When considering the number line, I see it as an excellent tool for student education if it is used carefully! I say carefully because we can easily cause students difficulty understanding the differences between operations if we move to quickly and don’t explain clearly why differentiating certain concepts matters. When I say certain concepts, I mean subtraction and the addition of a negative number. Also the subtraction of a negative number can be difficult to understand thoroughly. It is easy to make up rules and let students go before they truly understand what is happening in an operation.

3+5=8

2+3=5

These are relatively simplistic problems, but the concept of addition and the preservation of integers in the operation are key. A great number line game for basic addition and understanding the order of the addition that I found is here

http://www.sheppardsoftware.com/mathgames/earlymath/fruit_shoot_NumberLine.htm

The game is great because it emphasizes the importance of order within the addition.

4+2=6

So a student may see the starting point of the addition as 2 and add 4, but this will run them into trouble when they begin working with operations that are not commutative.

2+4=6, yes, but if a student does not understand the commutative property and that it does not always hold they will have trouble when they find 2^5 does not equal 5^2. Clarifying commutativity when using the number line is essential for developing a students’ future mathematical understanding.

Another game that I found involved an airplane and had addition and subtraction with positive and negative integers. The game had an airplane which dropped a paratrooper to their final destination. This seemed like a somewhat real life context so I liked it better than the game involving shooting fruit, but it still lacked some specificity. Airplane and Paratrooper Number Line Game

The game did not display the difference between negative integers and subtraction that I hoped it would.

4+(-3)=1

4-3=1

I think it is useful for students to slow down and think about the difference between these two equations in their operations and components. A good extension question would be, how would you write a story for each of these equations? It is important that as teachers we emphasize the distinction on the number line to avoid future troubles in other operations. Also since physics is no so commonly required understanding the difference between going backwards and turning around as far as position goes is great for students to come accustomed to early. This little distinction can go a long ways.

While doing my search for useful number line games online, I did not encounter any that did as effective a job displaying the important distinction between subtraction and a negative number as we can do as teachers in class. I think that it is key for teachers to team up with students and close the gaps in the classes numerical understanding and the number line seems to be the best and most intuitive way to do that.

**To truly make our time as teachers valuable to students we need to come up with ways to further understand numbers beyond the rules. They can just go to a robot or the internet for those. Enhancing student understanding beyond the numbers to deeper conceptual understanding takes work, but it can be done through careful observation and support of student effort and questioning!**

“In mathematics the art of proposing a question must be held of higher value than solving it.”

I LOVE the games that you found. The fruit one was a good way to get the students comfortable with the number line and then the airplane one was working with negative number. I only wish that the fruit one had problems that involves negative numbers. This is because I like the visual of the fruit game and how it shows the path that it takes to get from one integer to the other.

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Rules, rules, rules. Who needs them? Oh, that’s right. We think we do. Totally agree with you that we need to teach beyond the rules. Being told to perform an operation means nothing if we do not teach the concept(s) hidden behind the rules. Our students will be able to apply the correct rules to a specific problem that may be used as an example within the classroom, but they will have much difficulty applying them to a generalized problem. Questioning is the best way to guide one’s understanding. One is able to understand where the confusion lies which can be addressed and change. Enjoyed this piece. Well done Kevin.

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clear, content, complete +

consolidated, coherent – almost. It feels like your central theme is to go beyond the rules. How does what you wrote before address that? Do the games do what you mean? The alternate representation on the number line? Then you close with the rules with which we struggled in class… it’s a good post, but there’s something here to finish up.

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Kevin, this is an excellent line of reasoning you employed regarding the crucial rules that students need to grasp when using operations on integers. Specifically, I enjoyed interacting with the games you posted above and then reading your thoughts on them. I concur that these games do not have as many of the “rules” for performing operations on integers as we (as teachers) would like, but I wonder if there are any games that do provide a good play at incorporating and correctly applying all rules? Personally, I think this is where a high level of creativity is required, or we can simply go default mode and simply instruct that certain rules are just what they are i.e. negative multiplied by a negative is positive. Overall though, I was pleased in reading your thoughts on this subject.

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