this week in tutoring: starting small, thinking big

This is the first of many posts in my tutor triumphs series, where I expound on the tutoring sessions I engage in each week.

This week, it was all about algebra.

Equations, inequalities, signed numbers, oh my! Such was the beat of Saturday’s sessions, which saw me tutoring two middle school students (an 8th-grade girl and a 6th-grade boy) in two different boroughs! I love that I can challenge my tutees with progressively difficult problems – sending them off to the races to solve multistep linear equations and observing how they approach the problems. One thing I noticed with both of my tutees was a common mistake most students their age make – right numbers, wrong signs!

One of the reasons I believe so many kids struggle with signs is that there are different rules for addition and subtraction versus multiplication and division. Let’s dive in:

Signed Number Rules – Addition and Subtraction

1) For any two numbers, if the signs are the same, we add the numbers and keep the sign.

Examples: (+9)+(+6)=+15; (-10)+(-14)=-24.

Put another way, adding two positive numbers gives us a bigger positive number, and adding two negative numbers gives us a bigger negative number.

2) For any two numbers, if the signs are different, we subtract the numbers and keep the sign of the number with the greater absolute value.

Uh-oh – absolute value? This might be confusing for some, so I often call this the “bigger number” to make things easier for my tutees.

Example: If we have (-36)+(+24) – two numbers with opposite signs – we subtract; 36 minus 24 is 12. However, since |-36| is bigger than |+24|, we keep the negative sign, so the answer is -12.

Now, let’s talk about absolute value. For any number or expression n:

|n| = {+n, n≥0; OR –n, n<0

In English, the absolute value of a number or expression is the same number or expression, but always positive.

Then we have the rules for multiplying or dividing signed numbers:

Signed Number Rules – Multiplication and Division

1) For any two numbers, if the signs are the same, the result is always positive.

Examples: (-9)×(-8)=+72; (+108)÷(+12)=+9

2) For any two numbers, if the signs are different, the result is always negative.

Examples: (+8)×(-11)=-88; (+120)÷(-5)=-24.

Note the differences between these rules and those for adding and subtracting signed numbers. I can understand why signs are a common source of confusion for many! That said, clearing up such confusion is part of my mission!

On Sunday, I began my free weekly Communal Tutoring workshop; though several potential tutees spoke of attending, I met with only one today. That’s OK though – the time spent navigating the murky waters of mathematics, whether with one or many, is always worth it to me. I even picked up a new Professional Tutoring client after the free session ended! All in all, a great weekend. Can’t wait for what next week brings!

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