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Exploring “off the peg” geoboard shapes

Exploring “off the peg” geoboard shapes . Adapted from Matt Ciancetta Western Oregon University . Geoboard’s variety of uses . As a brief review/ introduction , let’s brainstorm all the ways you have used geoboards . . Explorations with S ets of P olygons .

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Exploring “off the peg” geoboard shapes

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  1. Exploring “off the peg” geoboard shapes Adapted from Matt Ciancetta Western Oregon University

  2. Geoboard’svariety of uses • As a brief review/introduction, let’s brainstorm all the ways you have used geoboards.

  3. Explorations with Sets of Polygons • Creating sets of polygons to use for exploring certain topics or concepts is one style of geoboard activity. • How about activities that launch by creating all the squares on a 5-pin by 5-pin geoboard? What are some possibilities for where this exploration can lead?

  4. Exploring Sets of Squares • When constructingthesetofallnon- congruent geoboardsquares, there istypicallyasetofsquaresthatisobvious and studentsusuallyfindthem withnoproblem. • Then thereisasetofsquaresthatarelessobviousbut, still,manystudentsfindwithenoughtime tothink andexplore.

  5. Exploring Sets of Squares

  6. Exploring Sets of Squares

  7. Exploring Sets of Squares • And finallythereisasetofsquaresthatisnot obvious tomoststudents.

  8. Exploring Sets of Squares

  9. Exploring Sets of Squares • How doyouknowthatallyoursquaresare non-congruent? Can youshoworproveit? • Classify thesquaresbyarea. • Volunteers?

  10. Classify Squares by Area

  11. Classify Squares by Area

  12. Classify Squares by Area

  13. Classify Squares by Area

  14. Classify Squares by Area

  15. Classify Squares by Area • So wefound squares ofareas: 1, 2, 4, 5, 8, 9, 10, 16 • Wait asecond...do wereallyknow/believe that all theseareactuallysquares? Even those “tilted” figures?

  16. Finding Squares Extension • Let’s revisitthegeneralgeoboard“rules.” The mainruleisthatlinesegmentsare formed bystretchinggeobandsaroundpegs. • So, forpolygons, theverticescanonlybe located onthepegs. • This isimportantespeciallywhenstudents transitiontosketchinggeoboardpolygonson paper –you cannotjustplunkdownavertex between pegs!

  17. Finding Squares Extension • Hold everything! • What doweknowabouttwolinesthatintersect? • Let’s lookbackatsomeofyourshapescreated from playingaround(at thebeginningofthe investigation.) Didyouidentify anypointsoff the pegs? Could those points beverticesofa polygon?

  18. Finding Squares Extension • Use yourphysicalgeoboardstocreateoffthe peggeoboardsquares. Can youfindsquareswith • only 1vertexoffapeg? • 2 verticesoff the pegs? • 3 verticesoff the pegs? • 4 verticesoff the pegs? • Share yourresults…aretheyreallysquares?

  19. Finding Squares Extension • There aremultiplenon-congruentsquares whose sidesareorientedsimilarly, i.e., with the sametwoslopes. • For example, let’s lookassomeofour onthe pegs squares. Wepreviouslyconstructedtwo non-congruentsquaresthatutilizeslopesof+1 and –1.

  20. Finding Squares Extension

  21. Finding Squares Extension • Are theremoreoffthepegssquaresthathave sides withslopes+1 and –1? • Can youmakeaconstructiontoshowallof these squares?

  22. Finding Squares Extension

  23. Finding Squares Extension • Review whatyouhaveandthenlet’s determine allthepossible“slope familiesof squares.” The familiesare: !0 and undefined✓ !1 and –1✓ !2 and –1/2✓ ! 3 and –1/3✓ !4 and –1/4✓ ! 3/2 and – 2/3✓ !4 /3 and –3/4✓

  24. Finding Squares Extension • For eachslopefamily, how can we sketch asingle constructiontoshowallofthesquaresinthat family? • We alreadyhaveconstructionsforthe0& undefinedand the 1 & –1 families.

  25. 0 & undefined slope family of squares

  26. 1 & -1 slope family of squares

  27. Slope families of squares Constructions of the other families? Any volunteers?

  28. 2 & -1/2 slope family of squares

  29. 3 & -1/3 family

  30. 4 & -1/4 slope family of squares

  31. 3/2 & -2/3 slope family of squares

  32. 4/3 & -3/4 slope family of squares

  33. Finding Squares Extension • Now thatwehavewhatwehopearecomplete families ofsquares, wemayaskourselves, – How manynon-congruentsquaresinallare possible? – Are allthesquaresreallynon-congruent? – Whatare the areasofthosesquares? • Look forandsharesomepatterns youseein each family. • Try todetermineareaswithoutconverting to coordinate geometry.

  34. Area Patterns • Areas within each family increase by multiplying by perfect squares. • 0 & undefined; 1 & -1: area of smallest is multiplied by 1, 4, 9, 16 • 2 & -1/2: area of smallest is multiplied by 1, 4, 9, 16, 25 • 4 & -1/4: area of smallest is multiplied by 1, 4, 9

  35. Area Patterns • 3 & -1/3: area of smallest multiplied by 1, 4, 9, 16, 36, 100 • 3/2 & -2/3: area of smallest multiplied by 1, 4, 9, 16, 64 • 4/3 & -3/4: one square (area multiplied by 1)

  36. Area of Squares • Great! So now all we need to do is find the area of the smallest square in each family and then multiply it appropriately to determine all the areas. • 0 & undefined family: Areas: 1, 4, 9, 16 • Let’s look at the 1 & -1 family.

  37. 1 & -1 family

  38. 2 & -1/2 family

  39. 3 & -1/3 family

  40. 4 & -1/4 family

  41. 3/2 & -2/3 family

  42. 4/3 & -3/4 family

  43. Summary • We constructed 28 non-congruent squares on a 5-pin by 5-pin geoboard. • Now the world is our oyster! • Investigating octagons might be fun…

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