Developing Decimal Number Sense

Before you read on, think about how you would respond if someone asked you what it means to have decimal number sense. 

If you’re struggling, don’t worry, number sense is hard to define. According to the National Council of Teachers of Mathematics (NCTM, 1989), there are 5 components that characterize number sense, and these components are the same for decimal number sense:

• Number Meaning
• Number Relationships
• Number Magnitude
• Representations for Numbers and Quantities
• Operations Sense

When I try to explain to someone what decimal number sense is, I think about what it enables students to do. Someone with decimal number sense is able to understand numbers and has a sense of a number’s relative size compared to other numbers. Most importantly, number sense helps students make reasonable estimates when they compute, and their intuition about numbers and operations guides them as they solve problems. 

 

Number Meaning, Number Relationships, and Number Magnitude

When a student with number sense looks at a decimal number, they have a sense of how big or small that number is, and the place value of each digit. Take 0.425 and 0.45 for instance. A student with decimal number sense would know that 0.45 is greater because four hundred and fifty hundredths is greater than four hundred and twenty-five hundredths. Without that place value understanding, a student might use whole number logic and think “425 is greater than 45.”  

Describing decimal numbers using accurate mathematical language can reinforce and signal an understanding of place value. Rather than saying ‘Zero point forty-five,’ it’s more helpful to say, ‘Zero and forty-five hundredths.’ This language clues students into a number’s place values and helps them as they think about the relative size of decimal numbers. 

Students with number sense have extended their understanding of place value into working with numbers less than one. They know that there are 10 tenths in one whole, and 10 hundredths in one tenth. 

In one of Marilyn Burns’ blog posts, she shows students a problem with the answer provided and asks, “Where should the decimal points go?” 

        123 + 47 + 9 = 2.60

I love this question! Asking a student to explain their reasoning can provide a window into their number sense. When the sum is turned into a decimal number, how would a student think about number size and the place values of the digits in the addends? Would their decisions about where to place the decimal points be reasonable?

What about this question: “When you multiply 15.24 and 4.5, the answer is 6858, but the decimal point is missing. Where do you think the decimal point should be placed?” 

These are good number sense questions, and to answer them effectively, students need to have a sound understanding of place value and operation sense.

In addition to knowing what decimal numbers mean, a student with decimal number sense also understands how big or small a decimal number is in relation to other numbers. For example, given an open number line, someone with a strong number sense could estimate where different numbers would be located. About where would 0.50 go on the number line below? How about 0.10, 0.15, 0.30? For a challenge, have students choose their own decimal number, place it on the number line, and justify why it belongs there.  

 

Representations for Numbers and Quantities

Referents such as fractions and money, visual models like decimal grids (Tenths & Hundredths, Thousandths) and base ten blocks all help students develop and signal decimal number sense. Being able to interpret models such as equations also indicate evidence of decimal number sense. 

We have found that decimal grids (see below) are especially effective in helping students develop decimal sense because they provide a clear visual referent or model.

Helping students use what they know about fractions to build decimal sense is key because decimals are fractions written in a different way. They are a special collection of fractions that have denominators of 10, 100, 1000…, the powers of 10. 

In a terrific Mix & Math blog post I recently read, the author emphasizes the role that fractions play as a referent for learning about decimals by noting, “So when students struggle with knowing how many 0.01 are in 0.1, we have to go back to what they know about fractions-how many 1/100 are equivalent to 1/10?” Using visual models like the decimal grids (or base ten blocks) as referents or models are effective for helping students see the connections between fractions and decimals as the following visuals demonstrate. 

 

 

Operation Sense

Decimal number sense enables students to interpret what it means to add, subtract, multiply, or divide decimal numbers. They use this operation sense to effectively estimate the answer to math problems. For example, without calculating, think about whether 1.2 ÷ 0.5 is greater than or less than 2. 

Now let’s look at how Greta, a fifth grader, responded.

“I know that 1.2 ÷ 0.5 is greater than 2 because there are more than 2 groups of fifty cents in one dollar twenty.”

Greta knows what the problem is asking her to do, and this is key to making reasonable estimates. Asking questions like, “How many groups of 0.5 are there in 1.2?” can help develop operation sense, and providing sentence frames such as, “There are _______ ____s in ____.” can help students talk about their learning. 

Without having operation sense, our estimates can be way off, even for adults!  Having a sense of the magnitude of an answer when we add, subtract, multiply, or divide is important because it allows you to quickly check if your calculated answer is reasonable. This week I was having breakfast with two friends, and I asked them to estimate the answer to 100 ÷ 0.5. When they looked at me funny, I knew I’d thrown them into a state of disequilibrium (breakfast might not be the best time to discuss math). 

“Okay,” I said, “Do you think the answer will be more or less than 100?” They both immediately said, “Less.” I asked them why, and they told me that when you divide, things get smaller.” For those of us who learned about decimal numbers by only memorizing procedures (remember lining up the decimal points?), problems like the one I posed to my friends can be challenging, especially if one hasn’t reasoned about what happens to decimal numbers when you divide. When we ask students to estimate, we are giving them a chance to develop their intuition about numbers and operations. 

 

Experiences that Develop Decimal Number Sense

 On the Math Transformations website under resources, you’ll find a variety of activities that help students develop decimal number sense. Here are three of our favorites.  

 

Tell Me All You Can

This number sense routine helps students make estimates in different ways and gives them a ‘sense’ of the answer without necessarily computing or finding an accurate response. Try this one:

3.04 x 5.3

These sentence frames can help students talk about their estimates:

        The product will be less than ____ because__________.

        The product will be greater than ____ because___________.

        The product will be about ____ because _____________.

        The product will be between ____ and ____ because _________.

 

Patterns on the Hundredths Chart

This One Hundredths Chart is a variation on the 1-100 chart. Show students the chart below and ask them what patterns they notice. The chart can be used to help students see patterns and relationships between decimal numbers.

 

Decimal Maze

Decimal Maze is one of my favorite activities and one that my colleague Caren Holtzman and I included in our book, Math Workshop Essentials: Developing Number Sense (Math Solutions, 2018). In the game, partners work together (or compete) to end up with the largest score. They move through a maze of choices, with the ability to add, subtract, multiply, or divide along the way to the finish line. The game is accessible to a range of learners because players use a calculator after they make estimates. 

The focus isn’t to find accurate answers, but to make estimates and keep track of their score as they make their way through the maze. The goal is for students to realize what happens to decimal numbers when you operate on them. As they keep track of their score, the hope is that they see that when we divide by decimal numbers, the result isn’t always to “make smaller” as with whole numbers. The inverse is true for multiplication; try multiplying 6 x 0.5. How much is 6 groups of 0.5? 

The game engages students in strategic thinking, problem solving, and making conjectures. Using the calculator as a tool frees them up to discover patterns and ultimately develop their operation and decimal number sense. 

Players start with 100 points and move a cube after choosing a path to move forward. Here are the  directions:

1. Begin with a value of 100 on your calculator. When you choose a segment to cross, you must estimate the result. Then cross the segment, perform the indicated operation on your calculator, and record the equation at the bottom of the game board. If you end up with a repeating decimal, round to the nearest hundredths or whole number. 
2. Move down or sideways (never up) through the maze from start to finish. You may not retrace your steps.
3. The goal is to choose a path that results in the largest value when you reach the finish line.

 Here’s the Decimal Game Board including a chart to keep track of their score:

 

Helping Students Think Flexibly and Intuitively About Decimal Numbers

We know students are bringing number sense to a task when they can visualize decimals, have a sense of their size in relation to other numbers, and can make reasonable estimates when they compute.

When we offer engaging number sense-related tasks, ask students to make estimates, and provide visual representations of decimal numbers, we are helping students develop their number sense. Paul Trafton who was a professor at the University of Northern Iowa once said, 

“A person who possesses number sense might be said to have a well-integrated mental map of a portion of the world of numbers and operations and is able to move flexibly and intuitively throughout the territory.” 

We hope that the ideas in this post will help you help your students develop their number sense about decimals and move flexibly and intuitively throughout this portion of the world of numbers and operations.