheet already arranged by 0 whites, ___ blues; 1 white, ___ blues; etc. This format should help them organize their thinking and found most (if not all) combinations.