Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Monthly Update sign up
Mailing List signup
Search
Follow The Annenberg Learner on LinkedIn Follow The Annenberg Learner on Facebook Follow Annenberg Learner on Twitter
MENU
Learning Math Home
Number and Operation Session 3, Part C: Place-Value Representation in Base Ten and Base Four
 
Session 3 Part A Part B Part C Homework
 
Glossary
number Site Map
Session 3 Materials:
Notes
Solutions
Video

Session 3, Part C:
Place-Value Representation in Base Ten
and Base Four
(40 minutes)

In This Part: Examining Base Four | Operations in Base Four

In Part C, we shift our focus to the base four number system. You will learn how to interpret whole numbers, common fractions, and decimals using this system.

In base ten, 123 means (1 • 100) + (2 • 10) + (3 • 1), and 1.23 means (1 • 1) + (2 • [1/10]) + (3 • [1/100]). Or, to put it another way:

123ten= (1 • 102) + (2 • 101) + (3 • 100)

and

1.23ten= (1 • 100) + (2 • 10-1) + (3 • 10-2)

We can represent each of the place values in the base ten number 123 with pieces of 100 units (102), 10 units (101), and one unit (100). They are called flats, longs, and units respectively.

The base four number system uses these place values:

44

43

42

41

40

4-1

4-2

256

64

16

4

1

1/4

1/16


So in base four, 123 means:

and 1.23 means:

(1 • 16) + (2 • 4) + (3 • 1),

(1 • 1) + (2 • [1/4]) + (3 • [1/16]),

or 

or 

123four = (1 • 42) + (2 • 41) + (3 • 40),

1.23four = (1 • 40) + (2 • 4-1) + (3 • 4-2).

We can represent each of the place values in the base four number 123 with pieces of 16 units (42), four units (41), and one unit (40). They are called flats, longs, and units respectively.

Problem C1

Solution  

Write the base four numbers 123four and 1.23four in expanded notation and complete the base ten value of the number.


 

Problem C2

Solution  

Find the base ten fractions represented by the following:

a. 

0.1four, 0.2four, and 0.3four

b. 

0.01four, 0.02four, and 0.03four

c. 

0.11four, 0.12four,and 0.13four


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
What number is represented by "11" in base four? It is not what we would call 11!   Close Tip

 

Problem C3

Solution  

Find the base four representation for these base ten fractions:

a. 

1/2

b. 

5/8

c. 

7/8

d. 

1/64

Note 2


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Play with these fractions to get them into the desired form x/4 + y/16 + z/64.... Remember that in base four, the face values of x, y, and z can only be digits 0 though 3.   Close Tip

 

Problem C4

Solution  

a. 

If you were counting in base four, what number would you say just before you said 100? (Read this number as "one-zero-zero," not "one hundred.")

b. 

What number is one more than 133? (Read this number as "one-three-three.")


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Use the base four blocks diagram above.   Close Tip

 
 

c. 

What is the greatest three-digit number that can be written in base four? What numbers come just before and just after this number?



video thumbnail
 

Video Segment
In this video segment, Ben and Liz work with manipulatives to represent numbers in base four and solve some arithmetic problems. They realize that they need to move to the next place value in base four. Watch this segment after you've completed Problems C1-C4.

Did you find it necessary to think about what each place value means in order to solve these problems in base four?

If you are using a VCR, you can find this segment on the session video approximately 16 minutes and 15 seconds after the Annenberg Media logo.

 

 

Problem C5

Solution  

a. 

Count by twos to 30four.

b. 

In base four, how can you tell if a number is even?


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Look for a pattern in the results of the first question to help you answer the second.   Close Tip

 
 

c. 

Count by threes to 30four.


Next > Part C (Continued): Operations in Base Four

Learning Math Home | Number Home | Glossary | Map | ©

Session 3: Index | Notes | Solutions | Video

© Annenberg Foundation 2014. All rights reserved. Legal Policy