1.5 symbols and pictures w

Symbols and Pictures

http://www.lahc.edu/math/frankma.htm

Besides numbers, we also use letters and pictures to help us to
keep track of quantities and relations. Let’s start with using
letters to track different types of items.

For example, let’s use the letter “A” to represent an apple, i.e.
is A or 1A

is A or 1A
then

is 2A,

is 5A, and no apple is 0A or 0,

is A or 1A
then

is 2A,


Addition or subtraction operation may be recorded accordingly,
for example:

+

–

is A or 1A
then

is 2A,


for example:

+

is simply recorded as 3A + 2A → 5A

–

is A or 1A
then

is 2A,


for example:

+


–
as

3A – 2A → A

is A or 1A
then

is 2A,


for example:

+


–
as

3A – 2A → A

We may extend this notation to addition and subtraction of
apple-arithmetic.


Example A. On the 1st day, Farmer Andy picked 34 apples.
He sold 16 of them and used another 5 to bake a pie. On the
2nd day, he picked 16 more apples, then traded 25 apples for a
block of cheese with his neighbor. Record these
apple-transactions with addition and subtraction operations
using “A” for
. How many apples are left after 2 days?


using “A” for
We may record the transactions as 34A – 16A – 5A + 16A – 25A.


using “A” for
Let’s compute the outcome in two different ways.


using “A” for
I. In the order of the transactions:
34A – 16A – 5A + 16A – 25A


using “A” for
34A – 16A – 5A + 16A – 25A
= 18A – 5A + 16A – 25A


using “A” for
34A – 16A – 5A + 16A – 25A
= 18A – 5A + 16A – 25A
= 13A + 16A – 25A


using “A” for
34A – 16A – 5A + 16A – 25A
= 18A – 5A + 16A – 25A
= 13A + 16A – 25A
= 29A – 25A = 4A


using “A” for
I. In the order of the transactions: lI. Group it into two groups,
the apples that came in vs
34A – 16A – 5A + 16A – 25A
apples that went out:
= 18A – 5A + 16A – 25A
= 13A + 16A – 25A
= 29A – 25A = 4A


using “A” for
34A – 16A – 5A + 16A – 25A
= 18A – 5A + 16A – 25A
34A – 16A – 5A + 16A – 25A
= 13A + 16A – 25A
= 29A – 25A = 4A


using “A” for
34A – 16A – 5A + 16A – 25A
= 18A – 5A + 16A – 25A
34A – 16A – 5A + 16A – 25A
= 13A + 16A – 25A
= 29A – 25A = 4A

=

50A


using “A” for
34A – 16A – 5A + 16A – 25A
= 18A – 5A + 16A – 25A
34A – 16A – 5A + 16A – 25A
= 13A + 16A – 25A
= 29A – 25A = 4A

=

50A

– 46A


using “A” for
34A – 16A – 5A + 16A – 25A
= 18A – 5A + 16A – 25A
34A – 16A – 5A + 16A – 25A
= 13A + 16A – 25A
= 29A – 25A = 4A

=

50A

– 46A

= 4A


using “A” for
34A – 16A – 5A + 16A – 25A
= 18A – 5A + 16A – 25A
34A – 16A – 5A + 16A – 25A
= 13A + 16A – 25A
= 29A – 25A = 4A

– 46A = 4A
=
50A
So we have 4 apples left. Note that method lI tells us that Andy
picked 50 apples in total and 46 apples are “spent.”


Using “A” to represent an apple and “B” to represent a banana,
we may record 5 apples and 4 bananas as 5A + 4B:


5A + 4B


5A + 4B
The expression A + B

+

means 1 apple + 1 banana.


5A + 4B
+
Note that while A + A is 2A, B + B is 2B, the expressions
A + B or B + A may not be condensed as AB or BA.


5A + 4B
+
(An AppleBanana is not an Apple nor a Banana.)


5A + 4B
+
We will reserve AB or BA for other purposes.


5A + 4B
+
We also use the verb “combine” for resolving an addition or
subtraction problems.
.


5A + 4B
+
.
Example B. Combine
a. 2A + 3B + 5A – B


5A + 4B
+
.
Example B. Combine
a. 2A + 3B + 5A – B
=

7A


5A + 4B
+
.
Example B. Combine
a. 2A + 3B + 5A – B
=

7A + 2B


5A + 4B
+
.
Example B. Combine
a. 2A + 3B + 5A – B
=

7A + 2B

(This answer may not be shorten.)

When combining multiple transactions of apples and bananas
by hand, collect them in the following organized manner.

b. 16A + 9B + 5A – B – 4A + 3A + 4B – 4A – 2B + 3B

1. Combine all the apples that came in, the ones with a “+” sign
in the front.

b.

–
16A +
–

9B +
24A

–
5A –
–

B – 4A +

–
3A +
–

4B – 4A – 2B + 3B

in the front.
2. Combine all the apples that went out, the ones with a “–”
sign in the front.

b.

–
16A +
–

9B +
24A

–
5A –
–

B–

–
4A
–

– 8A

+

–
3A +
–

4B –

–
4A –
–

2B + 3B

in the front.
sign in the front.
3. Combine the above results to obtain the number of apples left.

b.

–
16A +
–

9B +

–
5A –
–

24A

B–

–
4A
–

– 8A
16A

+

–
3A +
–

4B –

–
4A –
–

2B + 3B

in the front.
sign in the front.
4. Repeat steps 1–3 for the bananas.
–
–
–
–
–
b. 16A + 9B + 5A – B – 4A + 3A + 4B – – – 2B + 3B
4A
–
–
–
–
24A

– 8A
16A

in the front.
sign in the front.
–
–
–
–
–
–
–
–
–
–– + 3B
b. 16A + – + 5A ––B – 4A + 3A + 4B – – – 2B –
9B –
4A
–
–
–
–
24A

– 8A
16A

16B

in the front.
sign in the front.
–
–
–
–
–
–
–
–
–
–– + 3B
b. 16A + – + 5A ––B – 4A + 3A + 4B – – – 2B –
9B –
4A
–
–
–
–
24A

– 8A
16A

16B

– 3B

in the front.
sign in the front.
–
–
–
–
–
–
–
–
–
–– + 3B
b. 16A + – + 5A ––B – 4A + 3A + 4B – – – 2B –
9B –
4A
–
–
–
–
24A

– 8A
16A

16B

– 3B
13B

in the front.
sign in the front.
–
–
–
–
–
–
–
–
–
–– + 3B
b. 16A + – + 5A ––B – 4A + 3A + 4B – – – 2B –
9B –
4A
–
–
–
–
––
24A

– ––
8A
16A

= 16A + 13B

–
16B
–

– ––
3B
13B


Suppose we have two quantities A and B represented by two
line segments as shown.

A
B


line segments as shown. Just as 5m = 2m + 3m can be view
as gluing two sticks together and getting one stick of length
5 meters,
A
B
2m
3m


5 meters,
A
B

5m

2m
3m

2m

3m


5 meters, we may view the sum S = A + B as the line
segments formed by joining A and B into one piece.
A
A + B = S (Sum)
B
2m

A

B

5m

3m

2m

3m


A
A + B = S (Sum)
B
2m

A

5m

B

3m
3m
2m
Note the addition relation 2m + 3m = 5m may be phrased as
subtractions: 5m – 3m = 2m or 5m – 2m = 3m.


A
A + B = S (Sum)
B
2m

A

5m

B

3m
3m
2m
Note the addition relation 2m + 3m = 5m may be phrased as
subtractions: 5m – 3m = 2m or 5m – 2m = 3m.
Likewise, the sum and differences
S = A + B, S – A = B, S – B = A
describe the same relation between A, B and S.

So if the total is given, we may recover the parts by subtraction.

For example, if we have the following pictures:
100
A

100
A
then this = 100 – A,

80
100
A

B

80
100
A

B

and this = 80 – B.

80
100
A

B

If we know that
S
100


T

80

80
100
A

B

If we know that
S
100


T

80
then this = S – 100,

80
100
A

B

If we know that
S


T

100

80

and this = T – 80.

80
100
A

B

If we know that
S
100


T

80

Often in real life, we are interested in tracking a quantity that
is composed of two separate parts which can be represented
by pictures shown.

80
100
A

B

If we know that
S
100


T

80

Often in real life, we are interested in tracking a quantity that
is composed of two separate parts which can be represented
by pictures shown. The importance of the pictures is to clarify
the order of subtraction visually.

Example B.
With the given information, answer the following questions.
i. Which number or letter represents the total?
Which numbers or letters represent the parts?
ii. Draw and label a line picture with the given number(s)
and letters.
iii. List all the addition and subtraction relations.

Example B.
and letters.
a. Andy and Beth went to lunch, the bill came to $18 out of
which Andy paid A dollars and Beth paid B dollars.

Example B.
and letters.
Ans. i. The total is the $18 bill with A and B as the parts.

Example B.
and letters.
ii. Representing them with
line segments, we have
18
A

B

Example B.
and letters.
ii. Representing them with
line segments, we have
18
A

B

iii. The addition and subtraction
relations are:
18 = A + B,
18 – A = B, and
18 – B = A.


b. Let M be the number of males and F be the number of
females. After a survey, out of a group of P people, we found
that there are 16 males in the group.


Ans. i. The total is the P with M and F as the parts.


ii. In picture,
P
F

16 (=M)


ii. In picture,
P
relations are: P = F + 16,
F

16 (=M)

P – F = 16, and P – 16 = F.


ii. In picture,
P
F

16 (=M)

P – F = 16, and P – 16 = F.

c. Let S be the number of sunny days and C be the number of
cloudy (or rainy days). From previous records, on the average,
LA has 263 days with sun in a year.


ii. In picture,
P
F

16 (=M)

P – F = 16, and P – 16 = F.

Ans. i. The total is 365 days with S and C as the parts.


ii. In picture,
P
F

16 (=M)

P – F = 16, and P – 16 = F.

Ans. i. The total is 365 days with S and C as the parts.
ii. In picture,
365
C

263 (=S)

Example C. A bag of 100 pieces of mixed candies contains
three different types: Apple-drops, Butter-scotch and Chocolate.
Let A be the number of apple-drops, B be the number of butterscotches, and C be the number of chocolates. We counted that
there are 26 pieces of chocolates. Answer the following.
ii. Draw and label a line picture with the given numbers and
letters. List all the addition and subtraction relation.

Ans. i. The total is the 100 with A, B, and C as the parts.

ii.
100

A

B

26 (=C)

ii.
100
B
26 (=C)
A
The ± relations are: 100 = A + B + 26

ii.
100
B
26 (=C)
A
100 – A = B + 26, 100 – B = A + 26, 100 – 26 = A + B,

ii.
100
B
26 (=C)
A
100 – A = B + 26, 100 – B = A + 26, 100 – 26 = A + B,
100 – A – B = 26, 100 – A – 26 = B, 100 – B – 26 = A

We note that each relation listed in part ii. may be viewed as a
physical procedure.

physical procedure. For example
“100 – B – 26 = A” says that
“take away from 100 the number of butter-scotches and the
number of chocolate; we get the number of apple-drops”.

physical procedure. For example
“100 – B – 26 = A” says that
“take away from 100 the number of butter-scotches and the
number of chocolate; we get the number of apple-drops”.
“100 – 26 = A + B (= 74)” says that
“take away from 100 the 26 pieces of chocolates,
the remaining 74 pieces, are butter-scotches and apple-drops,
or that there are 74 non-chocolates.

1.5 symbols and pictures w

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