Polygon Patterns. 1. From any vertex of a 4 sided-polygon, one diagonal can be drawn. Draw this shape.

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1 Plygn Patterns 1. Frm any vertex f a 4 sided-plygn, ne diagnal can be drawn. 2. Frm any vertex f a 5 sided-plygn, tw diagnals can be drawn. 3. Frm any vertex f a 6 sided-plygn, three diagnals can be drawn. 4. Frm any vertex f a 7 sided-plygn, fur diagnals can be drawn. 5. Create an equatin that will generate the number f diagnals fr n- sided plygns. 6. State the slpe and interpret it in the cntext f this prblem. 7. State the y-intercept and interpret it in the cntext f this prblem. 8. Hw many diagnals can be drawn frm any vertex f a 20-sided plygn? 9. Hw many sides des a plygn have if yu can draw 42 diagnals? Surce: Algebra I NCSCOS Indicatr

2 Plygn Patterns (These were the respnses t the 3 prmpts frm the grup cmpleting this task). What s the imprtant mathematics? Objective 1.02 patterns Objective 4.01 tables/charts T make linear functins What scafflding culd make the task accessible t all? Manipulative fr figures Suggest students set up a table Help them label the independent and dependent variables Hw wuld yu extend? Discuss dmain/range Verificatin methds Surce: Algebra I NCSCOS Indicatr

3 The Hneycmb Prject Three friends, Clay, May, and Jay, are using a drawing prgram t create a prject in art class. Their class has been studying varius tilings and their jb is t create an example f tiling that ccurs in the real wrld. They ve decided t draw a hneycmb and fund that the easiest way t d this with their sftware requires using rtatins. The bad news is they must knw the angle measures in a regular hexagn and unfrtunately, they were all ddling that day in gemetry class and cannt remember the frmula they need t use. Each f them started scribbling n paper, trying t cme up with a technique t find the measure f a hexagn s interir angle. Surprisingly, they all arrived at the same answer but used slightly different methds. Each f their methds is shwn belw. Use each methd belw t see if yu can determine the interir angle sum f a hexagn. Clay's methd May's methd Jay's methd Clay, May, and Jay were s impressed with their prblem slving ability that they wndered if they culd find a frmula that wuld always prduce the angle sum fr any regular plygn. They agreed that a gd strategy wuld be t cnsider ther regular plygns, rganize the infrmatin in a table, and lk fr a pattern. They knew that a triangle s angles summed t 180º s they began with a plygn having 4 sides. [Hint: Write the expanded frm f each cmputatin and then the ttal.] N. f sides Clay s methd May s methd Jay s methd n D their methds always prduce the crrect sum? Why r why nt? Adjust their frmulas t find the measure f ne angle f any regular plygn. SITE Gemetry - Summer 2007 Authred by Eleanr Pusey

4 The Hneycmb Prject (These were the respnses t the 3 prmpts frm the grup cmpleting this task). What s the imprtant mathematics? Patterns Angle sum f a triangle Plygn Angle Sums 360º in a circle 180º in a straight line/straight angle Radii are cngruent Inductive Reasning What scafflding culd make the task accessible t all? Read the first paragraph alud t set up the cntext f the prblem Hw wuld yu extend? Verify all 3 frmulas are equivalent Ask them fr anther methd t write the frmula Draw a picture t g with it D the methds extend t nn-regular plygns? Why? Nte the frmula fr any angle f a regular plygn using May s methd 180n! 360 results in: = 180! 360. Hw is this related t the exterir angles n n f the plygn? SITE Gemetry - Summer 2007 Authred by Eleanr Pusey

5 Heart Rates Suppse a patient is given a 250mg injectin f a therapeutic drug that has a side effect f raising the Heart rate. Table A gives the relatinship between Q, the quantity f drug in the bdy (in milligrams, r mg) and r, the persn s heart rate (in beats per minute). Table A. Heart rate r as a functin f drug level Q Q, drug level (mg) R, heart rate (beats per min) Hw des the heart respnd t higher drug levels? Over time, the patient s bdy metablizes the drug, and the level f the drug in the bdy falls. Table B gives the drug level, Q, as a functin f t, the number f hurs since the injectin was given. Table B. Drug level Q, as a functin f time, t T, time (hurs) Q, drug level (mg) Since the patient s heart rate depends n the drug level, and the drug level depends n the time, then the heart rate depends, via the drug level, n time. 2. What is the heart rate 1 hur after the drug is administered? 3. Estimate the heart rate fr each time in Table C belw. Table C. Heart rate r, as a functin f time, t T, time (hurs) R, heart rate (beats/min) 4. Verify yur estimates by using a graphing calculatr t evaluate a cmpsitin f functins frmed by the first tw tables. What is the rate f the heart after 24 hurs? Surce: adapted frm Functins Mdeling Change, Wiley & Sns, 1998

6 Heart Rates (Slutins) 1. Hw des the heart respnd t higher drug levels? The Heart rate increases as the drug level increases. The actual equatin fr the mdel in Table A is linear and given by: R = 1 5 Q What is the heart rate 1 hur after the drug is administered? Frm table B, we bserve that ne hur after the drug is administered the remaining amunt f the drug is 200 mg. Frm table A we see that the heart rate with 200 mg f the drug wuld be 100 beats per minute. The equatin fr the mdel in Table B is " 4 % expnential and can be apprximated by: Q = 250! # $ 5 & ' 3. Estimate the heart rate fr each time in Table C. T Lk up time in Table B t find the amunt f drug. Find the heart rate in Table A fr the given amunt f drug. Fill in the values fr Table C. Time mg f the drug remains heart rate is 110 beats per minute (0, 110) Time mg f the drug remains heart rate is 100 beats per minute (1, 100) Time mg f the drug remains heart rate is??? beats per minute (2,? ) Since 160 mg is nt in Table A, we must apprximate. We can use a prprtin (linear interplatin) t make the apprximatin. Find the tw values s that 160 is between them: Drug Level Heart rate ??? The slpe f a line is cnstant s the slpe calculated using pints 2 and 3 must be the same as the slpe calculated with pints 1 and 3. Slve the prprtin D/ ( ) = (100-90)/ ( ) r D/10 = 10/50. Hence D =2. Nte D is the difference in the y-values (Heart Rate). The Heart Rate wuld be apprximately r 92 beats per minute. Time = 3, Drug Level = 128 Time = 4, Drug level = 102 Drug Level Heart rate Drug Level Heart rate ??? 102??? D/ ( ) = (90-80)/ ( ) D/ ( ) = (90-80)/ ( ) D = 28(10/50) D = 2(10/50) D = 5.6 D = 0.4 Heart Rate Heart Rate Surce: adapted frm Functins Mdeling Change, Wiley & Sns, 1998

7 Table C. Heart rate r, as a functin f time, t T, time (hurs) R, heart rate (beats/min) Verify yur estimates by using a graphing calculatr t evaluate a cmpsitin f functins frmed by the first tw tables. What is the rate f the heart after 24 hurs? Find a best fit functin fr Table A. Enter Q int List1 and R int List2. Linear regressin shws a perfect fit: Y1 =.2X + 60 Find a best fit functin fr Table B. Enter T int List3 and Q int List4. An expnential functin fits very well (R squared is ): Y2 = ( ) ^X r Y2 = 250(0.8) ^X Frm the cmpsitin s time is the input and heart rate is the utput. In ther wrds, cmpute R! Q ( )(T ) = R(Q(T )) = 50( 4 5 )T + 60 Y3 = Y1 (Y2) and lk at the tables t cmpare values. This cmpsitin can als be dne at the hme screen. Manual calculatin: X 250(0.8) ^X.2(X) (0.8) ^4 r (102.4) + 60 r The rate after 24 hurs is: 250(0.8) ^ (1.2) (These were the respnses t the 3 prmpts frm the grup cmpleting this task). What s the imprtant mathematics? Interpret the table data Mdeling linear, expnential functins Graphical and algebraic representatins f data Use data t predict What scafflding culd make the task accessible t all? Handut fr putting the data int calculatr t get regressin ut. Chsing the independent and dependent variables, apprpriate mdel Hw wuld yu extend? Pster display f the mathematics Algebraic cmpsitin f the functins Make a new prblem with new Tables A, B Surce: adapted frm Functins Mdeling Change, Wiley & Sns, 1998