Selected Problem: POW : Kick It!

This problem allows students to work with different combinations of numbers. It is especially important to note that properties of even numbers, odd numbers, and a combination with even and odd numbers are very important to understand. This problem will help students to explore relationships with even and odd numbers.

POW: KICK IT
Problem Statement:
The Free Thinkers Football League is able to score 5 points for every field goal and 3 points for every touchdown. The football league can score with a combination of a field goal and a touchdown, or just one event at a time (one field goal or one touchdown). One team member on the Free Thinkers Football League notices that not every score is possible. For example, a score of 1 or 2 is not possible because a touchdown yields 3 points while a field goal yields 3 points. The Free Thinkers Football League wants to know if there is a score that represents the highest impossible score they won’t be able to get. Is there a highest impossible score?
The Football League then wants you to come up with other scoring systems using whole numbers and see if there are scores that are impossible to make. Is there a highest impossible score with each scoring system that you created? Is there ever a situation where there is not a highest impossible score? Are you able to find a rule or condition that explains what scoring combinations yield highest impossible scores and what combinations do not?
In combinations that do yield a highest impossible score, can you find any patterns or rules to help people figure out what the highest impossible score is?







Process:
When I went to solve this problem, I first identified that a field gold represents 5 while a touchdown represents 3. After stating my givens, I created a chart to help document my findings. I then started a tally of impossible scores that I found while I was writing out possible combinations in my chart:


At first I thought that 17 was the highest impossible value because I thought to myself; “5+5+5=15, and I can’t add 3 to that to make 17”. I did not think about the combination of multiplying 3 by 4 to yield 12, and then adding 5 to get a score of 17. I did not come to this conclusion until after I presented my findings to the class. After realizing that 17 does work out, I realized that 7 was my highest impossible score that can be achieved with the given constraints. I was able to realize that 7 was the highest impossible score because with every score after 7, you can keep adding 10 to get all possible whole numbers greater than 7.

I then created an additional scoring tally which the instructions request:














After creating this diagram, I realized 1 was my highest impossible score because I can still add 10 (represents 5 touchdowns) like the first score combination. I also realized that in my work I noted that 3 was an impossible score, not taking in to account that 3 can represent one field goal.


Solution:
(First combination, field goal is 5 points and touchdown is 3 points)
After creating the chart, I stopped at the combination of 30. I stopped here because I realized that when you look at an increment of 10 that is possible, let’s say from 20-30, I concluded that you can add 10 to each number in the increment to make the next set of 10. You can keep repeating this process to get every single other whole numerical possibility. I chose to add 10 because 10 is a possible point combination for this constraint since 10 is composed of two field goals, or in equation terms 5+5=10. For example:
1. 20+10= 30 (30 is composed of 6 field goals)
2. 21+10= 31 (31 can be created with 7 touchdowns, yielding 21, and two field goals, yielding 10. 21+10= 31)
3. 22+10= 32 (32 can be created with 4 touchdowns, yielding 12, and 4 field goals, yielding 20. 20+12=22)
4. 23+10= 33 (33 can be created with 1 touchdown, yielding 3, and 6 field goals, yielding 30. 30+3=33)
5. 24+10= 34 (34 can be created with 8 touchdowns, yielding 24, and 2 field goals, yielding 10. 24+10= 34)
6. 25+10= 35 (35 can be created with 7 touchdowns, yielding 35)
7. 26+10= 36 (36 can be created with two touchdowns, yielding 6 points, and 6 field goals, yielding 30. 30+6=36)
8. 27+10= 37 (37 can be created with 9 touchdowns, yielding 27 points, and two field goals, yielding 10. 27+10=37)
9. 28+10= 38 (38 can be created with 1 touchdown, yielding 3, and 7 field goals, yielding 35. 35+3=38)
10. 29+10= 39 (39 can be created with 3 touchdowns, yielding 9 points, and 6 field goals, yielding 30. 30+9= 39.
11. 30+10= 40 (40 can be created with 5 touchdowns, yielding 15, and 5 field goals, yielding 25. 25+15=40
Every single number in this process just added 10, which represents two field goals. This process works because 10 is a possible combination, being two field goals.
(Second combination, field goal is 3 points and a touchdown is 2 points)
After using this score combination, I realized that I could again add 10 within a 10 increment like in the scoring above because in this scoring system, 10 represents 5 touchdowns. I also realized though that instead of just using an increment of 10 like the first combination, I can use a smaller increment. I can take an increment of 6, for example, because 6 represents a possible score, which translates to 2 field goals or 3 touchdowns.
(Other possible combinations)
After going over two separate combinations, I realized that if both the touchdown and the field goal yield the same score, then there is no highest possible amount because all possible scores have to be divisible by the chosen score. For example, let’s say that a field goal represents 2 points while a touchdown represents 2 points as well. Using 2 means that all score possibilities will be even numbers. This is because any number multiplied by 2 will yield an even result. This means that all odd numbers represent impossible scores and there is no definite highest impossible score that can be calculated. All possible combinations of field goals and touchdowns that use even numbers will never yield a highest impossible score. This is because anytime an even number is multiplied, added, or subtracted with another even number, the answer will always yield an even number. This leave the odd numbers to represent impossible scores. I was able to use my prior knowledge to help come to this conclusion. Odd numbers can result when two even numbers are divided, but this problem is not asking for division. This means that again, with the 2 and 2 combination, all odd numbers will represent the highest impossible scores and thus there is no highest impossible score that can be recorded. I realized that when you are looking at the combinations of any score, you can simply look at the highest score value, whether touchdown or field goal, and multiply that by two. The product will yield a testable increment with your score combinations. (For example: Touchdown is 3 points and Field Goal is 5 points. The increment used was 10 because the highest score was 5, and 5 multiplied by 2 yields 10. After 7, the increment of 10 from 8-18 can be proven possible because you can add 10, which is again composed of two field goals, to every single number to get your desired value. You can keep adding 10 to get every number after 7. When using the combination of the touchdown representing 2 points and the field goal represents 3 points, I used an increment of 6 because 3 multiplied by 2 equals 6. The highest impossible value was 1 because the increment of 6 after 1, from 2-7, was able to be calculated. I was able to take this increment and add 6 to each number to yield the next increment of 6. You can keep repeating this process to get every single number above your highest impossible value of 1. )


Extensions:
This problem can be extended to further knowledge regarding properties of numbers. This problem required knowledge of combinations of numbers, specifically what happens when you have combinations of two even numbers, two odd numbers, and both an even and odd number. Possible strategies to help extend thinking include:
1. Are there any score combinations that yield a highest possible score of 10 or more? (Allows students to work with and experiment with greater numbers)
2. List three separate score combinations. 1 score combination needs to represent an even and an odd number, 1 score combination needs to represent two even numbers, and 1 score combination needs to represent two odd numbers. (This prompt is helpful to drive students right to the catch that two combinations of even numbers can’t yield a highest impossible score and the other two combinations can yield a highest impossible score.)
Evaluation:
Option B: Personal Reaction to the Problem
When I first encountered this problem I thought it was a mouthful. I was first unsure of how to approach the problem. Because I am a visual learner, I decided to start creating a chart of score combinations that were possible and hopefully I will be able to see the highest impossible score, if any. This problem was educationally worthwhile because it reminded me of how important it is to understand the properties of odd and even numbers and the possible combinations they can make. For this problem a combination of two even numbers, two odd numbers, and a combination with one even number and one odd number was very important to understand, in regards to addition and multiplication. I was able to access and apply my prior knowledge of these different combinations to help come to my conclusion in this problem. I enjoyed working on this problem because it challenged me to recall prior knowledge and it allowed me to apply that prior knowledge. This problem was enjoyable to me because again it only proves that mathematics constantly builds on itself, which is very enjoyable to me. I found that even though in my first chart I wrote numerical combinations a lot higher than what I needed to, I enjoyed doing it because it was something I understood. It was almost like a puzzle to me in trying to find the missing piece, as the highest impossible score represented that missing piece. Eventually I was able to stop and try to look for patterns, which allowed me to conclude that the highest impossible value I found was in fact true. This problem at first was hard for me because it was not a problem that could be solved using one or two steps. After taking the problem piece by piece I was able to break it down and make it easier. I did this through creating the diagram and listing the possible score combinations. This is a great problem to use in math classrooms because it encompasses several different math practices; persevering in solving problems, attending to precision, and looking for structure and regularity in repeated reasoning. (retrieved from: http://www.corestandards.org/Math/Practice )