My philosophy is that students need a variety of models and experiences to learn and conceptually understand multiplication. This diagram from helps show how all the facets of understanding our connected.

external image conceptual-understanding-pentagon.png?w=300&h=257

Continuum of Understanding
Students go through a general continuum of understanding for multiplication, division, and other operations (see anchor chart we created in class). (Direct Modeling --> Counting --> Derived Numbers --> Known Facts) The value in this is encouraging metacognition about math strategies, helping students understand that these stages are important for their own understanding and necessary to go through in order to reach that end goal.

multiplication strategies.jpg

Also notable in any new conceptual understanding, new information should presented Concrete --> Verbal -->Abstract/Sybmolic.

Multiplication Construct 1

Multiplication Construct 2

  • Lemonade Stand
  • Loops and Groups (Bridges Math Curriculum, Copyrighted) similar as Circle and Stars by Marilyn Burns - create your own game!
  • Structured Array (see below)
    • Arrays fully visible
    • Arrays partial screen showing group markers and items in each group
    • Arrays partially screened showing only group markers
    • Verbal tasks

Multiplication Construct 3

  • 100 Dot Structured Array - 100 Dot Array and Screen - Use this and a screen to build any size array that you want. Fantastic tool for multiplicative thinking.
  • 100 Dot Array, Student Version (print and make an L-shaped screen) - Using the structured array, you can make any desired array. You can then use the screens to gradual screen the number in each group and group markers. Always watch for how students are counting: are they perceptual replacing the dots? Are they stress or skip-counting? Are they using something they know and building on?

Activities and Games to Support Verbal Sequences of Multiples (provide quantitative support in form of dot cards, dice patterns, etc. to begin)

  • Class Count Arounds (verbal, BUZZ, Pop, etc.) -Get creative. Remember to count both forwards and backwards.
  • Treasure Hunt (see AVMR Book for instructions, Course 2) - This is a great independent math activity. You can use the same cards to play Garbage.
  • Disappearing Numerals(red MR Book) - Write the multiples of any given number. Read the numberals forward and backward. Then, cross/cover one numeral. Read forward and backwards again. Continue process until all numerals are covered. Great independent sequencing practicing.
  • Multiple Card Games - Use for Quick Draw, Trio for Multiples (AVMR/Green Book)
  • Numeral Tracks - Practice counting forward and backwards by any numeral. You can do Disappearing Numerals with this as well.

SMART Board for Numeral Tracks and Dot Strips


Multiplication Construct 4

These students would benefit from relational tasks now. Use the structured array and question to really stretch student thinking!

Using Multiplicative Language Ideas in Class Instruction
  • Are ten 8s the same as eight 10s. How do you know?
  • Fluently double a number –doubling up to 100
  • Fluently half a number –halving numbers up to 100
  • How are 10 facts related to 5 facts?
  • Using arrays to show 8 x 3 = 6 x 4
  • How can I use 9 x 8 to help me solve 18 x 4? (Construct 4)
  • What non-count by strategies an I use to solve 8 x 6? (Use array model and students can find and partition out 5s, etc.
  • How can I use 2 x 8 = 16 to solve 4 x 8?
  • If 14 x 3 = 42, what is 15 x 3 =?
  • Commutative Property – “flip flops”
  • Inverse Property
  • Multiplication Property of One
  • Multiplication Property of Zero

Multiplication Construct 5
Multi-digit Work and continued practice for fluency.