Learning Math: Number and Operations
Fractions and Decimals Homework
a. Find the fractional equivalent for 0.142857.
b. Find the fractional equivalent for 0.142857142857142857….
Shigeto and Consuela were computing the decimal expansion of 1/19. Since Shigeto used scratch paper, he had only a little room to write his answer. He continued writing the digits on the next line, and in the end, his answer looked like this:
Shigeto noticed a pattern in these numbers. Describe his pattern.
Look at the columns of digits. What do you notice?
Consuela did her computation on a narrow notepad. Her answer looked like this:
After looking at Shigeto’s pattern, Consuela tried to find a pattern in her answer. What observations can you make about Consuela’s pattern?
Look at the columns of digits. What do you notice?
David started to compute the decimal expansion of 1/47. He got tired after computing this much of the expansion:
Shigeto had no trouble finishing the expansion using his pattern. How about you? Can you finish the expansion and explain your answer?
What are the possible periods for the decimal expansion of 1/47? Can you predict the actual period based on how many digits there are in the decimal expansion so far?
How would you arrange those digits so that the columns add up to 9?
Does the length of the period of your expansion make sense? Explain why or why not.
Consuela looked at David’s work and knew immediately that her method would not be helpful. Explain why not.
Mr. Teague asked the class to compute a decimal expansion with period 42. Unfortunately, his dog spilled paint on Shigeto’s and Consuela’s answers. Use the visible information in Shigeto’s and Consuela’s answers and the patterns you have seen to find the complete decimal expansion:
Find the fraction with this particular decimal expansion.
Is it possible to represent the number 1 as a repeating decimal?
Think about the decimal expansion for 1/3 = 0.333333…. What would 2/3 be? What fraction would be closest or equal to 1?
Is it possible to predict the period of 1/14 if you know the period of 1/7 (i.e., six)?
Is it possible to provide a convincing argument to prove that the decimal expansion of 1/n has a period that is less than n?
Problems H2-H11 adapted from Findell, Carol, ed. Teaching with Student Math Notes, Volume 3. p. 121. © 2000 by the National Council of Teachers of Mathematics. Used with permission. All rights reserved.
a. 0.142857 = 142,857/1,000,000.
b. F = 0.142857142857. . . . There are six digits repeated, so multiply by 1,000,000 (106) to get 1,000,000F = 142,857.142857142857. . . . Subtracting F from each side gives you 999,999F = 142,857, so F = 142,857/999,999, which can be reduced to F = 1/7.
Adding the two groups of nine numbers gives you 99,999,999.
Note that if the period length is even, you can always break the number into two rows where columns of digits will add up to 9.
Adding the three groups of six numbers gives you 999,999.
If the period length is a multiple of three, you can always break the number into three rows where each column will either add up to 9 or will be a two-digit number ending in 9. Note that this works only if we break a number into two or three rows. No other break-up of the number will work all the time.
Since the sum should be all 9s, the pattern would continue:
Note that the repetition starts after 46 decimal places, so the period is 46.
Yes, it makes sense. As with other prime numbers, the period is a factor of one less than the number itself. In this case, with the prime number 47, the period would need to be either 46, 23, 2 or 1. We already have 23 digits and they haven’t repeated, so the period must be 46.
Consuela’s method is not helpful because 3 is not a factor of the period, 46. Her method relies on being able to arrange the non-repeating digits into three rows of identical lengths.
We can primarily refer to Consuela’s numbers, starting from the right side. If the sum is going to be all 9s, the right-most covered digits must be 775 51 (don’t forget to carry). We can then use Shigeto’s numbers 408 163 (uncovered) to find the rest of Consuela’s covered bottom row: 346 938 (the 34 is visible on Shigeto’s page). Finally, we can complete Consuela’s covered top row: 020. The completed number is
As a check, you might verify that these numbers work by using Shigeto’s rule; i.e., that they add (in pairs) to 9s.
Going on the assumption that the number is in the form 1/n, n must be less than 50 (since 1/50 = 0.02) but greater than 47 (1/47 = 0.021276. . ., as seen in Problem H3). So n is either 48 or 49; 1/48 = 0.208333. . ., which is too large, and 1/49 is just right.
If we could not make this assumption, we could use the multiply-and-subtract method from Part B, but we’d need a really accurate calculator!
Yes, 1 can be represented as 0.99999…. The technique of Part B can be applied to this decimal; if F = 0.999…, then 10F = 9.999…, and, subtracting F from each side, 9F must equal 9. This means that F = 1!
Another way to convince yourself that this is true is that 0.333… represents 1/3. Then 2/3 is represented as 0.666…, and 3/3 is represented as 0.999. …Since 3/3 = 1, 1 and 0.999… are the same number.
All terminating decimals have an alternate representation in this form. For example, 0.25 can also be represented as 0.24999999….
Yes. Because 2 is a factor of 10, it will have no effect on the period; it will only delay the repetition by one decimal place. So 1/14 = 0.0714285714285. . ., which, like 1/7, has a period of six. Similarly, 1/28 will also have a period of six, delayed by two decimal places.
One convincing argument is that when dividing by n, there are only n possible remainders: the numbers (0, 1, 2, 3, 4, …, n – 1). If we divide and get a remainder of 0, then the division is complete, and the resulting decimal is terminating. If we divide and get a remainder that we have seen earlier in our division, this means that the decimal is about to repeat. So if we want the longest possible period for our decimal, we want to avoid 0 but run through every possible number before repeating. To do this, we would hit all the non-zero remainders (1, 2, 3, …, n – 1) while missing 0 — a total of n – 1 possible remainders. Since each remainder corresponds to continuing the decimal by one place, there is a maximum of n – 1 decimal places before repetition begins.
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.