Decomposing the Number 5: A Comprehensive Mathematical Study Guide

Core Concepts of Decomposing

• Definition of Decomposing: Decomosing is described in the lesson as the process of "breaking stuff down" or taking a whole number and breaking it into "part, part."

• Key Characteristics:     * Decomposed parts do not necessarily consist of the same number.     * The parts are distinct from the starting whole but must add back up to that whole.

• Class Chant and Mnemonics:     * "Decompose, decompose, break it on down, break it on down. Decompose, decompose, break it on down, break it on down."     * "Start with a whole, part, part. Start with a whole, part, part."

• Affirmations and Mirroring:     * "I can take the whole number $5$ and break it into part, part."     * "I can use objects to show the part, part."

Mathematical Patterns and Predictions

• Review of Previous Numbers: The class historically reviewed numbers $1$, $2$, $3$, and $4$.     * For the number $1$: There is only $1$ way to decompose it.     * For the number $2$: There are $2$ ways to decompose it (using specific finger gestures).     * For the number $3$: There are $4$ ways to decompose it.

• Elle’s Pattern Prediction: Student Elle identified a numerical pattern in decomposition possibilities:     * The number of ways to decompose a number is usually the number itself plus $1$.     * Elle predicted that for the number $5$, there would be exactly $6$ ways to decompose it.

• Logical Constraints: Mathematicians look for patterns. For any whole number $X$, the parts must be lower than $X$. For example, if the whole is $5$, all parts must be less than or equal to $5$.

The Movie Theater Simulation ($Whole = 5$)

• Scenario: Five girls (Isabella, Molly, Elle, Tyler, and Faith) have movie passes and enter a theater with five chairs. The theater is divided into two parts (left side and right side).

• Sequential Decompositions of 5:     1. Stage 1: $0$ people on the left, $5$ people on the right ($0$ and $5$).     2. Stage 2: Molly moves to the left. Now there is $1$ person on the left, $4$ on the right ($1$ and $4$).     3. Stage 3: Isabella feels the chair "is not right" and moves. Now there are $2$ people on the left, $3$ on the right ($2$ and $3$).     4. Stage 4: Tyler moves because there is "gum on the seat." Now there are $3$ people on the left, $2$ on the right ($3$ and $2$).     5. Stage 5: Faith moves because her seat is "too sticky." Now there are $4$ people on the left, $1$ on the right ($4$ and $1$).     6. Stage 6: Elle moves to talk with her friends during a "really good part" of the movie. Now there are $5$ people on the left, $0$ on the right ($5$ and $0$).

• Conclusion: Elle’s prediction was correct. There are $6$ ways to decompose $5$. The resulting pattern is:     * Left side: $0, 1, 2, 3, 4, 5$     * Right side: $5, 4, 3, 2, 1, 0$

Storytelling and the "Quackulator"

• Book Title: Little Quack by Lauren Thompson.

• Materials: Students use a "Quackulator" (a tower of $5$ linking cubes) and a story mat to track the movement of ducklings.

• Characters: Mama Duck and her five ducklings: Widdle, Waddle, Piddle, Puddle, and Little Quack.

• Narrative Decomposition Steps:     * The Nest: Initially, all $5$ ducklings are in the nest ($5$ in nest, $0$ in pond).     * Widdle: Jumps in first. Result: $4$ in nest, $1$ in pond.     * Waddle: Jumps in next. Result: $3$ in nest, $2$ in pond.     * Piddle: Feels brave and jumps. Result: $2$ in nest, $3$ in pond.     * Puddle: Leaps in. Result: $1$ in nest, $4$ in pond.     * Little Quack: Eventually plunges in. Result: $0$ in nest, $5$ in pond.

• Supplemental Song: "Five Little Ducks Went Out One Day."     * Process: Mama duck calls, and one by one, ducklings return to the water.     * Goal: Tracking parts as they move from "over the hills" to the pond.

The Grocery Store Problem

• Problem Statement: Ms. Stafford and her mother went shopping and bought $5$ items. They have $2$ grocery sacks. Students must determine how many ways the items can be distributed between the two bags.

• Student Work Procedures:     * Use cubes to represent grocery items.     * Use a recording sheet with two circles (representing the sacks) to draw the items (dots or circles).     * Record combinations of parts that equal the whole of $5$.

• Partner Observations:     * Maddie and Riley: Found a way where the mother has $2$ and the daughter has $3$. They also found $4$ and $1$.     * Jordan and Tyler: Explored "all for Ms. Stafford ($5$) and none for mom ($0$)," and then vice-versa.     * Elle and Dalton: Logged $3$ on one side and $2$ on the other.

Learning Centers and Rotations

Students rotate through specialized stations to practice decomposing $5$:

• Jewels Table: Using colored beads hooked together.     * Faith used $5$ blue beads and $0$ of other colors.     * Kennedy used $3$ black beads and $2$ green beads.

• Number Bracelets: Sliding beads on a pipe cleaner to show part-part relationships.     * Brantley showed $2$ and $3$, then $3$ and $2$.     * Elle showed $1$ and $4$.

• Two-Sided Beans Table: Tossing beans with two different colored sides (red and white).     * Riley recorded $2$ white and $3$ red.     * Tyler recorded $5$ red and $0$ white.

• Small Group (Spider Web Activity):     * Metaphor: A spider web and an "egg sac."     * Action: Spiders crawl from the egg sac onto the web.     * $5$ 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 (Zero means "none," "nothing," or "not any").

Questions & Discussion

• Question: "Joseph, if you'll raise your hand if you can remember what we've been talking about this week?"

• Response: "Decomposing."

• Question: "Faith, what did you and Brantley talk about when you were talking about decomposing?"

• Response: "We were talking about start with a whole, part, part."

• Question: "Tyler, what did you and Jordan talk about?"

• Response: "They're not the same numbers… they were different parts."

• Question: "Isabella and Molly, what did you talk about?"

• Response: "We talked about breaking stuff down by putting it in parts and breaking it."

• Question: "What did we find out [about the number three] Elle?"

• Response: "That we can decompose… it's kind of like a pattern… one's two, and two's three, and three's four, and four's five. And I think five is going to be six."

• Question: "What is my whole that I'm starting with [at the movie theater]?"

• Response: "Five."

• Question: "What do we know zero means Logan?"

• Response: "It means nothing."

• Question: "What is that that good mathematicians are always looking for in numbers?"

• Response: "A pattern."

• Question [to Emma]: "How many is in the water Miss Emma?"

• Response: "Four."

• Question [to Jordan]: "Jordan, how many do you have in the water now?"

• Response: "Two… Oh, another one just came back… Three."

• Question: "Why do you call it a 'handy dandy' [clipboard]?"

• Response: "Cause it's right here. It's close and handy. It's got everything we need right there."

The observation video highlights the importance of hands-on learning, which engages students more deeply in the subject matter. It showcases how visual aids, such as diagrams and objects, enhance understanding by making abstract concepts more tangible. Additionally, the video demonstrates the effectiveness of peer collaboration, where students learn from each other through discussion and teamwork. In an elementary classroom, these practices can be applied by incorporating interactive activities and group projects that encourage student participation and deepen comprehension of the material.