Optimization explained to a middle school kid

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Let's say you play the game of collecting the maximum amount of coins along a path in the $(x,y)$ plane. The total amount of coins collected $S$ is given by $S = 5x + 7y$ where $5$ and $7$ represents the number of coins collected by walking respectively one unit in the $x$ and $y$ direction. However I can only go as far as one unit of distance ($x²+y²=1$).



It turns out that the solution to the problem is so such that $y/x = 7/5$ or said in other words, the ratio of how much you have to walk in the $y$ direction compared to in the $x$ direction is the same as the ratio of the rates at which you collect the coins in the $y$ direction compared to in the $x$ direction.



My question : suppose I have to explain this result to a middle school student who does not know any calculus of trigonometry. He is also quite shaky on describing equations of lines in the plane so I would not rely on that. How can I convince him with intuition that the two ratios are the same? (I'm not looking for a series of algebraic equations but rather for some kind of a visual proof).






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    Let's say you play the game of collecting the maximum amount of coins along a path in the $(x,y)$ plane. The total amount of coins collected $S$ is given by $S = 5x + 7y$ where $5$ and $7$ represents the number of coins collected by walking respectively one unit in the $x$ and $y$ direction. However I can only go as far as one unit of distance ($x²+y²=1$).



    It turns out that the solution to the problem is so such that $y/x = 7/5$ or said in other words, the ratio of how much you have to walk in the $y$ direction compared to in the $x$ direction is the same as the ratio of the rates at which you collect the coins in the $y$ direction compared to in the $x$ direction.



    My question : suppose I have to explain this result to a middle school student who does not know any calculus of trigonometry. He is also quite shaky on describing equations of lines in the plane so I would not rely on that. How can I convince him with intuition that the two ratios are the same? (I'm not looking for a series of algebraic equations but rather for some kind of a visual proof).






    optimization intuition







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      up vote
      5
      down vote

      favorite
      1









      up vote
      5
      down vote

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      1






      1





      Let's say you play the game of collecting the maximum amount of coins along a path in the $(x,y)$ plane. The total amount of coins collected $S$ is given by $S = 5x + 7y$ where $5$ and $7$ represents the number of coins collected by walking respectively one unit in the $x$ and $y$ direction. However I can only go as far as one unit of distance ($x²+y²=1$).



      It turns out that the solution to the problem is so such that $y/x = 7/5$ or said in other words, the ratio of how much you have to walk in the $y$ direction compared to in the $x$ direction is the same as the ratio of the rates at which you collect the coins in the $y$ direction compared to in the $x$ direction.



      My question : suppose I have to explain this result to a middle school student who does not know any calculus of trigonometry. He is also quite shaky on describing equations of lines in the plane so I would not rely on that. How can I convince him with intuition that the two ratios are the same? (I'm not looking for a series of algebraic equations but rather for some kind of a visual proof).






      optimization intuition







      share|cite|improve this question













      Let's say you play the game of collecting the maximum amount of coins along a path in the $(x,y)$ plane. The total amount of coins collected $S$ is given by $S = 5x + 7y$ where $5$ and $7$ represents the number of coins collected by walking respectively one unit in the $x$ and $y$ direction. However I can only go as far as one unit of distance ($x²+y²=1$).



      It turns out that the solution to the problem is so such that $y/x = 7/5$ or said in other words, the ratio of how much you have to walk in the $y$ direction compared to in the $x$ direction is the same as the ratio of the rates at which you collect the coins in the $y$ direction compared to in the $x$ direction.



      My question : suppose I have to explain this result to a middle school student who does not know any calculus of trigonometry. He is also quite shaky on describing equations of lines in the plane so I would not rely on that. How can I convince him with intuition that the two ratios are the same? (I'm not looking for a series of algebraic equations but rather for some kind of a visual proof).






      optimization intuition




      optimization intuition






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      Corvinus

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          This is related in an interesting way to the method of Gradient Descent:




          • Why is it that if you find yourself on a 2D-plane tilted by a slope of $a$ in the $x$-direction and a slope of $b$ in the $y$-direction, the steepest uphill direction will be toward $(a,b)$?



          I have no idea how you would convince someone without going into at least some mathematical details, but from my perspective I have the following reasons:




          1. If one moves perpendicular to the $v=(5,7)$ vector, which would be toward $hat v=(-7,5)$ you would gain zero score from the expression $5x+7y$. Simply we have $5cdot(-7)+7cdot 5=0$.

          2. Any direction can be decomposed into two parts in terms of the two perpendicular directions $v=(5,7)$ and $hat v=(-7,5)$, and only the part in direction of $v$ will increase the score.



          Good luck breaking this down and coming up with a visual representation of it. I am positive it can be done, but I am less sure how you would omit any of the non-intuitive details.



          enter image description here



          At least you can say that the part of "some direction" pointing in the direction of $v$ is shorter than "some direction" itself, and so it achieves the same score while traversing a shorter distance. The red dotted line represents the part of "some direction" not contributing to the score at all.






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            I'd probably start with easier numbers to visualise. For example $10$ and $5$, you can explain that every step in $1$ direction you pick up $£5$, and every step in the other you pick up $£10$. Therefore to collect the same amount of money in the first direction as the second you have to walk twice as fast (or far). Hopefully this can start the intuitive thinking about the relationship between the ratios.






            share|cite|improve this answer



























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              Consider the iso-$S$ curves, i.e. the combinations of $x$ and $y$ giving the same amount.



              If you increase $x$ by $7$, you'll have to decrease $y$ by $5$ to ensure status-quo. As this is true whatever $x,y$, the iso-$S$ are straight lines (of direction $(7,-5)$).



              Now it should be clear that the optimal solution is found on the iso-$S$ that is tangent to the circle.



              And as a tangent at a point is perpendicular to the corresponding radius, the point of contact is found in the perpendicular direction. Rotate the vector $(7,-5)$ to get $(5,7)$, giving the slope of the perpendicular.






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                3 Answers
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                active

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                3 Answers
                3






                active

                oldest

                votes









                active

                oldest

                votes






                active

                oldest

                votes








                up vote
                2
                down vote













                This is related in an interesting way to the method of Gradient Descent:




                • Why is it that if you find yourself on a 2D-plane tilted by a slope of $a$ in the $x$-direction and a slope of $b$ in the $y$-direction, the steepest uphill direction will be toward $(a,b)$?



                I have no idea how you would convince someone without going into at least some mathematical details, but from my perspective I have the following reasons:




                1. If one moves perpendicular to the $v=(5,7)$ vector, which would be toward $hat v=(-7,5)$ you would gain zero score from the expression $5x+7y$. Simply we have $5cdot(-7)+7cdot 5=0$.

                2. Any direction can be decomposed into two parts in terms of the two perpendicular directions $v=(5,7)$ and $hat v=(-7,5)$, and only the part in direction of $v$ will increase the score.



                Good luck breaking this down and coming up with a visual representation of it. I am positive it can be done, but I am less sure how you would omit any of the non-intuitive details.



                enter image description here



                At least you can say that the part of "some direction" pointing in the direction of $v$ is shorter than "some direction" itself, and so it achieves the same score while traversing a shorter distance. The red dotted line represents the part of "some direction" not contributing to the score at all.






                share|cite|improve this answer


























                  up vote
                  2
                  down vote













                  This is related in an interesting way to the method of Gradient Descent:




                  • Why is it that if you find yourself on a 2D-plane tilted by a slope of $a$ in the $x$-direction and a slope of $b$ in the $y$-direction, the steepest uphill direction will be toward $(a,b)$?



                  I have no idea how you would convince someone without going into at least some mathematical details, but from my perspective I have the following reasons:




                  1. If one moves perpendicular to the $v=(5,7)$ vector, which would be toward $hat v=(-7,5)$ you would gain zero score from the expression $5x+7y$. Simply we have $5cdot(-7)+7cdot 5=0$.

                  2. Any direction can be decomposed into two parts in terms of the two perpendicular directions $v=(5,7)$ and $hat v=(-7,5)$, and only the part in direction of $v$ will increase the score.



                  Good luck breaking this down and coming up with a visual representation of it. I am positive it can be done, but I am less sure how you would omit any of the non-intuitive details.



                  enter image description here



                  At least you can say that the part of "some direction" pointing in the direction of $v$ is shorter than "some direction" itself, and so it achieves the same score while traversing a shorter distance. The red dotted line represents the part of "some direction" not contributing to the score at all.






                  share|cite|improve this answer
























                    up vote
                    2
                    down vote










                    up vote
                    2
                    down vote









                    This is related in an interesting way to the method of Gradient Descent:




                    • Why is it that if you find yourself on a 2D-plane tilted by a slope of $a$ in the $x$-direction and a slope of $b$ in the $y$-direction, the steepest uphill direction will be toward $(a,b)$?



                    I have no idea how you would convince someone without going into at least some mathematical details, but from my perspective I have the following reasons:




                    1. If one moves perpendicular to the $v=(5,7)$ vector, which would be toward $hat v=(-7,5)$ you would gain zero score from the expression $5x+7y$. Simply we have $5cdot(-7)+7cdot 5=0$.

                    2. Any direction can be decomposed into two parts in terms of the two perpendicular directions $v=(5,7)$ and $hat v=(-7,5)$, and only the part in direction of $v$ will increase the score.



                    Good luck breaking this down and coming up with a visual representation of it. I am positive it can be done, but I am less sure how you would omit any of the non-intuitive details.



                    enter image description here



                    At least you can say that the part of "some direction" pointing in the direction of $v$ is shorter than "some direction" itself, and so it achieves the same score while traversing a shorter distance. The red dotted line represents the part of "some direction" not contributing to the score at all.






                    share|cite|improve this answer














                    This is related in an interesting way to the method of Gradient Descent:




                    • Why is it that if you find yourself on a 2D-plane tilted by a slope of $a$ in the $x$-direction and a slope of $b$ in the $y$-direction, the steepest uphill direction will be toward $(a,b)$?



                    I have no idea how you would convince someone without going into at least some mathematical details, but from my perspective I have the following reasons:




                    1. If one moves perpendicular to the $v=(5,7)$ vector, which would be toward $hat v=(-7,5)$ you would gain zero score from the expression $5x+7y$. Simply we have $5cdot(-7)+7cdot 5=0$.

                    2. Any direction can be decomposed into two parts in terms of the two perpendicular directions $v=(5,7)$ and $hat v=(-7,5)$, and only the part in direction of $v$ will increase the score.



                    Good luck breaking this down and coming up with a visual representation of it. I am positive it can be done, but I am less sure how you would omit any of the non-intuitive details.



                    enter image description here



                    At least you can say that the part of "some direction" pointing in the direction of $v$ is shorter than "some direction" itself, and so it achieves the same score while traversing a shorter distance. The red dotted line represents the part of "some direction" not contributing to the score at all.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited 1 min ago

























                    answered 23 mins ago









                    String

                    13.3k32753




                    13.3k32753




















                        up vote
                        1
                        down vote













                        I'd probably start with easier numbers to visualise. For example $10$ and $5$, you can explain that every step in $1$ direction you pick up $£5$, and every step in the other you pick up $£10$. Therefore to collect the same amount of money in the first direction as the second you have to walk twice as fast (or far). Hopefully this can start the intuitive thinking about the relationship between the ratios.






                        share|cite|improve this answer
























                          up vote
                          1
                          down vote













                          I'd probably start with easier numbers to visualise. For example $10$ and $5$, you can explain that every step in $1$ direction you pick up $£5$, and every step in the other you pick up $£10$. Therefore to collect the same amount of money in the first direction as the second you have to walk twice as fast (or far). Hopefully this can start the intuitive thinking about the relationship between the ratios.






                          share|cite|improve this answer






















                            up vote
                            1
                            down vote










                            up vote
                            1
                            down vote









                            I'd probably start with easier numbers to visualise. For example $10$ and $5$, you can explain that every step in $1$ direction you pick up $£5$, and every step in the other you pick up $£10$. Therefore to collect the same amount of money in the first direction as the second you have to walk twice as fast (or far). Hopefully this can start the intuitive thinking about the relationship between the ratios.






                            share|cite|improve this answer












                            I'd probably start with easier numbers to visualise. For example $10$ and $5$, you can explain that every step in $1$ direction you pick up $£5$, and every step in the other you pick up $£10$. Therefore to collect the same amount of money in the first direction as the second you have to walk twice as fast (or far). Hopefully this can start the intuitive thinking about the relationship between the ratios.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 32 mins ago









                            MRobinson

                            85115




                            85115




















                                up vote
                                0
                                down vote













                                Consider the iso-$S$ curves, i.e. the combinations of $x$ and $y$ giving the same amount.



                                If you increase $x$ by $7$, you'll have to decrease $y$ by $5$ to ensure status-quo. As this is true whatever $x,y$, the iso-$S$ are straight lines (of direction $(7,-5)$).



                                Now it should be clear that the optimal solution is found on the iso-$S$ that is tangent to the circle.



                                And as a tangent at a point is perpendicular to the corresponding radius, the point of contact is found in the perpendicular direction. Rotate the vector $(7,-5)$ to get $(5,7)$, giving the slope of the perpendicular.






                                share|cite
























                                  up vote
                                  0
                                  down vote













                                  Consider the iso-$S$ curves, i.e. the combinations of $x$ and $y$ giving the same amount.



                                  If you increase $x$ by $7$, you'll have to decrease $y$ by $5$ to ensure status-quo. As this is true whatever $x,y$, the iso-$S$ are straight lines (of direction $(7,-5)$).



                                  Now it should be clear that the optimal solution is found on the iso-$S$ that is tangent to the circle.



                                  And as a tangent at a point is perpendicular to the corresponding radius, the point of contact is found in the perpendicular direction. Rotate the vector $(7,-5)$ to get $(5,7)$, giving the slope of the perpendicular.






                                  share|cite






















                                    up vote
                                    0
                                    down vote










                                    up vote
                                    0
                                    down vote









                                    Consider the iso-$S$ curves, i.e. the combinations of $x$ and $y$ giving the same amount.



                                    If you increase $x$ by $7$, you'll have to decrease $y$ by $5$ to ensure status-quo. As this is true whatever $x,y$, the iso-$S$ are straight lines (of direction $(7,-5)$).



                                    Now it should be clear that the optimal solution is found on the iso-$S$ that is tangent to the circle.



                                    And as a tangent at a point is perpendicular to the corresponding radius, the point of contact is found in the perpendicular direction. Rotate the vector $(7,-5)$ to get $(5,7)$, giving the slope of the perpendicular.






                                    share|cite












                                    Consider the iso-$S$ curves, i.e. the combinations of $x$ and $y$ giving the same amount.



                                    If you increase $x$ by $7$, you'll have to decrease $y$ by $5$ to ensure status-quo. As this is true whatever $x,y$, the iso-$S$ are straight lines (of direction $(7,-5)$).



                                    Now it should be clear that the optimal solution is found on the iso-$S$ that is tangent to the circle.



                                    And as a tangent at a point is perpendicular to the corresponding radius, the point of contact is found in the perpendicular direction. Rotate the vector $(7,-5)$ to get $(5,7)$, giving the slope of the perpendicular.







                                    share|cite












                                    share|cite



                                    share|cite










                                    answered 4 mins ago









                                    Yves Daoust

                                    115k667210




                                    115k667210



























                                         

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