PART 2 Marking the Tasks Focusing on the Competencies: Uses Mathematical Reasoning and Communicates by Using Mathematical Language

Transcription

1 PART Marking the Tasks Focusing on the Competencies: Uses Mathematical Reasoning and Communicates Using Mathematical Language 8. Marking Ke Applications 1. GOING TO SCHOOL Use a rational function to determine the speed needed to get to school in five (5) minutes. Theme: Arithmetic & Algera : k Rational function of the form x Representing and interpreting the inverse function Oserving patterns Finding the rules Rule of the function representing the duration of the trip according to the speed: x: Speed (km/h) : Duration of Trip (hours) k The product of the variales is constant; therefore it is a rational function of the form x k k = 3 The distance from the house to the school is 3 km. 3 The rule of the function is therefore: x Check if this rule is valid for the other silings values: Michael (x = 10): Manoli (x = 5): hours 0.1 hours Speed for a 5 minute journe: 1 5 minutes hour x 36 km / hour Conclusion: The father must drive at a speed of 36 km/hour to get to school in 5 minutes. Page 1

2 . COMMON FACTOR Simplifing algeraic expressions to find a common factor and evaluating the resulting monomial. Theme: Arithmetic & Algera manipulating numerical and algeraic expressions performing context-related calculations with integral exponents (rational ase) and fractional exponents finding the common factor The following expressions are simplified as follows: x 3x 4x x x 5x 3 10 (4x )(5x ) 0x 7 3 0x 4x x The greatest common factor that can e factored from these four resulting monomials 3 is: 5x Given that x= 4. Given that = 3. The numerical value of the greatest common factor is: 3 3 5x Page

3 3. MYSTIC AQUARIUM Finding the dimensions of an equivalent solid Theme: Geometr & Arithmetic Equivalent solids o cue root Old aquarium Volume of the rectangular-ased right prism V = l x w x h = x 1.4 x 1.5 = 4. m 3 New aquarium h =.6r Volume of the circular right clinder V= r h Since the capacities of these two aquariums are equivalent, the volume of the circular right clinder will also e 4. m 3 : 4. = r (.6r) 4. = r = r = r The radius of the new aquarium is 0.8 m, which means that the diameter is 1.6 m and the height is.08 m. Page 3

4 4. LAWN CARE Find the area of grass to calculate the cost of fertilizer. Step 1: Height of trapezoid Find the ase of the right-angled triangle: 0.4 m 9.6 m = 10.8 m Appl Pthagorean Theorem to find height of the trapezoid. a c h h h h h 14.4 m Step 3 : Solving for the value of x (use information from BBQ): Appl Pthagorean Theorem to find the value of x. a c 0.54x x 0.916x x x x x 5 Step 5: Area of Flower Bed (Area of rectangle) A w A A A 1 m Step 7: Cost to fertilize grass $ m $ m Note: Accept $159.3 ecause some students will argue that usinesses alwas round up. Theme: Geometr & Algera Multipling/dividing algeraic expression Pthagorean theorem Area of decomposale solid Step : Area of trapezoid B h A A A 16 m Step 4: Area of BBQ (Area of triangle) h A A 9.7 A A 4.86 m Step 6: Area of grass Area of trapezoid Area BBQ Area Flower Bed = m Page 4

5 5. FUNDRAISING Determine numer of iccles to sell in order to make a greater profit than the girls. Theme: Algera Polnomial function degree 1 o partial variation o direct variation Sstem of first degree equation Step 1: Determine the amount of mone raised Group A Let x 1 e the numer of pieces of jeweller sold and e the net profit from the jeweller sales We have points given: (85,480) and (10,760) Use the two data points to calculate the rate of change: a 8 x Since each piece of jeweller is sold at a fixed price the relation can e expressed as a linear function in the for ax Using the slope and one other point, solve for. ax Therefore the rule of this relation is: 8x 00 Use the function rule to determine mone raised when 90 pieces of jeweller are sold: 8x $50 Page 5

6 Step : Minimum numer of iccles sold Let x e the numer of iccles sold and e the net profit from iccle sales From the graph we have two points: (0,0) and (4,104) Use the two data points to calculate the rate of change: a 6 x The graph illustrates a direct linear function therefore the rule can e expressed as Therefore the rule of this relation is: 6x ax Use the function rule to determine the numer of iccles sold if the mone raised exceeds $50: Group B must sell more than 0 iccles. OR Group B must sell at least 1 iccles. Page 6

7 6. MAKING THE POOL SAFE Determining the cost of gating a pool Algeraic area of the pool (x + 4)(x + 1) = x + 6x + 4 Dimension of exterior rectangle L: x = (x + 8) W: x = (x + 4) Theme: Geometr & Algera Area of figures that can e split into sectors Direct variation Multipling/dividing polnomials Total area with gravel path (x + 8)(x + 4) = x + 16x + 3 Area of gravel (x + 16x + 3) (x + 6x + 4) = 10x + 8 Value of x 10x + 8 = 48 x = Perimeter of the pool (x + 4) + (x + 1) = 6x + 10 = 6() + 10 = m Cost of the fence 45 = $990 Page 7

8 7. NEW YORK CITY Comparing two linear functions Theme: Arithmetic & Algera comparing situations Step 1: Find the rule for option 1 Let x e the numer of nights in NYC Let e the total cost of the trip Using two points from the tale of values, (6, 750) and (4, 500), calculate the slope: a x Use the slope and one point to solve for ax x 0 6 Add $50 for transportation and option 1 can e represented the equation: 15x 50 Page 8

9 Step : Find the rule for option The transportation and memership fees are the fixed costs(). The $150 per night is the rate therefore option can e represented the equation 150x 175 Numer of nights Total cost for option 1 ($) Total cost for option ($) If Luka stas in NYC for 1 or nights, option is cheaper than option 1. If Luka stas in NYC for 3 nights, oth options are equal since the total cost is the same. If Luka stas in NYC for 4 or more nights, option 1 is cheaper than option. Page 9

10 8. CHARITY REPORT Correcting statements aout charit donations for a written report. Theme: Statistics and Proailit Graphs: ox-and-whisker Plot Measures of central tendenc: mode, median and weighted mean Measures of dispersion: range of each part of a ox-and-whisker plot (including interquartile range) o Calculating measures of central tendenc and of dispersion Comparing distriutions The following can e deduced from the information provided: 008: Mean was $ The same numer of people donated etween $300 and $5000 as donated etween $5 and $75. The maximum donation was in the range of [750, 3000[, ut the exact maximum value is unknown. 009: We do not have enough information from just the ox and whisker chart to determine the mean. The $100 indicates the median donation. The ox and whisker plot is divided in quartiles, therefore the same numer of people donated etween $5 and $75 as the numer of people that donated etween $300 and $5000. The maximum donation was $5000. The following are examples of correct statements: 1. The mean donation from 008 is $318.9, not $300. We do not have enough information from the chart given to find the mean donation from 009, ut we know that $100 is the median, and not the mean.. The same numer of people made donations etween $300 and $5000 as made donations from $5 to $ In 008, the maximum donation was in the range [750, 3000[. In 009, the maximum donation was $5000. Page 10

11 9. THE AREA OF REGULAR POLYGONS Make a conjecture comparing the numer of sides of a regular polgon and its area If the perimeter is set to then: For the triangle: Let x e the length of the sides: P x 4 3x c 3 8 units 8 units c x x a units Theme: Geometr 8 units a = (1/)8 = 4 units Pthagorean theorem Area of regular polgons Perimeter of regular polgons ase height A A A 7.71 units For the square or diamond: Let x e the length of the sides: P 4x 4 4x x x 6 units 6 units 6 units 6 units A ase height A 6 6 A 36 units Page 11

12 For the hexagon: Let x e the length of the sides: P x 4 6x x x A hexagon is made up of six (6) equilateral triangles: There the apothem can e calculated using the Pthagorean Theorem: c a c = units a = 1/(4) = units A A A triangle triangle triangle ase height units A A A Hex Hex Hex A triangle Summar: Shape Perimeter Numer of sides Area Triangle units Square 4 36 units Hexagon units Conjecture: For regular polgons of equal perimeter, as ou increase the numer of sides the area will also increase. -The Amazing Race Board Game Student Profile