Math 221 – F07
Activities, page 14
Classification of Addition and Subtraction Word Problems
1. Van de Walle describes four types of addition/subtraction problems. What are they? Each
of these main types has two or three subtypes. What distinguishes the subtypes?
For problems each problem of 2 – 12 below, determine its type and subtype, and write one or
more number sentences associated with the problem, putting a box around the unknown in your
number sentence. Your number sentences should include the sentence you feel is most “natural” to
the problem as well as a number sentence in computational form. Indicate which number sentences
are in computational form.
2. Al is 49 inches tall. Bea is 52 inches tall. How much taller is Bea than Al?
3. Deb had 7 cookies. She gave some to Cy. Now Deb has three cookies. How many did she give
to Cy?
4. Ed has eight cousins. Three of them are boys. How many girl-cousins does Ed have?
5. After Flo lost five of her new back-to-school pencils, she had 7 pencils left. How many did
Flo start with?
6. Hal had two sweaters. Grandma knitted some more for him. Now Hal has 8 sweaters. How
many did he get from Grandma?
7. Ida has three more brothers than Jose. Ida has five brothers. How many does Jose have?
8. Kendra had 6 hats. Lea gave her 3 more. How many hats does Ken have now?
9. Maria has 5 trucks and 8 cars in her sandbox. How many vehicles does she have in all?
10. Nadia had twelve wine glasses, but Oliver broke four of them. How many wine glasses does
Nadia have left?
11. Pa has five fewer teeth than Ma has. Pa has 8 teeth. How many does Ma have?
12. Rolando had done some of his homework problems, but was stuck on some others. His friend
Syd helped him on four more problems. Now Rolando has 11 problems finished. How many
did he have finished before getting help from Syd?
13. Did all of Van de Walle’s subtypes occur above? If not, write your own problems for any subtypes that did not occur. Identify the type and subtype of each, and write number sentences
for you problems just as you did for problems 2 – 12.
14. Which of the subtypes did you find most natural to express in their computational form?
Math 221 – F07
Activities, page 15
Meaning of Multiplication and Division
Van de Walle identifies four categories of multiplication/division problems, among which two
are especially important: equal groups and multiplicative comparison. Each of these categories has
three subcategories, based on what piece of information is unknown.
1. If there are two categories and each has three subcategories, how many classifications by
category and subcategory are there? Explain how you arrived at your answer.
2. What type of problem is the problem above?
3. For each type of problem (category and subcategory):
a) write a brief story problem of that type, making sure that no number occurs more than
once in the problem or its solution,
b) solve the problem using an array model and one other model (make a sketch of each
model on your paper)
c) write down a number sentence for your problem and put a box around the unknown
information,
d) write down any related number sentences (again with a box around the missing information).
4. Which of the problems above were hardest to invent?
5. Which of the problems above were hardest to model?
Math 221 – F07
Activities, page 16
Basic Facts
How do you do your basic facts? For each basic fact below, identify which strategies listed in
Van de Walle you use. If different people in your group use different strategies, record all strategies
used. If no one uses any of the strategies from Van de Walle, describe some other way of arriving
at the basic fact in the space below the table.
Here is a list of strategies for each operation on this sheet and a shorthand notation for each.
Addition:
1-2: 1 & 2 more than
ND: Near Doubles
0: Zero facts
E10: Elevator (make 10)
D: doubles
Subtraction:
1-2: 1 & 2 less than
D: Doubles/Near Doubles 0: Zero facts
10F: Ten frame facts
S10: Subtract 10 and adjust
B10: Bridge through 10
Multiplication:
D: Doubles
9: Nifty Nines
Fact
9+8
7+5
6+6
4+6
7+6
8+1
8+4
2+4
5+9
3+0
Strategy
Z: Zero and One facts
5: 5 facts
H: Helping facts (list the helping fact)
Fact
6−1
7−2
4−0
9−5
10 − 7
8−3
15 − 6
13 − 8
9−6
9−5
Strategy
Fact
9×6
6×6
0×7
5×3
8×2
7×5
6×8
4×1
2×4
3×7
Strategy
Math 221 – F07
Activities, page 17
Pentimal Pieces
In the land of Pentima a different numeration system is used. Despite the beauty, simplicity
and power of the pentimal system, it takes Pentimal children a while to learn it. Often teachers
make use of models called Pentimal Pieces to help children learn the place-value system involved
in pentimal numbers. (There are similar models, called Dienes blocks or multibase pieces, for other
place-value number systems as well, including the decimal system.) The model consists of premade
pieces of several types: there are tiny cubes, called units; sticks made up of fen (P10) units stuck
together, called longs; thick squares made up of fen (P10) longs joined side by side, called flats; and
large cubes made up of fen (P10) flats placed one on top of the other, called blocks.
1. Complete the following chart:
Type of piece
number of
units in piece
(decimal)
number of
units in piece
(power of 5)
number of
units in piece
(pentimal)
25
52
P100
Unit
Long
Flat
Block
number of
units in piece number
(power of P10)
word
P102
funner
2. For each number in the table below, (a) form the number using pentimal pieces, (2) record
how many pieces of each type you used, and (3) write down the pentimal number word. Be
sure to actually form the number with the blocks.
blocks
flats
longs
units number word
P32
P302
P320
P2030
P1004
P2013
3. How many different ways can you model P321? Record at least 10 different ways by sketching
the model for each one. Note: You may not have enough pieces in your set for some of
them, but you can still sketch them.
Math 221 – F07
Activities, page 18
4. For each of the following, make any trades necessary to show the number using as few pieces
as possible. Record how many pieces of each type you used in your most economical representation and then write the corresponding pentimal number.
blocks
flats
longs
units
Pentimal number
one flat, two units
two flats, three longs
alf flats, three longs
alf flats, four longs
alf longs, bet units,
four flats
one block, three flats,
thirfen longs, two units
four flats,
thirfen longs, twofy-one units
5. For trading purposes, it is important that those who normally use the pentimal system also
have some facility with the decimal system. Model each of the following numbers, then
exchange everything for units and record the number of units as a decimal number. (Why
didn’t I have you record the number of units as a pentimal number as well?)
(a) P23
(b) P410
(c) P2304
(d) P2
(e) P21
(f) P2033
6. Of course, one must also convert in the other direction: Suppose you have the number of units
indicated by the decimal numbers in the chart below, if you perform all possible exchanges
that reduce the number of pentimal pieces, how many of each type of piece will you have?
(You may use your pentimal blocks.)
decimal number
17
56
145
329
503
blocks
flats
longs
units
Pentimal number
0
2
1
1
P211
Math 221 – F07
Activities, page 19
Free Trade and FUNN
Due to the recent free trade agreement among nations with various systems, there has been a
much greater interest in converting between these systems of late.
• In addition to the pentimal systems, some trading partners use other systems, most notably
heximal (base 6) and octal (base 8), but other systems also occur, including binary (base 2),
hexadecimal (base 16), and base 12.
• In order to reduce the number of errors due to employees using the wrong system or forgetting
which system uses which base, the Fully Universal Numbering Notation (FUNN) has been
developed. In this notation, instead of using whichever conventions the locals use to represent
their numbers (like the P used in pentimal, or nothing at all used in decimal) the base is
explicitly noted as a subscripted word. For example
Local notation FUNN notation
150
150ten or 150twofy
P24
24five or 24fen
• Due to the dominance of the United States in world politics, and the fact that the decimal
system uses more numerals than the pentimal system, the arabic numerals of the decimal
system have become the standard – indeed they were adopted some time back by the local
users of pentimal, as we have seen – and the notation numberten is usually just written as
number. This causes some difficulty when a system with a larger base than ten is used, since
there are not enough numerals for this. The convention is to use upper-case roman letters
instead, in order. Thus, ten can be represented by A, eleven by B, etc. when additional
numerals are needed. For example
ABtwelve = (10 × 12) + 11 = 120 + 11 = 131 = 131ten .
Now that the FUNN system is in place, it is possible to work in any system without learning lots
of new notation.
Math 221 – F07
Activities, page 20
Practicing with FUNN
Try your hand at FUNN notation. Make the following conversions. You may use models if that
is helpful, but be sure to record something on paper to justify your answers. Use FUNN notation
throughout.
1. Convert 345 to pentimal. Remember to use FUNN notation.
2. Convert 345 to octal (base eight).
3. Convert 543six to base ten.
4. Convert 2104five to base ten.
5. Convert 3020five to base four.
6. Convert 274eight to base five.
7. Convert A9Btwelve to base ten.
8. Convert 143ten to base twelve.
9. Convert 43ten to base two.
10. Convert 10011two to base ten.
11. Convert 43eight to base two.
12. Convert 10011two to base eight.
Math 221 – F07
Activities, page 21
Addition Algorithms
1. Solve the following problems using models any way you like. (You might like to see if you
can use different methods on each problem.) Have one person manipulate the models. A
different person should record what is going on. Take turns in each role.
a) There were 46 first-graders from Adams Elementary School at the picnic. Then 37
first-graders from Lincoln School arrived. How many first-graders were at the picnic all
together?
b) There are 355 boys and 187 girls in the youth soccer program. How many children are
participating in the youth soccer program?
c) Jo’s bank account had $1328 in it. Then Jo deposited a check for $576. How much was
in the account after the deposit?
2. Do then write. Do the following sums using the traditional “right to left” algorithm.
Manipulate the model and record your work as you go, as before, each person taking
a turn as manipulator and as recorder.
Redo the last one using a “left to right” algorithm.
55
+ 48
248
+ 165
706
+ 395
3. Write then do. Do the following additions first without using models, recording each
step as it occurs. Then read your steps and simulate the work with a model.
875
+ 204
234
+ 567
4. Do the following base five sums. Use the models as you see fit. Is there more than one way
to do these?
23five
+ 34five
104five
+ 401five
+
1432five
23five
1234five
+ 1432five
5. Do the following sums in other bases.
23six
+ 34six
103four
+ 321four
1234nine
+ 5678nine
101101two
+ 100111two
Math 221 – F07
Activities, page 22
Subtraction Algorithms
1. Exploration. Solve the following problems using models any way you like. (You might like
to see if you can use different methods on each problem.) Have one person manipulate the
models and another person record what is going on. Take turns in each role.
a) Last week, Jo earned $423, but $176 was taken out for taxes, health insurance, etc. After
this withholding, how much was Jo’s takehome pay?
b) Craig said, “I have 53 keychains in my collection.” Diane said, “Oh, I have only 37
keychains.” How many more keychains than Diane does Craig have?
By the way, how would each of the problems above be categorized by Van de Walle?
2. Standard Algorithm. Do the following differences using the models but following the
“standard” algorithm (i.e., working from little to big). Manipulate the model and record your
work as you go.
82
− 57
−
127
59
430
− 234
403
− 246
3. Other Algorithms. Can you figure out what is going on in each of the algorithms below?
Try to explain both what is happening (procedural description) and why it gives the correct
result.
243
− 68
243
− 68
243
− 68
4. Base Five. Do the following base five differences. Use the models as you see fit. Try using
a variety of algorithms.
41five
− 14five
32five
− 24five
−
1421five
123five
4321five
− 1234five
5. Do the following sums in other bases.
32six
− 14six
−
101four
22four
507nine
− 463nine
101101two
− 100111two
Math 221 – F07
Activities, page 23
Multiplication Algorithms
1. Use base 5 pieces to model the following multiplication problems. For each sketch the model
and indicate your regrouping.
22five
× 2five
12five
× 3five
12five
× 13five
12five
× 23five
2. Now try to do the following multiplications without the pieces. Try various algorithms,
including the “standard algorithm,” the “intermediate algorithm” (what I also call “all pairs”;
this has two flavors, depending on whether or not you record the zeros), and the “lattice
algorithm”. Do each one at least 2 of the three ways. How do your two solutions compare?
×
32five
3five
23five
× 31five
×
432five
23five
3. Here are some multiplication problems in other bases for you to try. Use any algorithm you
like.
×
22four
3four
23four
× 32four
23eight
× 16eight
×
110two
11two
Math 221 – F07
Activities, page 24
Division Algorithms
1. Use base 5 pieces to model the following division problems.
22five ÷ 4five
121five ÷ 4five
344five ÷ 3five
1113five ÷ 3five
2. Now try to do the following divisions without the pieces. (But you may certainly think about
what you would be doing with the pieces. You may even want to use the pieces to check your
work.)
3five )234five
2five )1432five
4five )1403five
12five )1410five
3. Here are some division problems in other bases for you to try.
2four )122four
3eight )675eight
4six )1403six
12six )1410six
Math 221 – F07
Activities, page 25
Algorithm Review
1. Basic Facts Tables. One of the difficulties in doing pencil-and-paper algorithms in other
bases is our lack of knowledge of the basic facts expressed in those bases. But we could make
a table of these facts and see if that helps us. Fill in the tables below with basic facts for
base 6.
+six
0
1
2
3
4
5
0 1 2 3
4
×six
0
1
2
3
4
5
5
12
0 1 2 3 4
5
23
How large would these tables be in base eight? Why?
2. Addition. Use the addition facts table and the standard algorithm to compute the following
sums. (You should not need to do any reasoning in base ten.)
243six
+ 121six
344six
+ 214six
1234six
+ 333six
3. Subtraction.
a) Explain how to use the addition fact table to find 13six − 4six .
b) Do the following subtractions twice each, using two different algorithms. [(Algorithms
you should know: standard, “9’s” complement,1 elevator (add the same to each number),
and the left-to-right algorithm we looked at.] You may want to consult the addition fact
table as you do these.
43six
111six
1432six
− 14six
− 24six
− 333six
1
Note that in other bases, ‘9’ must be changed to one less than the base. Why?
Math 221 – F07
Activities, page 26
4. Multiplication. Multiply using the basic facts tables and the indicated algorithm.
a) Use the “all-pairs” algorithm to compute 21six × 34six .
b) Use the lattice algorithm to compute 23six × 314six .
c) Use the standard algorithm to compute 324six × 23six .
5. Division.
a) Explain how to use the multiplication fact table to determine 23six ÷ 3six .
b) Use any method you like to do the following division problems.
3six )234six
4six )1322six
6. You can make up additional problems for your self with any base, any operation and any
algorithm.