How to Teach the Making Ten Strategy

In first grade, children meet sums like 9 + 4 that go past ten, and subtractions like 13 − 6 that undo them. In the United States, the Common Core names the strategy for this directly: “making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14)” (Common Core State Standards, Grade 1, Operations and Algebraic Thinking).

In Japan, where I live, first graders draw it as a small diagram. A number splits into two branches under it, and because the shape looks like a pair of cherries, it is known as “cherry calculation” (sakuranbo keisan). The name is not in Japan’s official curriculum, but the idea behind it is described in detail in the Ministry of Education’s commentary on the elementary mathematics course of study (Elementary School Course of Study Commentary, Arithmetic, 2017, in Japanese). In this post I read that commentary and put together how to help at home.

What the diagram shows

Here is 9 + 4 as a cherry diagram.

 9 + 4
    / \
   1   3

 9 + 1 = 10   10 + 3 = 13

Split 4 into 1 and 3. 9 and 1 make 10. 10 and 3 make 13. All the diagram does is split a number so that a ten can be made.

The commentary explains that for sums greater than ten, children focus on seeing numbers as “ten and some more”, and by splitting numbers to add or subtract, they can use calculations they already know. In other words, making ten is not a new kind of calculation. It is a tool for taking a new problem back to things a first grader already knows: sums up to ten, and ten and some more.

The foundation: the pairs that make ten

That is why the pairs that make ten need to be solid first. 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5. A child should be able to say these without counting.

The commentary lists, as part of first grade number work, seeing 8 in relation to 10, as “2 less than 10”, and says this way of seeing numbers is an important basis for understanding carrying and borrowing. When a child struggles to draw the cherry diagram, the problem is usually not the drawing. It is that “how many more to make ten” is still uncertain.

At home, the best thing is to check groups of ten with eyes and hands. For example:

• Lay out ten counters, cover some with your hand, and ask “How many are hiding?”
• Use a ten frame, two rows of five, and ask “How many more to fill it?”
• Hold up ten fingers, fold some down, and ask “How many are folded?”

Once the five pairs come out without counting, a child is ready for the diagram.

Which number to split

The commentary uses 8 + 7 to show two ways. Split 7 into 2 and 5, and add 2 to 8 to make ten. Or split 8 into 5 and 3, and add 3 to 7 to make ten.

Japanese textbooks seem to start most often with splitting the second number to make the first one into ten. I think the more important thing is knowing that either number can become the ten. For 3 + 9, making 9 into ten is easier than making 3 into ten. Split 3 into 2 and 1: 9 and 1 make 10, and 10 and 2 make 12. Making the larger number into ten keeps the number you split small.

If a child splits differently from the way school taught, and the answer is right and they can say what they did, I take that as a sign they understand.

Subtraction

The same diagram works for subtraction. For 13 − 6, split 13 into 10 and 3.

1. Split 13 into 10 and 3
2. 6 cannot come out of 3, so take 6 from 10, leaving 4
3. 4 and 3 make 7

The other way is to split 6 into 3 and 3: 13 − 3 is 10, and 10 − 3 is 7. The Common Core calls this “decomposing a number leading to a ten”.

On which to teach first, the commentary says the principle is to let children respond flexibly to the size of the numbers, while teaching to fit the children in front of you. When the number taken away is small, as in 12 − 3, counting back to ten is easier. When it is large, as in 13 − 9, taking it from the ten is easier. Both rest on breaking ten apart.

Should children have to draw it?

Opinions on the cherry diagram are divided. Some say that having to draw it every time, when a child can already find the answer in their head, is a detour.

I see the diagram as a way to put thinking on paper, not as a goal in itself. It helps a teacher or parent see what a child was thinking. Early on, drawing it makes the steps clear. Once a child can answer quickly and correctly without it, I do not think they need to draw it at home.

If the class or tests ask for the diagram, though, I think it is being used as practice in explaining one’s thinking. Telling a child “you don’t need to draw it” at home can confuse them. It may work better to separate the settings: draw it at school, and skip it during practice at home if they like.

From diagram to mental math

I find it helpful to think in four stages.

1. Make ten with objects. Move counters or blocks, shifting one onto 9 to make 10
2. Draw the diagram. Write down with numbers what the hands were doing
3. Say it. Without drawing, say “9 needs 1, take 1 from 4 to make 10, 3 left, 13”
4. Do it in your head. No words, only the answer

Moving back and forth between stages is normal. When the numbers get bigger, or subtraction gets in the way, going back one stage is enough.

What MathCal can do

MathCal is an arithmetic drill app I make, where children write their answers by hand on the screen.

To see groups of ten, there are the Addition Beads and Subtraction Beads lessons, where children move beads. In Addition Beads, when ten unit beads gather, a tap trades them for one ten bar. In Subtraction Beads it goes the other way, breaking a ten bar into ten loose beads.

For practice, use the Basic Calculation Drill. Turn off subtraction, multiplication and division, switch on No Carrying or Borrowing, and only sums up to ten appear. Once that is comfortable, switch it off to mix in problems like 9 + 4. Enable Dots on Number shows dots for each number, so a child can count what is missing to make ten. The Fill-in-the-Blank Drill gives arithmetic problems where the child writes the number that belongs in the box.

Today’s Drill, Today’s Challenge and Today’s Lesson on the home screen are free every day. If no birthday is set, Today’s Lesson is chosen from Addition Beads and Subtraction Beads. Choosing any drill or lesson from the full lists requires a one-time purchase. There are no ads.

For what comes after, from carrying to column arithmetic in grade 2, see How to Teach Carrying and Borrowing (Regrouping). For an overview of all the arithmetic in grades 1 to 6, see What Kids Learn in Arithmetic, Grade by Grade.

I think of the cherry diagram as a small map for making ten. Once the route is learned, the map can be put away. It is still good to keep it somewhere easy to reach, for the days a child gets lost.