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1 GregTangMath.com copyright © Gregory Tang
Clever Connections: Computations & Word Problems
www.gregtang.com
www.gregtang.com
Greg Tang’s
Cabarrus K–5 Workshop February 15, 2016
Concord, NC
GregTangMath.com [email protected]
2 GregTangMath.com copyright © Gregory Tang
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?
3 GregTangMath.com copyright © Gregory Tang
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?
4 GregTangMath.com copyright © Gregory Tang
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 _____________________________
5 GregTangMath.com copyright © Gregory Tang
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 ____________
6 GregTangMath.com copyright © Gregory Tang
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 = ________________________ = _____________
7 GregTangMath.com copyright © Gregory Tang
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
8 GregTangMath.com copyright © Gregory Tang
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
9 GregTangMath.com copyright © Gregory Tang
Multi-digit Addition Assessment
Solve 378 +259 using (1) a concrete model, (2) a partial sums algorithm, and (3) the Standard Algorithm.
10 GregTangMath.com copyright © Gregory Tang
Multi-digit Subtraction Assessment
Solve 832 - 376 using (1) a concrete model, (2) a partial differences algorithm, and (3) the Standard Algorithm.
11 GregTangMath.com copyright © Gregory Tang
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.
12 GregTangMath.com copyright © Gregory Tang
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.
13 GregTangMath.com copyright © Gregory Tang
Multi-digit Multiplication Assessment
Solve 34 x 68 using (1) an area model, (2) the partial products algorithm, and (3) the Standard Algorithm.
14 GregTangMath.com copyright © Gregory Tang
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.”
15 GregTangMath.com copyright © Gregory Tang
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.
16 GregTangMath.com copyright © Gregory Tang
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.”
17 GregTangMath.com copyright © Gregory Tang
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)
18 GregTangMath.com copyright © Gregory Tang
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?!
19 GregTangMath.com copyright © Gregory Tang
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)!!
20 GregTangMath.com copyright © Gregory Tang
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?
21 GregTangMath.com copyright © Gregory Tang
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?
22 GregTangMath.com copyright © Gregory Tang
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?
23 GregTangMath.com copyright © Gregory Tang
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?*
24 GregTangMath.com copyright © Gregory Tang
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
25 GregTangMath.com copyright © Gregory Tang
3.NF.3 & 4.NF.1. Find Equivalent Fractions
Show each fraction 3 different ways.
1. 1/4 =
2. 1/3 =
3. 3/5 =
26 GregTangMath.com copyright © Gregory Tang
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
27 GregTangMath.com copyright © Gregory Tang
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
28 GregTangMath.com copyright © Gregory Tang
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?
29 GregTangMath.com copyright © Gregory Tang
Fraction Problems
31. 15 is 1/3 of what number?
32. 18 is 2/5 of what number?
33. 28 is 4/3 of what number?
30 GregTangMath.com copyright © Gregory Tang
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?
31 GregTangMath.com copyright © Gregory Tang
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?
32 GregTangMath.com copyright © Gregory Tang
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?
33 GregTangMath.com copyright © Gregory Tang
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?
34 GregTangMath.com copyright © Gregory Tang
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