Have the students circle the repeating remainders as shown in the above graphic. Answer: Question: What part of your calculations causes the decimal to repeat? When a remainder repeats, the calculations that follow must also repeat in a cyclical pattern, causing the digits in the quotient to also repeat in a cyclical pattern. Instead, we indicate that the decimal has a repeating pattern by placing a bar over the shortest sequence of repeating digits (called the repetend). We cannot possibly write the exact value of the decimal because it has an infinite number of decimal places. 8 Students notice that since the remainders repeat, the quotient takes on a repeating pattern of 3 s.
Because of this pattern, the decimal will go on forever, so we cannot write the exact quotient. This repeating remainder causes the numbers in the quotient to repeat as well. The remainders repeat, yielding the same dividend remainder in each step. Date: 4/8/14 138Ĥ Example 4 (5 minutes): Converting Rational Numbers to Decimals Using Long-Division (Part 2: Repeating Decimals) Example 4: Converting Rational Numbers to Decimals Using Long-Division Use long division to find the decimal representation of. Scaffolding: For long division calculations, provide students with graph paper to aid their organization of numbers and decimal placement. Answer: Exercise 1 (4 minutes) Exercise 1 Students convert each rational number to its decimal form using long division. Dividing by gives us, but we know the value must be negative. We know that, so we use our rules for dividing integers. Scaffolding: ELL Learners Review vocabulary of long division, i.e., algorithm, dividend, divisor, remainder. The fraction is a negative value so its decimal representation will be as well. Date: 4/8/14 137ģ Example 3 (3 minutes): Converting Rational Numbers to Decimals Using Long-Division (Part 1: Terminating Decimals) Example 3: Converting Rational Numbers to Decimals Using Long-Division Use the long division algorithm to find the decimal value of. What do these fractions have in common? Each denominator contains at least one factor other than a or a. Terminating Non-terminating What do these fractions have in common? Each denominator is a product of only the factors and/or. Example 2 (4 minutes): Decimal Representations of Rational Numbers Example 2: Decimal Representations of Rational Numbers In the chart below, organize the fractions and their corresponding decimal representation listed in Example 1 according to their type of decimal. Therefore, other fractions must be represented by decimals that do not terminate. We have seen already that fractions with powers of ten in their denominators (and their equivalent fractions) can be represented as terminating decimals. All rational numbers can be represented in the form of a decimal. All quotients have decimal representations but some do not terminate (end). Question: Did you find any quotients of integers that do not have decimal representations? No. What two types of decimals do you see? Some of the decimals stop and some fill up the calculator screen (or keep going). Record your fraction representations and their corresponding decimal representations in the space below. Using the division button on your calculator, explore various quotients of integers 1 through 11. Example 1: Can All Rational Numbers Be Written as Decimals? a. Question: What do you notice about the quotient? It does not terminate and almost all of the decimal places have the same number in them. Use the division button on your calculator to divide 1 by 6. Since is a fraction, we can divide the numerator by the denominator. Question: Is there another way to convert fractions to decimals? A fraction is interpreted as its numerator divided by its denominator. The equivalent fraction method will not help us write as a decimal.
This means we cannot write the denominator as a product of only s and s therefore, the denominator cannot be a power of ten. There are no factors of in the numerator, so the factor of has to remain in the denominator. Classwork Example 1 (6 minutes): Can All Rational Numbers Be Written as Decimals? Question: Can we find the decimal form of by writing it as an equivalent fraction with only factors of and/or in the denominator?. Students interpret word problems and convert between fraction and decimal forms of rational numbers.
Students represent fractions as decimal numbers that either terminate in zeros or repeat, and students represent repeating decimals using, a bar over the shortest sequence of repeating digits. 1 Lesson 14: Converting Rational Numbers to Decimals Using Long Division Student Outcomes Students understand that every rational number can be converted to a decimal.