Representing Numbers in Different Ways
College Mathematics | Math for Teachers | Part I | Math for Teachers I
Models, Words, Symbols & Number Lines
A number is an idea. The symbols we write, the words we say, the blocks we build, and the points we mark on a number line are all representations of that idea. For example, the quantity one-half can be represented as 1/2, 0.5, 50%, half of a square, or a point halfway between 0 and 1. These representations look different, but they describe the same quantity.
Learning to move among representations is an important part of mathematical understanding. Different representations can make different features of a number easier to see. In this lesson, we will explore symbolic, verbal, visual, and spatial representations of numbers and think about how teachers can use them to deepen understanding.
What You'll Learn
By the end of this lesson, you should be able to:
Explain the difference between a number and its representation.
Represent the same quantity in multiple ways.
Connect symbolic, verbal, expanded, visual, and number-line representations.
Use decimal squares to represent decimal quantities.
Use base-ten blocks flexibly by changing which block represents one whole.
Explain why the value represented by a model depends on the unit that has been defined.
Locate and interpret numbers on a number line.
Use increasingly precise number lines to represent decimal quantities.
Choose a representation that highlights a particular mathematical idea.
Analyze student representations for evidence of mathematical understanding.
Start Here: A Number Is More Than Its Written Symbol
Consider 0.5. What is 0.5? You might say:
five tenths,
one-half,
50 hundredths,
halfway between 0 and 1,
or half of one whole.
All of these describe the same quantity. The written symbol 0.5 is only one way to represent it. This distinction is important.The number is the mathematical quantity. The representation is how we choose to show that quantity. Strong mathematical understanding includes being able to move flexibly among representations.
One Number, Many Representations
Consider the number 2.35. We can represent it in several ways.
Standard Decimal Form - 2.35
Words - two and thirty-five hundredths
Expanded Form - 2 + 0.3 + 0.05 or 2 + 3/10 + 5/100
Place-Value Form - 2 ones + 3 tenths + 5 hundredths
Fraction Form - 2 35/100
Visual Model - Two complete wholes, three tenths of another whole, and five hundredths.
Number Line - A point between 2.3 and 2.4.
None of these representations is the number itself. Each gives us a different way to see the same quantity.
Why Use More Than One Representation?
Different representations make different mathematical ideas easier to notice. A symbolic representation such as 0.25 is compact and efficient. A decimal square can make it easier to see that 0.25 = 25/100. A number line can make it easier to see that 0.25 lies between 0 and 1 and is halfway between 0 and 0.5. A base-ten model can emphasize the relationship among ones, tenths, and hundredths.
No single representation is always best. The useful question is: What do I want this representation to help me see?
Seeing Decimals with Decimal Squares
A decimal square typically divides one whole into 100 equal parts. If the entire square represents 1 then each small square represents 0.01 or 1/100. So shading 37 small squares represents 0.37, which is 37 hundredths and 37/100.
Connecting the Model to Place Value
Suppose we shade:
4 full rows,
and 6 additional small squares.
That represents 46 hundredths or 0.46. We could also see 4 tenths + 6 hundredths because four complete rows contain 40 hundredths. The visual model helps connect 0.46 with 0.4 + 0.06 and 46/100.
Try It
Imagine a 10 × 10 decimal square.
Model A
65 squares are shaded. Write the quantity as a decimal, a fraction, and as a sum of tenths and hundredths.
Model B
8 complete rows are shaded. Write the quantity as a decimal and a fraction. What equivalent decimal could you write using hundredths?
What Counts as One?
This is one of the most important questions in mathematics. What are we calling one whole? A mathematical model does not have a fixed value by itself. Its value comes from how we define the unit. That becomes especially clear with base-ten blocks.
When the Flat Is One
Many students first encounter base-ten blocks with these values:
small cube = 1
rod = 10
flat = 100
But those values are not permanently attached to the blocks. Suppose instead that we declare that the flat represents 1 whole. Since the flat contains 10 rods, one rod represents 1/10 = 0.1 And since each rod contains 10 small units. So one small unit represents 1/100 = 0.01. So the same physical materials can now represent decimals.
The Model Didn't Change - the Unit Did
This is a powerful mathematical idea. The physical block did not somehow become smaller. We changed the quantity assigned to the whole. Once the whole changed, the values of all the related pieces changed with it. This connects directly to fractions. A fraction only has meaning when we know a fraction of what whole? Similarly, the value represented by a manipulative depends on what we have defined as the unit.
Scaling the Model Again
Now suppose we change the unit again. What if the large cube - not the flat - represents 1 whole? Then the flat represents 0.1. The rod represents 0.01. And the small cube represents 0.001. The relationships among the blocks remain the same. Each smaller type is one-tenth the value of the larger type. What changes is the scale.
The Relationships Stay Constant
This is worth emphasizing. Suppose the blocks have this relationship 10 small units = 1 rod and 10 rods = 1 flat. That relationship stays true no matter which block we define as one whole. What changes is the numerical value assigned to each piece. This mirrors the place-value pattern:
1 → 0.1 → 0.01 → 0.001
The physical model can therefore help make powers-of-ten relationships visible.
Try It
Suppose a rod represents 1 whole. What does the small unit represent? What does the flat represent? Explain how you know.
Now suppose the small unit represents 1 whole. What does the rod represent? What does the flat represent? What changed? What stayed the same?
A Teacher's View: Models Need Defined Units
Imagine a teacher places a base-ten flat on the table and asks “What number is this?” A student answers “100.” Is the student necessarily correct?
Not without more information. The important question is: What has been defined as one? If the small cube represents 1, the flat represents 100. If the flat represents 1, the flat represents 1. If the large cube represents 1, the flat might represent 0.1. A model does not carry a universal numerical value. Its interpretation depends on the defined unit.
Numbers Live on a Number Line
Visual area models and blocks help us see quantities as collections or parts of wholes. A number line provides a different perspective. It represents numbers as locations. For example 0.4 is not simply four tenths. It is also a specific point between 0 and 1.
Zooming in on the Number Line
Suppose we begin with a number line from 0 to 1. Divide the interval into ten equal parts. Now we can locate 0.1, 0.2, 0.3, ... 0.9. But suppose we want to locate 0.37. We can zoom in on the interval from 0.3 to 0.4 and divide that smaller interval into ten equal pieces. Now we see 0.31, 0.32, 0.33, ... 0.39. So 0.37 has a precise location.
And We Can Keep Zooming
Suppose we want 0.372. Start with 0.37 to 0.38. Divide that interval into ten equal parts. Now we can locate thousandths. This process can continue. There is always room between two different numbers for more numbers. That is one reason the number line is such a powerful model.
Different Models Highlight Different Ideas
Consider 0.37.
Decimal Square - Highlights 37 hundredths
Base-Ten Blocks - Highlights 3 tenths + 7 hundredths
Expanded Form - Highlights 0.3 + 0.07
Fraction Form - Highlights 37/100
Number Line - Highlights the location of 0.37 between 0 and 1.
Every representation is useful - but for a different reason.
Which Representation Would You Choose?
Suppose you want to help a student understand:
Why 0.4 = 0.40 - A decimal square may be especially useful.
Why 0.37 is greater than 0.35 - A number line or place-value model may help.
Why 0.37 = 37/100 - A decimal square makes the connection visible.
Why 0.3 + 0.07 = 0.37 - Base-ten blocks or expanded form may be helpful.
Good teaching is not simply about using manipulatives. It is about choosing a representation that makes the mathematical relationship you want students to notice more visible.
Moving Between Representations
Consider 0.62. Can you represent it as:
Words
Expanded Form
Fraction
Tenths and Hundredths
Decimal Square - Describe what would be shaded
Number Line - Between which two tenths would it lie?
Being able to make these connections is evidence of deeper mathematical understanding.
Student Thinking: What Do You Notice?
Situation A
A student sees a decimal square with 35 shaded cells and says “That's 35.”
Think About:
What may the student be treating as one?
What question could clarify the unit?
How could you connect 35 shaded squares to 0.35?
Situation B
A student says “A base-ten flat is always worth 100.”
Think About:
Why might this belief develop?
What important idea about units is missing?
How could changing the defined whole help?
Situation C
A student represents 0.4 with four rods when a flat represents one whole.
Think About:
Is the representation correct?
What does each rod represent?
How might the student also represent 0.40?
Situation D
A student places 0.38 to the left of 0.4 on a number line.
Think About:
Is the placement correct?
How might writing 0.4 as 0.40 help?
What does the number line show that a symbolic comparison might not?
Situation E
A student says “0.35 and 35/100 can't be the same because one is a decimal and one is a fraction.”
Think About:
What distinction is the student making?
How could a decimal square connect the representations?
Common Misconceptions
“The representation is the number.”
A written symbol, model, or drawing represents a quantity. It is not the quantity itself.
“A base-ten block always has the same value.”
The value depends on which block has been defined as the unit.
“The whole is always obvious.”
It is not. Whether working with fractions, decimals, or manipulatives, we need to identify what represents one whole.
“Decimals are just digits after a decimal point.”
Decimals represent quantities. Models such as grids, blocks, and number lines help connect the symbols to those quantities.
“A number line is just for whole numbers.”
Fractions, decimals, irrational numbers, and many other numbers can all be represented on a number line.
“If I zoom in far enough, eventually there will be no numbers between two decimals.”
Between any two distinct real numbers, there are always more real numbers. At this level, repeatedly zooming in on a number line can help students begin developing that intuition.
“Manipulatives make the mathematics automatically clear.”
Models can support understanding, but teachers still need to connect the model explicitly to mathematical language and symbols.
Check Your Understanding
Try answering these without looking back.
What is the difference between a number and a representation of a number?
Give four different representations of 0.25.
If a base-ten flat represents 1, what does a rod represent?
If a large base-ten cube represents 1, what does a flat represent?
Why can the same base-ten block represent different numerical values?
What does a decimal square help make visible?
Where would 0.63 lie on a number line?
How could you use a number line to locate 0.637 more precisely?
Why might one representation be more useful than another for a particular teaching goal?
What question should you always consider when interpreting a visual model?
Teaching Challenge
Choose the number 0.36. Represent it in at least five ways.
Standard Decimal Form
Words
Expanded Form
Fraction Form
Decimal Square
Base-Ten Model (Assume the flat represents 1.)
Number Line
Now Choose the Best Representation.
Suppose your goal is to show 0.36 = 36/100. Which representation would you choose? Why?
Now suppose your goal is to show 0.36 < 0.4. Which representation would you choose? Why?
Now suppose your goal is to show 0.36 = 0.3 + 0.06. Which representation would you choose? Why?
The goal is not to find one universally “best” representation. The goal is to choose a representation that supports the mathematical idea you want students to see.
Reflect
Think about a number as simple as 0.5. How many different ways can you represent it? Which representation feels most natural to you? Which one makes its relationship to 1/2 easiest to see? Which one makes its location between 0 and 1 easiest to see?
Now think from a teaching perspective. Why might showing a student several representations be more powerful than repeatedly explaining the same symbolic procedure? And perhaps most importantly, how does asking “What is the whole?” change the way you think about mathematical models?