Another way proof $1/2-1/3=1/6$ by using picture?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











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We know that $dfrac12 -dfrac13 =dfrac16$. I proved it by picture
enter image description here
What is (are) another way (ways) by using picture?




algebra-precalculus proof-writing alternative-proof




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  • 3




    I mean... instead of a rectangle you could pick a circle and cut it in a 6-piece pie
    – b00n heT
    Jul 23 at 11:44






  • 1




    Directly from the definition of addition and subtraction of fractions is another way.
    – Arthur
    Jul 23 at 11:49










  • Or have $6$ dots (arbitrary object) in total. Isolate $3$, subtract $2$, you get $1$.
    – RayDansh
    Jul 23 at 12:20










  • Instead of using a rectangle I suppose you could always use a regular hexagon and take advantage of its 6-fold rotational symmetry.
    – omegadot
    Jul 24 at 1:56














up vote
3
down vote

favorite
1












We know that $dfrac12 -dfrac13 =dfrac16$. I proved it by picture
enter image description here
What is (are) another way (ways) by using picture?




algebra-precalculus proof-writing alternative-proof




share|cite|improve this question

















  • 3




    I mean... instead of a rectangle you could pick a circle and cut it in a 6-piece pie
    – b00n heT
    Jul 23 at 11:44






  • 1




    Directly from the definition of addition and subtraction of fractions is another way.
    – Arthur
    Jul 23 at 11:49










  • Or have $6$ dots (arbitrary object) in total. Isolate $3$, subtract $2$, you get $1$.
    – RayDansh
    Jul 23 at 12:20










  • Instead of using a rectangle I suppose you could always use a regular hexagon and take advantage of its 6-fold rotational symmetry.
    – omegadot
    Jul 24 at 1:56












up vote
3
down vote

favorite
1









up vote
3
down vote

favorite
1






1





We know that $dfrac12 -dfrac13 =dfrac16$. I proved it by picture
enter image description here
What is (are) another way (ways) by using picture?




algebra-precalculus proof-writing alternative-proof




share|cite|improve this question













We know that $dfrac12 -dfrac13 =dfrac16$. I proved it by picture
enter image description here
What is (are) another way (ways) by using picture?





algebra-precalculus proof-writing alternative-proof





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 23 at 12:18









Omnomnomnom

121k784170




121k784170









asked Jul 23 at 11:41









minhthien_2016

259110




259110







  • 3




    I mean... instead of a rectangle you could pick a circle and cut it in a 6-piece pie
    – b00n heT
    Jul 23 at 11:44






  • 1




    Directly from the definition of addition and subtraction of fractions is another way.
    – Arthur
    Jul 23 at 11:49










  • Or have $6$ dots (arbitrary object) in total. Isolate $3$, subtract $2$, you get $1$.
    – RayDansh
    Jul 23 at 12:20










  • Instead of using a rectangle I suppose you could always use a regular hexagon and take advantage of its 6-fold rotational symmetry.
    – omegadot
    Jul 24 at 1:56












  • 3




    I mean... instead of a rectangle you could pick a circle and cut it in a 6-piece pie
    – b00n heT
    Jul 23 at 11:44






  • 1




    Directly from the definition of addition and subtraction of fractions is another way.
    – Arthur
    Jul 23 at 11:49










  • Or have $6$ dots (arbitrary object) in total. Isolate $3$, subtract $2$, you get $1$.
    – RayDansh
    Jul 23 at 12:20










  • Instead of using a rectangle I suppose you could always use a regular hexagon and take advantage of its 6-fold rotational symmetry.
    – omegadot
    Jul 24 at 1:56







3




3




I mean... instead of a rectangle you could pick a circle and cut it in a 6-piece pie
– b00n heT
Jul 23 at 11:44




I mean... instead of a rectangle you could pick a circle and cut it in a 6-piece pie
– b00n heT
Jul 23 at 11:44




1




1




Directly from the definition of addition and subtraction of fractions is another way.
– Arthur
Jul 23 at 11:49




Directly from the definition of addition and subtraction of fractions is another way.
– Arthur
Jul 23 at 11:49












Or have $6$ dots (arbitrary object) in total. Isolate $3$, subtract $2$, you get $1$.
– RayDansh
Jul 23 at 12:20




Or have $6$ dots (arbitrary object) in total. Isolate $3$, subtract $2$, you get $1$.
– RayDansh
Jul 23 at 12:20












Instead of using a rectangle I suppose you could always use a regular hexagon and take advantage of its 6-fold rotational symmetry.
– omegadot
Jul 24 at 1:56




Instead of using a rectangle I suppose you could always use a regular hexagon and take advantage of its 6-fold rotational symmetry.
– omegadot
Jul 24 at 1:56










4 Answers
4






active

oldest

votes

















up vote
8
down vote



accepted










This is a nice picture, using an equilateral triangle, because the shape has both 2-way and 3-way symmetry. Each large right triangle is 1/2, and each kite is 1/3.
enter image description here



Is this the kind of thing you were looking for?






share|cite|improve this answer

















  • 1




    An isoceles triangle with base on one side of the larger triangle and apex at the center of the larger triangle also 1/3 the area of the large triangle; same result.
    – David K
    Jul 23 at 12:44

















up vote
5
down vote













Primary school teachers generally use pattern blocks.



Show that the green [1/6] plus the blue [1/3] equals the red [1/2].



enter image description here






share|cite|improve this answer




























    up vote
    0
    down vote













    The assertion is true in any commutative ring where the multiplicative inverses exist, even though you can't always draw a picture.



    $1/a$ is the solution to the equation $ax = 1$ so



    $$
    6 left( frac12 - frac13 right)
    =
    2 times 3 times left( frac12 right) - 2 times 3 timesleft( frac13 right)
    = 3 - 2 = 1
    $$
    so the expression on the left in the question is the multiplicative inverse of $6$.



    (This is the core of the rule for adding fractions.)






    share|cite|improve this answer

















    • 1




      I want to proof by using picture. Please read my question.
      – minhthien_2016
      Jul 23 at 12:06






    • 3




      @Ethan any mention of commutative rings in the context of fractions makes me think of this
      – Omnomnomnom
      Jul 23 at 12:11

















    up vote
    0
    down vote













    I think that an other useful picture could be the disk.
    We can divide in two pieces the disk, of course at 180° and in the same way we can divide it in 3 pieces at 120°. At this point is simple to show what remains, and also the computation of the difference.






    share|cite|improve this answer





















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      4 Answers
      4






      active

      oldest

      votes








      4 Answers
      4






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      8
      down vote



      accepted










      This is a nice picture, using an equilateral triangle, because the shape has both 2-way and 3-way symmetry. Each large right triangle is 1/2, and each kite is 1/3.
      enter image description here



      Is this the kind of thing you were looking for?






      share|cite|improve this answer

















      • 1




        An isoceles triangle with base on one side of the larger triangle and apex at the center of the larger triangle also 1/3 the area of the large triangle; same result.
        – David K
        Jul 23 at 12:44














      up vote
      8
      down vote



      accepted










      This is a nice picture, using an equilateral triangle, because the shape has both 2-way and 3-way symmetry. Each large right triangle is 1/2, and each kite is 1/3.
      enter image description here



      Is this the kind of thing you were looking for?






      share|cite|improve this answer

















      • 1




        An isoceles triangle with base on one side of the larger triangle and apex at the center of the larger triangle also 1/3 the area of the large triangle; same result.
        – David K
        Jul 23 at 12:44












      up vote
      8
      down vote



      accepted







      up vote
      8
      down vote



      accepted






      This is a nice picture, using an equilateral triangle, because the shape has both 2-way and 3-way symmetry. Each large right triangle is 1/2, and each kite is 1/3.
      enter image description here



      Is this the kind of thing you were looking for?






      share|cite|improve this answer













      This is a nice picture, using an equilateral triangle, because the shape has both 2-way and 3-way symmetry. Each large right triangle is 1/2, and each kite is 1/3.
      enter image description here



      Is this the kind of thing you were looking for?







      share|cite|improve this answer













      share|cite|improve this answer



      share|cite|improve this answer











      answered Jul 23 at 12:20









      G Tony Jacobs

      25.6k43483




      25.6k43483







      • 1




        An isoceles triangle with base on one side of the larger triangle and apex at the center of the larger triangle also 1/3 the area of the large triangle; same result.
        – David K
        Jul 23 at 12:44












      • 1




        An isoceles triangle with base on one side of the larger triangle and apex at the center of the larger triangle also 1/3 the area of the large triangle; same result.
        – David K
        Jul 23 at 12:44







      1




      1




      An isoceles triangle with base on one side of the larger triangle and apex at the center of the larger triangle also 1/3 the area of the large triangle; same result.
      – David K
      Jul 23 at 12:44




      An isoceles triangle with base on one side of the larger triangle and apex at the center of the larger triangle also 1/3 the area of the large triangle; same result.
      – David K
      Jul 23 at 12:44










      up vote
      5
      down vote













      Primary school teachers generally use pattern blocks.



      Show that the green [1/6] plus the blue [1/3] equals the red [1/2].



      enter image description here






      share|cite|improve this answer

























        up vote
        5
        down vote













        Primary school teachers generally use pattern blocks.



        Show that the green [1/6] plus the blue [1/3] equals the red [1/2].



        enter image description here






        share|cite|improve this answer























          up vote
          5
          down vote










          up vote
          5
          down vote









          Primary school teachers generally use pattern blocks.



          Show that the green [1/6] plus the blue [1/3] equals the red [1/2].



          enter image description here






          share|cite|improve this answer













          Primary school teachers generally use pattern blocks.



          Show that the green [1/6] plus the blue [1/3] equals the red [1/2].



          enter image description here







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Jul 28 at 6:38









          John Joy

          5,88511526




          5,88511526




















              up vote
              0
              down vote













              The assertion is true in any commutative ring where the multiplicative inverses exist, even though you can't always draw a picture.



              $1/a$ is the solution to the equation $ax = 1$ so



              $$
              6 left( frac12 - frac13 right)
              =
              2 times 3 times left( frac12 right) - 2 times 3 timesleft( frac13 right)
              = 3 - 2 = 1
              $$
              so the expression on the left in the question is the multiplicative inverse of $6$.



              (This is the core of the rule for adding fractions.)






              share|cite|improve this answer

















              • 1




                I want to proof by using picture. Please read my question.
                – minhthien_2016
                Jul 23 at 12:06






              • 3




                @Ethan any mention of commutative rings in the context of fractions makes me think of this
                – Omnomnomnom
                Jul 23 at 12:11














              up vote
              0
              down vote













              The assertion is true in any commutative ring where the multiplicative inverses exist, even though you can't always draw a picture.



              $1/a$ is the solution to the equation $ax = 1$ so



              $$
              6 left( frac12 - frac13 right)
              =
              2 times 3 times left( frac12 right) - 2 times 3 timesleft( frac13 right)
              = 3 - 2 = 1
              $$
              so the expression on the left in the question is the multiplicative inverse of $6$.



              (This is the core of the rule for adding fractions.)






              share|cite|improve this answer

















              • 1




                I want to proof by using picture. Please read my question.
                – minhthien_2016
                Jul 23 at 12:06






              • 3




                @Ethan any mention of commutative rings in the context of fractions makes me think of this
                – Omnomnomnom
                Jul 23 at 12:11












              up vote
              0
              down vote










              up vote
              0
              down vote









              The assertion is true in any commutative ring where the multiplicative inverses exist, even though you can't always draw a picture.



              $1/a$ is the solution to the equation $ax = 1$ so



              $$
              6 left( frac12 - frac13 right)
              =
              2 times 3 times left( frac12 right) - 2 times 3 timesleft( frac13 right)
              = 3 - 2 = 1
              $$
              so the expression on the left in the question is the multiplicative inverse of $6$.



              (This is the core of the rule for adding fractions.)






              share|cite|improve this answer













              The assertion is true in any commutative ring where the multiplicative inverses exist, even though you can't always draw a picture.



              $1/a$ is the solution to the equation $ax = 1$ so



              $$
              6 left( frac12 - frac13 right)
              =
              2 times 3 times left( frac12 right) - 2 times 3 timesleft( frac13 right)
              = 3 - 2 = 1
              $$
              so the expression on the left in the question is the multiplicative inverse of $6$.



              (This is the core of the rule for adding fractions.)







              share|cite|improve this answer













              share|cite|improve this answer



              share|cite|improve this answer











              answered Jul 23 at 12:01









              Ethan Bolker

              35.7k54199




              35.7k54199







              • 1




                I want to proof by using picture. Please read my question.
                – minhthien_2016
                Jul 23 at 12:06






              • 3




                @Ethan any mention of commutative rings in the context of fractions makes me think of this
                – Omnomnomnom
                Jul 23 at 12:11












              • 1




                I want to proof by using picture. Please read my question.
                – minhthien_2016
                Jul 23 at 12:06






              • 3




                @Ethan any mention of commutative rings in the context of fractions makes me think of this
                – Omnomnomnom
                Jul 23 at 12:11







              1




              1




              I want to proof by using picture. Please read my question.
              – minhthien_2016
              Jul 23 at 12:06




              I want to proof by using picture. Please read my question.
              – minhthien_2016
              Jul 23 at 12:06




              3




              3




              @Ethan any mention of commutative rings in the context of fractions makes me think of this
              – Omnomnomnom
              Jul 23 at 12:11




              @Ethan any mention of commutative rings in the context of fractions makes me think of this
              – Omnomnomnom
              Jul 23 at 12:11










              up vote
              0
              down vote













              I think that an other useful picture could be the disk.
              We can divide in two pieces the disk, of course at 180° and in the same way we can divide it in 3 pieces at 120°. At this point is simple to show what remains, and also the computation of the difference.






              share|cite|improve this answer

























                up vote
                0
                down vote













                I think that an other useful picture could be the disk.
                We can divide in two pieces the disk, of course at 180° and in the same way we can divide it in 3 pieces at 120°. At this point is simple to show what remains, and also the computation of the difference.






                share|cite|improve this answer























                  up vote
                  0
                  down vote










                  up vote
                  0
                  down vote









                  I think that an other useful picture could be the disk.
                  We can divide in two pieces the disk, of course at 180° and in the same way we can divide it in 3 pieces at 120°. At this point is simple to show what remains, and also the computation of the difference.






                  share|cite|improve this answer













                  I think that an other useful picture could be the disk.
                  We can divide in two pieces the disk, of course at 180° and in the same way we can divide it in 3 pieces at 120°. At this point is simple to show what remains, and also the computation of the difference.







                  share|cite|improve this answer













                  share|cite|improve this answer



                  share|cite|improve this answer











                  answered Jul 23 at 12:39









                  Cuoredicervo

                  287210




                  287210






















                       

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