Clever Connections: Computations & Word Problems division (grouping/measurement): ... Partitive Division (sharing) 8 divided into 2 groups Quotative Division (grouping) ... From CCSS,

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Clever Connections: Computations & Word Problems

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Greg Tang’s

Cabarrus K–5 Workshop February 15, 2016

Concord, NC

GregTangMath.com [email protected]


1-100 Chart

Is this a good visual tool for building number sense?

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Food for thought. Compare 22 and 42. Now compare 22 and 29. Any thoughts?


0-99 Chart

Focus on Place Value and Base 10.

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

Are there any advantages or disadvantages compared to a 1-100 chart?


Number Nicknames

Generalizable names are easier to learn.

10 one ten zero

11 one ten one

12 _____________________________

19 _____________________________

20 two ten zero

30 _____________________________

55 _____________________________

128 _____________________________

413 _____________________________

1,596 _____________________________


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 88 ____________

10 shi 93 ____________


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 = _____________ = _____________ = _____________

2 7 + 3 5 = “fifty-twelve” = 62!

50+12!

4 8 + 2 6 = “sixty-fourteen” = 74!

60+14!

36 + 36 = ________________________ = _____________

47 + 47 = ________________________ = _____________

29 + 34 = ________________________ = _____________

68 + 29 = ________________________ = _____________


Funny Numbers

Partial Differences lay the foundation for the Standard Algorithm.

1. 53 – 17 = 36!

40 13 10 7

2. 96 – 38 = 36!

40 13 10 7

3. 74 – 25 = 36!

40 13 10 7

4. 82 – 57 = 36!

40 13 10 7


Funny Numbers

A bridge to the Standard Algorithm.

tens ones

3 6 + 3 6

6 12 7 2

tens ones

4 7 + 4 7

tens ones

2 9 + 3 4

tens ones

5 8 + 2 9

tens ones

7 15 8 5 - 3 8

4 7

tens ones

6 3 - 1 7

tens ones

7 2 - 5 9

tens ones

9 4 - 2 5

85 is the same as

______________

63 is the same as

______________

72 is the same as

______________

94 is the same as

______________ seventy-fifteen

2-digit Addition with Regrouping

2-digit Subtraction with Regrouping


Multi-digit Addition Assessment

Solve 378 +259 using (1) a concrete model, (2) a partial sums algorithm, and (3) the Standard Algorithm.


Multi-digit Subtraction Assessment

Solve 832 - 376 using (1) a concrete model, (2) a partial differences algorithm, and (3) the Standard Algorithm.


Example: 4 x 6 = ? a. b. c. d. Q: Which shows the commutative property best? Distributive?

6 6 6 6

Array

Area

Groups

Bar Model

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 = ?

4 4

Partitive Division (sharing) 8 divided into 2 groups

Quotative Division (grouping) 8 divided into groups of 2

2 2 2 2

Represent and solve problems involving multiplication and division.


Multi-digit Multiplication Assessment

Solve 34 x 68 using (1) an area model, (2) the partial products algorithm, and (3) the Standard Algorithm.


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.”


Multi-digit Division Assessment

Solve 1,072 ÷ 16 using (1) an area model, (2) the partial quotients algorithm, (3) the Standard Algorithm and (4) Greg’s Algorithm.


Multi-digit Division Assessment

Solve 25.72 ÷ 4 using (1) an area model, (2) the partial products algorithm, and (3) the Standard Algorithm. Please do not “ignore the decimal.”


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 0.

÷11 Subtract sums of even and odd digits. (1000 = 1001-1, 100 = 99+1)

÷12 The number is divisible by 3 and 4.

÷13 What is a divisibility rule for 13? (39b is divisible by 13)


Add to (join)

Take from (separate)

Put together/ Take apart

(part-whole)

Compare with more

Result Unknown Change Unknown Start Unknown

Total Unknown Addend Unknown Both Addends Unknown

Difference Unknown Bigger Unknown Smaller Unknown

*101.!Tammy!has!two!

apples.!Greg!has!five!

apples.!How!many!more!

apples!does!Greg!have!

than!Tammy?!

131.!Tammy!has!two!

apples.!Greg!has!five!

apples.!How!many!fewer!

apples!does!Tammy!

have!than!Greg?!

81.!Five!apples!are!on!

the!table.!Three!are!red!

and!the!rest!are!green!

How!many!apples!are!

green?!

111.!Greg!has!three!more!

apples!than!Tammy.!

Tammy!has!two!apples.!

How!many!apples!does!

Greg!have?!

*142.&Tammy&has&three&fewer&apples&than&Greg.&Tammy&has&two&apples.&How&many&apples&does&Greg&have?&

1K.!Two!bunnies!sat!on!

the!grass.!Three!more!

bunnies!hopped!there.!

How!many!bunnies!are!

on!the!grass!now?!

4K.!Five!apples!were!on!

the!table.!I!ate!two!

apples.!How!many!

apples!are!on!the!table!

now?!

21.!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?!

51.!Five!apples!were!on!

the!table.!I!ate!some!

apples.!Then!there!were!

three!apples.!How!many!

apples!did!I!eat?!

32.&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?&

62.&Some&apples&were&on&the&table.&I&ate&two&apples.&Then&there&were&three&apples.&How&many&apples&were&on&the&table&before.&

*122.&Greg&has&three&more&apples&than&Tammy.&Greg&has&five&apples.&How&many&apples&does&Tammy&have?&

151.!Tammy!has!three!

fewer!apples!than!Greg.!

Greg!has!five!apples.!

How!many!apples!does!

Tammy!have?!

K-2 Word Problems

Compare with fewer

From CCSS, p. 88, which is based on Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, National Research Council, 2009, pp. 32–33.

7K.!Three!red!apples!

and!two!green!apples!

are!on!the!table.!How!

many!apples!are!on!the!

table?!

9K.!Grandma!has!five!

flowers.!How!many!can!

she!put!in!her!red!vase!

and!how!many!in!her!

blue!vase?!


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.

3.OA.3 Equal

Groups

3.OA.3 Arrays Area

4.OA.2. Compare

Product Unknown Group Size Unknown “How many in each group?”

partitive or sharing

Group Number Unknown “How many groups?” quotative or grouping

a x b = ? a x ? = p, p ÷ a = ? ? x b = p, p ÷ b = ?

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 x 6 = ? 3 x ? = 18, 18 ÷ 3 = ? ? x 6 = 18, 18 ÷ 6 = ?

General

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?!!

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?!!

Common Multiplication

& Division Situations

1Adapted!from!Box!2Y4!of!MathemaTcs!Learning!in!Early!Childhood,!NaTonal!Research!Council!(2009,!pp.!32,!33)!!


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?


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?


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?


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?*


Find Equivalent Fractions

Use a visual model to find each equivalent fraction. Explain intuitively – not arithmetically – why your answer makes sense.

1. 1 2

= 6 4

2. 1 3

= 3 4

4. 1

12 = 2

3 3.

6 8

= 6 4


3.NF.3 & 4.NF.1. Find Equivalent Fractions

Show each fraction 3 different ways.

1. 1/4 =

2. 1/3 =

3. 3/5 =


3.NF.3 & 4.NF.2. Compare Fractions

No pictures, cross-multiplying, or common denominators.

a.

b.

c.

d.

e.

f.

g.

4 7

5 7

3 4

3 5

5 6

4 7

2 5

4 7

4 5

6 7

3 8

4 9

3 34

2 23


Fraction Comparison Assessment

Use any strategy you like!

Order from smallest to largest.

1 7

2 13

1 6

2 11

Order from smallest to largest.

7 9

4 5

3 4


Fraction Problems

28. What number is 1/4 of 24?




Fraction Problems

31. 15 is 1/3 of what number?




Percent Problems

Solve using both a tape diagram and double number line. 34. 16 is 25% of what number? 35. 18 is 24% of what number?


Percent Problems

Solve using both a tape diagram and double number line. 36. 51 is 60% of what number? 37. 63 is 210% of what number?


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?





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?


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Clever Connections: Computations & Word Problems division (grouping/measurement): ... Partitive Division (sharing) 8 divided into 2 groups Quotative Division (grouping) ... From CCSS,

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