Learning Math: Number and Operations
Place Value Homework
Write the base five numbers 1234five and 1.234five as base ten numbers.
Find the base ten fractions represented by the following:
a. a. 0.1five, 0.2five, 0.3five, and 0.4five
b. b. 0.01five, 0.02five, 0.03five, and 0.04five
c. c. 0.12five, 0.23five, 0.34five, 0.43five
Find the base five representation for these base ten fractions:
a. If you were counting in base five, what number would you say just before you said 100?
b. In base five, what number is one more than 344?
c. What is the greatest three-digit number that can be written in base five? What numbers come just before and just after this number?
What number in base five behaves the way 3 does in base four?
a. Count by twos to 30five.
b. In base five, how can you tell if a number is even?
c. Count by threes to 30five.
TAKE IT FURTHER
Find the base five representation for the base ten fraction 1/2.
How might you tell if a number is even or odd in bases two, three, four, five, six, seven, eight, nine, and ten? Can you generalize to base n?
In order to use base sixteen, we need 16 digits. However, we only know 10 digits — 0, 1, 2, … , 8, and 9 — so to represent 10, 11, 12, 13, 14, and 15 in base sixteen, we’ll use A, B, C, D, E, and F, respectively. This gives us the following representation for the base sixteen digits:
Remember that 16 in this base is written as 10 (one-zero). So, for example, the number A6sixteen becomes (10 • 16) + 6, or 166, in base ten. The number 123 in base ten is (7 • 16) + 11 , or 7B, in base sixteen.
Now translate these base sixteen numbers into base ten numbers:
Using the same system, translate these base ten numbers into base sixteen numbers:
1234five = (1 • 53) + (2 • 52) + (3 • 51) + (4 • 50) = 125 + 50 + 15 + 4 = 194 in base ten.
1.234five = (1 • 50) + (2 • 5-1) + (3 • 5-2) + (4 • 5-3) = 1 + 2/5 + 3/25 + 4/125 = 194/125 = 1 69/125 in base ten.
a. The base ten fraction for 0.1five is 1/5. The others are 2/5, 3/5, and 4/5.
b. The base ten fraction for 0.01five is 1/25. The others are 2/25, 3/25, and 4/25.
c. The base ten fraction for 0.12five is 7/25 (1/5 + 2/25). The others are 13/25 = 2/5 + 3/25, 19/25 = 3/5 + 4/25, and 23/25 = 4/5 + 3/25.
a. 9/25 = 5/25 + 4/25 = 1/5 + 4/25 = 0.14five
b. 23/125 = 20/125 + 3/125 = 4/25 + 3/125 = 0.043five
a. The number just before 100 is 44. The count goes …40, 41, 42, 43, 44, 100….
b. The number that is one larger than 344 is 400. Carrying a 1 to the next digit is required for both the ones digit and the fives digit.
c. The greatest three-digit number that can be written in base five is 444. Just before this number is 443, and just after it is 1000.
Four. In base four, 3 is the greatest digit; adding 1 to a 3 requires regrouping. In base five, this is true of the number 4. In general, in any base n, the number n – 1 will be the greatest digit and will require regrouping when 1 is added to it.
a. The count goes 2, 4, 11, 13, 20, 22, 24. Note that 30five is not a multiple of 2.
b. It is more difficult to decide this in base five. The easiest way is to look for the sum of the digits of a number. If the sum is even, the number is even.
c. The count goes 3, 11, 14, 22, 30.
We know that in base five, we write a fraction as (A • 50) + (B • 5-1) + (C • 5-2) + (D • 5-3) + …, where A, B, C, and D can only be 0, 1, 2, 3, or 4. To tackle this problem, refer to the algorithm we used in Problem A2. There, we found the largest number we could make as a power of 2, then subtracted to find a remainder, then continued to the next power of 2. Here, we’ll do this with powers of 5.
First we find A: 1/2 = A • 50 + …. Since 50 = 1, which is larger than 1/2, we cannot make 50 from 1/2. Therefore, A = 0.
Now find B: 1/2 = B • 5-1 + …. Since 5-1 = 1/5 = 0.2 and 1/2 = 0.5, we can make two 5-1s from 1/2. Therefore, B = 2. Now subtract 2/5 from 1/2 to leave 1/10, the remainder.
So far, our base five decimal is 0.2____. Now find C using the remainder from the previous step: 1/10 = C • 5-2 + …. Since 5-2 = 1/25 = 0.04 and 1/10 = 0.1, we can make two 5-2s from 1/10. Therefore, C = 2. Now subtract 2/25 from 1/10 to leave 1/50, the remainder.
At this point, our base five decimal is 0.22___. Now find D using the remainder from the previous step: 1/50 = D • 5-3 + …. Since 5-3 = 1/125 = 0.008 and 1/50 = 0.02, we can make two 5-3s from 1/50. Therefore, D = 2. Now subtract 2/125 from 1/50 to leave 1/250, the remainder.
You might have noticed a repeating pattern. This is caused by the fact that our remainder has been one-fifth of the previous remainder at each step, and that we are using base five. This means that the pattern will continue indefinitely, and 1/2 = 0.2222222…, a repeating decimal, in base five. This is a perfect case to support the saying “A picture is worth a thousand words.” Here is this proof in visual form:
|Each piece is one-fifth
of the whole.
|This line splits the
whole into two halves.
|Half of the whole is two-fifths and half of a third fifth.|
The third fifth can be further divided into five 25ths, and the drawings will look like the ones above, except that every piece will be 1/25 instead of one-fifth. This process continues.
For each place value, we need two of the five parts (i.e., two fifths, two 25ths, two 125ths, and so on). The decimal is 0.222222….
Here is a count of the first seven multiples of 2, from base two to base ten:
Base two: 10, 100, 110, 1000, 1010, 1100, 1110
Base three: 2, 11, 20, 22, 101, 110, 112
Base four: 2, 10, 12, 20, 22, 30, 32
Base five: 2, 4, 11, 13, 20, 22, 24
Base six: 2, 4, 10, 12, 14, 20, 22
Base seven: 2, 4, 6, 11, 13, 15, 20
Base eight: 2, 4, 6, 10, 12, 14, 16
Base nine: 2, 4, 6, 8, 11, 13, 15
Base ten: 2, 4, 6, 8, 10, 12, 14
An important note is that the numeral 10 is a multiple of 2 only when the base is an even number. When the base is an even number, the units digits of even numbers repeat, so we need only look at the units digit of a number to determine if it is odd or even. If the units digit is even, the number is even.
If the base is an odd number, the units digit is not enough information to determine if a number is even. In odd bases, it is the sum of the digits that determines whether a number is even — if the sum is even, the number is even.
a. 6Dsixteen = (6 • 16) + 13 = 109ten
b. AEsixteen = (10 • 16) + 14 = 174ten
c. 9Csixteen = (9 • 16) + 12 = 156ten
d. 2Bsixteen = (2 • 16) + 11 = 43ten
a. 97 = (6 • 16) + 1 = 61sixteen
b. 144 = (9 • 16) = 90sixteen
c. 203 = (12 • 16) + 11 = CBsixteen
d. 890 = (3 • 256) + 7 • 16 + 10 = 37Asixteen
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.