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Counting Dots in Arrays


Alignments to Content Standards: 2.OA.C.4

Task

Which of the following are equal to the number of dots in the picture below? (Choose all that apply.)

Dots_a89d8328578dc271275c0e535563fbda

  1. 3 + 3 + 3
  2. 3 + 4
  3. 4 + 4 + 4
  4. 4 + 4 + 4 + 4
  5. 3 + 3 + 3 + 3

IM Commentary

Students who work on this task will benefit in seeing that given a quantity, there is often more than one way to represent it, which is a precursor to understanding the concept of equivalent expressions. This particular question also lays a foundation for students to understand the commutative property of multiplication in third grade. This task would be much more valuable if included in an appropriate place in an instructional sequence than as an isolated task.

Solution

We can see 3 rows with 4 dots in each row, so (c) 4+4+4 can represent the number of dots in the array. We can also see 4 columns with 3 dots in each column, so (e) 3+3+3+3 can represent the number of dots in the array.

Judy Maury says:

over 4 years

Using arrays for multiplication is a great visualization, but I think using "plates of cookies" is a more specific way of illustrating set theory. For example, in third grade, Core 1A for operations indicates that the multiplication fact for four plates of three cookies would be four groups of three which is 4 times 3 or 4X3. Facts for threes would be one group of three, two groups of three, three groups of three, four groups of three, five groups of thee, etc. Written facts for threes would be1X3, 2X3, 3X3, 4X3, 5X3, etc. However, many of the commercially available multiplication charts and practices show facts for threes as 3X1,3X2, 3X3, 3X4, 3X5, etc. which would be illustrated as three groups of one, three groups of two, three groups of three, three groups of four, three groups of five, etc. I think this is why young students get confused. Young students need to have a consistent vocabulary so that mathematics becomes a language for numbers.