## Multiplying Integers: Why is -3 x -4 = +12?

Here are

*‘The Rules’ and ‘The Reasons’,*

*‘The How’ and ‘The Why’*

for Multiplying Integers.

I uploaded a couple pages of my Math Model Book for a ‘Pair-a-Dimes’ post, “Assessment & Rote Learning: Math Conundrums“… and thought I would share these very practical resources here.

The first page has The Rules for Multiplying and Dividing Integers.

Next, using counters, I look at Why the Rules for Multiplying Integers Work*. I call this lesson **“Why is a negative times a negative a positive?”** and slowly build up to this at the end of the lesson. I enjoy seeing the a-hah moments in students when they finally understand this concept.

*It is very important to have pre-taught the concept of zero *before* this lesson, (the same negative and positive number together cancel each other out: together -4 and +4 = 0).

But what about division you might ask? I find this harder to show with counters so I usually explain that every multiplication question has two equivalent, related division questions:

If 3 x 4 = 12 Then 12 ÷ 4 = 3 and 12 ÷ 3 = 4

So if, -3 x -4 = +12 Then +12 ÷ – 4 = -3 and +12 ÷ -3 = -4

This makes further sense to students when they realize that multiplying two integers with opposite signs = negative, and they can see that the same rings true for division as well.

## 18 Comments

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Hello,

I found your site while searching for math websites, and would like to propose a link exchange with you. I am creating a site that explains math concepts that are usually the most difficult for students. I am adding material to it regularly, with a goal of creating several ‘lessons’ that span numerous math topics, and I am trying to increase my site’s visibility. My site is at http://sk19math.blogspot.com. Of course, I would be more than happy to link back to your site.

Thanks for the consideration!

Comment by Shaun | April 13, 2007

Shaun,

You do have a great site for Math concepts, thank you for pointing to it. You link above does not work because of the period at the end of the link, here it is:

http://sk19math.blogspot.com

I am about to introduce my Grade 8 students to Functions right now (actually a Grade 9 concept) and I am going to point some of the more advanced ones to your site next week.

I must be honest with you, my site here drives very little traffic! I have another blog that I spend time developing, it has a more philosophical bent to it. This site is just a dumping ground for more pragmatic thoughts/ideas/links that can be helpful to me in my classroom.

That said, I will add your site to my blogroll and I congratulate you on contributing meaningfully to the blogosphere. I am a fan of anyone who has a passion for educational concepts and wants to share what they know in a earnest way. Thanks!

Dave.

ps. If you come across any good Numeracy Tasks, be sure to send them my way!

My Numeracy Task post in my Pair-a-Dimes blog:

http://eduspaces.net/dtruss/weblog/149817.html

Numeracy Task items on this blog:

https://datruss.wordpress.com/tag/numeracy/

Comment by David Truss | April 14, 2007

Thank you for visiting my site and the lovely feedback.

Though I have nothing in common with the math, but your formulas have captured me indeed: (-) X (-) =+ that’s wow!!!

When we the killing sense of the needlessness becomes multiplied, the positive feeling appear – I am sorry for the possible ambiguity of my Lithuanian-English vocabulary, but I hope you will understand me right:

Prior the multiplication I could recognize myself as the

(-), but at a moment the action took place, the (+) appears in the result. That’s great. The numbers show us the roots of our unhappiness and give the outcome. Humanity needs to recognize her spiritual oneness that transforms all forms of the flesh diseases into the joy of being grateful.

Comment by Tomas | July 8, 2007

i just love your website it explains math very well like adding integers i never thought i would understand it but now i do thank you mutches i can now do my integers homework without having to ask my dad for help now i can do it on my own!!!

♥*~Gabby~*♥

Comment by gabby | December 5, 2007

Hey,

I’m Amber I have finals during the last week of school this week in 6 th grade..and before knowing that we had an integer test that i had to ask for more weekend on to study for it..see my teacher has always looked down on me for being sharp (all a’s) in her s.s. class but gets straight B’s in math class.

I’ve had straight a’s since kindergarden! But then entering middle school my math grades changed real quick like ! She’s not a very good teacher almost everone in her class in flunking! I have a 99 in her math class right now but thats only b/c we just finishing fractions and probabilty and i’m good at that!☺ But with her being the teacher i’m not teachable. Thanks to your website and me and my dad staying up until 1:00 in the morning trying to get it down pat ,i might pass the test!

My dad is 45 years old he’s been out of school for 28 years! Teaching him simple math and refreshning his memory is like teaching an old dog new tricks!!Thanks!:)

– Amber ♥☻☺

Comment by Amber | May 18, 2008

thank you so much!

Comment by trixie | March 25, 2009

OMGG!!!!! thnxx so much! this website has helped alot! =]]

Comment by brianna | June 2, 2009

wagamama ra shu wa sa makantukashu

Comment by shunawa | August 24, 2009

Heeeey, thanks alot!

Comment by BillieJo | November 18, 2009

Could you help me?

I have home work from my teacher. he talked to me to answer

the question, why 3 x 4 = +12 is and why 3 x 4 = -12 is.

please halp me. . .

.sS’^_^’Ss.

Comment by Ghea Nurkartika Sari | December 13, 2009

It’s wonderful that you focus on the “why” in addition to the “how” of integer multiplication. It’s so important to help students understand the meaning behind the rules. This can be quite challenging with multiplying integers, but your counter model definitely helps. For reinforcement try this online counter model interactive applet (virtual manipulative):

http://www.brainingcamp.com/resources/math/integer-multiplication/interactive.php

Comment by Mary | June 21, 2010

[…] Example: When multiplying integers I teach the ‘rules’, the algorithm, but I also teach ‘Why?’. A student who has rote understanding of their times tables will see within my Multiplying Integers lessons that multiplication is repeated addition… a student lacking basic multiplication skills usually cannot go beyond the ‘rules’ since the multiplication itself is a neuron-taxing challenge to them. You need an understanding of basic skills before you can move on to more challenging tasks. […]

Pingback by David Truss :: Pair-a-dimes for Your Thoughts » Assessment & Rote Learning: Math Conundrums | November 9, 2010

math is hard and i really nead healp

Comment by sadai | November 23, 2010

Why must we start with zero though? I know that everything starts out as nothing, but then I’m thinking why can’t I do it without zero. When you do do it without zero you get the wrong answer.

Comment by Tiffany | April 14, 2011

[…] on how to follow an algorithm or are they learning why that algorithm works? Here is a small example to illustrate my point: I can give students the ‘rules’ for multiplying positive and […]

Pingback by 3 keys to a flipped classroom | Connected Principals | April 24, 2011

[…] first teach the concept of zero, then addition of integers, then subtraction, before moving on to multiplication and division… in this case, multiplication of integers was where the class was, but I would […]

Pingback by David Truss :: Pair-a-dimes for Your Thoughts » The Stickiness Factor | March 8, 2012

I am still having difficulties with a negative times a negative concept that the first – means something different than the second -. I look forward to hearing your comments

Comment by Rich | August 31, 2012

[…] on how to follow an algorithm or are they learning why that algorithm works? Here is a small example to illustrate my point: I can give students the ‘rules’ for multiplying positive and negative […]

Pingback by 3 keys to a flipped classroom - Teachers with Apps | October 6, 2012