A.7.a.
Tony and I were comparing our baseball card collections. Tony gave me 6 cards and then we both had 10. How many cards did I have before he gave some to me?
A.7.b.
How many cards did Tony start with?
A.8.a.
Tony and I were comparing our baseball card collections. I had 4 cards and Tony had 16. Tony promised to give me some cards so we could have the same number of cards. How many did I end up with?
A.8.b.
How many did Tony give me?
A.9.
Tony and I were comparing our baseball card collections. Tony gave me 6 cards and then we had the same number. I started with 4. How many cards did Tony start with?
First Step: Identify the givens and the question
A.7.
Givens:
-
- Tony gave me 6 cards
-
- We both have 10 after he gave me the cards
Question:
-
- How many cards did I have before he gave some to me?
-
- How many cards did Tony start with?
A.8.
Givens:
-
- I had 4 cards
-
- Tony had 16 cards
-
- Tony gave me cards so we both had the same number
Question:
-
- How many cards did I end up with?
-
- How many did Tony give me?
A.9.
Givens:
-
- Tony gave me 6 cards
-
- I had 4 cards to start
-
- We both had the same number of cards after he gave me 6
Question:
-
- How many cards did Tony start with?
Second Step: Problem Analysis
A.7.a. Tony gave me 6 cards and then I had 10. I had 6 less cards before he gave any to me. If I subtract 6 from 10 the amount of cards that I started with is found.
A.7.b.
Tony gave me 6 cards and then we both had 10. He had 6 more cards before he gave any away. The 6 cards must be added to the 10 he has in the end to find the total number of cards that Tony had before he gave any away.
A.8.a.
In this problem we know that we both end up with the same amount of cards. To find that sum total all of the cards should be combined, adding 4 to 16. Then since there are two of us who are splitting the cards, the sum should be divided into 2 groups.
A.8.b.
If we find out that both parties end up with 10 cards after Tony gives some away, then we know that Tony had 10 in the end. He started with 16 and 10 should be subtracted from that amount to find the number that he gave away.
A.9.
We know that Tony gave me 6 cards to add to my 4 cards. After his donation I have 10 cards and the problem states that we then both own the same amount of cards. 10 should be multiplied by 2 since there are two groups of 10 cards. The product of 20 is our combined number of cards. Since I had four to begin with and we are looking for Tony’s original number, we can subtract my original four from the collective total of 20 cards.
Third Step: Identify the operation(s) needed to find the answer and then solve
A.7.a.
Subtraction
10 - 6 = 4 cards
A.7.b.
Addition
10 + 6 = 16 cards
A.8.a.
Addition, division.
4 + 16 = 20 cards (altogether)
20 / 2 = 10 cards
A.8.b.
Addition, division, subtraction
4 + 16 = 20 cards (altogether)
20 / 2 = 10 cards
10 - 4 = 6 cards
A.9.
Addition, multiplication, subtraction
4 + 6 = 10 cards (each)
10 x 2 = 20 cards (altogether)
20 - 4 = 16 cards
Fourth Step: Identify the connection between both problems and reasoning.
All of the problems are about the same situation, but they give different pieces of information about that situation. We must reconstruct other information using the relationships given in the problem. In A.7. the student must notice that “I” will have 10 after Tony gives “me” 6, so to find “my” original total, 6 must be subtracted from 10. In A.8. it must be observed that both persons will have the same number of cards after Tony gives some away. To find how many both have after the trade, the total of all the cards must be found, 4 + 16 = 20 cards altogether. Then by dividing by two (the number of people the cards are split among) the number of cards that each person has can be found, 20 / 2 = 10 cards each. Finally, by taking the number of cards “I” have at the end, which is 10, and subtracting what “I” had to start, which is 4, it can be found how much Tony gave to “me,” which is 6 cards. Finally, by using the inverse operations used in A.8., A.9. can be solved. The first step to A.9. is adding the two numbers together, because Tony gave “me” 6 cards in addition to the original 4. That gives “me” 10 cards total (the teacher must be very explicit about the observation that the total number of cards remains the same). The problem also stated that both people had the same amount of cards after Tony gave some away, so if 10 is multiplied by 2 (the number of people that each have 10 cards), it can be found that there are 20 cards altogether. Finally, to find out how many Tony had to start, the number of cards that “I” had (4) can be subtracted from the total number of cards between both people (20) to equal the 16 cards that Tony had to start.
