Learning Math: Number and Operations
What Is a Number System? Part B: Comparing Number Systems (15 minutes)
How does the number system you’ve been working with in Part A relate to the real number system you usually use? Through analyzing a small, finite system, you’ve gained some understanding of number systems in general. Using units digit arithmetic, you’ve created an independent number system which was relatively easy to manage.
You’ve considered only the units digit for any one answer and in so doing have limited the size of your number system to just 10 numbers — 0 through 9. You’ve seen that the finite system has its own addition and multiplication tables, additive and multiplicative inverse and identity elements, and computational rules. You’ve seen that the finite system does not have unique inverse elements for multiplication, as does the system of real numbers. Furthermore, you’ve seen that the computational rules sometimes coincide with those of the real number system. Let’s explore some of the similarities and differences between the two systems.
a. In what ways does addition act the same way in the finite system as it does in the infinite, real number system? Which rules are the same, and which are different?
b. What about subtraction?
a. In what ways does multiplication act the same way in the finite system as it does in the infinite, real number system? Which rules are the same, and which are different?
b. What about division?
a. Does the distributive law act the same way in the finite system as it does in the infinite, real number system?
b. Why do we need a distributive law?
In your opinion, what are the most important rules that apply to both this finite system and our infinite system?
a. The rules are very similar. Addition is closed, commutative, and associative; there is always exactly one answer to an addition problem. One difference is that the result of adding two numbers is not necessarily “larger” than the original numbers in the finite system; for example, 6 + 7 = 3. This means that the use of “larger” is not applicable to this system. Another difference is that this system has only 10 elements, whereas the real number system has an infinite number of elements.
b. The rules are very similar; there is always exactly one answer to a subtraction problem. One difference is that there are no “negative” numbers in the finite system. However, each number has an additive inverse, so if we interpret -b as the inverse of b, we can say that (a – b) is the same as (a + (-b)).
a. The rules are similar. Multiplication is closed, commutative, and associative. There is always exactly one answer to a multiplication problem. Multiplying by 0 results in 0, and multiplying by 1 results in the original number. One difference is that sometimes, in a finite system, you can multiply the same (non-zero!) number by two different numbers and get the same answer (for example, 4 • 2 and 4 • 7 both equal 8).
b. The rules are not as similar. In the finite system, some division problems have no solution (74, for example), though this is not very different from whole numbers. However, some division problems have more than one solution (84, for example, which in units digit arithmetic can be solved by both 2 and 7). Therefore, you cannot divide by 0, 2, 4, 5, 6, or 8 in this system. In the real number system, we can divide by all non-zero numbers. Also, in the finite system, as in the real number system, you can’t make sense of 0 divided by 0 because there are too many solutions, and you can’t make sense of number a divided by 0 for any other number a, because there are no solutions.
a. Yes, it acts in the same manner.
b. The distributive law ties together addition and multiplication: a • (b + c) = (a • b) + (a • c). To consider a system with two operations, we must have a property that tells us how the two operations are related. Otherwise, we would not know how to compute an expression that contains both addition and multiplication.
Similarly, in the real number system, the distributive law allows us to see how these operations relate to each other. It also allows us to use different methods to compute products. For example, if you need to multiply 25 by 99, the distributive law allows you to do the computation as 25 • (100 – 1), which you can do in your head.
Answers will vary.
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.