# Multiplication Sequence 3rd grade working outline

1. Overview
2. Process goals or outcomes
3. Mathematical content standards or outcomes
4. Assessment
5. Multiplication as represented as arrays
6. Distributive property
7. Associative and Communitive property
8. To do
9. Definitions

### Overview

The information is a first draft of ideas to include in 3rd grade with respect to Multiplication.
We encourage others to comment and make suggestions so we all might share ideas to benefit our students.

### Mathematical Processes

Problem Solving - Find solutions to the presented arrays, and boxes

Representation - Use manipulatives, pictures, diagrams, symbols, actual objects, and writing to represent their problems and solutions.

Communication – Discuss what they have discovered, write their thoughts in a journal, draw their solutions, prompt further inquiry of situations

Connections - Experiences of fruit in arrays at grocery stores, objects in boxes of various sizes and the total number of objects (eggs, donuts, candy …), carnival games, and patterns of décor

Reason & Proof - Representations to support descriptions of how it works, what makes them confident of its accuracy, and generalize the solution for (all) other similar examples.

### Mathematical content - (Numbers reference - Nebraska math standards approved 10/08/09)

Number Sense - Operations - Multiplication as represented by Arrays
MA 3.1 Students will communicate number sense concepts using multiple representations to reason, solve problems, and make connections within mathematics and across disciplines.
MA 3.1.2 Operations: Students demonstrate the meaning of multiplication with whole numbers.
MA 3.1.2.a Represent multiplication as repeated addition using objects, drawings, words, and symbols (e.g., 3 x 4 = 4 + 4 + 4)
MA 3.1.2.d Use drawings, words and symbols to explain the meaning of multiplication using an array (e.g., an array with 3 rows and 4 columns represents the multiplication sentence 3 x 4 = 12)
MA 3.1.4 Students will estimate and check reasonableness of answers using appropriate strategies and tools.
MA 3.1.4.a Estimate the two-digit product of whole number multiplication and check the reasonableness

Geometric/Measurement – discover area of arrays, construct arrays by first laying out perimeter
MA 3.2 Students will communicate geometric concepts and measurement concepts using multiple representations to reason, solve problems, and make connections within mathematics and across disciplines.
MA 3.2.4. Students will create two-dimensional shapes and three-dimensional objects.
MA 3.2.5 Measurement: Students will apply appropriate procedures and tools to determine measurements using customary and metric units.
MA 3.2.5.a Select and use appropriate tools to measure perimeter of simple two-dimensional shapes (e.g., triangle, square, rectangle)

Algebraic –
MA 3.3 Students will communicate algebraic concepts using multiple representations to reason, solve problems, and make connections within mathematics and across disciplines.
MA 3.3.1 Relationships: Students will represent relationships.
MA 3.3.1.a Identify, describe, and extend numeric and non-numeric patterns
MA 3.3.1.b Identify patterns using words, tables, and graphs
MA 3.3.2 Modeling in context: Students will create and use models to represent mathematical situations.
MA 3.3.2.a Model situations that involve the addition and subtraction of whole numbers using objects, number lines, and symbols
MA 3.3.3 Procedures: Students will identify and apply properties of whole numbers to solve equations involving addition and subtraction.

Data Analysis –
MA 3.4.1 Students will organize, display, compare, and interpret data.

### Assessment –

Journals, Conference, Gallery Walks, & Class Congress:
Describe their solution, describe their thought process, explain reasons for a solution's likelihood of being acceptable, recognize other students misconceptions about their solution and help others understand it, and move their specific solutions to general proofs and prompt further inquiry.
Observe others explanations, ask necessary and sufficient questions to gain enough information to critically evaluate a suggested solution for its completeness, ask for further information as needed, and to suggest further inquires. Critical evaluation -

### Multiplication as represented by Arrays

Students will ….
Count by one
skip count
double
double/halve
unitize
discover perimeter/area
construct solutions for arrays and use arrays for solutions

Activities
Simple Arrays
Discuss how much fruit are in boxes arranged 3x5, 6x4 etc.
Partial Arrays
Discuss how many objects are shown, and construct a method to decide how many aren't
Ex: How many spots are on a wall, how many spots on the opposite wall if it is exactly alike
Unequal Arrays
Display several areas with various parts filled. Find how much is shaded/how much is not?
Ex: Carnival game: balloons popped and yet to be popped, 3 equal arrays to start, pop several rows of balloons, and solve to figure out the remaining balloons

### Distributive Property

Partially covered arrays
Display a picture with several arrays of different sizes (using unit squares)
Use several pieces of furniture to partially cover the arrays
Challenge students to discover how many 'squares' are covered by furniture

Decompose wholes into groups
Write a larger, whole number multiplication problem on the board, ask students to solve without working the whole problem (ex. 18 x 18)
Ask, "What are all the different ways we could break this multiplication up? What are the parts connected to the whole?"
Discuss methods and solutions

Friendly Numbers (extension)
Discuss what numbers are simpler to multiply (50, 100)
Estimate product, then subtract
"Friends, more or less"

Students will ….
Count by one
skip count
double
double/halve
unitize
part/whole relationships
discover perimeter/area
construct solutions to using arrays
mental computation

### Associative & Commutative Property

Arrays
Display an array such as 3 x 6
Ask if anyone notices anything about the rows and columns and the math problem
What is 3 x 6?
What is 6 x 3?
How can this be?
Discuss

Boxes
Investigate how many different ways 24 toys can fit into different sized boxes (toys are the same size, are one unit square)
Prompt students to build different size boxes/outline base of box on graph paper
Encourage students to represent box by writing the base in parentheses followed by the number of layers
Prompt inquiry by encouraging students to turn the boxes so that a side becomes the base
Discuss various methods/solutions

Students will ….
Count by one
skip count
double
double/halve
part/whole relationships
discover perimeter/area
construct solutions to using arrays
mental computation

### TO DO -

Multiplication identity property and discuss the role of zero.
Expand to include the fourth grade.
Creating a 'connecting multiplication to division' outline.