Making Sense of Two-Digit Multiplication and Much More!

Making Sense of Two-digit Multiplication and Beyond

            24

      x    35

Take a moment to think about how you would perform the computation. How you probably learned to perform this computation involved using an algorithm driven by speed and accuracy. Most of us never were asked by our teachers or considered why the algorithm worked. We simply computed. Our goals are different now. If we want students to perform the computation with speed and accuracy we can simply hand them a calculator. Our desire now is for something more from our students than performing steps to a memorized algorithm.  We want them to understand the purpose behind the steps and the meaning associated with the steps of the algorithm. We want them to understand the structure of multiplication. We also want students to understand the relationship between place value and the computations performed to complete a two-digit (or more) by two-digit multiplication algorithm. The goal is to make a clear connection between the computations performed with a visual, concrete representation and the abstract representation. Base-Ten Blocks are an effective visual, concrete representation for modeling a two-digit multiplication problem such as 35 * 24. Making a clear link to place value ideas, the numeral ‘35’ represents 3 tens and 5 ones and the numeral 24 represents 2 tens and 4 ones. These two numerals are represented by the outer boundaries shown in red. The computation shown on the right is the Partial Product Algorithm whose results are also shown in the Base-Ten Block representation.

The top Partial Product Algorithm computations are written in terms of the units shown in the four quadrants of the Base 10 representation {20 ones, 10 tens, 12 tens, and 6 hundreds}. Three of the four computations can be re-expressed (i.e. 20 ones is also 2 tens, 10 tens is also 1 hundred, and 12 tens is also 1 hundred and 2 tens) yielding an answer of 8 hundreds and 4 tens (840). Interestingly, when you write the numerals representing each of these partial products as shown in the bottom Partial Product Algorithm (20, 100, 120, 600) both interpretations are visible. For example, the numeral ‘20’ can be interpreted as either 20 ones or 2 tens and the numeral '120' can be interpreted as either 1 hundred, 2 tens, or 12 tens.

A less concrete, visual model than Base 10 Blocks is an area model of multiplication. Each numeral is separated by the place value of each of the numbers. The area model representing 35 * 24 is shown with 35 as 30 (3 tens) and 5 (5 ones) and 24 as 20 (2 tens) and 4 (4 ones).

One of the key points of making sense of multiplication through visual representations is the reasoning and sense-making potential these representations afford. Although not an exhaustive list of the potential reasoning and sense-making possibilities the three examples shared exemplify the potential of these models to help students prove conjectures generated from analyzing number pattern relationships, to explain why a mental arithmetic strategy works and to make sense of algebraic relationships.

Proving conjectures generated from analyzing number pattern relationships

Suppose that you introduced the following pattern to students and asked them to reason about how one could use the given information to determine the value of 85².

15² = 225

25² = 625

35² = 1225

45² = 2025

A typical response from students is that 85² is 7225 because you take the tens digit {8} and multiply it by the next consecutive tens digit {8*9} to find the last two digits and drop 25 down to represent the first two digits {7225}. Reasonable follow-up questions are 1) Does the algorithm always work when the units digit is a 5? 2) Are there other multiplication facts in which the algorithm will work?, and 3) Why does the algorithm work? Understanding the structure of the multiplication in terms of visual representations might enhance students’ ability to reason about each of these questions. Using an area model one of the instances in the pattern, 25², is shown.

      

Recall that the student’s algorithm involved multiplying the tens digit {2} and multiplying by the next consecutive tens digit {2 * 3 = 6} and placing this value to determine the number of hundreds {6__ ___}. It is important to note that mathematically the student was actually multiplying 20 * 30 or 20 groups of 30, or 2 tens and 3 tens.

 Moving the rectangle representing 20 groups of 5 to the right of the figure it can be seen that the two rectangles together represent 20 groups of 10 which means there will be a total of 20 groups of 30 {600} and 5 groups 5 {25} or 625.

Modeling multiplication using an area model provides the means for students to reason and make sense of why not only this particular student’s algorithm works but also why other potential algorithms generated by students might work. The area model also provides the means to reason about why the last two digits are always a 2 and a 5 and why problems such as 23 * 27 {621} can be computed using the student’s algorithm but other problems such as squares not ending in 5 (e.g. 27²) or those in which the units digits do not add up to ten {e.g. 23 * 26} will not work.

There is also the potential to make a connection between the concrete representation shown with the area model and the abstract representation that shows in general why the student’s algorithm works. The abstract representation of the length of each side of any square can be represented by the expression 10n + 5 in which n represents the number of tens. For example, 25² can be written as 10(2) + 5 in which the 2 represents the number of tens. Squaring the expression 10n + 5 results in the expression (10n + 5)² = 100n² + 100n + 25 = 100(n(n+1)) + 25. The first term 100(n(n+1)) yields the number of hundreds and is analogous to the student’s step of taking the numeral representing the number of tens and multiplying it by the numeral representing the next consecutive tens digit (e.g. 100(2(2 +1) = 100(2 * 3)) = 600). The abstract representation 100(n(n+1)) + 25 also reveals why the last two digits are a 2 and a 5 since the first term reveals the number of hundreds and the last term reveals the number of tens and ones (e.g. 100(2*3) + 25 = 600 + 25 = 625).

Making sense of why a mental arithmetic strategy works

A mental arithmetic strategy for computing a multiplication problem such as 18 * 22 is to square the number that is in the middle of the two numbers {20² = 400}. Subtract that number {400} from the difference each number is from the middle number squared {2²}. In other words, or symbols, 18 * 22 can be computed by taking 20² and subtracting 2² {400 – 4 = 396}. Check it out! Pretty cool! Try it for 33 * 27! Remember in an earlier problem it was discussed that 25² = 625. We can apply that knowledge and the mental arithmetic strategy just discussed to also find 24 * 26 or 23 * 27. The big question is, why does this mental arithmetic strategy work? As before, refer to an area model for 18 * 22, but this time the area model is represented using a Cartesian plane.

The model is representing 18 groups of 22 or 18 * 22. Consider cutting the last two columns (18 groups of 2) and moving the piece to the location shown to make a larger square with a smaller square (shown in black) missing. The area of the larger square is 400 {20 * 20} and the area of the smaller square is 4 {2 * 2}. This results in an area of 396. 

This mental arithmetic strategy can be connected to an important algebraic idea, difference of squares. The generalized version of difference of squares is (x + y)(x – y) = x² – y².  Now think about this idea in terms of computing 18 * 22. We can represent 18 * 22 equivalently as (20 – 2)(20 + 2) = 20² – 2² = 400 – 4 = 396. Once again making a connection between what we are doing arithmetically with an important algebraic ideas.

Making sense of an algebraic relationship 

A mistake that many algebra students make in expanding the expression (x + y)² is to square each term resulting in the expression x² + y².  Using an area model to represent this action not only exemplifies the meaning of “squaring” but also why the expression  (x + y)² is not equivalent to x² + y² but is equivalent to x² + 2xy + y².

Findings from brain research suggests that vision is the primary way that most of us receive stimuli and that analogous thinking is a strong learning mechanism. (Almarode, 2016, personal conversation). It makes sense then that teachers can significantly impact student learning by providing opportunities for students to reason about important mathematical ideas, such as multiplication, using visual representations. It also makes sense that student learning can be greatly impacted by providing them opportunities to reason about the relationship between what he/she “sees” in the visual, concrete representations of those ideas and what he/she “sees” in the abstract symbols, as well as making connections between these two representational forms.