In this investigation, we will check the number of different halftime scores for when one of the teams consistently scores a 0 or 1 or 2, etc. For this particular investigation we will use 2-5, 2-4, 2-3, 2-2, 2-1, and 2-0
There are two methods that I used to find the half time scores. First is drawing the diagram and listing out all the possible scores where the line intersects.
There are two methods that I used to find the half time scores. First is drawing the diagram and listing out all the possible scores where the line intersects.
Table 1: Possible Half-Time Score when there one team consistently get the same score (2). (Team A – Team B)
Then, I listed the final results out in another table to make the data clearer and easier to understand.
If the first difference between successive terms is always the same, then we can determine the formula for a linear sequence.
T = 3n + 3
Where T = Total number of possible Half time score
N = score of the team that do not consistently get the same scores; in this case it’s Team B
T = 3n + 3
Where T = Total number of possible Half time score
N = score of the team that do not consistently get the same scores; in this case it’s Team B
Check the equation and compare it to table 2
T = 3n + 3
T = 3(3) + 3
T = 9 + 3
T = 12 (correct)
T = 3n + 3
T = 3(5) + 3
T = 15 + 3
T = 18 (correct)
The equation T = 3n + 3 works if the constant score of the Team A is 2. What if the constant of the Team A is 3? Will this equation work?
Let’s try if the equation works for other final scores
3-0, 3-1,3-2,3-3,3-4,3-5
- If the final score is 2-3 (Team A - Team B)
T = 3n + 3
T = 3(3) + 3
T = 9 + 3
T = 12 (correct)
- If the final score is 2-5
T = 3n + 3
T = 3(5) + 3
T = 15 + 3
T = 18 (correct)
The equation T = 3n + 3 works if the constant score of the Team A is 2. What if the constant of the Team A is 3? Will this equation work?
Let’s try if the equation works for other final scores
3-0, 3-1,3-2,3-3,3-4,3-5
The first difference between these successive terms is always 4. This means the equation
T = 3n + 3 is not applicable for this sequence. The equation for this sequence is T = 4n +4. We have to find a common equation that we could use to solve the possible halftime score when one team is consistently getting the same score. I noticed that when the score of each team is added 1, and multiplied together, the answer will be the number of different possibilities of half time scores.
Let T = Total number of possible Half time score
A = final score of team A
B = final score of team B
T = (A + 1) (B + 1)
T = 3n + 3 is not applicable for this sequence. The equation for this sequence is T = 4n +4. We have to find a common equation that we could use to solve the possible halftime score when one team is consistently getting the same score. I noticed that when the score of each team is added 1, and multiplied together, the answer will be the number of different possibilities of half time scores.
Let T = Total number of possible Half time score
A = final score of team A
B = final score of team B
T = (A + 1) (B + 1)
Check your equation: (compare answers with table 3 and 9)
T = (2 + 1) (3 + 1)
T= (3) (4)
T = 12 (correct)
T = (3 + 1) (5 + 1)
T = (4)(6)
T = 24 (correct)
Conclusion: If a game ends with one winner, you can find the number of possible half-time scores by using the equation
T = (A + 1) (B + 1).
- If the final score is 2-3
T = (2 + 1) (3 + 1)
T= (3) (4)
T = 12 (correct)
- If the final score is 3-5
T = (3 + 1) (5 + 1)
T = (4)(6)
T = 24 (correct)
Conclusion: If a game ends with one winner, you can find the number of possible half-time scores by using the equation
T = (A + 1) (B + 1).