Place Value, Fractions & Decimals

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1 Cabarrus 3-5 Workshop February 28, 2017 Concord, NC Greg Tang s Place Value, Fractions & Decimals 1 gregtang@gregtang.com copyright Gregory Tang

2 Number Nicknames Focus on quantity. What is it? Nickname 11 ten and one ten one two tens two ten five tens and four five ten four

3 Chinese Number Names Asian students learn place value earlier. Why? 1 yi 11 shi yi 2 er 12 shi er 3 san 15 4 si 17 5 wu 20 er shi 6 liu 30 7 qi 40 8 ba 67 liu shi qi 9 jiu shi 93 3

4 NICKNAME CHART zero one two three four five six seven eight nine ten ten 1 ten 2 ten 3 ten 4 ten 5 ten 6 ten 7 ten 8 ten ten 2 ten 1 2 ten 2 2 ten 3 2 ten 4 2 ten 5 2 ten 6 2 ten 7 2 ten 8 2 ten ten 3 ten 1 3 ten 2 3 ten 3 3 ten 4 3 ten 5 3 ten 6 3 ten 7 3 ten 8 3 ten ten 4 ten 1 4 ten 2 4 ten 3 4 ten 4 4 ten 5 4 ten 6 4 ten 7 4 ten 8 4 ten ten 5 ten 1 5 ten 2 5 ten 3 5 ten 4 5 ten 5 5 ten 6 5 ten 7 5 ten 8 5 ten ten 6 ten 1 6 ten 2 6 ten 3 6 ten 4 6 ten 5 6 ten 6 6 ten 7 6 ten 8 6 ten ten 7 ten 1 7 ten 2 7 ten 3 7 ten 4 7 ten 5 7 ten 6 7 ten 7 7 ten 8 7 ten ten 8 ten 1 8 ten 2 8 ten 3 8 ten 4 8 ten 5 8 ten 6 8 ten 7 8 ten 8 8 ten ten 9 ten 1 9 ten 2 9 ten 3 9 ten 4 9 ten 5 9 ten 6 9 ten 7 9 ten 8 9 ten 9 10 ten 1 hundred 4

5 Decimal Names What is it? Name.15 1 tenth, 5 hundredths 15 hundredths.36 3 tenths, 6 hundredths 36 hundredths 1. What is confusing about decimal names? 2. What do students need to understand well to make sense of decimal names? 3. Why are you allowed to put any number of zeros on the end of any decimal? 4. What can we do when teaching whole numbers that would make decimals easier to understand? 5

6 Base 10: Blocks vs Discs vs Ten Frames tens ones Base 10 Blocks Proportional model size matters. Place does not matter. Base 10 Discs Non-proportional model size does not matter. Place does not matter. Base 10 Ten Frames Non-proportional model size does not matter. Place matters! 6

7 Use Visual Models to Represent Numbers to Hundredths 1. Model 3.68 with Base 10 Blocks. 2. Model 3.68 with Base 10 Discs and Ten Frames Model 3.68 with Base 10 Ten Frames. Ones Tenths Hundredths 7

9 3.NF.1. Develop Understanding of Fractions Show each fraction 3 different ways. 1. 3/9 2. 4/ /15 9

10 Number Line, Path or Bar? Number Line What is countable on a number line? When kids add 2 + 2, what is confusing? What represents the quantity 4? Number Path What is countable on a number path? Is adding more intuitive? What represents the quantity 4? Number Bar What is countable on a number bar? Is adding more intuitive? What represents the quantity 4? 10

11 3.NF.2. Represent fractions on a number line. 1. Locate 1/2, 1/3, 2/3, 1/4, 2/4 and 3/4 on the number line Locate 2/2, 4/4, 6/4, 12/8 and 16/8 on the number line

12 3.NF.3. Find Equivalent Fractions Simple Recognize and generate simple equivalent fractions. Explain why the fractions are equivalent, e.g., by using a visual fraction model. 1? 1. = = = =

13 4.NF.1. Find Equivalent Fractions General Explain why a fraction a/b is equivalent to a fraction (n a)/(n b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size... Find an equivalent fraction and explain why your answer makes sense in terms of the number and size of the parts. 3? 1. = = = =

14 TangaRow TM Fraction Puzzles Use 7 & 9 consecutive digits a TangaRow. 2 = 3 = 1 2 = 1 3 =

15 Compare fractions without using common denominators! a b c d e f g

16 Fraction Comparison Assessment Use any strategy you like! Order from smallest to largest Order from smallest to largest

17 Fraction Number Sense Addition Estimate the answer using only mental calculations. Circle A, B or C. 1. 4/7 + 4/9 =? A B C /5 + 2/9 =? A B C

18 Fraction Number Sense Subtraction Estimate the answer using only mental calculations. Circle A, B or C. 3. 1/4-3/11 =? A B C /11-9/37 =? A B C

19 Fraction Number Sense Multiplication Estimate the answer using only mental calculations. Circle A, B or C. 5. 6/7 x 11/15 =? A B C /8 x 9/5 =? A B C

20 Fraction Number Sense Division Estimate the answer using only mental calculations. Circle A, B or C /14 10/13 =? A B C /8 17/7 =? A B C

21 Common Denominators Essential, unnecessary, helpful! Addition Requires a common unit: 2 cats + 1 dog 3 cats 2 pets + 1 pet = 3 pets = = 6 Subtraction Requires a common unit: 2 cats - 1 dog 1 cat 2 pets - 1 pet = 1 pet 1-1 = = 12 Multiplication Common denominators make it worse: 1 x 1 = 3 x = 36 1 = 6 Division Common denominators can be ignored: = = = 3 21

22 Fraction Algorithms Understanding before efficiency! Addition a b a + b + = c c c a b ad + bc + = c d cd Subtraction a b a - b - = c c c a b ad - bc - = c d cd Multiplication a b a x b x = c d c x d Division a b a b = c d c d a b a d = x c d c b a b = a b c c 22

23 Mixed Numbers & Improper Fractions Algorithms or number bonds? Algorithm b a x c + b a = c c 3 2 x = = 4 Algorithm c R d a d b) a c b b 2 R ) Number Bonds Number Bonds = =

24 Multistep, reasoning, fluency & number lines! 1. Starting at 1/8, what number is 1/3 of the way to 1/2? Starting at 1/6, what number is 2/3 of the way to 1/2?

25 Multistep, reasoning, fluency & number lines! 3. Starting at 1/6, what number is 1/4 of the way to 2/3? Starting at 1/4, what number is 3/4 of the way to 2/3?

26 Multistep, reasoning, fluency & number lines! 5. Starting at 1/8, what number is 2/5 of the way to 1/4? Starting at 1/9, what number is 4/5 of the way to 1/8?

27 4.NF.5. Decimal Fractions Equivalence Express a fraction with denominator 10 as an equivalent fraction with denominator... Use a decimal grid to find an equivalent fraction and explain why your answer makes sense in terms of the number and size of the parts = = = 10 45? 2 4. = 10 27

28 4.NF.5. Decimal Fractions Addition... add two fractions with respective denominators 10 and. Use a decimal grid to solve each equation = = = = + 28

29 4.NF.5. Decimal Fractions Addition... add two fractions with respective denominators 10 and. Solve each equation = = = = 10 29

30 4.NF.6. Decimal Notation Use decimal notation for fractions with denominators 10 or... ones hundreds tens tenths hundredths oneths = 10 decimal = = 10 decimal = = 10 decimal = = decimal = 30

31 4.NF.6. Locate Decimals 1. Locate 0.6 on the number line Locate 0.03 on the number line Locate 1.75 on the number line Locate 4.36 on the number line

32 5.NBT.3B. Compare Decimals to Thousandths Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results. Choose a scale for the number line then locate and compare the 2 numbers. 1. Compare: Compare: Compare: Compare:

33 3.NBT.1 & 4.NBT.3: Round Whole Numbers 1. Round 98 to the nearest ten. 2. Round 132 to the nearest ten. 3. Round 996 to the nearest ten. 4. Round 9,954 to the nearest hundred. 5. Round 9,997 to the nearest ten. 33

34 5.NBT.4. Round Decimals to Any Place Use place value understanding to round decimals to any place. Choose a scale for the number line then locate and round each number. 1. Round.247 to the nearest tenth. 2. Round.136 to the nearest hundredth. 3. Round.999 to the nearest thousandth. 4. Round to the nearest hundredth 34

35 Common Multiplication & Division Situations Product Unknown Group Size Unknown How many in each group? partitive or sharing 3.OA.3. Use multiplication and division within to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Group Number Unknown How many groups? quotative or grouping 3 x 6 =? 3 x? = 18, 18 3 =?? x 6 = 18, 18 6 =? 3.OA.3 Equal Groups 1a.There are 3 bags with 6 plums in each bag. How many plums are there in all? 1b. Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether? 2a. If 18 plums are shared equally into 3 bags, then how many plums will be in each bag? 2b. Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be? 3a. If 18 plums are to be packed 6 to a bag, then how many bags are needed? 3b. Measurement example. You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have? 3.OA.3 Arrays Area 4a. There are 3 rows of apples with 6 apples in each row. How many apples are there? 4b. Area example. What is the area of a 3 cm by 6 cm rectangle? 5a. If 18 apples are arranged into 3 equal rows, how many apples will be in each row? 5b. Area example. A rectangle has area 18 square cenqmeters. If one side is 3 cm long, how long is a side next to it? 6a. If 18 apples are arranged into equal rows of 6 apples, how many rows will there be? 6b. Area example. A rectangle has area 18 square cenqmeters. If one side is 6 cm long, how long is a side next to it? 4.OA.2. Compare 1a. A blue hat costs $6. A red hat costs 3 Qmes as much as the blue hat. How much does the red hat cost? 1b. Measurement example. A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 Qmes as long? 2a. A red hat costs $18 and that is 3 Qmes as much as a blue hat costs. How much does a blue hat cost? 2b. Measurement example. A rubber band is stretched to be 18 cm long and that is 3 Qmes as long as it was at first. How long was the rubber band at first? 3a. A red hat costs $18 and a blue hat costs $6. How many Qmes as much does the red hat cost as the blue hat? 3b. Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many Qmes as long is the rubber band now as it was at first? General a x b =? a x? = p, p a =?? x b = p, p b =? 1Adapted from Box 2-4 of MathemaQcs Learning in Early Childhood, NaQonal Research Council (2009, pp. 32, 33) 35

36 Multistep Word Problem Progressions For Halloween, Emily and her sister Katie combined all of their candy into one big bowl. Together, the 2 sisters collected 84 pieces of candy. 2 nd Emily collected 48 pieces of candy. How many fewer pieces did Katie collect? 3 rd The sisters decided they would eat all of the candy in just one week. If they eat the same number of pieces each day, how many pieces will they have eaten after 4 days? 4 th Greg Jr. and his best friend Alex collected twice as much candy as Emily and Katie, and Greg Jr. collected twice as much candy as Alex. How many pieces of candy did Greg Jr. collect? 5 th The boys decided they would eat one-eighth of their candy that night and donate two-thirds of what was left to their school s fundraiser. How many pieces of candy did they donate? 36 14, 60, 160, 142

37 DigiCross Grade 3 37

38 Greg Tang s Tangy Tuesday weekly puzzle pack for grades K-5. 5 new puzzles each week. Visit store.gregtangmath.com. 38

39 Greg Tang s Tangy Tuesday weekly puzzle pack for grades K-5. 5 new puzzles each week. Visit store.gregtangmath.com. 39

40 Greg Tang s Tangy Tuesday weekly puzzle pack for grades K-5. 5 new puzzles each week. Visit store.gregtangmath.com. 40

41 Greg Tang s Tangy Tuesday weekly puzzle pack for grades K-5. 5 new puzzles each week. Visit store.gregtangmath.com. 41

42 Greg Tang s Tangy Tuesday weekly puzzle pack for grades K-5. 5 new puzzles each week. Visit store.gregtangmath.com. 42

43 Greg Tang s Tangy Tuesday weekly puzzle pack for grades K-5. 5 new puzzles each week. Visit store.gregtangmath.com. 43

44 Greg Tang s Tangy Tuesday weekly puzzle pack for grades K-5. 5 new puzzles each week. Visit store.gregtangmath.com. 44

45 Greg Tang s Tangy Tuesday weekly puzzle pack for grades K-5. 5 new puzzles each week. Visit store.gregtangmath.com. 45

Clever Connections: Computations & Word Problems

Cabarrus K 5 Workshop February 15, 2016 Concord, NC Greg Tang s Clever Connections: Computations & Word Problems www.gregtang.com 1 gregtang@gregtang.com copyright www.gregtang.com Gregory Tang 1-100 Chart