ｊ ｾ ｖ ｾ Ｔ ｾＬＬ - r ｾ Ｉ ｪ lha -; ｾ ｴ ｚ ｮ ､ ｾ -hft""" Ｎ Ｍ Ｑ ｾ ｾ ｾ ｾ ｾ I I I /2()() - : t> I I I ch ·ng Solutions ＭＺＮ ﾷ Ｚ Ｎ Ｍ ｾ Ｇ Ｂ Ｍ］ Ｑ ge rai c ations b Unit XIII I Math and the Mind's Eye Activities Sketching Solutions to Algebraic Equations D I II I I Sketching Solutions Sketches are used to solve standard algebra problems Sketching Quadratics, Part I Sketches are used to solve problems involving quadratic relationships Sketching Quadratics, Part II Further ways of using sketches to solve quadratics are discussed m ath and the Mind's Eye materials IUJ are intended for use in grades 4-9 They are written so teachers can adapt them to fit student backgrounds and grade levels A single activity can be extended over several days or used in part Equations Involving Rational Expressions Sketches are used to solve equations involving rational functions A catalog of Math and the Mind's Eye Irrational Roots The irrationality of J3 is established The method is extended to other roots able from The Math Learning Center, materials and teaching supplies is avail- PO Box 3226, Salem, OR 97302, 1-800 575-8130 or (503) 370-8130 Fax: (503) 370-7961 Learn more about The Math Learning Center at: www.mlc.pdx.edu Math and the Mind's Eye Copyright© 1997 The Math Learning Center The Math Learning Center grants permission to classroom teachers to reproduce the student activity pages in appropriate quantities for their classroom use ISBN J-886UI-43-0 Unit XIII • Activity Sketching Solutions Prerequisite Activity None Materials Copies of Puzzle Problems (see Comment 10) Actions Comments Ask the students to draw a rectangle on a blank sheet of paper Comment on the various rectangles drawn Students often have difficulty in drawing sketches that disclose the essential features of a problem situation Thus, the beginning Actions in this activity focus on drawing sketches that require few, if any, words and symbols to convey information If your students have had experience using diagrams and sketches to solve problems visually, you may want to skip to Action The Actions begin with one that almost every student will carry out rapidly Most students will draw a rectangle that is wider than it is tall Most, if not all, of the sketches will contain no words or symbols Generally, it is unnecessary to label a sketch of a rectangle for the students to identify what has been drawn Show the students the following sketch Ask them to describe what they see After the students have had an opportunity to respond, ask them what more they can say about the rectangle if its perimeter is 56 units Discuss the students' responses I I If the students simply reply, "A rectangle," ask them to tell you all they know about the rectangle Most of the students will recognize that one dimension is units longer than the other Some may ask if the unlabeled segments are of equal length If so, you can label the segments with the same letter, as shown in the sketch below d d d d I I 6 Continued next page Unit XIII ã Activity âCopyright 1997, The Math Learning Center Actions Comments Continued The perimeter of the rectangle is comprised of segments of length 6, and segments of unknown but equal length The sum of the lengths of these latter segments is 56- 12 or 44 Hence, each segment is 11 and the dimensions of the rectangle are 11 and 11 + or 17 Alternately, a student may decide that half the perimeter is 28 Hence of the segments total 28 - 6, or 22, inches If one wants, one can paraphrase a student's thinking while recording their thoughts in symbolic shorthand, as in the following example: paraphrase: record: As I understand your argument, you say the perimeter consists of segments of length 6, and other segments all of the same length -let's call it d-and, since the perimeter is 56, these lengths total 56 12 + 4d= 56 So the segments have a total length of 56- 12 or44 4d =56- 12 = 44 Thus, the length of each segment is 44 + or 11 d=44+4=11 Hence, the dimensions of the rectangle are 11 and 11 + width= d = 11 length = d + = 17 Notice, in this case, the algebraic equations become a symbolic way of recording one's thinking In order to deal with the symbols, it is not necessary to have mastered a set of rules for their manipulation Rather, the equations reflect a chain of thought based on the thinker's knowledge and insight Ask the students to draw a sketch, using as few words and symbols as possible, that portrays a rectangle of unknown dimensions whose length is units longer than times its width Have several students replicate their drawings on the chalkboard Discuss whether the drawings adequately convey the information given about the rectangle and whether the words and symbols used are essential Having the students draw sketches of a situation before a problem is posed focuses their attention on creating a sketch that portrays the essential features of the situation Below are some possible sketches Notice that, in the last sketch shown, the essential information is carried in the symbols and not the sketch-that is, if the symbolic phrase "3w + 4" is erased, the distinguishing feature of the rectangle is lost X x Unit XIII • Activity : X : X : 11 +- r - - - - t - 3w + wJ _ _ _ _ - Math and the Mind's Eye Actions Comments Tell the students to suppose the perimeter of the rectangle they drew in Action is 48 inches Then ask them to determine the dimensions of the rectangle Ask for volunteers to describe their thinking The students will use various methods to arrive at the dimensions One way is to note that the perimeter of 48 inches consists of segments of length and other segments of equal length Hence, the lengths of the segments total40 inches, so each is Thus, the dimensions of the rectangle are inches and x + 4, or 19 Repeat Action for a rectangle whose length is inches less than twice its width Then ask the students to determine the dimensions of the rectangle if its perimeter is 32 inches Have several students show their sketches and describe their thinking in determining the dimensions of the rectangle Here is one sketch: w w ｾ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ ｲ Ｍ Ｍ ｾ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ I I w ｾ Ｕ ｾ Ｚ J·- - - - - I The extended rectangle shown above has a perimeter of 42 inches-10 inches longer than the original rectangle These 42 inches are composed of equal lengths So each of these lengths is inches The width of the original rectangle is one of these lengths, or inches; the length of the rectangle is inches less than of these lengths, or inches Ask the students to sketch a square Then have them sketch an equilateral triangle whose sides are feet longer than the sides of the square Then ask the students to determine the length of the side of the square if the square and the triangle have equal perimeters Ask for volunteers to show their sketches and describe their thinking Unit XIII • Activity The perimeter of the square, in the following drawing, contains segments of length s; that of the equilateral triangle contains segments of length s and of length Thus, the segments of length must sum to s So sis Math and the Mind's Eye Actions Comments Ask the students to draw diagrams or sketches which represent a number and that number increased by Show the various ways in which students have done this Then ask the students to use one of the sketches to determine what the numbers are if their sum is 40 The students may find this Action more difficult than the previous Actions in which they are asked to draw geometric figures Since numbers have no particular shape, the students must invent a way of portraying number They might this in a variety of ways, e.g., as a length or as an area or as an amorphous blob D I ｾ Three sketches of a number and that number plus Looking at the sketch on the left above, the sum of the lengths of the segments portraying the numbers is 40 The small segment has length Hence, the sum of lengths of the other segments is 34 Since these segments are congruent, the length of each is 34 + 2, or 17 Hence the numbers are 17 and 17 + 6, or 23 Ask the students to draw sketches that represent two numbers such that times the smaller number is less than the larger Then ask them to use their sketches to determine the numbers if their sum is 36 Ask for volunteers to show their sketches and explain how they were used to arrive at their conclusion The students will represent the numbers in various ways If the sum of the two numbers is 36, in the sketch shown below, the sum of the lengths of the congruent segments ( in the smaller number and in the larger number) is 36- 1, or 35 Hence, the length of each is 35 + 5, or Thus the numbers are and 4(7) + 1, or 29 smaller number larger number Unit XIII • Activity Math and the Mind's Eye Actions Comments Tell the students that Mike has times as many nickels as Larry has dimes Ask them to draw sketches representing the value of their money Then ask them to use their sketches to determine how much money Mike has if he has 45¢ more than Larry Ask for volunteers to show their solutions Again, the students' sketches will vary In the following sketch, the value of Mike's and Larry's coins are represented by stacks of boxes, all of which have the same value Since Mike has times as many coins as Larry, his stack of boxes is times as high as Larry's Larry's stack is twice the width of Mike's since each of Larry's coins is worth twice as much as each of Mike's Mike's stack contains more box then Larry's Since Mike has 45¢ more than Larry, this box is worth 45¢ Thus Mike has x 45¢ or $1.35 value of Mike's coins 10 Ask the students to use sketches or diagrams to solve problems selected from the attached collection of puzzle problems ONE Separate 43 people into groups so that the first group has less than times the number in the second group Sketch Sketch GRP + GRP2 Oo 0 value of Larry's coins 10 You may wish to select a problem or two for the students to work on in class, asking for volunteers to present their solutions Others can be assigned as homework or the students can be asked to choose the problems they wish to work on One way to involve students in reflecting on other students work is to show a sketch a student used to solve a problem, omitting explanations, and ask the other students how they think the sketch was used to solve the problem For example, shown on the left are three different sketches students used in problem to determine that there are 31 people in the first group and 12 in the second The explanations they gave are omitted Examples of solutions for the other problems are attached 48 Sketch 2nd group Unit XIII • Activity I 1st group Math and the Mind's Eye Puzzle Problems Sample Sketches TWO There are numbers The first is twice the second The third is twice the first Their sum is 112 What are the numbers? _1_-' J 1st number D 2ndnumber 3rd number!L _ Each box represents 112 + 7, or 16 Hence, the numbers are 32, 16 and 64 J. _L -1 _J,_ THREE The sum of numbers is 40 Their difference is 14 What are the numbers? Solution Solution 40 larger number larger smaller I I I difference ｾ - - - - - - 1smaller ' - - - - - - -1 ｾ ｾ Ｍ 14 Ｍ Ｍ Ｍ Ｍ Ｍ ｾ ｾ x smaller Ｍ Ｍ Ｍ =40 - Ｍ Ｍ Ｍ ｾ smaller number ' 14 =26 The area of the shaded region is the sum of the numbers; the area of the unshaded region is the difference The combined area of the shaded and unshaded regions is twice the larger number Hence, the larger number is (40 + 14) + 2, or 27 The smaller number is 27- 14, or 13 The smaller number is 26 + 2, or 13 The larger number is 40- 13, or 27 FOUR The sides of one square are inches longer than the sides of another square and its area is 48 square inches greater What is the length of the side of the smaller square? In each of the following, s is the side of the smaller square Solution Solution s Solution s S+1 s s s s The area of the unshaded border is 48 Hence, the area of each of the two x s rectangles is (48 - 4) + 2, or 22 Thus, s is 11 s Hence, the area of each of the four x s rectangles is (48- 4) + 4, or 11 Thus, s is 11 S+1 S+1 S+1 The area of the unshaded border is 48 Hence, the area of each of the four x (s + 1) rectangles is 48 + 4, or 12, and sis 11 The area of the unshaded border is 48 Unit XIII • Activity Math and the Mind's Eye Puzzle Problems Sample Sketches continued FIVE Melody has $2.75 in dimes and quarters There are 14 coins altogether How many of each does she have? SIX Find consecutive integers such that the product of the first and second integers is 40 less than the square of the third integer third integer first integer no of quarters no of dimes 14 coins second integer The value of each shaded bar is x 14, or 70¢ Hence, the value of each unshaded bar is (275- 140) + 3, or 45¢ So, there are quarters and dimes The area of the shaded rectangle is the product of the first two integers The area of the unshaded region is the difference between that product and the square of the third integer which is given to be 40 Hence, each of the unshaded rectangles has area (40- 4) + 3, or 12 Thus, the numbers are 12, 13 and 14 SEVEN Karen is times as old as Lucille In years, she will be times as old as Lucille How old is Lucille? EIGHT One pump can fill a tank in hours Another pump can fill it in hours If both pumps are used, how long will it take to fill the tank? ages now Karen I Lucille D Solution I pumpA ｾ ｾ ｾ ｾ Ｑ ｾ ｾ ｾ ｾ ｾ ｾ ｾ ｾ ｾ hour I ｾ I6 I Karen Lucille ｾ ' -y J pump B ages in years - - - - - - the tank _ _ _ _ I hour together I ｾ I I I ｾ hour Together, the pumps take Comparison of Karen's age in years with times Lucille's age in years: I Solution Time to fill tank: 22/s I hour I ｾ ｯ ｦ hr hours pump A hours Karen pump B Lucille (3 times) These have the same value if each box represents two 6's, or 12 Hence, Karen is now 48 and Lucille is 12 I Tanks filled in 12 hours: pump A pump B 1-l hours hours 4houm I hours 4houm I 4houm Together, pumps A and B fillS tanks in 12 hours; so they fill tank in 12/s or 22/s hours Continued next page Unit XIII • Activity Math and the Mind's Eye Puzzle Problems Sample Sketches continued EIGHT (cont.) Solution the tank: amount pump A fills in hour ｾ ｾ ｾ ｾ ｾ ｾ ｾ ｾ ｾ NINE A tank has drains of different sizes If both drains are used, it takes hours to empty the tank If only the first drain is used, it takes hours to empty the tank How long does it take to empty the tank if only the second drain is used? the tank: amount 1st drain empties in hour ｾ amount pump B fills in hour Pump A fills subdivisions in hour Pump B fills subdivisions in hour amount both drains empty in hour _j Together, they fill 10 subdivisions in hour: hour Working together, both drains empty subdivisions in hour The first drain empties 3, so the second drain empties 4: hour hour 01 hour Together, pumps A and B fill the tank in 2'Yw hours hour hour hour hour hour It takes the second drain 1/4 hours to empty the tank Unit XIII • Activity Math and the Mind's Eye Actions Comments Continued (d) %x- 4)- ¥x = 1/2 The sum of the values of the bases of regions A and B is 1/2 These regions can be combined and then adjusted to form a square, as shown A ｾ ｸ Ｍ ｸ Ｔ x(x-4) ｾ x(x-4) 3x -2x+8 x(x-4) X+ ｾ Ｑ Ｋ Ｍ Ｍ Ｍ 2Ｍ Ｎ Ａ ｾ Ｑ Ｑ Ｋ Ｍ Ｍ Ｍ 2Ｍ Ｎ Ａ ｾ Ｑ ｸＭＴｾＭＭＭＭＭｸＫｂ｟｟｟｟Ｎ _:-41. _2x-+16_ ._l-_2x+ _.81- 1x -2 X , -, - - - - - -' -1 24 x-41 x-4 24 24 x-4 ｾ x-2 x-4 x-3 x-4 25 x-3 =±5, x= or x= -2 x-3 Continued next page Unit XIII• Activity Math and the Mind's Eye Actions Comments Continued (e) (x + V12 =31(5- 2x) The values of the bases of regions A and B are equal Expanding the base of B by a factor of 12 produces region C The value of the base of Cis 12b, which equals the value, x + 6, of the region A D is obtained from C by multiplying the value of one edge of D by -2 and the other by 4, and hence the value of the region by -8 Then D can be converted to a square as shown A The figures are not drawn to scale c B ＵＭＲｘｄｾＵＭＲｸ X+ 12 ｾ Ｍ Ｍ ｾ Ｑ Ｒ ｾ ｢ Ｍ ］ Ｍ Ｍ ｘ Ｍ Ｋ ｾ Ｖ ｾ Ｍ Ｍ ｾ b b 36 D 4x-10 -288 4x-10 4X+ 24 I I I I -288 4x-10 17 17 17 4X+ 4x-10 -288 4x-10 17 4x+7 4x + =±1, 4x = -6 or 4x = -8, x =-3;2 or x =-2 Continued next page 10 Unit XIII • Activity Math and the Mind's Eye Actions Comments Continued (f) (3x- 3Y