Counting up to find a difference

When you reach for it

Hlengiwe’s family is holding a yard sale. They put 702 items out on the tables in the morning, and by the afternoon 386 have been sold. How many are still on the tables?

The answer is 702 - 386. You could line the numbers up and subtract by place value, but that needs a trade across the zero in 702 (see subtracting three-digit numbers by place value). There is another way that needs no trade at all. Ask a different question: how far is it from 386 up to 702? Start at 386 and count up, in a few big, easy hops, until you reach 702. The hops together are the answer.

This works because a subtraction and a missing addend are the same question: 702 - 386 = \square asks exactly what 386 + \square = 702 asks. That idea was first met with small numbers in subtraction as finding an unknown addend; here it is put to work within 1{,}000, on an open number line.

You reach for counting up when the two numbers are close together, when the larger number is a round one or sits just past one (such as 702, 800 or 1{,}000), and every time you hand back change. A cashier who counts change into your hand, “2.70, 3, 4, 5”, is counting up to subtract.

The lesson has three parts, one for each day. Part 1 is counting up on an open number line. Part 2 is choosing when counting up is the quicker way. Part 3 is making change.

The procedure

Part 1: hopping from the smaller number to the larger one

The picture. An open number line is a line with only the numbers you need marked on it, not drawn to scale (as in adding on an open number line, where you hop by tens and ones to a total). To find 702 - 386, mark 386 near the left end and 702 near the right end. The distance between them is the difference. Now cover that distance in hops, each one landing on a landmark: a multiple of ten or of a hundred, where the next hop is easy.

• From 386, the next ten is 390. That hop is 4.
• From 390, the next hundred is 400. That hop is 10.
• From 400, hop straight to 700, the hundred just below 702. That hop is 300.
• From 700, hop 2 to land on 702.

Every hop was easy because each one only fills up a single place: 386 to 390 fills the ones up to a ten, 390 to 400 fills the tens up to a hundred, and 400 to 700 moves only the hundreds digit. Now add the hops:

So 702 - 386 = 316. There are 316 items still on the tables.

Why the hops add to the difference. The hops run end to end, with no gaps and no overlaps, from 386 to 702. So together they cover exactly the distance from 386 to 702, which is 702 - 386. In symbols, 386 + 4 + 10 + 300 + 2 = 702, and grouping the four hops together (the associative property of addition: three or more numbers may be grouped either way and the sum is the same) gives

That last line is the missing-addend form of 702 - 386 = 316, and it doubles as the check.

Any hops will do. The landmarks are a choice, not a rule. Jabulani might hop 300 first, from 386 to 686, then 14 to 700, then 2 to 702. His hops are 300 + 14 + 2 = 316, the same answer, because they cover the same distance. Use whatever hops you can do in your head. A good habit is to hop to the next ten, then the next hundred, then by hundreds, then the rest, because each of those hops is a one-place jump.

Keeping track. The hops are the answer, not the landing points. Write each hop above its arc as you go, and add them at the end. The numbers you land on (390, 400, 700) are only places to stand.

Part 2: when counting up is quicker than taking away

Nonhle’s yard sale table had 523 comic books in the morning, and 478 have been sold. How many are left? By place value, 523 - 478 needs two trades: the ones column (3 - 8) runs short, and so does the tens column after that. But on a number line, 478 and 523 are close together.

Count up from 478:

• 478 to 480 is a hop of 2, to the next ten.
• 480 to 500 is a hop of 20, to the next hundred.
• 500 to 523 is a hop of 23.

So 523 - 478 = 45, and 45 comic books are left. Three short hops, no trades.

Close numbers, short distance. A difference is a distance on the number line. When the two numbers are close, the distance is small, and counting up crosses it in a hop or two. Taking away starts at the large number and has to remove a large amount, 478 here, to get down to a small answer. Counting up starts near the answer’s end and only walks the small gap.

Far-apart numbers, take away instead. The same picture says when not to count up. For 523 - 8, the two numbers are far apart: counting up from 8 to 523 is a long walk. Taking away is quicker, since only 8 comes off: 523 - 3 = 520, then 520 - 5 = 515. Taking away is the better choice when the number being taken away is small, and counting up is the better choice when the number being taken away is close to the number you start with.

Two more signs that counting up will be quick. The larger number is a round number, such as 800 or 1{,}000: then the last hop lands right on it and there are no ones to add at the end. Or the larger number sits just past a hundred, such as 702 or 603: then taking away would mean trading across a zero, while counting up only hops to that hundred and then a few ones more.

Another route. When the number being taken away is just under a round number, as in 402 - 197, compensation (take away 200, then give 3 back) is also quick. Counting up does the same job without any fixing at the end: 197 to 200 is 3, then 200 to 402 is 202, and 3 + 202 = 205. Both routes give the same difference, because both find the same distance.

Part 3: making change by counting up

Money as a number line. A dollar is 100 cents (see Money). A price such as 2.65 dollars means 2 dollars and 65 cents, which is 265 cents. The coins are the penny (1 cent), the nickel (5 cents), the dime (10 cents) and the quarter (25 cents), and bills come in whole dollars.

Change is a difference. Jabulani buys a board game at the yard sale for 2.65 dollars and pays with a 5-dollar bill. His change is 5 dollars less the price, a subtraction. In cents it is 500 - 265, which by place value would need a trade across both zeros of 500. Counting up from the price needs none:

• From 2.65 dollars to 2.70 dollars is 5 cents, to the next ten cents. Hand over a nickel.
• From 2.70 dollars to 3.00 dollars is 30 cents, to the next whole dollar. Hand over a quarter and a nickel (or three dimes).
• From 3.00 dollars to 5.00 dollars is 2 dollars. Hand over two 1-dollar bills.

The change is 5 cents + 30 cents + 2 dollars, which is 2 dollars and 35 cents, written 2.35 dollars. In cents:

Why a cashier counts up. Hlengiwe, at the cash box, says the price and then counts aloud as she hands over each coin or bill: “2.65; a nickel makes 2.70; a quarter and a nickel make 3; and 4, 5.” She never works out 2.35 on its own. She just counts from the price up to the amount she was handed, and the coins and bills in Jabulani’s hand are the change. It is the same open number line as Parts 1 and 2, with whole dollars playing the part of hundreds and ten cents playing the part of tens.

Landmarks in money. The handy landmarks are the next five or ten cents, the next quarter (25, 50 or 75 cents), and the next whole dollar. Hopping to them uses the coins you have: pennies to reach a five, a nickel or dimes to reach a quarter or a dollar, then bills.

Worked examples

Boundaries

• Count up from the number being taken away, to the number you start with. In 702 - 386 the walk starts at 386 and ends at 702. The hops measure the gap between the two numbers, so both ends have to be the two numbers in the subtraction.
• The answer is the sum of the hops. The landing points are only stepping stones. Every hop must be counted once, and the hops must meet end to end: a gap between two hops, or two hops that overlap, gives a wrong distance.
• Whole numbers, larger minus smaller. Within whole numbers, counting up finds a - b only when b is at most a, so that there is a walk upward from b to a. When the two numbers are equal there is no walk at all, and the difference is 0.
• It is quick only when the walk is short or the landmarks are near. When the number taken away is small and the other is large, as in 604 - 9, the walk is long and taking away is the better tool. Counting up always gives the right answer; it just stops being the quick one.
• Money: 100 cents to a dollar. In making change, the hops are counted in cents until they reach a whole dollar, and then in dollars. Adding the hops means adding cents to cents and dollars to dollars, and 100 cents make 1 dollar.

Where this goes next

Counting up is one of several ways to subtract within 1{,}000. It sits beside compensation and taking away by place value, and the habit it builds of asking “how far apart are these numbers?” is the start of estimating differences. Every count-up is also a ready check for a difference found another way, such as by the standard subtraction algorithm and subtracting across zeros, where the trades that counting up avoids are written out in full. Later the same hops work with larger numbers, with time on a clock (from 9{:}45 to 11{:}10 is 15 minutes and then 1 hour and 10 minutes), and with decimals.

Exercises

Part 1: counting up within 1,000

1. Find 512 - 268 by counting up on an open number line.
2. Find 900 - 456.
3. Find 731 - 295.
4. Find 1{,}000 - 625.
5. Fill in each hop: 157 \to 160 \to 200 \to 800 \to 834. Then find 834 - 157.
6. Find the missing number: 249 + \square = 610.
7. Jabulani’s book has 750 pages, and he has read 479 of them. How many pages are left?

Part 2: choosing the quicker way

8. Find 612 - 589.
9. Find 805 - 796.
10. Find 431 - 6.
11. Find 360 - 290.
12. For each, say whether counting up or taking away is quicker, then find it: (a) 940 - 30 (b) 940 - 912.
13. Nonhle’s family brought 418 items to the yard sale and sold 389 of them. How many were left?
14. Hlengiwe stacks 307 paper cups into a tower, and Jabulani stacks 279. How many more cups are in Hlengiwe’s tower?

Part 3: making change

15. A teapot costs 3.75 dollars. You pay with a 5-dollar bill. What is the change?
16. A picture frame costs 1.30 dollars, paid with a 5-dollar bill. What is the change, and which coins and bills could make it?
17. A basket costs 4.15 dollars, paid with a 5-dollar bill. What is the change?
18. A pair of skates costs 7.60 dollars, paid with a 10-dollar bill. What is the change?
19. A toy car costs 65 cents, paid with a 1-dollar bill. What is the change?
20. Nonhle buys a scarf for 1.45 dollars and a jar of buttons for 2.20 dollars, and pays with a 5-dollar bill. What is the change?
21. Jabulani pays for a 2.05-dollar kite with a 5-dollar bill. Count up to find his change.

Reasoning

22. To find 600 - 345, Hlengiwe hops 345 \to 350 \to 400 \to 600, and Jabulani hops 345 \to 545 \to 600. Explain why both ways must give the same answer, and find it.
23. One of 900 - 15 and 900 - 885 is quicker by counting up, and the other by taking away. Which is which? Explain, and find both.
24. Explain why counting up from the price to 5 dollars always gives exactly the right change. Use a price of 2.40 dollars to show it.