Among the ‘rules’ for working with negative numbers, the most counter intuitive is “negative times a negative is a positive”. It is easily forgotten especially if it was learned by rote. It is also not an easy ‘rule’ to make sense of so it needs to be learned with conceptual understanding. Here’s my proposed lesson for teaching multiplication of integers. This lesson takes from the lesson Subtracting integers using tables- Part 1 and Algebraic thinking and subtracting integers – Part 2. Note that this lesson like the rest of the lessons in this blog is not just about students learning the math but more about them engaging in mathematical thinking processes such as searching for patterns, making generalization, reasoning, making connections, etc.
Set the task
Fill up this table by multiplying the numbers in the first column to the number in the first row. Start filling up the rows or columns you think would be easier to do.
For discussion purposes divide the table into 4 quadrants. The top right quadrant is Quadrant 1, top left is Quadrant 2, bottom left is Quadrant 3, and bottom right is Quadrant 4. This is also one way of leading the students to consider filling-up the quadrants according to their number label.
Explore, Observe, Explain why
Students are more likely to fill-up Quadrant 1 because the numbers to be multiplied are both positive. The next quadrant they are more likely to fill-up is Quadrant 2 or 4. You may want to give the following questions to scaffold their thinking: What do you observe about the row of numbers in Quadrant 1? How can it help you fill up quadrant 2? Do the numbers you put in Quadrant 2 make sense? What does 3 x -2 mean? What about in Quadrant 4?
From Quadrant 2 students are more likely to fill up Quadrant 3 or Quadrant 4 by invoking the pattern. Questions for discussion: Do the numbers in Quadrant 4 make sense? What does -3 x 2 mean? This is one way of making the students be aware that commutativity holds in the set of integers. The problematic part are the numbers in Quadrant 3. None of the previous arguments are useful to justify why negative times negative is positive except by following the patterns. But this explanation will be enough for most students. You can also use the explanation below.
Revisit the rule when teaching another topic
We know that 8 x 8 = 64. This means that (10-2)(10-2)=64. By distributive property, (10-2)(10-2)= 100+-2(10) + – 2(10)+ ____ = 64. Previously students learned that -2(10)= 20. Hence, 100 + -40+___= 64. What should go in the blank must be 4. So (-2)(-2) = 4. This proof was first actually proposed by Maestro Dardi of Pisa in year 1334. In explaining this to students I suggest rewriting (10-2)(10-2) to (10+-2)(10+-2) to reinforce the distinction between the dash sign as minus and as symbol denoting ‘negative’.
Girolamo Cardano sometime in 1545 proposed a geometric interpretation of this operation. He argued that (10-2)(10-2) can be interpreted as cutting off 2 strips of 2 x 10 rectangles from the two sides of the 10 x 10 square. Cutting the rectangles like these meant cutting the 2 x 2 square twice so you need to return back the other square. The figure below shows this. This proof by Cardano is usually used to teach the identity square of a difference 
Reference to history is from the paper Historical objection to the number line by Albrecht Heefer.
Hi Erlina,
You shouldn’t redo your drawing. After cutting the first strip of 2×10 rectangles from one side there isn’t a full strip of 2×10 rectangles on the other side. So we should borrow the 2×2 square from the 8×8 square (64 – 4 = 60). The red line shows the “crooked” strip. Thus 60 isn’t the product of 8×8, it is the difference 64 – 4.
Thank you. That drawing is already the revised version. My first drawing Jeff G was referring to was really wrong.
My favorite way to demonstrate this is with manipulatives. When I worked with the Mortensen Math program integers were introduces as early as 5 years old and the wording that I will use below was actually figured out by a second grader who was playing around with the concept.
Imagine using blocks similar to base 10 blocks. (the Mortensen and MathUSee blocks have color coded blocks from 1 -10 as well as hundred squares with one side marked in units and the opposite side hollow.
Introducing the blocks we say that holding the unit side up means we HAVE that # of units. Turning it to the HOLLOW side means we OWE that many units.
Even five year-olds can understand that having and owing the same amount means that you have ZERO. Visually, it is shown by placing the positive and negative blocks on top of (or aligned with lengths adjacent). Students quickly see that they can build zero with any combos that are = lengths.
Jerry Mortensen also teaches students that Numbers have FIRST NAMES that tell HOW MANY and LAST NAMES that tell WHAT KIND. This is valuable for Algebra as well and helps students to avoid the common mistakes of combining unlike kinds.
Multiplication is introduced with the rectangle model. Using both of these ideas, we can verbalize as follows:
[I will use _______ to represent a positive 10-bar and ——— for a negative 10-bar]
Beginning with a clear space with nothing on it representing ZERO, we add or take away whatever the problem dictates.
(2)(10) means “I HAVE 2 tens” or +20 (so the student can place 2 tens on their space)
_______
_______
(2)(-10) means “I OWE 2 tens” equivalent to -20 (the student takes 2 negative 10 bars)
————
————
(-2)(10) means I OWE or must TAKE AWAY 2 ten bars. But how can we do that when we are starting with an empty space?
Students who have been working with the blocks soon realize that they can BUILD ZERO any way that might be useful. In this case we need to be able to take away 2 ten bars, so it makes sense to build ZERO with 2 ten bars and 2 negative ten bars.
________ (+10)
________ (+10)
————– (-10)
————– (-10)
Now it is possible to remove the 2 positive ten bars as the problem requires and the remaining amount is -20
Similarly, for (-2)(-10), we must build zero as in the previous problem, but this time we remove the 2 negative ten bars and the remaining amount is the 2 positive tens = +20.
This may sound complicated, but if you try it and allow students to have enough practice with it it becomes intuitive.
Related to this topic, I would love to see all students start working with integers in the early grades so that when they get to pre-algebra they can concentrate on the new material without getting confused, panicked and bogged down by problems with negatives for the months it takes them to get comfortable with them (if they ever do).
This is great. Thanks for sharing.
I’m going to try on the Cardano Geometric interpretation with my 6th graders. I get it verbally – but the picture has me boggled. It appears as though 8 x 8 = 60 … with the small square being 4, so that equals 64, except 8 x 8 ≠ 60. Yikes. Perhaps that part of the illustration should be put in a cloud – to show it illustrates the story, but is not the usual grid model for multiplication. Still, the lesson appears doomed.
On the other hand, the main lesson – following the sequence on the multiplication table is spot on, and many of my students will be able to follow the distributive property application.
I continue to be impressed – even delighted – with this blog.
you are right. it’s 64. I should redo my drawing:-). thanks
Students will find this lesson difficult if the lesson does not introduce addition and subtraction this way: https://math4teaching.com/subtracting-integers-using-tables/. Also, this is not intended for Year 6 but for Year 7.
You may want to check on the lessons about integers in this blog.
Hi Erllina,
You shouldn’t redo your drawing. After cutting the first strip of 2×10 rectangles from one side there isn’t a full strip of 2×10 rectangles on the other side. So we should borrow the 2×2 square from the 8×8 square (64 – 4 = 60). The red line shows the “crooked” strip. Thus 60 isn’t the product of 8×8, it is the difference 64 – 4.