*[Rescued from my old blog.]*

Would you like to introduce your students to negative numbers *before *they study them in pre-algebra? With a whimsical number line, negative numbers are easy for children to understand.

Get a sheet of poster board, and paint a tree with roots — or a boat on the ocean, with water and fish below and bright sky above. Use big brushes and thick poster paint, so you are not tempted to put in too much detail. A thick, permanent marker works well to draw in your number line, with zero at ground (or sea) level and the negative numbers down below.

## To Build Understanding, Tell Stories

Hang the number line on your wall, and then tell each other little story problems while you run your fingers up for the positive numbers and down for the negatives to count out your answers.

“I climbed up three meters into the tree, but then I decided I’d rather dig a hole to China. I went down a total of five meters from my lofty perch. How deep was my hole?”

Written as a math equation, that story looks like this: (+3) + (-5) = -2.

“Dad thought he could dig a better hole than that. He dug down one meter, and then he dug down two more meters, and then he dug three more after that! How deep did he go?”

That would be: (-1) + (-2) + (-3) = -6.

“A squirrel went into Dad’s hole looking for nuts. Then the dog ran outside to bark at him, so he scrambled up the tree as fast as he could go. From the bottom of the hole, the squirrel climbed ten meters. Did he make it safely out of Fido’s reach?”

This time: (-6) + (+10) = +4.

## Challenge Middle-Grade Students to Explore Further

Have fun exploring how negative numbers work. How are they different from positive numbers? You can see from stories like the ones above that adding a negative number will make your total smaller, because it puts you deeper in the hole. This is a great beginning for young children. If your children are old enough to enjoy the mental challenge, however, you might ask them to consider a few questions like these:

*What would happen if you subtracted a negative number?*

Remember that subtraction is the opposite of addition, so it would “undo” whatever addition does. If adding a negative number is like digging a hole, then subtracting it would be like filling the hole back in. So subtracting a negative number moves you up the number line. It works the same as adding a positive number. Cool!

*Can you figure out how to multiply negative numbers?*

Certainly, 1 x (-1) has to be -1, since anything times one is itself. What is

2 x (-2)? Think of starting at zero and counting -2 twice, to get -4.

*What about division? *

This is getting harder, but let’s try (-6) ÷ 2. If you cut -6 in half, what would it be? Oh, that is not so hard after all: it has to be -3.

## To Keep in Practice, Play Cards

My kids’ current favorite math game is *Zero *which they call “Hit Me.” It’s based on the game of Blackjack, but instead of trying to get a score of 21, your aim is zero. Use a deck of math cards, which are simply normal playing cards with the face cards and jokers removed. Black cards are positive numbers, red cards are negative. For each player, turn one card face down and one face up. Everyone can see the face-up card, but only the player gets to look at her face-down card (until the end of the game, when all cards are revealed).

Each player adds her cards together in her head. Then she may ask for up to 5 “hits” — extra cards which are dealt face up — for a maximum of 7 cards total. When everyone is done asking for hits, all cards are turned face up. Whatever each player’s cards add up to is her score, and whoever scores closest to zero (the lowest *absolute value*) when all the cards are revealed wins that hand. Winner becomes dealer for the next hand.

Negative numbers are fun. And, despite that third grade teacher who told us we couldn’t subtract six from four, negative numbers help clarify many real-life situations: winter temperatures, lost football yardage, or the bank account of someone who relies too much on credit cards.

**Edited to add:** This post is an excerpt from my book *Let’s Play Math: How Homeschooling Families Can Learn Math Together, and Enjoy It!*

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What a great card game idea. I just love it! My 5 year old is very interested in negative numbers lately so we’ll have to try this game. Thanks for sharing.

RUBBISH DIDNT HELP AT ALL! but i think that might be because im to old for this! so im sorry yer it would be great for 10-11 year olds!

thanks for your help

x

Hi, Elle,

If you have a specific question, you are welcome to send me an e-mail. Or you might look at some of the math lesson sites or help forums listed in my sidebar.

What a great idea! Do you have any more?

I’m glad you liked it, Karen. For more ideas, try one of the “Posts by Category” links in the sidebar. You might find something interesting listed under your child’s grade level or under Activities or Games.

Gooooooooooood!

I am working with a brain injured child and will try your ideas this week. She has a lot of difficulty understanding negative numbers and I am looking for something that will her her get the picture.

As an advanced algebra teacher in Boston, I can tell you this trouble with negative numbers follows kids all the way into high school. My kids can solve difficult quadratics and logarithmic equations, but if it comes down to “x + 13 = 7” where 13 needs to be subtracted from 7, I get “-20” more times than you’d think.

Why? Well, I have a beef with the elementary curriculum used in Boston and the fact that kids’ first introduction to negative integer problems is 7th grade (6th grade this year). But I told myself it’s easier to complain than to get up and do something, so that’s what I did.

I created the ZeroSum ruler (http://zerosumruler.wordpress.com/) that has helped my weaker math students to gain an understanding of the relationship between positive and negative numbers. Kids who once needed the ruler, now know to ask themselves, “ok, which is further from zero, -13 or 7?” and can take the problem successfully from there.

I actually have to do a project on positive and negative numbers for a college project for extra points. apparently it has to be creative, colorful, and so on, since its a design major we are expected to be really creative about this, with videos, games, paintings and so on. been searching for ideas for a while now, and nothing. i have no clue what i can come up with thats creative using negative and positive numbers. any ideas? and please go all out if u want. :)

Hi Diego. Do you have to create something from scratch? You’re welcome to use my ruler as part of your project. (http://zerosumruler.wordpress.com)

The other thing used in Boston is an elevator model of a building with “floor zero” as the ground floor. The problem with this model in the US is that we count “floor zero” as floor 1! So teachers have to work around that in the beginning. Can you make a building with floors of different colors, calling the ground “floor zero”? That’s the best I can come up with!

yes has to be from scratch and very creative. the elevator idea is not bad. and ty for the rules offer also. i was thinking maybe using the “longcat” idea… if u havent heard of it just google longcat and look in the images section. and u will see what im talking about :) let me kno what you think

That longcat is CRAZY! I did a google image search for him and boy is he long! In one photo he is made of fire. In another he is on his way to the moon!

I’m not familiar with this idea; in fact I had never even heard of it until you posted here. How would you work the longcat into negatives? It’s definitely a creative idea!

yea and if u search some more you will find even longer ones. and i mean loooooooong, anyways i was thinking of possibly buying a couple long papers, the longest i can find and attaching them together, draw a “longcat” and if u seen a couple u can see them from earth up to the universe and down towards what ever is under earth and so on. so i would draw my cat like that. the floor would be my 0 point. then just simply go up with + numbers and down with – numbers, showing how the positive and negative numbers are positioned in a fun way… at least i guess its fun.

look at this link, some of the images this person used for his “longcat” are kinda offensive, so be aware, but u can see how long it was made and the images surrounding it.

http://tabnirstorageplz.deviantart.com/art/Longcat-106641132

also notice he uses some math problems in the picture, this is something i can also integrate to give it that mathematical feeling, since this is for math class.

I attend an Art College so everything we do has to be shown as art, even math :P

let me kno what u think :)

Maybe you could name the longcats for their magnitude, so that longcat27 would be 27 units long. Then make up problems: If longcat27 was standing (on a vertical number line) at -15, what number would be the top of his head? If longcat39 stood at -23, would his head be higher or lower than longcat27’s head? Who is longer, a cat that can stretch from -99 to -56, or a cat that reaches from -14 to +25? Etc.

ok i like the idea. i got the last part, the first part i got a bit confused. so we are saying if longcat 27 is of course 27 postive and 27 negative numbers, then we can crate problems like that. for questions like u said, could be like… what part of his body is longer -99 to -56 or -14 to +25… something like that? about the comparing longcats i dont think i can do more then one. :P remember this will have to be drawn, painted with full color. unless we can just create problems with different longcat names and just use out 1 longcat we have. if thats what you meant, i apologize.

ps: im sorry this turned out to a complete different discussion then what it was meant for.

I was not thinking of making the cat = the number line, but rather putting the cat on the numberline. As if the numberline is like a ladder or a tree that longcat is climbing. So a longcat27 would be 27 units long, no matter where he stood, and you could move him around to make up different problems.

Are you trying to create a complete lesson (with sample problems), or just to make something like a poster or display? What class is this for?

Here is another blog post you might check for ideas (though it has nothing to do with cats): Tilting the Number Line.

i see what u are saying now, and yea its sort of a display picture or a poster, about creating problems for it im not sure, but i do think they will ask us to explain how it works. and its for math class. and u gave me some more ideas lol… man this is gona be a long project to finish. but what ever i decide to do, i will take a picture and post it here when im done. i still have a long time to do this. so no hurry

I’ll look forward to seeing what you come up with, Diego. Best wishes!

I don’t want to be offensive, but I teach advanced algebra to students who still have trouble with the concept of adding positives to negatives so am very aware of the misconceptions bad examples cause our students to have.

A problem written as: “I climbed up three meters into the tree, but then I decided I’d rather dig a hole to China. I went down a total of five meters from my lofty perch. How deep was my hole?” Would have an answer of “5 meters”. Holes are in the ground; we don’t start digging holes in a tree.

Maybe improving the problem to read “I left my lofty perch in the tree and began the first 5 meters of my journey to China. How far into the ground am I now?” would make the problem more clear.

Language is never as precise as math, so I would give credit to any student who could justify his or her answer, even if it was not what I intended when I posed the question.

The way I interpret the words, the distance is counted “from my lofty perch” — from the place where the decision to change direction was made. “I went down a total of 5 meters” means that the total downward motion, including both the climb and the digging, was 5 meters.

You friend borrows $13 from you. She pays you back $7 the next day. How much does she still owe you? Asked this way, it’s obvious she owes you $6. Give a kid the problem -13 + 7, and the answer mysteriously becomes, well, mysterious.

Why?

It could just be me, but if I let a friend borrow $13 and she paid me back $7, I count up from 7 until I get to 13 (“8, 9, 10, 11, 12, 13”), keeping a tally on my fingers of how many numbers I passed on my way to 13. But the key here is “on my way to 13”, not “from 13”.

Then why in school do we teach students to start on our rigid number lines at -13 and count up 7 spaces to get to the answer? This is the same as telling a person to count backwards from 13. “12, 11, 10, ….”, or worse, yet more accurate to the way we are taught, “-12, -11, -10, …” It’s amazing any of us get the correct change when we buy morning coffee! (Though, it’s cheaper to make at home.)

I created a video about adding numbers of opposite signs and how the way we think about it is much different from the way we are taught in school. If you have a minute (and about nine seconds!) I’d love for you to come check it out, leave comments or criticism, just watch, whatever you feel like doing. http://zerosumruler.wordpress.com/

p.s. though I don’t like the cold or snow, I DO really like your site’s snowflakes!

We did the blackjack game today at math club, and it definitely seemed to help a few of the kids understand the concept better. Even for the kids who already understood it well, this was a fun game. (It helped that I brought a bag of candy for the winners of each round to grab from). Thanks for the idea!

I’m glad your students enjoyed the game!