# Discovering Negative Numbers

Today I had a three hour Math Boot Camp where I reviewed basics and exposed new to our school students to math topics that our current students have already been taught.  I do not expect them to grasp these topics in one day.  But, it is great exposure for them and I get to see where they are mathematically so I can more accurately plan for these students this fall.  Again, this is exposure, not a full blown lesson.

One of the main topics today was the discussion of negative numbers.  Kids have such a hard time with this because they are taught so many “rules”.  Man, I hate rules kids have to memorize for math.  Today several kids even told me some of the rules they had learned, but didn’t know what they meant and could not apply them.  I actually prefer this to kids that just apply these rules because I can wipe the slate clean and start fresh with them.

I started with the idea of a negative number, showed them patterns so they could discover the answer, and then moved to a number line.  I also frequently referred back to the “concept” of a negative number often and pull finances in at every opportunity.  I mean seriously, what kid doesn’t understand simple money spending?  (As in, I want this awesome xbox game that cost \$50, but I only have \$30.  My Mom buys it for me and now I owe her \$20, thus -20).

To introduce them to the idea of negative numbers, I asked them to represent (on a whiteboard) numbers for the scenarios that I gave them.  You received \$45, 15 degrees below zero, you owe 27 dollars and 62 cents (decimals too), the depth of the titanic and the height of skydivers.  I told them they have been using negative numbers for a long time and didn’t even know it.

Then I moved on to patterns. The patterns I used were very simple and kids even laughed when we went through them.  This stuff is TOO easy and they loved that.  They got to see negative numbers come out and they were excited that they discovered the pattern.

• 4 – 1 =
• 4 – 2 =
• 4 – 3 =
• 4 – 4 =
• 4 – 5 =
• 4 – 6 =

After this I modeled subtraction on the number line, always bringing in parallels to money.  Then, I moved on to more patterns.

• 4 – 2 =
• 4 – 1 =
• 4 – 0 =
• 4 – (-1) =
• 4 – (-2) =
• 4 – (-3)=

This was a brief introduction today.  I am going to add more discovery and the need for working with numbers like -4 – 5  =  for an in-class lesson.  I didn’t hit them with multiplication and division of negative numbers as I didn’t want to confuse them.

After all of our pattern work today, one kid even said, “My IQ just doubled today.”  Did I mention how much I LOVE 6th graders?

I hate saying, “Subtraction of a negative”.  Or “minus” a negative for 4 – (-3).  What does everyone else say?  And how are you all teaching negative numbers?  I would love more ideas and discussion.

## 20 thoughts on “Discovering Negative Numbers”

1. I like your pattern idea. It’s easy, but a very nice visual. I do the same thing to show patterns when teaching exponents, since students don’t quite understand why everything to the power of 0 is 1. Instead of saying memorize the rule, students see how the pattern works and it sinks in. For your last question, I always say “take away” instead of “minus” or “subtraction of” when dealing with negative numbers.

• I use patterns for exponents too. It really helps w the zero and negative exponents. 🙂

2. I teach high school, so it’s a little different for me, but why don’t you like “minus a negative”? How would you say 4 – (-3)?

• I usually do say minus and subtract. But I would like for students to see numbers as positive and negative. I used to hate when kids said, “Is that a negative or subtraction?” I want them to see -4 – 5 as “negative four negative five”, not negative four minus five. I am wondering if anyone phrases this differently to reduce misunderstandings.

• I think you’re right–I work on helping the kids see the symbol as subtraction, negative, and opposite. When the kids are in Pre-Alg, we work a lot on using the idea of the symbol that fits best with the problem. We spend time on “combining” like-signs and “canceling” opposite-signs, as well. It ultimately leads to accuracy with Distrib. Prop. and accuracy when solving complex equations. This conversation would be so much easier in person!! 🙂

3. I have the kids think of positives and negatives as units. I would say “4 positives minus 3 negatives.” I use unifix cubes to show why subtracting a negative yields the same result as adding a positive by canceling out zero pairs. This leads into showing the patterns you mention above, etc. 🙂

• I have used integer counters for this as well but my 6th graders have more trouble with this. I will look into the cubes.

• The unifix cubes are great bc they actually connect (for zero pairs) and can be used again when introducing basic eqns.

4. I like to model subtraction of integers using “find the difference” in context.
Your example of the kid “owing” \$20. If his sister HAS \$25, then how much more does she “have” than he does? 20 – (-25) = 45 so she “has” \$45 more than he does.
It works well with temperatures. If it is 40 degrees in LA and -5 degrees in Anchorage, Alaska, how much warmer is it in LA? 40 – (-5) = 45 so it is 45 degrees warmer.
(I always start with a “both positive” before moving onto “one of each” then “both negative.”)
We also look at “how far apart they are” on a number line to reinforce “difference.”

I would say 4 – (-3) as “four minus negative three,” and -4 – 5 as “negative four minus five.” The word for the subtraction operation is “minus” just like the word for addition is “plus,” but the words for integers are “positive” and “negative.”

I have also used the “integer tiles,” but only as a different approach after reinforcing subtraction as finding the difference. The integer tiles use the “take away” model of subtraction, but you can only “take away” after you include the zero pairs, as Johanna mentioned.

I also abhor the “follow these rules” approach. I have eighth graders who are just grasping at straws trying to remember the “right rule” because they never developed the conceptual understanding 😦
(Well, maybe “abhor” is too strong of a word – but it drives me crazy!)

• No, abhor is the perfect word. I love your money example! Since I always use money this will be an excellent addition! Thanks so much!

5. I use a team game of Jeopardy at the beginning of the year to review subtracting with negatives. Using individual white boards to record responses, we play a quick game based on my syllabus and I make sure I include high dollar questions that they will get wrong (like “how many times does the word THE appear on the syllabus”) so that scores go into the negatives. After finding out if their response is correct, they to write an equation to show me the total number of dollars they now have. They know that if they have -400 dollars and get a 1000 dollar question wrong, they are now at -1400 dollars and will write “-400 – 1000 = -1400.” Then, throughout the game, I’ll “accidently” make a mistake in telling them their answers were incorrect and have to “take away” the penalty of losing dollars. So they will adjust their equations to read “-400 – -1000 = 600. It’s a quick way to review two things with one activity: integers and the syllabus 🙂

I also relate subtracting a negative to having a double negative when speaking. We talk about how “I don’t have no pencil” (a phrase that makes my eyes water and ears burn) really means you have a pencil. We say that problems like 9 – (-8) is a mathematical “don’t have no” statement and needs to be cleaned up before we can work with it, just like their grammar needs to be cleaned up before I can respond to it.

• Oh! Love the jeopardy scoring into the negatives!! Awesome idea. I play many games and thus we can easily review negatives all year.

I use double negatives too but we say, “I’m not NOT going to the mall.”. Lol!

• Love the jeopardy! 🙂 In addition to the double negative idea (relating to language arts classes), you can talk about proportions as number analogies. I sometimes have my 6th graders write number analogy stories to make the language connection…

6. I love the idea of showing the pattern, starting with a positive answer and decreasing by one as you move to negative answers. This concept is so difficult for many students. Thanks for giving me another way to approach this concept for the coming school year! 🙂

• So glad I could help! I use so many ways to show negatives. I’m trying to find the best “route” for the kids, from introduction to mastery. It’s a complicated idea for students at first.

For subtracting, I like to talk about owing money. If you’re at the restaurant, and you have a bill for \$45 (-45), and I come along and say, “Hey, let me pay that for you.”, I would take away that negative: -45 – (-45) = 0.
You can do similar things for multiplying.
3(5) means I give you three 5 dollar bills: net result, add 15 (+15)
3(-5) means I give you three IOU slips (each owing \$5): net result, lose 15 (-15)
-3(5) means I take away three of your 5 dollar bills: net result, lose 15 (-15)
-3(-5) means I take away three IOU slips: net result, gain 15 (+15)

OR

getting something good is good