CHAPTER 17 Developing Concepts of Decimals and Percents Elementary and Middle School Mathematics Teaching Developmentally Ninth Edition Van de Walle, Karp and Bay-Williams Developed by E. Todd Brown /Professor Emeritus University of Louisville

Big Ideas The base-ten place-value system extends infinitely in both directions: to very small and to very large values. Decimal (also called decimal fractions) is a way of writing fractions within the base-ten system. The decimal point is a convention that has been developed to indicate the unit’s position. Addition and subtraction with decimals are based on the fundamental concept of adding and subtracting the numbers in the like position values – an extension of whole numbers. Multiplication and division of two numbers will produce the same digits, regardless of the positions of the decimal point. Percents are simply hundredths and as such are a third way of writing both fractions and decimals.

Common Core State Standards 4th grade- understand decimal notation to hundredths 5th grade- performing operations with decimals to hundredth and expanding to thousandths and rounding decimals 6th grade- developing standard algorithms for all four operations with decimals to tenths and hundredths 7th grade- unified understanding of number so they can move fluently between decimals, fractions, and percents

10-to-1 Relationship Now in Two Directions

Role of Decimal Point Placement of the decimal point indicates the position of the unit The decimal point looks up at the units.

Connecting Fractions and Decimals Say decimals correctly “five and two tenths” not “five point two” Visual models for decimal fractions Translation of a fraction to a decimal using models.

Developing Decimal Number Sense Familiar fractions connected to decimals Use the base-ten models to have students show common fractions Translations of a fraction to a decimal using a number line and circular model.

Calculator Important Tool for Decimal Concepts Finding decimal equivalents with a calculator can produce interesting patterns. Here are some to explore- Which fractions have decimal equivalents that terminate? For a given fraction, how can you tell the maximum length of the repeating part of the decimal? Explore all of the ninths- 1/9, 2/9, 3/9. How can you find what fraction produces this repeating decimal 3.454545…?

Common Misconceptions with Comparing and Ordering

Density of Decimals Try these Activity 17.10 Close Decimal Activity 17.11 Zoom Materials- Clothesline or cash register tape Directions- Ask students to mark 0.75 and 1.0 on the line. Then ask students to “zoom in” and find and record three more values between those two.

Computation with Decimals Make whole-number estimates of the following: 4.907 + 123.01 + 56.1234 459.8 – 12.345 24.67 X 1.84 514.67 ÷ 3.59 SHARE YOUR THINKING AND DISCUSS STRATEGIES (front-end, rounding, compatible)

Multiplication of Decimals Problems with a context Encourage estimation- Is it more than 12 liters? What is the most it could be? Physical models- example of a ten-by-ten grid and number line

Where Does the Decimal Go? Multiplication Multiplying 3.4 x 1.7 is the same as 34/10 x 17/10 = 578/ 100 Rewritten as a decimal fraction, 5.78 corresponds to moving the decimal two places to the left. When the focus is on rote procedures the students lose out on understanding and meaning of operations.

Division of Decimals Estimate Quotients 45. 7 ÷ 1.83 Think of what times 1 8/10 is close to 46. Because 1.83 is close to 2, the estimate is near 23. Because 1.83 is less than two the answer must be greater than 23, perhaps 25 or 26. Actual quotient is 24.972677

Where does the Decimal Go? Division Ignore the decimal and divide as if you were using whole numbers. Then place the decimal using estimation.

Percent another name for hundredths Physical models and terminology link fractions, decimals, and percents ¾ 75/100 0.75 75%

More Physical Models Fractions, Decimals and Percents Part-whole fractions Bar diagrams to solve percent problems.

Estimation of Percent Problems The 83,000-seat stadium was 73 percent full. How many people were at the game? The treasurer reported that 68.3 percent of the dues had been collected, for a total of $385. How much money could the club expect to collect if all dues are paid? Use ¾ and 80,000 – about 60,000 Use 2/3 and $380; will collect 1/3 more- about $190