Hi Folks,
As mentioned in my previous post, we have been very busy leading professional learning with our district. Recently, we focused on the Progression of Multiplication. Here's a snippet and a link if you're interested in reading further to better understand that pesky "Standard Algorithm" for multiplication those of us in North America would have likely learned when in school.
Have a great final week before Christmas!
Kyle
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Did you know that the words “array” and “area model” appear in the Grade 1-8 Math Curriculum a combined 22 times?
Not only do arrays and area models help to support the development of proportional reasoning when we formally introduce multiplication in primary, but they also help us understand how to develop strategies that lead to building number flexibility and the automaticity of math facts.
Arrays and area models should be used as a tool and representation for many big ideas in mathematics including, but not limited to:
- Multiplication
- Distributive Property with Whole Numbers
- Finding Area with Whole Number Dimensions
- Perfect Squares & Square Roots
- Multiplying a Binomial by Monomial
- Multiplying a Binomial by Binomial (aka FOIL)
- Factoring (Common, Simple/Complex Trinomials)
- Completing the Square
For many, the term “array” is not a familiar one. Luckily, the definition is fairly straightforward:
In mathematics, an array is a group of objects ordered in rows and columns.
Seems pretty simple, but they are extremely powerful in building a deep conceptual understanding as students learn multiplication and begin applying that knowledge to more abstract ideas requiring fluency with procedures.
Where Multiplication Begins In The Ontario Curriculum
In grade 3, students are asked to:
- relate multiplication of one-digit numbers and division by one-digit divisors to real life situations, using a variety of tools and strategies (e.g., place objects in equal groups, use arrays, write repeated addition or subtraction sentences);
- multiply to 7 x 7 and divide to 49 ÷ 7, using a variety of mental strategies (e.g., doubles, doubles plus another set, skip counting);
- identify, through investigation, the properties of zero and one in multiplication (i.e., any number multiplied by zero equals zero; any number multiplied by 1 equals the original number) (Sample problem: Use tiles to create arrays that represent 3 x 3, 3 x 2, 3 x 1, and 3 x 0. Explain what you think will happen when you multiply any number by 1, and when you multiply any number by 0.);
Consider 3 x 2, or “3 groups of 2”:
This may seem like a simple and maybe even unnecessary representation, but having a visual that multiplying two numbers will always yield a rectangular array is an important concept that not only shows the interconnectedness of the Number Sense and Numeration strand to Measurement, but also implicitly provides students with insight into why more abstract mathematics works later on in grade 9 and 10.
In grade 3, it is reasonable to believe that students are still working on counting and quantity including unitizing in order to skip count more fluently. By working with arrays, we can allow students to continue developing their ability to unitize and work with composing and decomposing numbers.
Arrays and Perfect Squares
Not only is it beneficial for students to understand that multiplying two quantities will yield an array covering a rectangular area, but it is also useful for students to discover without explicitly stating that when we make an array where the number of groups and number of items in each group are equal, the array is now a special rectangle; a square!
While we don’t specifically discuss perfect squares until intermediate when students represent perfect squares and square roots using a variety of tools in grade 7 and estimate/verify the positive square roots of whole numbers in grade 8, I think it would be much easier for students to identify a perfect square and estimate the square root of a number if they have four years of concrete and visual work with perfect squares.
How Multiplication With Arrays Lead to Area
As we have witnessed in the previous examples, with every array a student builds, we are implicitly providing them with a window into the measurement strand with opportunities to think about perimeter and area. Not only can we easily make connections between an array and the area of a rectangle, but we can also better serve specific expectations involving estimation such as this grade 3 Measurement expectation:
- estimate, measure (i.e., using centimetre grid paper, arrays), and record area.
With the use of arrays in the Number Sense and Numeration strand, I can better serve my Measurement strand with problems that force students to estimate using visuals and then improve their predictions using concrete manipulatives.
Something worth noting is the wording of the specific expectation in the grade 3 curriculum related to multiplication and division which states students are to multiply to 7 x 7 and divide to 49 ÷ 7, using a variety of mental strategies without any reference to memorization or automaticity. While I agree that knowing multiplication tables for intermediate and senior level math courses is a huge asset, I think working with multiplication early and often with concrete manipulatives is a great way to get there over other more traditional and/or rote strategies.
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