In one of the middle school sessions at TMC12 we were talking about integers and how students struggle especially with subtraction. We were brainstorming different strategies (chips, number lines, rules, etc.) This is the distance strategy I shared with the group.

When working with students it is important for them to understand that subtraction is more than just "take away". Yes, we can figure 25 - 3 as "You have 25 candies in your hand and you eat 3. What do you have left?" but that limits us when we expand to include integers.

Using a number line students can also recognize that the distance between 25 and 3 is also 22. Continue building on what students know.

13 - 4 = 13 take away 4 but also the distance between 13 and 4

So what happens when you switch it?

4 - 13

Many students will intuitively notice the result would have to be negative. What is the distance between 4 and 13? It is the same as 13 and 4 only it is negative because the smaller number is first.

*Students have to understand the above to be able to move forward.

9 - (-2) = the distance between 9 and -2, notice 9 is larger than -2 = 11

-2 - 9 = the distance between -2 and 9, notice the -2 is smaller than 9 = -11

-3 - (-8) = the distance between -3 and -8, notice the -3 is larger than -8 = 5

If students have trouble determining the distance, provide a number line. Counting on to determine how far apart numbers are is another strategy that some students might need practice with.

-135 - 124 =

(the distance from -135 to 0 is 135, the distance from 0 to 124 is 124 so the distance from -135 to 124 is 135 + 124)

Hmmm... I never thought of it this way! Thanks for the share!

ReplyDeleteCool, thanks for sharing! If you use "distance" to describe subtraction, how do you distinguish this from the definition of absolute value? So, for example, when you say 4 - 13 means finding the distance between 4 and 13, how would students see the difference between that question and abs(4 - 13)?

ReplyDeleteWhen we were first talking about it at TMC12, I was saying distance and direction but that seemed to get confusing. From my experience, absolute value is something that students continue to struggle with and the more opportunities to discuss the better. I can absolutely see this conversation of absolute value naturally following the subtraction. Any other suggestions are welcome and appreciated. Thanks for commenting.

DeleteAnother analogy for subtracting integers that I've found helpful is the "Hot and Cold Cubes" story from IMP, Year 1. http://bit.ly/M821Xq has a copy of the story, but the basic idea is that you can use a hot cube to represent a positive number and a cold cube to represent a negative number. In this analogy, one hot cube raises the temperature 1 degree and one cold cube decreases it by 1 degree. So both adding hot cubes and removing cold cubes raises the temperature while removing hot cubes or adding cold cubes decreases the temperature.

DeleteWell. It does. Since absolute value is just the pure distance, no sign, it shows why abs(4-13) is the same as abs(13-4), because they are the same distance, just different directions. I don't think the "directions" are confusing. In fact, I think they're fairly intuitive, especially when you need to reverse directions.

DeleteThis video - https://www.teachingchannel.org/videos/teaching-subtracting-integers - addresses the hot cube / cold cube thing that Anna mentioned. I think the teacher in the video does a great job with the lesson - but I find it sort of amusing that The Teaching Channel refers to this chef's endeavors as a "Real-World Scenario." Right... because we all use hot cubes in our kitchens all the time.

DeleteI use the distance in my classroom too and the students do pick up on it! Whenever I teach adding and subtracting integers I have them all sing row your boat too

ReplyDelete"same signs add and keep"

"different signs subtract"

"keep the sign of the bigger number"

"then it will be exact"

An arithmetic operation in which the difference between two numbers is calculated; "the subtraction of three from four leaves one"; "four minus three equals one".

ReplyDelete