PRINT VERSION - Subtraction As Distance

We usually consider subtraction a “take-away” or “removal”, but what if we look at it as a distance? This view of subtraction promotes flexible thinking and provides another strategy for solving subtraction problems using a number line and is a critical precursor to the thinking required when students encounter integer subtraction and calculating distance on a coordinate grid.

Goals and Standards

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Colorado Academic Standards:

• 1.OA.C.6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g.,8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 − 4 = 13 − 3 − 1 = 10 − 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 − 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Mathematical Practice Standards:

• MP1. Make sense of problems and persevere in solving them. 
• MP7. Look for and make use of structure.

Learning Goal:

• To extend the understanding of subtraction to include distance between two numbers.
• Communicate verbally how a strategy is used to solve a subtraction problem.

Next Section:

Progression of Skills (outline)

Before the lesson:

• Check students' understanding of subtraction, what can they tell you about subtraction?

During this toolkit lesson:

• Create a number line on the floor.  Use it to explore subtraction as a distance. Bridge to writing subtraction problems to match what was modeled on the number line.

After the lesson:

• Students will now have another strategy to use as they think about subtraction. Students should be allowed to practice and choose a strategy of choice for problems given. Students will develop flexible thinking to solve subtraction problems with a variety of strategies.

Lesson

Directions for Intervention Activity:

*If space is an issue, you could use a number rack to accomplish the same goal.

1. With help from your student(s), use blue tape to make a number line on the floor. Tiled floors are extremely helpful in making equal parts on your number line.  Aim to make a number line with at least 15 tick marks and large enough for students to stand on the tick marks and have some room. Be sure to include arrows at the ends of your number line to show that it goes on forever in each direction.
2. Ask your students, “Where is zero on this number line?” Let them point to places or stand where they think zero is.  
3. Say, “Did you know that we get to choose where zero is? When we make a number line it is up to us to label it!  Today we will pick a zero that gives us space to work with numbers up to 10.  Where do you think the best spot for zero is on our number line?” 
   1. Students may want to have zero the left-most tick, it is the beginning of the tick-marks on the number line.  You may want to ask if there are numbers before zero. If they say no, make a connection to a thermometer and temperature.  “If we turn our number line we have a model of a thermometer.  In the winter do we have temperatures below zero sometimes?”
   2. Once you have zero labeled, have the students tell you what the other tick marks should be labeled. Have the students help label all the tick-marks with the appropriate numbers.  To check the students' work, have them hop from zero to their number on the number line.  “I marked 5, I made five hops to get to 5.”
4. Say,” We will use this number line to talk about subtraction in a new way.  Next, I’m going to ask you to stand on some of the numbers.” Depending on the size of your group you can do pairs first, you may need to stand on one of the numbers.
5. For each set of numbers check that your students are on the correct spot and ask them how they would know if the number line only had a zero. They may say something like, “I hopped to my number from zero.” Then ask, “How far apart are we?  What is the distance between us?”  Be careful as students may count tick marks and they should be counting the space between the tick marks.  For example, 5 and 9 are 4 away from each other, but there are only three tick marks between them (6, 7, and 8). Students may be counting tick-marks and need to hop to see the actual distance. You can ask, “How many jumps are there from 9 to get to 5?”
   1. Repeat these pairs: 5 and 9, 6 and 4, 10 and 7, 0 and 3
   2. You can also have the kids pick random spots and tell you where they are and how far apart they are. 
   3. If you choose more pairs be sure to keep the numbers within the range of your number line for this activity or be ready to make your line longer.
   4. You may want to print a copy of the number pairs document for ease of use during the lesson.
6. Say, “We are now going to use this number line to talk about subtraction. I’m going to ask you to stand on numbers again but we are going to change our discussion.”  
   1. Use this model for a variety of problems.  Have students once again stand on 5 and 9, and ask how far apart they are.  Then ask them, “What is 9 - 5? Interesting… let's try the other pairs we did.” Write the subtraction problem to go with a pair of numbers. Repeat with all of the pairs. You may add additional pairs or challenge your students to create other problems.
   2. Look back at your web from the introduction and pre-assessment discussion and talk about removal and adding on, and the words the students came up with.  End with, “Let’s add another word! What did we notice about how far apart we were and our answers?”  Try to get the students to connect the answers to the distance apart from each other on the number line. You want to add the word distance to your web. 
7. Next, “Let’s change it up.  Now I will tell you a distance and I want you to tell me  subtraction expressions that have that distance.  Then we will use the number line to model them. Once we agree we will record them.”  
8. Start with the distance between two numbers (the subtraction answer) and have students come up with subtraction equations that have that distance. Be sure to have them stand on the numbers of their equations to show they are that distance from each other on the number line.  You can continue writing expressions until you feel your students are ready to move on.
   1. Here are a few examples.  
      1. Say, “The distance is 5. What could the equation be?” Students may stand at 9 and 4 to model 9 - 4 = 5. There are many pairs of numbers that are a distance of 5 apart. Another possibility is a student at 7 and another at 2 to model 7 - 2 = 5.
      2. Say, “The distance is 7. What could the equation be?” Students may stand at 8 and 1 to model 8 - 1 = 7. 
      3. Say, “Are there other subtraction equations with these answers?  Let's add those to our list.” 
   2. Be sure to keep coming back to the idea of distance. The difference between two numbers (answer to a subtraction problem) is the same as the distance between them on the number line. 
9. Next, “What if my distance is 4 and one person is already on 5? How can I write an equation matching this information?’
   1. For the above students may say 9 or 1, it will depend on where the 5 is in the problem. 9 - 5 = 4 or 5 - 1 = 4.  
   2. Do a few more problems like this. Here are a few combinations to get you started. 
      1. The distance is 3 and a person is on 6. 6 - 3 = 3 or 9 - 6 = 3. 
      2. The distance is 6 and a person is on 3. 9 - 3 = 6. 
      3. The answer is 4 and a person is on 4, 8 - 4 = 4 and 4 - 0 = 4.

Anticipate Misconceptions:

• Misconceptions at this level will include the idea that your answer is always smaller than the largest number in the subtraction problem, formally called the minuend. In later grades when working with other forms of numbers this may not be true. For example 11 - (-5) = 16. If a student asks, you can tell them they will eventually learn about integers (positive and negative whole numbers) and this will not always hold true. Also a simple example of 9 - 0 = 9 shows that it’s not always less.
• Students may think that “distance” and “difference” represent different values. While the word “distance” is used throughout the intervention, it is important for students to understand that the word “difference” could also be used in this context.
• If a student has a very limited amount of answers you may want to consider going back to the toolkit for Creating and Using Partitions of Tens.

Progress Monitoring:

• The conversation should be continuous; ask, wait and listen before jumping in to help your student.  If you have a small group you can ask others to help.
• Be sure to focus on language, use the graphic organizer to circle back to things the students already know about subtraction.

• Free number line tools are available to use if you would like to explore number lines differently.  This number line tool from PHET may require you to make a free account.
• Desmos has an activity with a frog jumping to get to bugs. You may need to set up a free account to access Grade 1: Leaping Lilly Pads
• Another idea is to turn the number line vertically and do this lesson again. This connects well with a thermometer and temperature problems.
• Using dice to pick the number used is also a way to make the lesson more fun and engaging.
• 1st graders will need to subtract numbers from 10 to 90. Once the number line has been established you can extend it to larger numbers.