VIEWS: 10 PAGES: 12 POSTED ON: 7/8/2011
Patterns and Algebra The case of the disappearing M & Ms and other interesting stuff MANSW Annual Conference 21st Sept 2008 George Anderberg Mathematics Consultant The Mathematics Connection Ph: 0421151043 Fax: 02 49538320 Web: www.mathcon.com.au EXPONENTIAL DECAY Syllabus Ref: PAS3.1a/4.2/Stage 6 Ext 1 Log and Exp functions 1. Students take a 250 gm pack of M & Ms and spread them over a sheet of butcher’s paper. 2. Students count all the M & Ms and plot this on the graph at “eating No.” zero. They should also complete the table on the next page. 3. Students take note of which of the M & Ms have the M facing up and they eat or remove these. 4. Students count all the remaining M & Ms and plot this on the graph at “eating No 1” and they should also complete the table on the next page. 5. The remaining M & Ms are put into the paper cup and shaken before again being spread on the butcher’s paper. Students take note of which of the M & Ms have the M facing up and they eat or remove these. 6. Students count all the remaining M & Ms and plot this on the graph at “eating No.” 2 and they should also complete the table on the next page. 7. This process is repeated until no M & Ms remain. 8. When no M & Ms remain and all the points are plotted students attempt to draw a smooth curve joining, as far as possible, all the points. 9. If time permits students can calculate the theoretical values (to the nearest whole M & M.). Tables of Values Record of M&Ms after each eating. # of eatings 0 1 2 3 4 5 6 7 8 9 10 # of M&Ms Theoretical Record of M&Ms after each eating (what you think should have happened) # of eatings 0 1 2 3 4 5 6 7 8 9 10 # of M&Ms 2 Graph of the decay in M & Ms after each eating 310 290 270 250 230 210 190 Number of M & Ms 170 150 130 110 90 70 50 30 10 0 0 1 2 3 4 5 6 7 8 9 10 Number of Eatings 3 SPAGHETTI MATHS Syllabus Ref: PAS 3.1a/4.2 Rich Task– Linear Relationships. This unit incorporates: Measurement Algebra/equations Graphs/number plane Problem solving AIM: 1. To investigate the relationship between the number of strands of spaghetti and the load they can withstand without breaking. 2. To estimate the number of strands of spaghetti required to suspend a 50 kg student. EQUIPMENT: paper clip plastic cup nails graphical calculator dry strands of spaghetti METHOD: A plastic cup is suspended from a strand of spaghetti by a paper clip. Nails gently placed into the cup until the spaghetti breaks.. The number of nails the strand of spaghetti could hold before breaking is recorded. Strand(s) of spaghetti desk desk paper clip gap betw. desks approx. 10cm nails plastic cup The gap between the desks is held constant and the experiment Repeated for 2, 3, 4 or 5 strands of spaghetti. The number of strands of spaghetti and the number of nails are recorded and graphed on a graphical calculator. The constants of the resulting equation are optimized by trial and error to obtain the “line of best fit”. 4 Sample set of Results: Strands of Number of nails spaghetti The mass of one nail was 1 12 calculated by averaging the mass of 2 18 10 nails. 3 30 4 34 mass of 10 nails = 64 g 5 42 mass of 1 nail = 64 ÷ 10 = 6·4 g Number of nails (dependent variable) 50 y y = 8x + 2 40 X X 30 X 20 X X 10 x 1 2 3 4 5 Number of strands of spaghetti (independent variable) student mass = 50 kg = 50 000 g and if 1 nail = 6·4 g then student mass ≡ 50 000 = 7813 nails 6·4 therefore y = 7813 and substituting this in y = 8x +2, we get x = 7811 ÷ 8 = 976 i.e. the number of sticks (x) to support a 50 kg student is approx. 1000 CONCLUSION: 1. The graph was a straight line i.e. the relationship between the number of strands of spaghetti and the mass they can support is a linear relationship. 2. Based upon this experiment, approximately 1000 strands of spaghetti would be needed to support a 50 kg student. During the optimizing of the equation it was noted that the number multiplying the x (8) changed the slope of the line and that the number at the end of the equation (+2) was where the graph line crossed the y axis. 5 Tables of Values Record of the number Theoretical Record of of nails supported by the number of nails the spaghetti supported by the spaghetti # of # of nails # of # of nails strands of strands of spaghetti spaghetti 1 1 2 2 3 3 4 4 5 5 6 6 6 Graph of the number of nails that are supported by the strands of spaghetti 150 140 130 110 100 90 Number of Nails 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 Number of Strands of spaghetti 7 LESSON PLAN FOR PATTERNS AND ALGEBRA, “FINDING A RULE” Kath’s Hot Bread Rolls Date: The Following NSW Mathematics 7-10 Syllabus outcomes will be addressed. Patterns and Algebra Strand 3.1a/4.2 (PAS 3.1a/PAS 4.2) Students learn to “create, record, analyse and generalise number patterns using words and algebraic symbols in a variety of ways. Goals: Students use algebraic methods to explore, model, and describe patterns. Students learn how to develop a table, a graph and an algebraic rule from a word description of a function. Students develop the problem solving skills, of understanding the problem, questioning and investigating, reasoning and reflection. They also learn some techniques such as trial and error, working backwards and systematic lists. Standard Strategy Questions you might ask. Duration #2 Select 6 students to be the Who can tell me; 15mins This narrator and actors. How many bread rolls did Kath start section Give the students the scripts at out with? uses least 5 minutes before the the play lesson. Kath’s Sit back with the rest of the class How could we find out? hot and watch them act out the Is there another way we could act this bread scene. out? rolls At the end of the play pose the How many bread rolls did Kath bring question in bold opposite and home. The allow a discussion to develop. How many did her friend Adriana get? script is If students want to do a trial and How did you work that out? attache error process give them some How many did Matt the surfer get? d counters or blocks to represent Is there a way to work how many she bagels and let them get on with started out with? QT 1.6, it. 2.2, 3.1. Give the class a copy of the How would Kath explain to her 20 mins QT 1.4 script and ask them to rewrite it daughter Kim how the rolls (or rethink it) in such a way that disappeared? WMS they will get the answer. If Kim ran next door to Adriana’s 4.2, 4.4. Ideally you want them to do the house and invited her to breakfast and play backwards starting with asked her to bring the rolls with her Kylie discovering that she only how many would they have? had 2 rolls left. They could do a If Kath only gave each person half of flashback. what she had left how many would If students are having difficulty she have started out with? suggest that they try the scenario without Kath giving out 8 the two extra rolls each time. Only do At this point the students have How about we rewind the tape a bit? 10 mins this if no probably worked out that Kath If Kath got home with two rolls how student started out with 44 rolls. If not many did she have before she gave has give the whole class some Adriana the extra two? come up concrete materials (blocks, How many did she have before she with the markers, pennies etc) and gave Adriana half? answer. encourage them to model the Can you keep working backwards process backwards. from here? QT 2.3 Students need to recognize that the answer to the problem (the PAS dependant variable) is the 4.2.1 number of rolls that Kath starts with. In this section students If Kath got home with a different 25mins QT 1.4, investigate other possible number of rolls could you work out 1.6 outcomes. Such as Kath ending how many she started out with? up with zero, 1, 3, 4, 5, _, _, _, Kath and Kim like to have 2 rolls each PAS rolls. for breakfast, how many would Kath 4.2.1, Make sure students record their need to start out with for this to 4.2.2 results in a systematic fashion. happen? An in – out or function table Try some other cases, for example WMS would be best. Kath gets home with none, 3, 6 and 4.5 In Out so on. (ending rolls) (starting rolls) How should we keep track of our 0 ___ findings? 1 ___ 2 44 3 ___ 4 ___ 5 ___ At this point students should be Kath is a very generous person and 25 mins encouraged to go from the always gives to the needy, but she QT 1.2 specific to the general. wants to be sure she has enough rolls They should be able to describe when she gets home for her needs. WMS a way of answering the problem Can you describe a way Kath can 4.2, 4.4, for any number of rolls. easily calculate how many rolls she 4.5 Students should also have needs to buy so she can have the emphasized that a function can exact amount she will need for that be described in four ways: day? Assume that she will always As a story, meet 3 needy people on her way as a table, home. as a graph, and as an algebraic rule. This could be a good time to introduce the graphics calculator. Students should discover that the general rule is: #Of starting rolls = 8 times the # of rolls left + 28. Or R = 8r + 28 9 Extension: Can any one work out a general rule if Students could try seeing what the number of needy people Kath happens when they alter the meets is four instead of three? 15 mins number of people given rolls. They could add or delete What would be the outcome if Matt characters from the original was still surfing and did not meet Total time script. Kath? 85 to 110 mi Quality teaching elements that are promoted: 1.1? 1.2, 1.3? 1.4, 1.5? 1.6. 2.2, 2.3, 2.6? 3.1, 3.3?, 3.4, 3.6? 10 ACT IT OUT Kath’s Hot Bread Rolls A play about a lady who likes hot bread rolls and people. Cast: Narrator Kath, a lady who goes shopping for bread rolls for her and her daughter. Dorothy, a social worker who runs a homeless shelter, Matt, a fanatical surfer who rides his surfboard in all weather. Adriana, Kath’s neighbor an old lady with arthritis. Kim, Kath’s daughter. Scene: The road from the bakery to Kath’s house. Kath is walking along with a LARGE shopping bag. Narrator: Kath is walking home from the hot bread shop. She has bought a LOT of hot rolls for her and her daughter’s breakfast. Now who is this, why it is Dorothy, the social worker who runs the homeless shelter. Dorothy waves to Kath and stops her to chat. Dorothy: My, Kath, those bread rolls sure do smell good. Can you spare some for the people in my homeless shelter Kath: Why sure I can Dorothy, here have half of them and two more for good measure Narrator: So Kath gives Dorothy half of her rolls plus two more Kath waves goodbye to Dorothy and walks on still carrying her shopping bag, Dorothy walks off stage with her rolls. Narrator: As Kath walks on she walks past a beach and sees her friend Matt who has been out for an early morning surf in the cold, cold ocean Matt comes running up to Kath and gives her a hug. Matt: My, Kath, those bread rolls sure do smell good. Can you give me some; it’s so cold that I am very hungry. Kath: Why sure I can Matt, here have half of them and two more for good measure. Narrator: So Kath gives Matt half of her rolls plus two more. Kath waves goodbye to Matt and walks on still carrying her shopping bag, Matt walks off stage with his rolls. 11 Narrator: Kath is nearly home, as she walks past the house next door she sees her elderly neighbor, Adriana Appleby sitting on her porch, she waves and goes up to her and gives her a hug. Kath: How are you this morning Adriana. Adriana: (In a quivery voice) Not so good today darl. My knees are acting up again; you young people don’t know what it’s like to be old. (She smells the bread rolls) My, Kath, those bread rolls sure do smell good. Can you spare me some I don’t think I will be able to go shopping today and they smell so good. Kath: Why sure I can Adriana, here have half of them and two more for good measure. Narrator: So Kath gives Adriana half of her rolls plus two more. Kath waves goodbye to Adriana and walks next door to her house still carrying her shopping bag, Adriana hobbles inside/offstage with her rolls. Narrator: Kath walks up her front path and her daughter Kim runs out to meet her. Kim runs towards Kath Kim: Mum, Mum, your home at last, I am soooo hungry. Have you got the rolls? Kath hands the shopping bag to Kim who eagerly peers inside her expression turning to one of dismay. Kim: Mum !!! There are only two rolls here; you said that you were going to buy heaps! Kath looks surprised. Narrator: Poor Kath, only two bread rolls left, I wonder how many rolls she started with. 12