Little Brains, Big Concepts: Understanding 10

This post is part of the Little Brains, Big Concepts series. Check out the previous posts: Introduction, Number, Counting Strategies, Part-Whole Relationships.

One of the most important things you can do as a parent is help your child develop an understanding of ten.

Ten is the foundation of our number system, so a deep and flexible understanding of ten is quite literally the foundation for your child’s future math learning. Your child will rely heavily on their understanding of ten to learn new concepts, like addition and subtraction, decimals and percents.

If you grew up during the age of memorization and procedures like me, this focus on ten may seem a little over the top. There is a reason for your confusion: Research shows that “it takes time and diverse opportunities to experience and understand” our number system (Dougherty 2010). Unfortunately, instead of getting to explore our number system, many of us were told to memorize facts and follow procedures. If you struggled to understand why on Earth we need to borrow when we subtract, this is why.

Your child does not have to experience the frustration of memorizing arbitrary phrases. Meaningful math for your child starts here, with an understanding of ten.

The Big Concept

The number 10 gets its powers from our base 10 number system, which is all about groups of ten. We create new place value units by grouping sets of ten:

  • 10 ones form one ten
  • 10 tens form one hundred
  • 10 hundreds form one thousand

The idea that “ten” can be both ten individual units and one whole unit is complex. Children go through a series of understandings before they fully understand this flexible nature of 10.

Ten as Individual Units

In the beginning, children see 10 as ten individual units. Ten is nothing more than the number that comes after 9 and before 11.

At this level of understanding, when asked to add 10 + 13, children will count on from 10 by ones: 10…11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23.

Ten as a Whole Unit

In this phase, children are able to think about ten as individual units or one whole unit, but not as both. It’s as if there is a small ten and a big ten.

A child at this stage can count by tens, but they struggle to count by both ones and tens. Take a look at this example:

Instead of counting-on from 20 by ones (twenty-one, twenty-two, twenty-three), a child might continue counting by tens (thirty, forty, fifty). Fascinating, right? Counting by tens and ones requires the brain to shift from thinking about ten as a whole to thinking about ten as individual units. Developing this flexible thinking is no easy task!

Eureka: Ten!

At the final stage, children see ten as a whole that is composed of ten smaller units. Wait a minute, is this related to part-whole relationships? Why, it sure is! Isn’t the interconnectedness of math beautiful?

Once a child develops this understanding of ten, they can fluently switch between thinking about ten as a whole and individual units. They can easily count by tens and ones, and they start to think about numbers as being composed of tens and ones: 13 is one 10 and 3 ones.

This understanding of ten is the difference between memorizing algorithms and understanding algorithms. When we “carry the one,” we’re not just writing a little 1 on top of the next column – we’re combining 10 ones to make one ten. Thus, we’re flexibly switching from thinking about ten as individual units to thinking about ten as a whole.

Experiences that Build Understanding

Our number system may seem natural to us, but we’re not born with the innate desire to count by tens. In fact, children will group by twos and fives before they start grouping by 10.

The best way for your child to build a strong understanding of ten is to experience making ten and breaking it apart. Physically composing and decomposing ten gives your child the opportunity to see and feel ten as a whole unit and as individual units.

Linking Cubes

Linking cubes, like Unifix cubes or Legos, are great for composing and decomposing numbers.

  • Give your child a pile of cubes and have them create a tower of ten.
  • Have your child break the tower of ten into two groups. How many are in each group?
  • Have your child put the groups back together. Can you break the tower into two different groups? How many are in each group now?

Ten Frames

A ten frame is a 2 by 5 rectangular frame that helps children develop a mental model of ten. Its structure encourages children to think about numbers in relation to 5 and 10, an important strategy for computing mental math and developing number sense.

In the ten frame below, notice how the structure encourages you to see 6 as 5 + 1.

Ten frames are great for thinking about the different ways to make 10. You can make it a game by incorporating a dice for randomization.

  • Print out a ten frame or make one yourself. Give your child some items to use as counters.
  • Roll a die to see how many counters to put on the ten frame. Have your child put that many counters on the ten frame. They should fill the top row before moving to the bottom row.
  • Ask, How many more do we need to make 10?

Take Action

  • Make 10 the center of attention this week! Choose one of the above activities (linking cubes or ten frames) to practice making and breaking apart ten.

References

Feikes, David., Schwingendorf, Keith. and Gregg, Jeff. (2018) Children’s Mathematical Learning. Retrieved from this website.

Dougherty, et al. (2010). Developing Essential Understanding of Number and Numeration: Pre-K-Grade 2. National Council of Teachers of Mathematics.

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