Clever Connections: Computations & Word Problems

Transcription

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

2 1-100 Chart Is this a good visual tool for building number sense? Food for thought. Compare 22 and 42. Now compare 22 and 29. Any thoughts? 2

3 0-99 Chart Focus on Place Value and Base Are there any advantages or disadvantages compared to a chart? 3

4 Number Nicknames Generalizable names are easier to learn. 10 one ten zero 11 one ten one two ten zero ,596 4

5 Chinese Number Names Learn Chinese in 2 minutes! 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 5

6 Funny Numbers The secret s adding left to right when pen and paper aren t in sight! 34 = three ten four = two ten fourteen = twenty-fourteen 42 = four ten two = three ten twelve = thirty-twelve 56 = = = 61 = = = 70 = = = = fifty-twelve = 62! 50+12! = sixty-fourteen = 74! 60+14! = = = = = = = = 6

7 Funny Numbers Partial Differences lay the foundation for the Standard Algorithm = 36! = 36! = 36! = 36!

8 Funny Numbers A bridge to the Standard Algorithm. 2-digit Addition with Regrouping tens ones tens ones tens ones tens ones digit Subtraction with Regrouping 85 is the same as 63 is the same as 72 is the same as 94 is the same as seventy-fifteen tens ones tens ones tens ones tens ones

9 Multi-digit Addition Assessment Solve using (1) a concrete model, (2) a partial sums algorithm, and (3) the Standard Algorithm. 9

10 Multi-digit Subtraction Assessment Solve using (1) a concrete model, (2) a partial differences algorithm, and (3) the Standard Algorithm. 10

11 Represent and solve problems involving multiplication and division. Example: 4 x 6 =? a. Groups b. Array c. Area d Bar Model Q: Which shows the commutative property best? Distributive? 11

12 Represent and solve problems involving multiplication and division. Partitive division (sharing): divisor is group number. Quotative division (grouping/measurement): divisor is group size. Example: 8 2 =? Partitive Division (sharing) 8 divided into 2 groups Quotative Division (grouping) 8 divided into groups of

13 Multi-digit Multiplication Assessment Solve 34 x 68 using (1) an area model, (2) the partial products algorithm, and (3) the Standard Algorithm. 13

14 Multi-digit Multiplication Assessment Solve 2.7 x 3.6 using (1) an area model, (2) the partial products algorithm, and (3) the Standard Algorithm. Please do not ignore the decimal. 14

15 Multi-digit Division Assessment Solve 1, using (1) an area model, (2) the partial quotients algorithm, (3) the Standard Algorithm and (4) Greg s Algorithm. 15

16 Multi-digit Division Assessment Solve using (1) an area model, (2) the partial products algorithm, and (3) the Standard Algorithm. Please do not ignore the decimal. 16

17 Why do divisibility rules work? 2 The last digit is even. 3 The sum of the digits. Apply recursively. 4 The last 2 digits form a number that can be cut in half twice. 5 The last digit is a 5 or a 0. 6 The number is divisible by both 3 and 2. 7 Double the last digit and subtract it from the rest. (10a + b 21b) 8 The last 3 digits form a number that can be cut in half 3 times. 9 The sum of the digits. Apply recursively. 10 The number ends in Subtract sums of even and odd digits. (1000 = , 100 = 99+1) 12 The number is divisible by 3 and What is a divisibility rule for 13? (39b is divisible by 13) 17

18 K-2 Word Problems Result Unknown Change Unknown Start Unknown Add to (join) 1 K.!Two!bunnies!sat!on! the!grass.!three!more! bunnies!hopped!there.! How!many!bunnies!are! on!the!grass!now?! 2 1.!Two!bunnies!were! sifng!on!the!grass.! Some!more!bunnies! hopped!there.!then! there!were!five!bunnies.! How!many!bunnies! hopped!over!to!the!first! two?! 3 2.&Some&bunnies&were& siang&on&the&grass.&three& more&bunnies&hopped& there.&then&there&were& five&bunnies.&how&many& bunnies&were&on&the& grass&before?& Take from (separate) 4 K.!Five!apples!were!on! the!table.!i!ate!two! apples.!how!many! apples!are!on!the!table! now?! 5 1.!Five!apples!were!on! the!table.!i!ate!some! apples.!then!there!were! three!apples.!how!many! apples!did!i!eat?! 6 2.&Some&apples&were&on& the&table.&i&ate&two& apples.&then&there&were& three&apples.&how&many& apples&were&on&the&table& before.& Total Unknown Addend Unknown Both Addends Unknown Put together/ Take apart (part-whole) 7 K.!Three!red!apples! and!two!green!apples! are!on!the!table.!how! many!apples!are!on!the! table?! 8 1.!Five!apples!are!on! the!table.!three!are!red! and!the!rest!are!green! How!many!apples!are! green?! 9 K.!Grandma!has!five! flowers.!how!many!can! she!put!in!her!red!vase! and!how!many!in!her! blue!vase?! Difference Unknown Bigger Unknown Smaller Unknown Compare with more *10 1.!Tammy!has!two! apples.!greg!has!five! apples.!how!many!more! apples!does!greg!have! than!tammy?! 11 1.!Greg!has!three!more! apples!than!tammy.! Tammy!has!two!apples.! How!many!apples!does! Greg!have?! *12 2.&Greg&has&three& more&apples&than& Tammy.&Greg&has&five& apples.&how&many&apples& does&tammy&have?& Compare with fewer 13 1.!Tammy!has!two! apples.!greg!has!five! apples.!how!many!fewer! apples!does!tammy! have!than!greg?! *14 2.&Tammy&has&three& fewer&apples&than&greg.& Tammy&has&two&apples.& How&many&apples&does& Greg&have?& 15 1.!Tammy!has!three! fewer!apples!than!greg.! Greg!has!five!apples.! How!many!apples!does! Tammy!have?! From CCSS, p. 88, which is based on Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, National Research Council, 2009, pp

19 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 100 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!centmeters.!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!centmeters.!if!one! side!is!6!cm!long,!how!long! is!a!side!next!to!it?!! 4.OA.2. Compare 1a.!A!blue!hat!costs!$6.!A! red!hat!costs!3!tmes!as! much!as!the!blue!hat.!how! much!does!the!red!hat! cost?!! 1b.$Measurement$example.! A!rubber!band!is!6!cm!long.! How!long!will!the!rubber! band!be!when!it!is! stretched!to!be!3!tmes!as! long?!! 2a.!A!red!hat!costs!$18!and! that!is!3!tmes!as!much!as!a! blue!hat!costs.!how!much! does!a!blue!hat!cost?!! 2b.$Measurement$example.! A!rubber!band!is!stretched! to!be!18!cm!long!and!that! is!3!tmes!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!$6.!how! many!tmes!as!much!does! the!red!hat!cost!as!the!blue! hat?!! 3b.$Measurement$example.! A!rubber!band!was!6!cm! long!at!first.!now!it!is! stretched!to!be!18!cm!long.! How!many!Tmes!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!2Y4!of!MathemaTcs!Learning!in!Early!Childhood,!NaTonal!Research!Council!(2009,!pp.!32,!33)!! 19

20 3.MD.1. Elapsed Time (result unknown) 1. Knaya s bus left school at 3:35 p.m. and dropped her at home 45 minutes later. What time did she arrive at home? 2. Katie s hockey practice started at 10:30 a.m. and the girls skated for 1 hour and 40 minutes. What time did their practice end? 3. Rico works the night shift and starts working at 9:30 every evening. If his shift is 8 hours long, what time does he get off work? 20

21 3.MD.1. Elapsed Time (change unknown) 4. The express train leaves Grand Central at 6:45 p.m. and arrives in Sleepy Hollow at 7:23 p.m. How long is the trip? 5. Layla punched the time clock at work at precisely 7:52 a.m. If she punched out at 3:17 p.m., how long was she at work? 6. John went to sleep at 10:36 p.m. and woke up the next morning at 7:10 a.m. How long did John sleep? 21

22 3.MD.1. Elapsed Time (start unknown) 7. It took Bill 23 minutes to walk to his friend's house. If he arrived at 4:10 p.m., what time did he start walking? 8. Dad spent 75 minutes on Saturday morning cutting the grass. He finished at 12:05 p.m. What time did he start? 9. Katie left the library at 1:10 a.m. after spending 3 hours and 25 minutes studying for her biology exam. What time did she begin studying? 22

23 4.MD.2. Elapsed Time (multistep) 10. Martha put the 15-pound turkey in the oven at 8:50 a.m. If it needs to cook exactly 13 minutes for each pound, what time should she take the turkey out of the oven?* 23

24 Find Equivalent Fractions Use a visual model to find each equivalent fraction. Explain intuitively not arithmetically why your answer makes sense = = = =

25 3.NF.3 & 4.NF.1. Find Equivalent Fractions Show each fraction 3 different ways. 1. 1/4 = 2. 1/3 = 3. 3/5 = 25

26 3.NF.3 & 4.NF.2. Compare Fractions No pictures, cross-multiplying, or common denominators. a b c d e f g

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

28 Fraction Problems 28. What number is 1/4 of 24? 29. What number is 2/3 of 27? 30. What number is 6/5 of 35? 28

29 Fraction Problems is 1/3 of what number? is 2/5 of what number? is 4/3 of what number? 29

30 Percent Problems Solve using both a tape diagram and double number line is 25% of what number? is 24% of what number? 30

31 Percent Problems Solve using both a tape diagram and double number line is 60% of what number? is 210% of what number? 31

32 Algebraic Foundations: 2 equations, 2 unknowns 1. Can you name 2 numbers that have a difference of 2 and a sum of 14? 2. Can you name two numbers that have a difference of 2 and a sum of 20? 3. Can you name two numbers that have a difference of 4 and a sum of 10? 4. Can you name two numbers that have a difference of 5 and a sum of 17? 5. Can you name two numbers that have a difference of 5 and a sum of 16? 32

33 Algebraic Word Problems 1. Blake has 14 more songs on his phone than Adam. If together they have 242 songs, how many songs does Blake have? 2. Feodor Vassilyev was said to have fathered 87 children. If his second wife gave birth to 51 fewer children than his first wife, how many children did his first wife have? 33

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