Subtracting multiples of ten

When you reach for it

A child who has met Tens and ones knows that 80 is 8 tens and no ones, and that 50 is 5 tens. A question such as 80 - 50 looks far bigger than anything within twenty. It is not. Eight tens take away five tens leaves three tens, for the same reason that eight things take away five things leaves three things. The thing being counted is a ten instead of a single bead, and nothing else changes. So the fact 8 - 5 = 3, which the child already knows, does all the work, and the answer is 3 tens, which is 30.

That is the whole idea of this lesson: a ten is a unit, a thing that can be counted, added and taken away like any other thing. This lesson keeps to numbers that are whole tens, from 10 up to 90, on both sides of the minus sign: 70 - 40, 90 - 60, 60 - 60. There are no loose ones anywhere, so nothing ever has to be broken apart.

The lesson takes four days.

• Day 1 builds the idea with things that come in tens: strings of ten beads, bags of ten animal crackers, boxes of ten paint brushes.
• Day 2 shows the same subtraction on the tens column of a hundred chart and as jumps of ten back along a number line.
• Day 3 turns a subtraction into an addition with a missing number, 50 + \square = 80, and puts the four related facts together.
• Day 4 uses all of it to solve story problems.

The child has already met ten more and ten less, which takes one ten away from any number. Here a whole group of tens goes at once, and the pictures from that lesson, the chart and the jumps, come back as ways to see it.

The procedure

The running problem for all four days is 80 - 50. Each day finds the same answer, 30, a different way, and each way shows why the answer has to be 30.

Day 1: tens are things you can count

Agnetha makes bead bracelets. Her beads come on strings, and every string holds exactly ten beads. She has 8 strings, so she has 80 beads. She gives away 5 strings. How many beads does she have left?

Count the strings, not the beads. There were 8 strings and 5 went, so 3 strings stay. That is the fact 8 - 5 = 3, used on strings instead of single beads. Each string still holds ten beads, so 3 strings are 30 beads:

Said in tens, it reads “8 tens take away 5 tens is 3 tens”. It is worth saying it this way aloud with the child every time, because the words carry the reason. The child can check by counting the dark beads by tens, “ten, twenty, thirty”, but should not need to count all eighty.

Why it works. A taking-away fact such as 8 - 5 = 3 does not care what is being counted. Eight cups take away five cups leaves three cups; eight strings take away five strings leaves three strings. As long as every string is the same size, ten beads, counting strings is just counting. So for any two whole tens, the tens digits subtract as a basic fact, and the answer is that many tens. The ones digit of the answer is 0, because neither number had any loose ones to begin with and none were taken.

Written as digits, the pattern is plain to see:

A child should be able to say why the zero is there (“no ones”), not just notice that it is. Tying the zero to “no loose ones” matters, because the same pattern later grows into subtracting two-digit numbers, where the ones are no longer zero and have to be taken away too.

Bundles of ten sticks, linking cubes snapped into trains of ten, or ten frames filled to the top all work the same way as the strings. What matters is that each group is one ten that can be picked up and moved as a single thing.

Day 2: the hundred chart and the number line

The tens column of a hundred chart. On a hundred chart every row holds ten numbers, so the square directly above any number is ten less (see Ten more and ten less). The right-hand column of the chart holds the whole tens, 10, 20, 30 and on down to 100.

To find 80 - 50, start on 80 and go up one square for each ten taken away. Five tens means five squares up: 70, 60, 50, 40, 30. The finish is 30. The chart shows the Day 1 idea a new way: each square up the column is one string of beads gone.

Jumps on a number line. A number line marked at every ten shows the same thing as a path. Each jump of ten to the left is ten less, so taking away 5 tens is 5 jumps back.

Five jumps back from 80 land on 30, so 80 - 50 = 30 again. Counting the jumps back by tens is the skill from Ten more and ten less used on purpose: the child says the landing numbers, keeps track of how many tens have gone, and stops after the fifth.

These two pictures agree with Day 1 because they show the same tens. The chart squares, the jumps and the strings are three drawings of one thing, and that is why they always give the same answer.

Day 3: think of the addition

Five jumps back is fine, but there is a shorter path that the child already knows from subtraction as a missing part. The question 80 - 50 asks: “what goes with 50 to make 80?”

Start on 50 and jump forward by tens until you reach 80: 60, 70, 80. That is three jumps, three tens, so \square = 30 and 80 - 50 = 30.

The basic fact helps here too: 5 + 3 = 8, so 5 tens and 3 tens make 8 tens, and 50 + 30 = 80. Counting up by tens is often quicker than counting back when the two numbers are close together, as in 90 - 70, which is two jumps up rather than seven back. The two directions always land on the same answer because they describe the same three numbers, as counting up to find a difference explains for numbers of every size.

Four facts from one picture. The three numbers 30, 50 and 80 make a fact family, exactly as 3, 5 and 8 do:

Each is the bead picture read a different way: three strings and five strings make eight; eight take away the three dark strings leaves the five light ones. A child who checks a subtraction by adding is using this family: 30 + 50 = 80 confirms that 80 - 50 = 30.

A missing number anywhere. The same thinking finds a missing number in any place of a subtraction, as in Finding the missing number in an equation. For 80 - \square = 30, ask “how many tens must go from 8 tens to leave 3 tens?” The fact 8 - 5 = 3 answers it: 5 tens, which is 50.

Day 4: stories with tens

A subtraction story with tens is a subtraction story within ten, told in groups of ten. Three kinds come up, the same three kinds the child met within ten:

• Taking away. Some tens are there, some go, how many are left? Agnetha’s strings are this kind.
• Parts and the whole. A whole is made of two parts and one part is known. If Agnetha’s 80 beads were 50 red and the rest blue, the blue beads are 80 - 50 = 30, or 50 + 30 = 80.
• Comparing. Two amounts are set side by side and the question is how many more one has. If Cuauhtli has 80 beads and Nezahual has 50, Cuauhtli has 30 more, because when Nezahual’s 5 strings are matched with 5 of Cuauhtli’s, Cuauhtli has 3 strings without a partner.

All three are the same subtraction, 80 - 50. What changes is the question the story asks, and drawing the story with strings, bags or boxes shows which number is missing. A story told with loose things in numbers of tens, such as 50 beads, can be pictured in tens: 50 beads are 5 strings of ten.

Worked examples

Read each problem aloud. If the child has bead strings, bundles of sticks or cube trains, let them build it.

Boundaries

• Both numbers are whole tens. The move “subtract the tens digits, write a zero” works because neither number has loose ones. For 85 - 50 the 5 ones are still there and must be written: 8 tens and 5 ones take away 5 tens is 3 tens and 5 ones, 35. Numbers like that belong to subtracting two-digit numbers, and taking a ten away from any number one ten at a time is ten less.
• The first number must be at least as big as the second. In counting things, 50 - 70 has no answer: five strings cannot have seven strings taken from them. The basic fact 5 - 7 has no answer among the whole numbers either, and the tens inherit that.
• Taking away all the tens leaves 0. 60 - 60 = 0, 10 - 10 = 0. The answer is the single digit 0, not a tens digit and a ones digit.
• Zero tens. Taking away 0 tens changes nothing, so 70 - 0 = 70. It is true, but 0 is not one of the numbers 10 to 90 this lesson practices with.
• The groups must all be ten. Strings, bags and boxes work only because each holds exactly ten. If one bag holds only 8 crackers, 7 bags are not 70 crackers, and counting bags no longer counts tens.

Where this goes next

The lessons that follow in the same unit turn the idea around. In adding two-digit numbers without regrouping, tens are added to tens and ones to ones, so 34 + 50 is 8 tens and 4 ones. In adding by making a ten, loose ones are gathered into a new ten. Subtracting tens is half of that work: subtracting two-digit numbers takes tens from tens and ones from ones, and when a place runs short a ten is broken into ones, which is Regrouping in subtraction.

The unit idea keeps growing. Hundreds subtract the same way, 800 - 500 = 300 because 8 hundreds take away 5 hundreds is 3 hundreds, which is how subtracting three-digit numbers begins. The same thinking, a basic fact on a bigger unit, is how multiplying by tens works later on.

Exercises

Day 1: tens are things you can count

1. 7 tens take away 2 tens is how many tens? What number is that?
2. 50 - 30 = \square
3. 90 - 50 = \square
4. 8 - 6 = 2. Use it to find 80 - 60.
5. 40 - 40 = \square
6. A shelf has 8 boxes of paint brushes, with 10 brushes in each box. 3 boxes are taken to the art room. How many brushes are still on the shelf?

Day 2: the hundred chart and the number line

7. Start on 80 in the tens column of a hundred chart and go up 2 squares. Where do you land? What subtraction did you do?
8. Start at 70 on a number line and jump back by ten 3 times. Where do you land?
9. 80 - 10 = \square
10. How many jumps of ten back take you from 60 to 20? Write the subtraction.
11. 30 - 20 = \square

Day 3: think of the addition

12. 20 + \square = 60. Use it to find 60 - 20.
13. 10 + \square = 50. Use it to find 50 - 10.
14. 90 - 80 = \square
15. 40 - \square = 10
16. Write the fact family for 20, 50 and 70.
17. \square - 10 = 80

Day 4: stories with tens

18. Nezahual has 60 beads on strings of ten. He uses 30 beads to make a bracelet. How many beads does he have left?
19. The art room has 9 boxes of paint brushes, 10 in each box. The children use 2 boxes. How many brushes are still in boxes?
20. Cuauhtli has 80 animal crackers. Agnetha has 40. How many more crackers does Cuauhtli have?
21. A bracelet has 60 beads. 10 beads are red and the rest are blue. How many beads are blue?
22. Agnetha had 90 animal crackers. She shared some with friends, and now she has 50. How many did she share?

Explain

23. Explain why the fact 7 - 6 = 1 tells you the answer to 70 - 60.
24. When you take one whole number of tens from another, the answer always ends in 0. Why?
25. Why does 50 - 70 have no answer when you are counting beads on strings?