Decomposing the Number Five: An Encyclopedic Guide

Definition of Decomposing:
- Decomposing is identified as a "big word" in mathematics.
- Student Definitions:
  - Faith and Brantley describe it as: "Start with a whole, part, part."
  - Isabella and Molly describe it as: "Breaking stuff down by putting it in parts and breaking it."
  - Tyler and Jordan note that the parts are "not the same number," referring to different combinations.
The Decomposing Chant:
- "Decompose, decompose, break it on down, break it on down."
- "Start with a whole, part, part."
Learning Targets (Mirror with Words Activity):
- "I can take the whole number $5$ and break it into part, part."
- "I can use objects to show the part, part."

Prior Knowledge Review:
- The class reviews decomposing the numbers $1$ , $2$ , and $3$ using their fingers.
- The Number $1$ : There are $2$ ways to decompose the number $1$ .
- The Number $2$ : There are $3$ ways to decompose the number $2$ .
- The Number $3$ : There are $4$ ways to decompose the number $3$ .
Elle’s Pattern Prediction:
- A student, Elle, identifies a numerical pattern in decomposition.
- She notes that for every whole number, the number of ways to decompose it is one greater than the number itself: $1 \rightarrow 2$ ways; $2 \rightarrow 3$ ways; $3 \rightarrow 4$ ways; $4 \rightarrow 5$ ways.
- Prediction for Today: Elle predicts that for the whole number $5$ , the class will find exactly $6$ ways to decompose it.

The Scenario: Using movie passes for a theater with two seating sections (two "parts").
The Whole: $5$ moviegoers (Isabella, Molly, Elle, Tyler, and Faith).
Process of Decomposing the Whole ( $5$ ):
1. Way 1: $0$ people in the left part; $5$ people in the right part. (Part, Part: $0, 5$ ).
2. Way 2: Molly moves over. $1$ person in the left part; $4$ people in the right part. (Part, Part: $1, 4$ ).
3. Way 3: Isabella moves over. $2$ people in the left part; $3$ people in the right part. (Part, Part: $2, 3$ ).
4. Way 4: Tyler moves over because of "gum on the seat." $3$ people in the left part; $2$ people in the right part. (Part, Part: $3, 2$ ).
5. Way 5: Faith moves over because her seat is "too sticky." $4$ people in the left part; $1$ person in the right part. (Part, Part: $4, 1$ ).
6. Way 6: Elle moves over to be with her friends. $5$ people in the left part; $0$ people in the right part. (Part, Part: $5, 0$ ).
Mathematical Observations:
- Validation: Elle’s prediction was correct; there are exactly $6$ ways to decompose $5$ .
- Definition of Zero: Logan defines zero as meaning "nothing" or "none."
- Patterns within the Parts:
  - The numbers in the first part follow a sequence of increase by $1$ : $0, 1, 2, 3, 4, 5$ .
  - The numbers in the second part follow a sequence of decrease by $1$ : $5, 4, 3, 2, 1, 0$ .
  - All parts of a number must be lower than or equal to the whole number itself.

Book Details: Little Quack by Lauren Thompson.
The "Quackulator": Students use a tower of $5$ interlocking cubes to represent the ducklings.
Narrative Decomposition Steps:
- Initial State: Mama Duck has $5$ ducklings (Widdle, Waddle, Piddle, Puddle, and Little Quack). All are in the nest. Nest: $5$ , Pond: $0$ .
- Step 1: Widdle jumps in. Out of water: $4$ , in water: $1$ .
- Step 2: Waddle jumps in. Out of water: $3$ , in water: $2$ .
- Step 3: Piddle jumps in. Out of water: $2$ , in water: $3$ .
- Step 4: Puddle jumps in. Out of water: $1$ , in water: $4$ .
- Step 5: Little Quack plunges in. Out of water: $0$ , in water: $5$ .
Extension Song: "Five Little Ducks"
- The song is used to practice fluctuating parts (e.g., when one duck waddles back).
- Example: "Three little ducks went out… one little duck came waddling back." This results in $3$ ducks in the water and $2$ ducks outside the water.

Scenario: Ms. Stafford and her mother buy $5$ items. They have $2$ grocery sacks.
Goal: Determine all the different ways the $5$ items can be carried using the $2$ sacks.
Student Work with Learning Partners:
- Students use clipboards, recording sheets, and cubes to show combinations.
- Documented Combinations:
  - Stafford carries $3$ ; Mom carries $2$ .
  - Stafford carries $4$ ; Mom carries $1$ .
  - Stafford carries $2$ ; Mom carries $3$ .
  - Stafford carries $1$ ; Mom carries $4$ .
  - Stafford carries $5$ ; Mom carries $0$ (Mom carries "none").
  - Stafford carries $0$ ; Mom carries $5$ .

Center 1: Spider Web (Egg Sac vs. Web)
- Materials: A diagram of a spider web with an "egg sac" at the bottom and $5$ plastic spiders.
- Procedure: Move spiders from the sac to the web one by one to see the parts of $5$ .
- Sequence: $5$ in sac ( $0$ on web), $4$ in sac ( $1$ on web), $3$ in sac ( $2$ on web), $2$ in sac ( $3$ on web), $1$ in sac ( $4$ on web), $0$ in sac ( $5$ on web).
Center 2: Number Bracelets
- Materials: Pipe cleaners with $5$ beads of various colors (Green, Blue, Red, Orange).
- Procedure: Pull beads to opposite sides of the bracelet to create "part, part."
- Recording: Students draw colored marks on a "handy dandy" recording sheet to show the split (e.g., $3$ blue beads on one side, $2$ blue beads on the other).
Center 3: Two-Sided Beans
- Materials: Plastic beans with one red side and one white side.
- Procedure: Shake and spill $5$ beans; record the number of red versus white.
- Example Result: Riley threw $2$ white and $3$ red; Tyler threw $5$ red and $0$ white.
Center 4: Jewel Table
- Materials: Connecting colored jewels.
- Example Result: Kennedy used $3$ black jewels and $2$ green jewels connected to make $5$ .

Summary Speech: Good mathematicians always look for patterns. The class successfully decomposed the number $5$ into various part-part combinations including $0+5$ , $1+4$ , $2+3$ , $3+2$ , $4+1$ , and $5+0$ .
The Log Ride Celebration: A "Rollercoaster cheer with a log ride" involves specific sound effects: "Chug, chug, chug, chug, chug, chug, woo, woo, woo, woo, woo ppsh."