Counting Up on an Open Number Line - Second Grade Mathematics
While subtraction is often described as taking away, it also has a second equally powerful meaning: finding the distance or difference between two numbers. When two numbers are relatively close together, or when the starting number requires messy borrowing, counting up from the smaller number to the larger number is often much faster and easier than counting backward. An open number line is the perfect vehicle for this strategy, called the counting-up or shopkeeper's method. In this chapter, you will master counting up on an open number line, learn how to hit friendly decade benchmarks, and calculate differences with zero borrowing.
Subtraction as Distance Between Numbers
Consider the problem 62 - 48. Instead of imagining 62 objects and removing 48 of them, imagine two runners on a track at positions 48 and 62.
Finding the Distance between 48 and 62:
+2 +10 +2
+--------> +-------------> +-------->
-----+--------+------------+-------------+-------+--------+--->
48 50 60 62
The Shopkeeper's Method
In traditional stores before digital registers, if an item cost 48 cents and a customer handed over 62 cents, the shopkeeper would count up the change: "48 plus 2 makes 50..." "50 plus 10 makes 60..." "60 plus 2 makes 62!" The total change given is 2 + 10 + 2 = 14 cents! Notice that 62 - 48 = 14. Counting up found the exact same difference without a single pencil mark of borrowing!
Step-by-Step Counting Up Strategy
To find the difference between any two numbers by counting up, follow this three-step guide.
Solving 83 - 57 by Counting Up:
Step 1: Place the smaller number (57) on the far left of the number line.
Place the larger number (83) on the far right.
Step 2: Hop from 57 to the very next friendly decade (60).
57 + 3 = 60 (Jump = +3)
Step 3: Make jumps of 10 from 60 toward 83.
60 + 10 = 70 (Jump = +10)
70 + 10 = 80 (Jump = +10)
Step 4: Hop from 80 to 83.
80 + 3 = 83 (Jump = +3)
Step 5: Add up all your jump sizes!
Total Distance = 3 + 10 + 10 + 3 = 26!
Therefore, 83 - 57 = 26.
Why Landing on Tens Makes It Easy
Multiples of ten (50, 60, 70, 80) act like stepping stones across a river. Jumping from 57 to 60 takes just 3 steps. Jumping from 60 to 80 is two clean tens (20). Jumping from 80 to 83 takes 3 steps. Adding 3 + 20 + 3 = 26 in your head takes seconds.
When Is Counting Up Better Than Subtracting Vertically?
Counting up is especially brilliant when the two numbers are close together or when the top number has zeros.
Scenario A: Numbers are close together: 72 - 68
Vertical Method: Requires borrowing 1 ten from 7, making 12 - 8 = 4.
Counting Up: Start at 68. Hop 2 to 70. Hop 2 to 72. 2 + 2 = 4! (Instant!)
Scenario B: Subtracting across a decade: 81 - 76
Counting Up: Start at 76. Hop 4 to 80. Hop 1 to 81. 4 + 1 = 5! (Done in 2 seconds!)
Selecting the Right Tool for the Job
A master mathematician does not use the exact same strategy for every single problem. When numbers are far apart (like 89 - 12), standard vertical subtraction without regrouping is fast. When numbers are close together (like 82 - 77) or cross decades, counting up on a number line is often much cleaner and avoids calculation mistakes.
Practice Exercises
Exercise 1: Solve 71 - 58 by counting up on an open number line. Describe each jump you make from 58 to 71 and sum your jumps.
Exercise 2: Solve 93 - 87 using the counting-up strategy. Explain why counting up is faster than borrowing for this problem.
Exercise 3: A student solved a subtraction problem by starting at 45 on a number line, jumping +5 to 50, jumping +20 to 70, and jumping +4 to 74. What subtraction problem was the student solving, and what is the difference?
Exercise 4: A runner has a goal to run 52 miles this month. So far, the runner has completed 36 miles. Use counting up to find how many more miles the runner must complete.
Solutions and Explanations
Solution to Exercise 1: Place 58 on the left and 71 on the right. Step 1: Hop from 58 to the next decade 60 (jump of +2). Step 2: Jump from 60 to 70 (jump of +10). Step 3: Jump from 70 to 71 (jump of +1). Total distance: 2 + 10 + 1 = 13. Therefore, 71 - 58 = 13.
Solution to Exercise 2: To solve 93 - 87 by counting up: Start at 87. Jump 3 to reach 90. Jump 3 more to reach 93. Total jumps: 3 + 3 = 6. Counting up is much faster here because 87 and 93 are very close together on the number line; counting up takes two tiny mental hops of 3 rather than unbundling tens and borrowing on paper.
Solution to Exercise 3: The student started at 45 (the subtrahend) and landed on 74 (the minuend). The total distance jumped was 5 + 20 + 4 = 29. Therefore, the student solved the subtraction problem 74 - 45 = 29.
Solution to Exercise 4: To find the remaining miles, find the difference 52 - 36 by counting up from 36 to 52. Jump 4 from 36 to reach 40. Jump 10 from 40 to reach 50. Jump 2 from 50 to reach 52. Total distance: 4 + 10 + 2 = 16 miles. The runner must complete 16 more miles.