Learning Math: Number and Operations
Place Value Part C: Place-Value Representation in Base Ten and Base Four (40 minutes)
In This Part: Examining 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)
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:
|So in base four, 123 means:||and 1.23 means:|
(1 • 16) + (2 • 4) + (3 • 1),
(1 • 1) + (2 • [1/4]) + (3 • [1/16]),
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.
Write the base four numbers 123four and 1.23four in expanded notation and complete the base ten value of the number.
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
What number is represented by “11” in base four? It is not what we would call 11!
Find the base four representation for these base ten fractions:
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.
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.”)
Use the base four blocks diagram above.
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?
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?
You can find this segment on the session video approximately 16 minutes and 15 seconds after the Annenberg Media logo.
a. Count by twos to 30four.
b. In base four, how can you tell if a number is even?
Look for a pattern in the results of the first question to help you answer the second.
c. Count by threes to 30four.
In This Part: Operations in Base Four
The following Interactive Activity provides you with some base four blocks you can use to visualize place value in base four operations. The pieces are called flats, longs, and units, and they represent 42, 41, and 40, respectively. Use the blocks to solve Problem C6.
Remember that there are four digits in base four (0 through 3). For addition problems, every time you have four blocks of the same type (a given place value), you need to trade them (or regroup them) for one block of the next-larger place value on the left. So, for example, four blocks representing 40 (four units) would be traded for one block representing 41 (one long). Use the regroup button in the activity to do this.
Similarly, for subtraction, every time you don’t have enough blocks in a given place value to do the subtraction, you need to take one larger block from the place value on the left and trade it for four blocks of the next-smaller place value on the right. So, for example, one block representing 41 (one long) can be traded for four blocks representing 40 (four units). Use the regroup button in the activity to do this. See Note 3 below.
For a non-interactive version of this activity, print and cut out several base four blocks PDF to use with Problem C6.
Complete the following calculations of these base four numbers:
a. 33four + 11four
b. 123four + 22four
c. 223four – 131four
d. 112four – 331four
Why is computing in different bases useful? In this segment, Mr. Glasgow and Mr. Marable explain how today’s computer technology relies on base sixteen numbers in order to compress complex computer operations.
Can you make a prediction about what number systems computers will use in the future?
You can find this segment on the session video approximately 22 minutes and 24 seconds after the Annenberg Media logo. The second part begins approximately 25 minutes and 14 seconds after the Annenberg Media logo.
We will learn more about converting fractions to decimals in Session 7.
Think of how the base four blocks help you understand operations with base four numbers. Similarly, contemplate how base ten blocks would have helped you when you learned to add, subtract, multiply, and divide in base ten.
123four = (1 • 42) + (2 • 41) + (3 • 40) = 16 + 8 + 3 = 27. The illustration demonstrates the number 27 divided into groups of 42, 41, and 40.
1.23four = (1 • 40) + (2 • 4-1) + (3 • 4-2) = 1 + 2/4 + 3/16 = 1 11/16.
a. 0.1four is equal to 1/4 in base ten. 0.2four is twice 0.1four, so it is 2/4 = 1/2. 0.3fouris three times 0.1four, so it is 3/4 in base ten.
b. 0.01four is equal to 1/16 in base ten. 0.02four is twice 0.01four, so it is 2/16 = 1/8. 0.03four is three times 0.01four, so it is 3/16 in base ten.
c. 0.11four = 0.1four + 0.01four. This sum is 1/4 + 1/16 = 5/16. Similarly, 0.12four = 1/4 + 2/16 = 6/16 = 3/8, and 0.13four= 1/4 + 3/16 = 7/16.
a. 1/2 is equivalent to 2/4, so as a base four decimal, it would be written as 0.2four.
b. 5/8 = 4/8 + 1/8 = 2/4 + 2/16 = 0.22 in base four.
c. 7/8 = 6/8 + 1/8 = 3/4 + 2/16 = 0.32 in base four.
d. 1/64 = 0.001 in base four.
a. Since the digits in base four are 0, 1, 2, and 3, the last digit before “rolling over” to the next place value is 3. The number just before 100 is 33 (. . ., 30, 31, 32, 33, 100).
b. One more than 133 is 200. Adding 1 to 133 gives us one 16, three 4s, and four 1s. Four 1s equals one 4, so this gives us one 16, four 4s, and zero 1s. Four 4s equals one 16, so we now have two 16s, zero 4s, and zero 1s — 200.
c. The greatest three-digit number is 333. Just before 333 is 332, and just after 333 is 1000.
a. The count goes 2, 10, 12, 20, 22, 30.
b. Even numbers in base four end in 0 or 2. This can be seen by the previous question; any number in the list formed from counting by twos is even.
c. The count goes 3, 12, 21, 30.
a. 33four + 11four =110four
b. 123four + 22four = 211four
c. 223four – 131four = 32four
d. 112four – 33four = 13four
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.