There space different possible answers to this question, depending on the standard of evidence one needs and also the background expertise one bring to the question.

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Mathematical consistency and also patterns

Try resolving each of these problems, paying attention to the previous collection of problems as you execute so. Look for trends to make solving the problems easier.

3 × 3 = ? 3 × 2 = ? 3 × 1 = ? 3 × 0 = ? 3 × -1 = ? 3 × -2 = ? 3 × -3 = ? 2 × -3 = ? 1 × -3 = ? 0 × -3 = ? -1 × -3 = ? -2 × -3 = ? -3 × -3 = ?

The answers to these problems are below but i really execute recommend taking the moment to resolve the problems above on your own first, so you acquire the feeling of how students can think through this collection of problems.

3 × 3 = 9 3 × 2 = 6 3 × 1 = 3 3 × 0 = 0

At this stage, many human being will notification the answers are 3 smaller sized each time and the number gift multiplied by 3 is one smaller each time, therefore they continue that sample to price the adhering to questions.

3 × -1 = -3 3 × -2 = -6 3 × -3 = -9

Now, us decrease the an initial number in the sample by 3 and also one has to make part deductions around what the answer need to be.

2 × -3 = -6 1 × -3 = -3 0 × -3 = 0

One might now an alert that the answers space going up by 3 every time as we rise the an initial number, and so that is reasonable to continue this pattern.

-1 × -3 = 3 -2 × -3 = 6 -3 × -3 = 9

While to some this pattern might seem obvious, as soon as someone is quiet in the middle of discovering this concept, they have actually less cognitive capacity obtainable to achieve the task at hand (multiplying numbers together) and achieve the extr task of looking for patterns in their answers, so this is wherein someone else prompting them to stop and look for fads in their work so much will be very useful.

Prerequisite knowledge: One needs to know what these symbols mean, what is meant by recognize one number times another, and also how an adverse numbers job-related in terms of counting down and subtraction.

Mathematical consistency and mathematical properties

Let’s look at a trouble that we can do in much more than one way, obtained from the cannes Academy.

5 × (3 + -3) = ?

If we include the numbers inside the parenthesis first, climate this is 5 time 0 which is 0, because 3 + -3 = 0.

5 × (3 + -3) = 0

But what if us distribute 5 with both state first?

5 × 3 + 5 × -3 = ?

Since distributing the 5 throughout the addition does not adjust the value of the expression, we understand this is still equal to 0.

5 × 3 + 5 × -3 = 0

But this method that 5 × 3 and also 5 × -3 are opposite signs, so due to the fact that 5 × 3 = 15, climate 5 × -3 is -15. Let’s look at one more example.

-5 × (3 + -3) = ?

We know that this is the same as -5 time 0, therefore this has actually a worth of 0.

-5 × (3 + -3) = 0

Similar come before, us distribute -5 with both terms.

-5 × 3 + -5 × -3 = ?

Again, the circulation of state does not change the worth of the expression top top the left-hand side of the equation, therefore the result is tho 0.

-5 × 3 + -5 × -3 = 0

We recognize from prior to that -5 × 3 is -15 so we have the right to substitute that worth for -5 × 3 in the left-hand side of the equation.

-15 + -5 × -3 = 0

Therefore -15 and also -5 × -3 space opposites because they add to 0, so -5 × -3 should be positive.

Nothing in what us did for the two examples above is details to the value of 5 × 3, therefore we deserve to repeat this dispute for every various other multiplication truth we want to derive, so this two principles can be generalized.

Prerequisite knowledge: One has to know what these signs mean, what is expected by detect one number times another, just how the distributive residential property works, and also how negative numbers can be identified as the opposites of optimistic numbers.

Representation on a number line

Imagine we stand for multiplication as jumps top top a number line.

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3 time 3 top top the number line

For 3 × 3, we draw 3 teams of 3 relocating to the right. Both the number of groups and the direction that each team are come the right.

But what about 3 × -3? currently we have actually 3 groups of the number still, however the number is negative.

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3 time -3 ~ above the number line

If we uncover -3 × 3, the size and also direction the the number us multiply space the same, yet now we space finding -3 teams of that number. One method to think the this is come think of taking 3 teams of the number away. Another is to think of -3 time a number as being a have fun of 3 times the same number.

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-3 times 3 on the number line

So -3 × -3 is, therefore, a enjoy of 3 × -3 across the number line.

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-3 times -3 top top the number line

In one sense though, this visual argument is simply mathematical consistency stood for using a number line. If multiplication by a negative is a reflection across 0 ~ above the number line, and also we think of negative numbers as being reflections across 0 of the number line, then multiplication of a an adverse number times a an adverse number is a double-reflection.

Context

Karen Lew has this analogy.

Multiplying through a negative is repetitive subtraction. Once we main point a negative number times a negative number, us are getting less negative.

This analogy between multiplication and enhancement and subtraction help students nicely attach the 2 concepts.

Joseph Rourke common this context.

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A gambler loses $10 every day. Exactly how much an ext money walk they have 5 job ago? Here, the loss every day is one an adverse and walking backwards gradually is another.