# Distance Model - Subtracting

Many students come to understand subtracting as taking away. For example, \(10 - 3\) becomes known as "ten take away three." But, this is not the only context subtraction can represent. It can also be used to find the distance or difference of two numbers.

Take-away models often lead to lots of regrouping. To solve \(3000-199\), most students (and frankly adults) do a lot of regrouping.

This is a complicated procedure that is hard for many student to make sense of. Teachers often use base-10 blocks to model the regrouping (for example 1 thousand becomes 10 hundreds) but my experience has been that many students still struggle to apply and justify this algorithm even in older grades.

While it is important to know how to regroup, modeling subtraction as a distance can provide students with an alternative method for solving subtraction problems. So, what does modeling subtraction as a distance look like?

When you think of the result of a subtraction problem as the distance (really it's a vector) between the two given numbers (minuend - subtrahend) it is easy to see that you can adjust both the minuend and subtrahend by the same value and the difference will not change.^{[7]}For visual proof of this, see the above gif. As the left and right end of the line segment increase or decrease by the same amount, the length of the line does not change. Put more simply, you can increase both the numbers in a subtraction problem by the same amount and the difference remains the same.^{[8]}Technically you are increasing the first number and subtracting more with the second number.

Algebraically, it looks like this:

Statements | Reasons |
---|---|

\(a-b=a-b\) | Given |

\(a-b+0=a-b\) | Additive Identity^{[9]}Any number plus 0 equals the given number. |

\(a-b+(1-1)=a-b\) | \(0 = 1-1\) |

\(a+1-b-1=a-b\) | Commutative Property^{[10]}By treating \(-1\) as \(+(-1)\) you can move subtraction operands around as well. |

\((a+1)-(b+1)=a-b\) | Factoring^{[11]}\(-b-1=-(b+1) \) because I factored out a \(-1\). I simply grouped \((a+1)\) to make the result easier to see. |

Being able to envision subtraction this way yields efficient ways of subtracting all sorts of numbers. Here is @bstockus demonstrating it on twitter for whole numbers:

```
```Just had impromptu number talk w/ some interventionists about how this subtraction strategy works. #mNTmTch pic.twitter.com/HBUFlpWl3L

— Brian Bushart (@bstockus) October 28, 2015

This process works well for whole numbers, but it really shines with fractions. Consider for a moment the challenge of this subtraction problem:

#### \(3\frac{1}{8}-1\frac{3}{4}\)

Solving this requires making common denominators, then regrouping from minuend by turning one whole into eighths or by making an improper fraction.

#### Either \(2\frac{9}{8}-1\frac{6}{8}\) or \(\frac{25}{8}-1\frac{6}{8}\)

If instead I increase both the minuend and subtrahend by the same amount, I can turn this into a much easier problem; one that does not require borrowing.

#### \((3\frac{1}{8}+\frac{2}{8})-(1\frac{6}{8}+\frac{2}{8})\)

#### \(3\frac{3}{8}-2=1\frac{3}{8}\)

The same move can be accomplished with decimals too. In summary, this strategy is really handy any time it is easier to do a bit of adding (or subtracting) instead of subtraction with lots of regrouping.

**Strengths**

- Supports student understanding of a lessor used interpretation of subtraction
- Provides an alternative to regrouping in subtraction algorithms
- Can be used with whole numbers, fractions, and decimals
- Requires introducing students to distance model of subtraction

**Weaknesses**

- Requires that students are familiar with distance model of subtraction
- Not as efficient in all circumstances

**Sources**^{[12]}Leave a note in the comments if you know of another source where this model appears.

- Ben shares that EMP and EngageNY use this model (and he shares some other great thoughts in the comments).

Ben

November 1, 2015@ 3:13 amEMP (http://elementarymathproject.com/) uses these strategies for current and future elementary school teachers. If you are looking for something implemented for elementary school students, then you could check the engage^ny Common Core materials. For example, G4M5TF (https://www.engageny.org/resource/grade-4-mathematics-module-5-topic-f-overview) is entitled: "Addition and Subtraction of Fractions by Decomposition" and includes the friendly number (compensation) methods that you've posted about here.

A couple of things to consider:

1) When you are dealing with (fraction) subtraction, would you rather make the minuend or the subtrahend friendly? Why? Does it matter? Can you make them both friendly? (Always? Sometimes? Never?)

2) What happens if your fixed distance continues to shift to the left? That is to say, given "a-b": what happens when b (the subtrahend) becomes negative? Does the distance interpretation still make sense? Why or why not?

And here are some places for additional (subtractional?) sense-making:

1) In your table of algebraic reasoning (quibbles: I might adjust headers slightly - at present, "statements" is pluralized and "reason" is not; also, strictly speaking, the associative property of addition is used at some point) what is lost by erasing "a-b" from the right hand side of each equation? (And changing the first Reason to "Given"...)

2) I have long thought it slightly confusing to see, written on a piece of paper, something like: -2384-1 = -2383. One alternative is to introduce parentheses: -(2384-1) = -2383. Another choice is to use a +1, and write: -2384+1 = -2383. I am, admittedly, partial to the latter approach: I think it helps to connect back with the idea of additive inverses (here: +1 and -1) not affecting the value of an expression, since 0 is the additive identity. Yet I see the logic behind the current presentation with respect to the distance interpretation. Tough call, but I think perhaps worth exploring!