Pat Larsen
TTL June 2002
Understanding By Design Unit
3rd Grade Multiplication

Students will develop and use number sense to investigate the characteristics of numbers in a variety of forms and modes of operation.
Indicator 2: Apply number operations with real numbers and other number systems.

3 - 5 Benchmarks:
a. model the operations of addition, subtraction, multiplication, and division on rational numbers

Third Grade
2. recall multiplication and division facts through the nines.

Stage 1:
Identify Desired Results

Stage 2:
Determine Acceptable Evidence

Stage 3:
Plan Learning Experiences and Instruction

Enduring Understandings

Performance Tasks


Sequence of Learning Experiences and Instruction

Students will

*  Multiplication is a critical step in their math learning stairs.

* Success in all future math learning is contingent upon acquisition of this skill.


Essential Questions

1)    Why is multiplication necessary?

2)    How is addition related to multiplication?

3)    How is multiplication related to division?

4)    What is the Commutative Property of Multiplication and why is it important?

5)    What essential vocabulary words are necessary to communicate multiplication mathematically?

6)    Why is it important to use a strategic approach for acquisition of the basic multiplication facts?

7)  Would you rather memorize 81 facts or 15 facts?

Knowledge and Skills

Students will develop and use number sense to investigate characteristics of numbers in a variety of forms and models of operation.

Students will model the operations of addition and multiplication on rational numbers. 

Through a variety of strategies, the student will recall multiplication facts through 9.

Enrichment activities will include acquisition of basic facts through 12.

Available Resources

Math Learning Stairs
Multiplication Table

Modified Multiplication Table

Multiplication Unit Checklist
PM Progress Graph
Sample Basic Facts Acquisition Test Probe
Sample Basic Facts Fluency Test Probe
Sample Learning Sheet

Each student will be able to:

* demonstrate concrete understanding of the concept of multiplication by: (a) building/drawing arrays representing problems using groups and objects in a group with concrete materials, drawing, and creating color coded tables in Microsoft Word.  

Tiered Activities to Address Individual Learning Differences: 

Math Stations

Math Station One:

(Less Advanced Learners)

Students build multiplication problems concretely:

(a) Make arrays using paper plates to represent the groups and beans to represent the objects in a group.

(b) Make arrays using baking cups to represent the groups and elbow macaroni to represent the objects in a group.

(c) Make arrays using string to represent the groups and paper clips to represent the objects in a group.

Math Station Two:

(For students performing at or near grade-level)

Students build multiplication problems semi-abstractly:

(a) Draw arrays depicting given multiplication problems using colored utensils (pencils, markers, etc.)

(b) Complete the Multiplication Learning Sheets that have blank boxes in groups where students have to write the multiplication problem that it represents, then color in the squares to depict the given array.

Math Station Three:

(More Advanced Learners)

Students build multiplication problems abstractly:

(a) Students locate arrays around the classroom (light fixtures, windows, work station icons, etc.). Students record their observations in a log to share with their peers.

(b) Using Microsoft Word, students create color-coded tables to represent given multiplication problems.  They print these tables, then compile their tables into an array book.

*  explain the “Math Learning Stairs”: (a) how each skill is sequential, and (b) how addition, subtraction, multiplication and division are related: inverse operations and multiplication is repeated addition (sums) and division is repeated differences.

*  write related addition and multiplication problems.

* analyze a multiplication table and apply the (a) Commutative Property of Multiplication, (b) the 0, 1, 2, 5, and 9 times rules.

* define factors, products, multiples, common factors and least common factors.

  * recite Miss Larsen’s strategies or other self-generated strategies to assist with recall and memorization of the 15  facts.

* acquire basic facts through the use of Miss Larsen’s Strategic Multiplication flashcards or other commercial  flash- cards. 

*  progress through the multiplication unit Learning Sheets.

* progress through leveled Precision Math tests.

* utilize Internet resources for learning activities. 

Quizzes, Tests, Prompts

Students will be evaluated in the following ways:

Academic Prompts:

     Students will:

(1)  show they can build arrays.

(2)  Explain the Commutative Property of Multiplication.

(3)  Explain why multiplication is necessary.

Tests & Quizzes:

(1)   Vocabulary Test:

common factor
least common factor
Commutative Property of
0 times rule
1 times rule
2 times rule
5 times rule
Magic Nines

(2)   Precision Math Tests to assess acquisition of basic facts through 9.

(3)   Precision Math Tests to assess fluency in basic facts through 9.

Unprompted Evidence

Observations during discussions.

Participation in class activities.

Progression through Precision Math tests.

Progression through Multiplication Unit Learning Sheets.


Self Assessment

Students will ask themselves:

“Am I progressing through the Precision Math Test unit at a reasonable rate?”

“Am I progressing through the Multiplication Unit Learning Sheets at a reasonable rate?” 

“Am I getting faster and faster (becoming more fluent) in the recall of my basic multiplication facts?”

“Have I practiced enough times that this new learning has become a part of my lifelong learning?  Will I be able to recall the facts quickly and efficiently forever?” 

Students will need to know:
(Declarative Knowledge)

  * key terms for multiplication.

  * basic addition facts.

  * what symbols are used to communicate multiplication mathematically?

  * how to create a table in Microsoft Word.

  * how addition is related to multiplication.

Students will need to be able to:
(Procedural Knowledge)

* recall multiplication facts through the 9s.

What teaching and learning experiences will equip students to demonstrate the target understandings? Note: This is “W.H.E.R.E.” from the students’ perspective.

Where are we going?

Multiplication is a critical step in our Math Learning Stairs.


How will the teacher hook us?  

(1) Through team teaching, the special and regular education teachers will share stories with the students of their personal experiences trying to learn their multiplication facts through the 9s while they were in 3rd Grade: stories may include examples of:

No Cool Materials
provided by the school

-         The only material provided to us students was a single copy of a multiplication table.

-         learning facts consisted of rote drill and practice…outside of school.  Assessment consisted of standing in front of the class reciting tables.

-        teachers often used fear, humiliation or embarrassment as motivating factors.

 (2) The teachers will show how to manipulate the multiplication table so students do not need to memorize all of the facts contained within the table if they remember the 6 Rules they will learn in the next lesson.

(3) Students will be given a set of fun multiplication materials consisting of strategy sheets, flashcards, learning sheets, and Precision Math folders.  An explanation of the materials will follow in subsequent lessons.

(4) All students will have an opportunity to earn their, “Multiplication Gold License.”


What events will the students experience to make the ideas and issues real?

(1)        Students will experience “9 Weeks in the Life of a 5th Grade Math Student.”

Coming soon on video!

(currently presented from the overhead projector.)

(2)        Learning Activities and Instruction:


  • manipulation of the multiplication ta ble.
  • introduction and practice of 6 key rules.
  • introduction and practice with key vocabulary terms.
  • use of Miss Larsen’s unit flashcards and strategy sheets.
  • use of PM tests.
  • use of learning sheets.
  • how to progress through instruction.
  • how to pace self
    through independent
    learning activities.

How will the students reflect, rehearse, revise and refine their work based upon their progress?

Reflect on the strategies in terms of their effectiveness.  If not effective: students will:

(1)             Rehearse the strategies.

(2)             Revise the strategies for effectiveness; create strategies of their own.

(3)             Refine their self-generated strategies.


How will the students guide themselves through self-evaluation?

(1)             Precision Math Progress Graph

(2)          Self monitor weekly progress.

Web Resources
go to Thematic Units/Mathematics/Multiplication and Division

Math Dictionary for Kids


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