Learning Math: Number and Operations
Number Sets, Infinity, and Zero Homework
Divide the number 1 by the numbers 1 through 10 consecutively. What conjectures can you make about rational numbers when represented as decimals?
If we think of division as a repeated subtraction, can you explain why it is impossible to divide by 0?
In a hotel with an infinite number of rooms and a counting number assigned to each, there is a “No Vacancy” sign outside. A traveler comes in and asks for a room for the night. How does the staff accommodate this traveler?
Think of the traveler as one more element to add to a countably infinite set. Is the new set also countably infinite?
In a hotel with an infinite number of rooms and a counting number assigned to each, there is a “No Vacancy” sign outside. An infinite marching band — one where each member has a unique number on his or her uniform — enters and asks for a room for the night for each musician. How does the hotel staff accommodate everyone?
Think of the band and the rooms as two infinite sets to be added together. What kind of set do you get? How can you put this new set into one-to-one correspondence with the counting numbers?
There’s an infinite chain of infinite hotels, each with a unique address on the street. All of them are full. But one night, very suddenly, all but one of them go out of business! How does the one remaining hotel accommodate all the stranded guests?
Think of this as adding together an infinite number of infinite sets. This will be similar to putting rational numbers into one-to-one correspondence with the counting numbers.
Read History and Transfinite Numbers: Counting Infinite Sets.
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History and Transfinite Numbers: Counting Infinite Sets.
1/1 = 1
1/2 = 0.5
1/3 = 0.333…
1/4 = 0.25
1/5 = 0.2
1/7 = 0.142857142857…
1/8 = 0.125
1/9 = 0.111…
1/10 = 0.1
Some rational numbers when expressed in decimal form will terminate, such as 0.5 or 0.125. Others will have repeating, non-terminating decimals where the repeating part can be a single digit, such as in 1/3 = 0.333…, or six digits, such as in 1/7 = 0.142857…
To further explore why this happens and why some rational numbers terminate and others don’t, go to Number and Operations, Session 7.
If we think of division as repeated subtraction, in essence, we are subtracting groups of 0 from the number we are dividing. It is easy to see that you could subtract groups of 0 infinitely many times and never exhaust the number you started with. Therefore, dividing by 0 is not defined.
The easiest solution would be to move everyone in the hotel one room over. This way, the first room would be freed up for the traveler. Notice that here you’ve added one element to an infinitely countable set. The result is still an infinitely countable set:
To put everyone into a room, you would alternate the guests (G) already there and the marching band guests (MB). In other words, the guests who were already in the hotel would be moved into rooms with odd numbers:
By combining the two infinite sets in this way, you still get a countably infinite set that can be put into one-to-one correspondence with the counting numbers.
By writing down all the rooms in all the hotels in an infinite two-dimensional matrix (see below), they can all then be put into one-to-one correspondence with the counting numbers, each of which corresponds to a particular room:
Notice that here you have multiples of infinitely countable sets combined into one set. The new set is also countably infinite, which you’ve shown by putting it into one-to-one correspondence with the counting numbers.
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.