Division, Multiplication, & Algebra Discovered by Playing with Blocks

Purpose: students will demonstrate multiplication, division, and algebra by arranging blocks, students will show a relationship between algebra and geometry with blocks.

Background: these lessons were proposed for a 1st-2nd grade class (too late in the year to be tried)

Reformatted: January 2010


Division Demonstrated with Blocks

[1] Give students a set of blocks. Have them make rectangles using as many blocks as possible. Have them describe their arrangement of blocks by the dimensions. How many blocks long are each side?


This arrangement is 3 blocks by 5 blocks with 2 left over, totaling 17 blocks,

or 3x5 remainder 2


This is 13 blocks arranged as 3 blocks by 4 blocks with 1 left over,

or 3x4 remainder 1.



  • by & x : these symbolize multiply, students will see them as references to geometry also
  • left over & remainder : these words from division will mean to students how many blocks did not fit into the rectangles.

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Factoring and Winning Demonstrated by Playing with Blocks

Purposes: to demonstrate factoring concepts, to demonstrate that more is not always a winning strategy

  • Have students work in pairs.
  • Tell students that they will arrange their blocks in rectangles. They must use all their blocks in each rectangle. The student who creates the most rectangles "wins."
  • From the following set of choices the first student may pick which number of blocks he wants to use:

    6 or 7

    9 or 10

    12 or 13

    15 or 16

    24 or 25

    The other student gets the other number.
  • In each case, the student with the smaller number of blocks will win by creating more rectangles that use all of his blocks.
  • Have students discuss what they learned about winning strategies for this game.

Comments: The number of rectangles that you may make depends on how many ways that each number can be factored. For example, in the first set: 6 = 1x6, 6=6x1, 6=2x3, 6=3x2 and 7=1x7, 7=7x1. Students should discuss whether they think that 2x3 is the same as, or different from 3x2.

Algebra Demonstrated by Playing with Blocks

Purpose: to represent some geometrical ideas that occur in algebra

  • Have students arrange blocks into a rectangle
  • Have them group the blocks to describe the rectangle as being made of parts
  • Have them try again to show a similar arrangement with a different size rectangle

blocks arranged

Here we see a square represented as

  • 4 corners, 4 sides made of 3 blocks each, and a center 3x3 square of 9 blocks
  • Corners + sides + center = total
  • 4 + 4x3 + 3x3 = 25

This anticipates the form:

N2 = A2 + 4AB+ 4B2

blocks arranged 5x5

Here we see a square represented as

  • a series of blocks forming a corner around a smaller square
  • 1+3+5+7+9 = 25

This anticipates the form:

N2 = Sum (odds)


What groupings can students think of to represent parts of rectangles? How would they show them?


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