We know from before that -5 × 3 is -15 so we can substitute that value for -5 × 3 in the left-hand side of the equation. Similar to before, we distribute -5 through both terms.Īgain, the distribution of terms does not change the value of the expression on the left-hand side of the equation, so the result is still 0. We know that this is the same as -5 times 0, so this has a value of 0. Since distributing the 5 across the addition does not change the value of the expression, we know this is still equal to 0.īut this means that 5 × 3 and 5 × -3 are opposite signs, so since 5 × 3 = 15, then 5 × -3 is -15. If we add the numbers inside the parenthesis first, then this is 5 times 0 which is 0, since 3 + -3 = 0.īut what if we distribute 5 through both terms first? Let’s look at a problem that we can do in more than one way, borrowed from the Khan Academy. Mathematical consistency and mathematical properties Prerequisite knowledge: One has to know what these symbols mean, what is meant by finding one number times another, and how negative numbers work in terms of counting down and subtraction. While to some this pattern may seem obvious, when someone is still in the middle of learning this concept, they have less cognitive capacity available to accomplish the task at hand (multiplying numbers together) and accomplish the additional task of looking for patterns in their answers, so this is where someone else prompting them to stop and look for patterns in their work so far will be very useful. One might now notice that the answers are going up by 3 each time as we increase the first number, and so it is reasonable to continue this pattern. Now, we decrease the first number in the pattern by 3 and one has to make some deductions about what the answer should be. The answers to these problems are below but I really do recommend taking the time to solve the problems above on your own first, so you get the sense of how students might think through this set of problems.Īt this stage, many people will notice the answers are 3 smaller each time and the number being multiplied by 3 is one smaller each time, so they continue that pattern to answer the following questions. Look for patterns to make solving the problems easier. Try solving each of these problems, paying attention to the previous set of problems as you do so. There are different possible answers to this question, depending on the standard of proof one needs and the background knowledge one brings to the question.
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