Decimals

1. Decimals Fourth Grade Decimal Teacher Resource Cluster 6

2. Fourth Grade NCSCOS Numbers & Operations - Fractions Understand decimal notation for fractions, and compare decimal fractions. NC.4.NF.6 Use decimal notation to represent fractions. • Express, model, and explain the equivalence between fractions with denominators of 10 and 100. • Use equivalent fraction to add two fractions with denominators of 10 and 100. • Represent tenths and hundredths with models, making connections between fraction and decimals. NC.4.NF.7Compare two decimals to hundredths by reasoning about their size using area and length models and recording their results of comparisons with the symbols >, =, or >. Recognize that comparisons are valid only when the two decimals refer to the same whole.

3. Decimal Numbers: Big Ideas • Decimal numbers are another way of writing fractions. • The base-ten place-value system extends infinitely in two directions. • “Each place has a value that is 10 times that of the place to the right (and 1/10 of the value of the place to the left)”. • The decimal point indicates the units position (to its immediate left).

4. Extending Place Value Patterns • Patterns that name periods of numbers are repeated for each set of 3 digits to the left of the decimal place. • The pattern continues in both directions. • The ones place anchors everything. Billions Millions Thousands Units Decimals hundred billions ten billions one billions ten millions ten thousands one thousands hundreds tenths hundredths thousandths one millions hundred millions hundred thousands tens ones

5. Decimals Related to Fractions • Decimals and fractions should not be taught in isolation from each other. • Students can learn decimals as a natural extension of what they understand about base ten and fractions. • Develop understanding of how the base-ten system extends to include numbers less than one. • If children have a basic understanding of fractions, then introducing decimals essentially involves introducing a new notation for familiar numbers.

6. Research about decimals says… • Students confronted with a new written symbol system such as decimals need to engage in activities (e.g., using base-ten blocks) that help construct meaningful relationships. • The key is to build bridges between the new decimal symbols and other representational systems (e.g., whole number place values and fractions) before “searching for patterns within the new symbol system or practicing procedures” such as computations with decimals (Hiebert, 1988; Mason, 1987). From Teaching and Learning Mathematics, Washington State Department of Public Instruction, 2000

7. Decimal Models • A variety of fractional models should be used in the instruction of decimal concepts. • Visual models include area and length models such as decimal grids, decimal circles, number lines, and meter sticks.

8. Models & Representations • Area/Region Models • Linear Models • Set Models

9. Introducing Decimals Partition each model into tenths and hundredths. How are the models alike? How are they different?

10. Decimal Circles • How can you partition the circle into tenths and hundredths?

11. Introduction to Decimals • Decimal numbers are like fractions. They identify quantities that are between whole numbers. • You can write numbers less than 1 by using a decimal point. • Our number system is based on tens. Decimal means 10.

12. What do you notice about the chart?

13. Decimal Circles Show the following numbers on the decimal circles. 0.4 0.65 0.08 *Which numbers are more than one-half? * What are some different ways to say each number? * How would you express each number as a fraction?

14. Decimal Circles Show 0.75 on the decimal circle. *Is the number more than one-half? *What are some different ways to say this number? *How would you express this number as a fraction?

15. Decimal Circles • Use the decimal circles to reinforce estimation. • With the blank side of the disk facing them, have students adjust the disk to show a particular number. Ex. 0.72 • Students then turn the disk over and see how close they were to the number. • Turn this into a game by calculating the difference between the actual number and the estimate.

16. Decimal Squares • How can you partition the square into tenths and hundredths?

17. Decimal Grids/Squares • What number is represented on the grid? • What part of the grid is shaded? • Give your answer as a fraction and as a decimal. • How many whole tenths are shaded? • How many extra hundredths?

18. Decimal Grids/Squares • What number is represented on the grid? • What part of the grid is shaded? • What part is not shaded? • Give your answer as a fraction and as a decimal.

19. Decimal Grid Art • Students create an artistic design on a 10 x 10 decimal grid and identify each color with decimals and fractions

20. Base Ten Blocks • How can you partition the blocks into tenths and hundredths? • If the flat is one, what is the value of the rod? What is the value of the unit? • Express the value as a decimal and as a fraction.

21. Decimal Models • With a given model, any piece could be chosen as the ones piece; thus the decimal point has the important role of designating the units. (ones) position (to the left of the decimal point) • Caution: Be certain the model has meaning for students. If the rod is one, what would the value of unit cube be? Of the flat? What if the rod was 100? What would the flat and unit cube be? What if the flat equals 10? How much would the rod and unit cube be worth?

22. Number Lines • How can you partition the number line into tenths and hundredths?

23. Meter Sticks & Tape Measures • How can you partition the meter stick into tenths and hundredths? • How are the meter stick and the base ten blocks related?

24. Decimals on the Meter Stick • If the meter is 1 whole, how would you represent 0.45 concretely using the meter stick and base ten blocks? • Represent each number using the meter stick and place value blocks. 0.64 0.88 1.45 0.25 0.40 0.4 0.04 4.00 • How can the model be used to show equivalence? Is 0.3 the same amount as 0.30? How can you tell? Are they the same length?

25. Money as a Model While money can be written in decimal notation, and children can relate decimal numbers to their understanding of money, it is not recommended as a model, but as an application. Why do you think this is the case? How can we use money as an application for decimals?

26. Representing Numbers Select a model and represent each number.

27. Representing Numbers with Tenth Strings • Give each student a meter stick and a piece of string that is exactly a meter in length. Have them mark off each tenth using a sharpie. They can also mark the hundredths for the first tenth. • Ask students to use the string to place their finger on various numbers. Ex. Mark .95 of a meter. Mark .32, .5, .81, etc. • Use this activity to reinforce decimal numbers are part of a whole, strengthen estimation skills, and compare decimals.

28. Decimals in the Real World In what real world situations do we use decimals? Complete a decimal hunt to find examples of where decimals are used in the world.

29. Decimals in the Real World

30. Decimal Experiences Bring decimals into the students’ world by: Time: • Use stop watches to help students understand decimal numbers less than one second. Have students try to start and stop the stopwatch as fast as they can. Write numbers on board and compare. • Have students run short distances outside and compare times on the board. How do the decimal numbers help? What do they mean? Distance: • Have students try to hit a target and measure within a tenth of a meter to see who was closest. You may have to measure to the nearest hundredth to break ties. (Targets can be hula hoops or pieces of paper. Objects can be bean bags tossed, Frisbees thrown, golf balls putted, etc.)

31. Fraction/Decimal Connection Decimals allow us to represent fractional quantities using our base ten number system. Tasks and multiple representations help students connect fractions and decimals and see how they represent the same concepts.

32. Decimals and Fractions NC.4.NF.6Use decimal notation to represent fractions. • Express, model and explain the equivalence between fractions with denominators 10 or 100. • Use equivalent fractions to add two fractions with denominators of 10 and 100. • Represent tenths and hundredths with models, making connections between fractions and decimals. To help children make connections… • Use familiar fraction concepts and models to explore tenthsand hundredths. • Help them see how the base-ten system extends to include numbers less than one. • Help children use models to make meaningful translations between fractions and decimals.

33. Decimals: Base-Ten Fractions • Model and represent: 73 100 • Is this fraction more or less than ½? ⅔? ¾? • Represent in different ways: 73or7 3 or .73 or .7 + .03 100 10 100 or .70 + .03 + 70 + 3 100 100

34. Fractions and Decimals • Use the fraction-decimal conversion key on the calculator to make connections and discover patterns • What patterns emerge? Fraction to Decimal

35. Is it a Match? • Play various matching games to make connections between fractions and decimals. forty-four hundredths .44 +

36. Modeling Decimals • Partner 1 models a decimal number using the decimal grids, decimal circles, meter stick, or base ten blocks. • Partner 2 records the modeled number as a decimal and as a fraction. • Switch roles after five rounds. • Variation: Partner 2 creates numbers that are less or greater than Partner 1 in the second round.

37. Race to a Meter Directions: 1. Players play on opposite sides of the meter stick. 2. Players begin at zero, and place the appropriate number of rods or cubes along the edge of the meter stick according to the number selected from the pile of cards. 3. When a player has 10 or more cubes, they should trade them for a ten rod. 4. After each round, each player should record the move on the recording sheet. 5. The winner is the player to reach the end of the meter stick. Player does not have to land exactly on one meter, but may finish beyond the end of the meter stick.

38. Decimal Place Value • The base-ten place-value system extends infinitely in two directions. • “Each place has a value that is 10 times that of the place to the right (and 1/10 of the value of the place to the left)”. • The decimal point indicates the units position (to its immediate left). Billions Millions Thousands Units Decimals hundred billions ten billions one billions ten millions ten thousands one thousands hundreds tenths hundredths thousandths one millions hundred millions hundred thousands tens ones

39. Decimal Place Value Several students were discussing different ways to name 245-hundredths in their mathematics class. Denae and Chen each proposed other possibilities. Evaluate each student’s claim. Denae said that she could use 2.45 to represent 245-hundredths. Chen said that 245-hundredths is the same as the number 24-tenths + 5-hundredths. What do students understand about the place value of decimals?

40. Decimals Place-Value • Use a calculator to count by 0.1 • What happens when you get to 0.9? • Why does it not count 0.8, 0.9, 0.10, 0.11…? • Does 0.8, 0.9, 1.0, 1.1… make sense? Why? • Count by 0.01 • How long does it take you to get to 1?

41. Decimal Place-Value • Rewrite these numbers. • Put in decimal points so that the 7 is in the given place. 4672 tenths7469 ones 4672hundredths 7469 ten thousands 467 tenths 7469 hundreds 47hundredths 187 hundredths

42. Decimal Place-Value • Rewrite these numbers. • Put in decimal points so that the 7 is in the given place. 46.72 tenths7.469 ones 4.672hundredths74,690.ten thousands 46.7tenths 746.9 hundreds .47 hundredths .187hundredths

43. Decimal Place-Value • Rewrite these numbers. Put in decimal points so that the 3 is in the given place. 4632 tenths3 hundredths 463hundredths346 ten thousands 463 tenths 3469 hundreds 43 hundreds183hundredths

44. Decimal Place-Value • Rewrite these numbers. Put in decimal points so that the 3 is in the given place. 46.32 tenths.03 hundredths 4.63hundredths34,600. ten thousands 46.3tenths 346.9 hundreds 4300. hundreds1.83hundredths

45. Sorting Decimals tenths digit is even hundredths digit is odd .15 .4 .45 .99 .24

46. The Place Value Game • Select a game board. _._ _ 0._ _ _ _._ _ • Players take turns rolling the die or spinning a spinner. • Each time a number comes up, every player writes it in one space on his or her game board. Once written, the number cannot be moved. • The winner has the largest (or smallest) number.

47. Find the Number • Find a number that comes between the two decimals that are given. • There may be more than one possible answers. .43.45 .5.6 3.193.21 .08.09 2.0 _2.1 .79.81 .23.25 .2.4 1.29 _ 1.31 .8_ .9

48. Where is the Decimal? • Place a decimal in each statement if needed to make the statement make sense. • Half of 9 is 45 • 75 is the same as three fourths  • My height is about 175 meters • 245 is a little less than two and one-half

49. In-Between Game • Player 1 picks a number (7) and writes it down. • Player 2 picks a second number (9) and writes it down. • Player 1 picks a new number that is between A and B, such as 8, announces that number aloud and crosses out 7. • Player 2 picks a number between 9 and 8, such as 8.5, announces this number aloud, and crosses out 8. • Play continues so that each player picks a new number between the two current numbers and then removes their previous choice. • Players continue for 8 to 10 turns.

50. Representing Decimals Can you name 5.15 in other ways? 5 + 1/10 + 5/1001 + 1 + 1 + 1 + 1 + .15 5.5 - .854 + 11/10 + 5/100 500 + 155.154 + 1.15 100 100 5 + .1 + .05 515 4 + 1.1 + .05 100