Working with negative numbers can be a challenge for students, whether they are just learning about integers or they are solving equations or factoring quadratics. Why?
There are nice clean rules when multiplying and dividing integers, but those nice rules don't apply when adding and subtracting integers. With adding and subtracting, the "rules" are a lot more abstract.
Example: 7 - 12
My students' ability to work with negative integers, even after the manipulative was removed, improved by 62% and my thesis was [eventually, after many, many edits] accepted in May 2011.
This integers partner scavenger hunt is a self-checking activity and also includes a link to an online interactive Google Slides version.
There are nice clean rules when multiplying and dividing integers, but those nice rules don't apply when adding and subtracting integers. With adding and subtracting, the "rules" are a lot more abstract.
I developed this integer operations manipulative as part of my thesis to help my students see the relationship between negative and positive numbers. A month after the tool was taken away, lasting improvement was measured.
Here's a quick video showing how the manipulative works:
Here's a quick video showing how the manipulative works:
The tool works with absolute value when adding integers of different signs. It allows students to "see" the relationship between the numbers being combined on the number line.
Example: 7 - 12
Thinking of this as 7 + -12, -12 is farther from zero than 7, so our sum will be negative.
Now, fold the manipulative in half and count how much farther away -12 is from zero than 7.
-12 is 5 more away from zero than 7, so -12 + 7 = -5.
My students' ability to work with negative integers, even after the manipulative was removed, improved by 62% and my thesis was [eventually, after many, many edits] accepted in May 2011.
You can download the integer operations manipulative here. To build the tool, you will need cardstock, a hole punch, brads, and lamination.
Later without the tool, I'll ask:
Later without the tool, I'll ask:
"Which number is farther from zero. -12 or 7?"
-12
"OK, so our answer will be negative. How much farther away is it?"
5
"OK, so our answer will be?..."
-5
This year, I made a set of practice slides to help students get used to the tool and to build confidence working with integers. You can find the slides free here.
More Integer Activities
Below are a few integers activities that I think your students will enjoy.
In this integers escape room, students must open 5 locks by answering 20 questions. Each question asks students to combine three integers. The digital version is self-checking in Google Forms. There's also a printable version included.
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| Integers math pennant activity |
For a fun integer operations activity that doubles as classroom décor, you may like this integers math pennant activity. Students evaluate three problems on each pennant.
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| Integers partner scavenger hunt |
This integers partner scavenger hunt is a self-checking activity and also includes a link to an online interactive Google Slides version.
I like the idea of the integer manipulative. How do you use the ruler when finding the same number? Ex. -8 - 8 on the ruler?
ReplyDeleteThere are directions on the back for same-sign problems. I explain it to my students this way... when we were kids and were adding 8 + 8, we didn't think about signs because they were the same. Now with -8 + -8 the signs are still the same, so we add and keep the sign like we did when we were younger.
DeleteFantastic idea. Have you done any further research with your hinged number line?
ReplyDeleteYou might be interested in a Stanford study that was done in 2015 in neuroscience regarding understanding negative numbers and how the brain uses symmetry to understand abstract concepts. https://news.stanford.edu/2015/07/06/symmetry-math-schwartz-070615/
Thanks for sharing this and your thesis!
Thank you for sharing, Linda! I concluded my research in 2010 and agree that symmetry is how our brains work with integers. This is an interesting article. Thanks again!
DeleteThank you for sharing. I love the idea with the ruler.
ReplyDeleteI'm gonna be doing a Math Madness Session soon with nothing but Math for two hours.
I'll even get them to make their own ruler.
Thanks again,
snmyl6@gmail.com
Having them make their own rulers is such a great idea. I feel it will help the kids become familiar with it before using it on integer problems. And Math Madness sounds so awesome! I would love to be a visitor in your class that day!
DeleteHow do you subtract a negative with the ruler? EX. 4 - (-6)?
ReplyDeleteI teach -(-6) as "the opposite of (-) negative 6 (-6), which then turns to +6. We'd then get the 4 back in as 4 + 6. There are sign examples on the back of the ruler for situations when signs are the same (or the same in disguise like your example).
Delete