Multiple Representations of Decimals
using Base-10 Blocks
Objective/Goal: Using base-10 blocks, students build a conceptual understanding of
decimal place value.
Standard: 4.1.2 Represent and compare fractions and decimals in real-world and
mathematical situations; use place value to understand how decimals represent quantities.
Benchmarks
Touched On
4.1.2.4
4.1.2.5
4.1.2.6
4.1.2.7
5.1.2.1
5.1.2.3
5.1.2.5
6.1.1.1
6.1.1.2
Benchmark Emphasized
4.1.2.4 Read and write decimals with words and symbols; use place
value to describe decimals in terms of thousands, hundreds, tens, ones,
tenths, hundredths and thousandths.
4.1.2.5 Compare and order decimals and whole numbers using place
value, a number line and models such as grids and base-10 blocks;
Instructional strategies may be used for students struggling with the
concept of decimals in 5th and 6th grade.
Locate positive rational numbers on a number line.
Compare positive rational numbers represented in various forms.
Misconceptions
Students may think:
 The word and can be used anywhere in a number and has no significance. They do not realize
the use of the word and implies the decimal point.
 The words tens and tenths, hundreds and hundredths, thousand and thousandths are
interchangeable.
 0.52 is greater than 0.8 because the whole number 52 is greater than the whole number 8.
 Decimals get smaller as the number of digits to the right of the decimal point increases. For
example, students think 0.9 is less than 0.233 because two hundred thirty-three thousandths
has more digits to the right of the decimal point.
 Misunderstanding of placement of decimal point in a whole number (3 could be represented as
3.)
Reference: Book Math Misconceptions pp. 43-48.
Teaching Student-Centered Mathematics by John A. Van de Walle pp. 222-229
Materials
 Base-10 blocks (5 flats, 25 longs, 50 units)
 Decimal points
 Decimal recording sheet
 Number line
 Math notebooks
 Dice (for independent partner work after flex group instruction)
Stages
1.
2.
3.
4.
5.
Build the decimal concretely using base-10 blocks and say the decimal correctly.
All of the above and draw the base-10 representation and write the decimal in words.
All of the above and represent by shading the whole grid to represent the quantity.
All of the above plus estimate on the number line.
All of the above and write the fraction ( 2 ).
10
6. Compare decimals

1
7. Equivalent decimals (.02 and 0.20)
-1
_____
______
_____
3
Directions: Representing Decimals Using Base-10 Blocks
Introduction to representation of decimals for tenths and hundredths:
 Explain the value of the base-10 blocks when used for decimals.
 Discuss how many tenths are needed to make one whole and how many hundredths are
needed to make one whole.
 Give the students two decimal points and base-10 blocks. Write the decimal 0.8. Students write
the decimal on the Decimal recording sheet and say eight-tenths.
 Students then create the representation of 0.8 using base-10 blocks.
 Once all students are finished, Student A explains the representation of the decimal using the
vocabulary tenths and hundredths in the explanation.
 Students then draw their base-10 representation of the decimal.
 Have students point and estimate the location of the decimal on the number line. When the
students estimate, relate the estimation to 1 . Is the decimal more than, less than or equal to 1 ?


2
2
Have students shade in the representation of 0.8 on the grid and record ‘eight tenths’ under the
grid. If there are different shading representations, discuss how they are different and how they
are the same.


8
Students record the fraction for 0.8 ( ) on the Decimal sheet.
10

8
10
0.8
eight tenths






Keeping the above base-10 representation on the table for 0.8, write the decimal 0.08 and have
students record the decimal on their sheet.
Students create this decimal representation using the base-10 blocks. Be sure students create
this second decimal representation with base-10 blocks below their first model so they visually
see the difference between tenths and hundredths.
Student B describes this concrete representation and then explains how the two decimals
differ.
Students draw the base-10 representation of 0.08.
Students shade in the representation of 0.08 on the grid and write eight hundredths under the
grid.
Have students point and estimate the location of the decimal on the number line. When the
students estimate, relate the estimation to 1 . Is the decimal more than, less than or equal to 1 ?

2
2
Also have students relate the location to 0.8. Is this decimal more than, less than or equal to
0.8?


2

Students record the fraction for 0.08 ( 8 ) on the Decimal sheet.
100

8
100
0.08
eight hundredths


Continue the process giving each student the opportunity to create concrete representations of
the decimals, explain the model, record the drawing in the math notebook on the Decimal sheet
and then compare the tenths and hundredths.
Flex Group Problems:
o First two problems are just tenths and hundredths, the next two problems include a
whole number:
 0.8, 0.08
 0.03, 0.3;
 1.05, 1.5
 1.6, 1.36

Student Math Notebook Samples (glue in the recording sheet for students to use).
Title written at the top of the page:
Decimals
Date/Time
Purposefulness of Chosen Problems:
 0.8 and 0.08 (one decimal is greater than 1 the other less than 1 .)
2
2
 0.03 and 0.3 (both decimal are less than 1 .)
2
1
 1.05 and 1.5 (One is closer to 1 andthe other one is equal
 to 1 .)
2
 1.36 and 1.6 (students might think 1.36 is greater than 1.6 if they look only at the digits.)

3
Directions for Partner Work
 When all students in the Flex Group have explained their thinking, partners continue the activity
on their own.
 Students create two models using base-10 blocks and then compare the two quantities.
 Partner A rolls the dice (2 dice for tenths and hundredths, 3 dice if students are using whole
numbers),
 Students write the decimal, create with their base-10 blocks and then complete the Decimal
sheet.
 Students record their estimated location on the number line.
 When finished with the first problem, Partner B rolls the dice and continues the process.
 Once two decimals are created, students use the Student Sentence Frames to compare the
models.
 Teachers can check student understanding by looking at their math notebooks.
Student Language Frames (what the students say to each other)
“Who is Partner A and who is Partner B?”
“Our decimal is ___________________.”
(tenths, hundredths)
“The decimals are different because _______________.”
Variation: Regrouping tenths to hundredths
 To help students understand the concept of 0.3 having the same value as 0.30, provide
practice of regrouping tenths to hundredths with students explaining their thinking.
 For example, create 1.2 using base-10 blocks. Then regroup the 2 tenths to the same quantity
in hundredths. Create this model below the first model and label as 1.20.
 Students shade two grids representing the two decimals to help them compare the quantities.
 Have each student explain their thinking as to why the decimals are equivalent.
Ordering Decimals for Real-world Problems
Directions: Use the decimal sheet to create the models. Then answer the questions below.
 For science class, Jose, Maria, Xia and Davante created paper airplanes. Their task was to
decide which model flew the farthest. Here are the distances
o Jose: 1.03 meters
o Maria: 1.1 meters
o Xia: 1.11 meters
o Davante: 1.08 meters
 Place the distances in order from shortest to longest. Have Student A explain their thinking for
this question: “Whose model flew the farthest? How do you know?”
 Have Student B explain his/her thing for this question: “Whose model flew the shortest
distance? How do you know?”
 Looking at the table below, place the bodies of water in oder from least amount of salt per liter
to most amount of salt per liter.
Salt Per Liter in Major Bodies of Water
Body of
Water
Salt per
Liter
Arctic Ocean
Dead Sea
Caspian Sea
Black Sea
0.03 kg
0.28 kg
0.01 kg
0.02 kg
4
 Once all students have placed the salt amounts in order from shortest to longest, have Student
C explain his/her thinking for this question: “Which body of water has the most salt? How do
you know?”
 Have Student D explain his/her thing for this question: “Which body of water has the least salt?
How do you know?”
Variation: Adding Decimals Using Base-10 Blocks
 Student A rolls two dice and creates a decimal in the tenths place using the digits (for example,
2.4). Student B rolls two dice and creates a second decimal in the hundredths place (for
example, 3.02).
 Students A and B both create a model for the two decimals and find the sum by joining the
concrete materials.
 Both students draw the representation of this process in their math notebooks labeling each
picture with the decimal.
 Students record the sum of the two decimals.
 After students have worked on this process, have them explain how they would abstractly add
the two decimals without using base-10 bocks. They should be able to explain the placement of
the decimal point by this time.
Whole Number Misconception
 Students often have difficulty deciding where the decimal point is located when there is just a
whole number. Be sure some of your problems are just whole numbers but have students place
the decimal point in their model to help with placement of the decimal misconception.
 For example, have students create a model of 1.31, 1.01 and 2. They are to use the base-10
blocks and a decimal point for each model.
 Observe how they create the model of 2. Have the students explain their thinking about the
decimal point placement.
Example: Model of 1.31
Example: Model of 1.01
5
Example: A model of 2
Variation: Decimal Number Line with Fractions and Coin Values




0
0.1
0.2
0.3
0.4
0.5
0
1/10
2/10
3/10
4/10
5/10
Continue
number
line to 2
Create a number line on the floor with sheets of paper with decimals written on them. Have
students use 0 to 2 and the decimals in between these two whole numbers. (0.1, 0.2, … 1.1.
1.2,) Students place the numbers on the floor and then create the decimals using base-10
blocks to represent the decimal. Place the base-10 blocks above each of the decimal sheets.
Have students read the decimals always using the word “and” not “point”.
Using the number line, have students stand at a decimal and then move forward or backward
as you add 0.1 or subtract 0.1. When adding, students walk forward. When subtracting,
students walk backwards. For example, I have 1.2 if I add 0.1 where am I located? Write the
addition model for students to see. (1.2 + 0.1 = 1.3)
Using the number line, have students create the corresponding fraction and place this paper
below the decimal.
Using the number line, place coins to represent the decimals. For example, 0.1 would be one
dime.
6
Student Language Frames for Comparing Decimals
Student Language Frames for Decimals
“Who is Partner A and who is Partner B?”
“Our decimal is ___________________.”
(tenths, hundredths)
“The decimals are different because _______________.”
Student Language Frames for Decimals
“Who is Partner A and who is Partner B?”
“Our decimal is ___________________.”
(tenths, hundredths)
“The decimals are different because _______________.”
Student Language Frames for Decimals
“Who is Partner A and who is Partner B?”
“Our decimal is ___________________.”
(tenths, hundredths)
“The decimals are different because _______________.”
Student Language Frames for Decimals
“Who is Partner A and who is Partner B?”
“Our decimal is ___________________.”
(tenths, hundredths)
“The decimals are different because _______________.”
Student Language Frames for Decimals
“Who is Partner A and who is Partner B?”
“Our decimal is ___________________.”
(tenths, hundredths)
“The decimals are different because _______________.”
Decimals
7
Complete recording the whole numbers below the number line. Label
Then record the location for each decimal created on the chart.
1
on the number line.
2

-1
Decimal
_____
Base-10 Representation
____
____
3
Part of a Whole Representation
Fraction
Decimal written in words: ______________________
Decimal written in words: ______________________
Decimal
Base-10 Representation
Part of a Whole Representation
Fraction
Decimal written in words: _____________________
Decimal written in words: ______________________
8