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Published on February 6, 2014

Author: Tzenma

Source: slideshare.net

Symbols and Pictures http://www.lahc.edu/math/frankma.htm

Symbols and Pictures 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.

Symbols and Pictures 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

Symbols and Pictures 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 then is 2A, is 5A, and no apple is 0A or 0,

Symbols and Pictures 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 then is 2A, is 5A, and no apple is 0A or 0, Addition or subtraction operation may be recorded accordingly, for example: + –

Symbols and Pictures 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 then is 2A, is 5A, and no apple is 0A or 0, Addition or subtraction operation may be recorded accordingly, for example: + is simply recorded as 3A + 2A → 5A –

Symbols and Pictures 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 then is 2A, is 5A, and no apple is 0A or 0, Addition or subtraction operation may be recorded accordingly, for example: + is simply recorded as 3A + 2A → 5A – as 3A – 2A → A

Symbols and Pictures 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 then is 2A, is 5A, and no apple is 0A or 0, Addition or subtraction operation may be recorded accordingly, for example: + is simply recorded as 3A + 2A → 5A – as 3A – 2A → A We may extend this notation to addition and subtraction of apple-arithmetic.

Symbols and Pictures 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?

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A.

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways.

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways. I. In the order of the transactions: 34A – 16A – 5A + 16A – 25A

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways. I. In the order of the transactions: 34A – 16A – 5A + 16A – 25A = 18A – 5A + 16A – 25A

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways. I. In the order of the transactions: 34A – 16A – 5A + 16A – 25A = 18A – 5A + 16A – 25A = 13A + 16A – 25A

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways. I. In the order of the transactions: 34A – 16A – 5A + 16A – 25A = 18A – 5A + 16A – 25A = 13A + 16A – 25A = 29A – 25A = 4A

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways. 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

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways. 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 34A – 16A – 5A + 16A – 25A = 13A + 16A – 25A = 29A – 25A = 4A

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways. 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 34A – 16A – 5A + 16A – 25A = 13A + 16A – 25A = 29A – 25A = 4A = 50A

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways. 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 34A – 16A – 5A + 16A – 25A = 13A + 16A – 25A = 29A – 25A = 4A = 50A – 46A

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways. 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 34A – 16A – 5A + 16A – 25A = 13A + 16A – 25A = 29A – 25A = 4A = 50A – 46A = 4A

Symbols and Pictures 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? We may record the transactions as 34A – 16A – 5A + 16A – 25A. Let’s compute the outcome in two different ways. 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 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.”

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

Symbols and Pictures 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

Symbols and Pictures 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 The expression A + B + means 1 apple + 1 banana.

Symbols and Pictures 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 means 1 apple + 1 banana. The expression A + B + 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.

Symbols and Pictures 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 means 1 apple + 1 banana. The expression A + B + 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. (An AppleBanana is not an Apple nor a Banana.)

Symbols and Pictures 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 means 1 apple + 1 banana. The expression A + B + 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. (An AppleBanana is not an Apple nor a Banana.) We will reserve AB or BA for other purposes.

Symbols and Pictures 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 means 1 apple + 1 banana. The expression A + B + 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. (An AppleBanana is not an Apple nor a Banana.) We will reserve AB or BA for other purposes. We also use the verb “combine” for resolving an addition or subtraction problems. .

Symbols and Pictures 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 means 1 apple + 1 banana. The expression A + B + 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. (An AppleBanana is not an Apple nor a Banana.) We will reserve AB or BA for other purposes. We also use the verb “combine” for resolving an addition or subtraction problems. . Example B. Combine a. 2A + 3B + 5A – B

Symbols and Pictures 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 means 1 apple + 1 banana. The expression A + B + 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. (An AppleBanana is not an Apple nor a Banana.) We will reserve AB or BA for other purposes. We also use the verb “combine” for resolving an addition or subtraction problems. . Example B. Combine a. 2A + 3B + 5A – B = 7A

Symbols and Pictures 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 means 1 apple + 1 banana. The expression A + B + 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. (An AppleBanana is not an Apple nor a Banana.) We will reserve AB or BA for other purposes. We also use the verb “combine” for resolving an addition or subtraction problems. . Example B. Combine a. 2A + 3B + 5A – B = 7A + 2B

Symbols and Pictures 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 means 1 apple + 1 banana. The expression A + B + 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. (An AppleBanana is not an Apple nor a Banana.) We will reserve AB or BA for other purposes. We also use the verb “combine” for resolving an addition or subtraction problems. . Example B. Combine a. 2A + 3B + 5A – B = 7A + 2B (This answer may not be shorten.)

Symbols and Pictures 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

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

Symbols and Pictures When combining multiple transactions of apples and bananas by hand, collect them in the following organized manner. 1. Combine all the apples that came in, the ones with a “+” sign 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

Symbols and Pictures When combining multiple transactions of apples and bananas by hand, collect them in the following organized manner. 1. Combine all the apples that came in, the ones with a “+” sign in the front. 2. Combine all the apples that went out, the ones with a “–” 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

Symbols and Pictures When combining multiple transactions of apples and bananas by hand, collect them in the following organized manner. 1. Combine all the apples that came in, the ones with a “+” sign in the front. 2. Combine all the apples that went out, the ones with a “–” sign in the front. 3. Combine the above results to obtain the number of apples left. 4. Repeat steps 1–3 for the bananas. – – – – – b. 16A + 9B + 5A – B – 4A + 3A + 4B – – – 2B + 3B 4A – – – – 24A – 8A 16A

Symbols and Pictures When combining multiple transactions of apples and bananas by hand, collect them in the following organized manner. 1. Combine all the apples that came in, the ones with a “+” sign in the front. 2. Combine all the apples that went out, the ones with a “–” sign in the front. 3. Combine the above results to obtain the number of apples left. 4. Repeat steps 1–3 for the bananas. – – – – – – – – – –– + 3B b. 16A + – + 5A ––B – 4A + 3A + 4B – – – 2B – 9B – 4A – – – – 24A – 8A 16A 16B

Symbols and Pictures When combining multiple transactions of apples and bananas by hand, collect them in the following organized manner. 1. Combine all the apples that came in, the ones with a “+” sign in the front. 2. Combine all the apples that went out, the ones with a “–” sign in the front. 3. Combine the above results to obtain the number of apples left. 4. Repeat steps 1–3 for the bananas. – – – – – – – – – –– + 3B b. 16A + – + 5A ––B – 4A + 3A + 4B – – – 2B – 9B – 4A – – – – 24A – 8A 16A 16B – 3B

Symbols and Pictures When combining multiple transactions of apples and bananas by hand, collect them in the following organized manner. 1. Combine all the apples that came in, the ones with a “+” sign in the front. 2. Combine all the apples that went out, the ones with a “–” sign in the front. 3. Combine the above results to obtain the number of apples left. 4. Repeat steps 1–3 for the bananas. – – – – – – – – – –– + 3B b. 16A + – + 5A ––B – 4A + 3A + 4B – – – 2B – 9B – 4A – – – – 24A – 8A 16A 16B – 3B 13B

Symbols and Pictures When combining multiple transactions of apples and bananas by hand, collect them in the following organized manner. 1. Combine all the apples that came in, the ones with a “+” sign in the front. 2. Combine all the apples that went out, the ones with a “–” sign in the front. 3. Combine the above results to obtain the number of apples left. 4. Repeat steps 1–3 for the bananas. – – – – – – – – – –– + 3B b. 16A + – + 5A ––B – 4A + 3A + 4B – – – 2B – 9B – 4A – – – – –– 24A – –– 8A 16A = 16A + 13B – 16B – – –– 3B 13B

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

Symbols and Pictures Suppose we have two quantities A and B represented by two 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

Symbols and Pictures Suppose we have two quantities A and B represented by two 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 5m 2m 3m 2m 3m

Symbols and Pictures Suppose we have two quantities A and B represented by two 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, 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

Symbols and Pictures Suppose we have two quantities A and B represented by two 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, 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 5m B 3m 3m 2m Note the addition relation 2m + 3m = 5m may be phrased as subtractions: 5m – 3m = 2m or 5m – 2m = 3m.

Symbols and Pictures Suppose we have two quantities A and B represented by two 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, 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 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.

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

Symbols and Pictures So if the total is given, we may recover the parts by subtraction. For example, if we have the following pictures: 100 A

Symbols and Pictures So if the total is given, we may recover the parts by subtraction. For example, if we have the following pictures: 100 A then this = 100 – A,

Symbols and Pictures So if the total is given, we may recover the parts by subtraction. For example, if we have the following pictures: 80 100 A B then this = 100 – A,

Symbols and Pictures So if the total is given, we may recover the parts by subtraction. For example, if we have the following pictures: 80 100 A B then this = 100 – A, and this = 80 – B.

Symbols and Pictures So if the total is given, we may recover the parts by subtraction. For example, if we have the following pictures: 80 100 A B then this = 100 – A, If we know that S 100 and this = 80 – B. T 80

Symbols and Pictures So if the total is given, we may recover the parts by subtraction. For example, if we have the following pictures: 80 100 A B then this = 100 – A, If we know that S 100 and this = 80 – B. T 80 then this = S – 100,

Symbols and Pictures So if the total is given, we may recover the parts by subtraction. For example, if we have the following pictures: 80 100 A B then this = 100 – A, If we know that S and this = 80 – B. T 100 80 then this = S – 100, and this = T – 80.

Symbols and Pictures So if the total is given, we may recover the parts by subtraction. For example, if we have the following pictures: 80 100 A B then this = 100 – A, If we know that S 100 and this = 80 – B. T 80 then this = S – 100, and this = 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.

Symbols and Pictures So if the total is given, we may recover the parts by subtraction. For example, if we have the following pictures: 80 100 A B then this = 100 – A, If we know that S 100 and this = 80 – B. T 80 then this = S – 100, and this = 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.

Symbols and Pictures 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.

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

Symbols and Pictures 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. a. Andy and Beth went to lunch, the bill came to $18 out of which Andy paid A dollars and Beth paid B dollars. Ans. i. The total is the $18 bill with A and B as the parts.

Symbols and Pictures 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. a. Andy and Beth went to lunch, the bill came to $18 out of which Andy paid A dollars and Beth paid B dollars. Ans. i. The total is the $18 bill with A and B as the parts. ii. Representing them with line segments, we have 18 A B

Symbols and Pictures 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. a. Andy and Beth went to lunch, the bill came to $18 out of which Andy paid A dollars and Beth paid B dollars. Ans. i. The total is the $18 bill with A and B as the parts. 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.

Symbols and Pictures 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.

Symbols and Pictures 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.

Symbols and Pictures 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)

Symbols and Pictures 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, iii. The addition and subtraction P relations are: P = F + 16, F 16 (=M) P – F = 16, and P – 16 = F.

Symbols and Pictures 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, iii. The addition and subtraction P relations are: P = F + 16, 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.

Symbols and Pictures 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, iii. The addition and subtraction P relations are: P = F + 16, 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. Ans. i. The total is 365 days with S and C as the parts.

Symbols and Pictures 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, iii. The addition and subtraction P relations are: P = F + 16, 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. Ans. i. The total is 365 days with S and C as the parts. ii. In picture, 365 C 263 (=S)

Symbols and Pictures 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. 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 numbers and letters. List all the addition and subtraction relation.

Symbols and Pictures 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. 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 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.

Symbols and Pictures 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. 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 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)

Symbols and Pictures 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. 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 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 B 26 (=C) A The ± relations are: 100 = A + B + 26

Symbols and Pictures 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. 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 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 B 26 (=C) A The ± relations are: 100 = A + B + 26 100 – A = B + 26, 100 – B = A + 26, 100 – 26 = A + B,

Symbols and Pictures 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. 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 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 B 26 (=C) A The ± relations are: 100 = A + B + 26 100 – A = B + 26, 100 – B = A + 26, 100 – 26 = A + B, 100 – A – B = 26, 100 – A – 26 = B, 100 – B – 26 = A

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

Symbols and Pictures We note that each relation listed in part ii. may be viewed as a 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”.

Symbols and Pictures We note that each relation listed in part ii. may be viewed as a 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.

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