{"id":61,"date":"2016-06-18T03:23:33","date_gmt":"2016-06-18T03:23:33","guid":{"rendered":"https:\/\/sites.udmercy.edu\/archeyde\/?p=61"},"modified":"2025-02-21T18:46:56","modified_gmt":"2025-02-21T23:46:56","slug":"designing-toys","status":"publish","type":"post","link":"https:\/\/sites.udmercy.edu\/archeyde\/2016\/06\/18\/designing-toys\/","title":{"rendered":"Designing Toys"},"content":{"rendered":"

My son has a set of construction toys called K’nex.\u00a0 In this problem I ask my Elementary Functions (Precalculus) students to think like the designers of these toys.\u00a0 I have written two versions of this activity.\u00a0 The file Knex-Pyth-Comp<\/a> contains the pdf of both versions.\u00a0 Contact me by email if you would like the .tex file.\u00a0 The file begins with a shared introduction.\u00a0 Problem 1 (which can stand alone) is just one application of the Pythagorean Theorem plus a little thinking.\u00a0 Problem 2 (which can stand alone) requires using the Pythagorean Theorem repeatedly, function composition, and (of course) thinking.<\/p>\n

 <\/p>\n

Introduction<\/p>\n

The main pieces of the set are sticks of various lengths and connectors that look like gears to hook the sticks together. To make interesting shapes, several different sizes of sticks will be needed.\u00a0 My son’s set of K’nex has 5 different lengths, pictured below.\u00a0 From smallest to largest, they are black, yellow, gray, red, and purple.\u00a0 When I do this problem with my class, I will bring the K’nex with me so that the students can measure and see if they are getting the correct answers.<\/p>\n

\"KnexSmallSticks\"<\/a><\/p>\n

 <\/p>\n

One of the basic shapes that will be used by children to construct a variety of objects is an isosceles right triangle.\u00a0 You can see an example of a bicycle built from K’nex in the picture below.\u00a0 Notice the two isosceles right triangles making the frame of the bike.<\/p>\n

\"KnexBike\"<\/a><\/p>\n

Problem 1 (just uses Pythagorean Theorem)<\/p>\n

If the length of the gray bars which will form the legs of the isosceles right triangle is 4.1 cm and if it is 0.6 cm from the end of the bar to the center of the connector (the vertex of the triangle), how long should the red bars be made which will form the hypotenuse of the triangle.\u00a0 See the picture below for reference.<\/p>\n

\"KnexGrayRed\"<\/a><\/p>\n

 <\/p>\n

Problem 2 (uses Pythagorean theorem and function composition)<\/p>\n

Let x<\/em> be the length of the legs of the smallest isosceles right triangle it will be possible to make with your construction toy.\u00a0 These legs will consist of a black stick with a connector on each end.\u00a0 The distance from the end of the stick to the center of the connector (the vertex of the triangle) is 0.6 cm.\u00a0 The hypotenuse of this smallest triangle will consist of a yellow stick attached to the connectors.\u00a0 See the picture below.\u00a0 Give all answers in a complete sentence with units.<\/p>\n

\"KnexBlackYellow\"<\/a><\/p>\n

a)\u00a0 (First lets try to design the set of toy’s the easy way.) The easiest thing to do would be to pick integer lengths for the lengths of the sticks.\u00a0 Suppose the set of toys is manufactured so that the black sticks has length 2 and the yellow sticks have length 3.\u00a0 Can an isosceles right triangle be built from such a set of toys?\u00a0 Why or why not?<\/p>\n

b) (Ok, so the easy way didn’t work.\u00a0 For the rest of the worksheet, lets try being more careful.) What is the length of the black stick?\u00a0 (Give a formula in terms of x<\/em>).\u00a0 For the rest of this problem, it will be important to keep track of whether we are discussing the length of a stick or the length of the whole side of the triangle.\u00a0 You might find it useful to record your answers in the table included at the end of the worksheet so that you can refer back to them as you work.<\/p>\n

c)\u00a0 Give a function f(x)<\/em> for the length of the hypotenuse of a triangle whose legs have length x<\/em>.<\/p>\n

d) Suppose you decide to make the black sticks 1.4 cm long.\u00a0 What is the length of the legs of the isosceles right triangle that can be made with them as shown in the picture?\u00a0 What is the length of the hypotenuse of that triangle?<\/p>\n

e) Still supposing you decide to make the black sticks 1.4 cm long, what is the length of the hypotenuse of the triangle described in the previous problem? Compute the length of the hypotenuse in two ways and make sure you get the same answer both times.\u00a0 %using f(x) and usin the Pythagorean theorem directly.<\/p>\n

f)\u00a0 Go back to assuming the length of the legs (built from black sticks) of the smallest triangle is the unknown x<\/em>.\u00a0 Recall that the length of the hypotenuse (built from a yellow stick) of this triangle is given by f(x)<\/em>.\u00a0 Suppose you wanted to allow bigger isosceles right triangles to be built.\u00a0 If the legs are built from yellow sticks, use the Pythagorean theorem to give a formula, in terms of x<\/em>, for the length of the hypotenuse of the triangle with legs made from yellow sticks.\u00a0 We will use gray sticks for this larger hypotenuse.\u00a0 The following picture shows the whole set of possible isosceles right triangles.<\/p>\n

\"KnexAllTriangles\"<\/a><\/p>\n

g)\u00a0 Compute $f \\circ f (x)$ and compare your answer to the formula in the previous problem.\u00a0 (You may need to do some simplification in order to compare well.)\u00a0 Explain why any similarities you noticed make sense.<\/p>\n

h) Suppose you want to make a third size triangle, this one with legs formed from the gray sticks and a hypotenuse which we will make red.\u00a0 Does $ f \\circ f \\circ f (x) $ have any bearing on this problem, if so what?<\/p>\n

i) Go back to assuming the black sticks will be 1.4 cm long?\u00a0 How long will the gray sticks be?\u00a0 How long will the red sticks be?<\/p>\n

j)\u00a0 Still assuming the black sticks are 1.4 cm long, how long will the hypotenuse be of the triangle whose legs are made from red sticks?\u00a0 How long should we make the purple sticks we will use to form this hypotenuse?<\/p>\n

 <\/p>\n

\"KnexTable\"<\/a><\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

My son has a set of construction toys called K’nex.\u00a0 In this problem I ask my Elementary Functions (Precalculus) students to think like the designers of these toys.\u00a0 I have written two versions of this activity.\u00a0 The file Knex-Pyth-Comp contains the pdf of both versions.\u00a0 Contact me by email if you would like the .tex […]<\/p>\n","protected":false},"author":154,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[12265],"tags":[12266,12270,12246,12253,12271],"_links":{"self":[{"href":"https:\/\/sites.udmercy.edu\/archeyde\/wp-json\/wp\/v2\/posts\/61"}],"collection":[{"href":"https:\/\/sites.udmercy.edu\/archeyde\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.udmercy.edu\/archeyde\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.udmercy.edu\/archeyde\/wp-json\/wp\/v2\/users\/154"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.udmercy.edu\/archeyde\/wp-json\/wp\/v2\/comments?post=61"}],"version-history":[{"count":3,"href":"https:\/\/sites.udmercy.edu\/archeyde\/wp-json\/wp\/v2\/posts\/61\/revisions"}],"predecessor-version":[{"id":218,"href":"https:\/\/sites.udmercy.edu\/archeyde\/wp-json\/wp\/v2\/posts\/61\/revisions\/218"}],"wp:attachment":[{"href":"https:\/\/sites.udmercy.edu\/archeyde\/wp-json\/wp\/v2\/media?parent=61"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.udmercy.edu\/archeyde\/wp-json\/wp\/v2\/categories?post=61"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.udmercy.edu\/archeyde\/wp-json\/wp\/v2\/tags?post=61"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}