2022.Workbook.Mathematical Literacy Grade 12

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 2022 SUBJECT WORKBOOK Grade 12 MATHEMATICAL LITERACY A joint initiative between the Western Cape Education Department and Stellenbosch University.

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 BROADCAST SESSIONS GRADE 12 BMI GRADE 12 MODELS Session Date Time Topic 1 04/05/2022 15:00 – 16:00 BMI 2 26/07/2022 15:00 – 16:00 Models Page 2

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 INTRODUCTION AND TOPICS BMI For every adult, there is a relationship between the persons height and weight, and it is called the persons body mass index. In short BMI. Models The following must be investigated: • The most cost-effective and convenient way to pack cans or boxes. • The packaging must be such that the space is optimally utilized. • The use of models to solve problems. Topics Description To determine a persons body mass index and to make predictions on a BMI persons state of health. The concepts and skills that will be mainly focused on are: • packaging of a rectangular box and cylindrical can in a Models rectangular container / large box • to use models to solve problems Page 3

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 TERMINOLOGY Term Definition A number calculated from an adult’s weight and height, Body mass index (BMI) expressed in units of kg/m2 The measurement between two points, in a straight line, Height e.g., from the top of a persons head to the floor on which the person is standing Measure Using an instrument to determine size, weight etc. Weight An indication of how heavy an object is. Page 4

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 SESSION 1 | BMI BODY MASS INDEX (BMI) A. What is Body Mass Index (BMI)? ∙ Body Mass Index (BMI) is a number calculated from an SUMMARY adult’s weight and height and is expressed in units of kg/m2. WHAT YOU SHOULD KNOW ∙ BMI is used for adults to determine whether the adult is of normal weight, underweight, overweight or obese. The learner should know how to ∙ This weight status is used by health practitioners to determine the height and provide appropriate advice. weight of a person. Must also ∙ Note, for children and teenagers’ growth charts are be able to do conversions between different units. used to determine their weight status. B. How is BMI calculated? BMI is calculated by the following formula: BMI = C. How is the weight status determined from the BMI value? The following BMI-Weight Status table is used to determine the weight status of a person. BMI - WEIGHT STATUS TABLE BMI Weight Status Below 18,5 Underweight (U) 18,5 – 24,9 Normal (N) 25 – 29,9 Overweight (OW) 30 and more Obese (O) 24,22 is in grey row 18,5 – 24,9. Weight Status: Normal(N) Page 5

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 SESSION 1 | BMI EXAMPLES: The table is used to calculate the values to confirm the correctness of it and the weight status of Learner 4: WORKSHEET Learner Mass in kg length in m BMI Weight status 4 70 1.70 24.2 N 2 • Work through all the examples with your calculator. • You must get the same answers and then you know that you used the calculator correctly. 1. How to determine BMI BMI = BMI = = 24,22 Weight Status – Go to the BMI-weight status table to determine in which row 24,22 fall. In this case it is in the ‘grey’ row 18,5 – 24,9. So, the weight status is Normal (N) 2. How to determine the length Length in m = length in m = = 1,70 3. How to determine the weight Weight in kg = BMI × (length in m)2 Learner 4: Weight in kg = 24,22 × 1.72 ≈ 70 kg Page 6

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 SESSION 1 | BMI ACTIVITY (BMI – BODY MASS INDEX) CLASS DISCUSSIONS During the health week a group of persons were selected to take their Body Mass Index (BMI). The information is shown below: Person Mass length BMI Weight in kg in m status 1 56 1.56 (A) N 2 60 1.63 (B) N 3 44 1.55 (C) (D) 4 70 1.70 24.22 N 5 52 (E) 22.51 N 6 60 (F) 23.73 N 7 45 1.30 26.63 (I) 8 (G) 1.55 25.39 OW 9 (H) 1.60 23.83 N 10 57 1.68 20.20 N BMI-WEIGHT STATUS TABLE BMI Weight Status Below 18,5 Underweight (U) 18,5 – 24,9 Normal (N) 25 – 29,9 Overweight (OW) 30 and more Obese (O) 1 pound = 0,4536 kg BMI = 1 inch = 2,54 cm 1. Determine the BMI - value (A) and (B). 2. Determine the BMI-value of (C) and (D), the weight status of Person 3 as well as that of Person 7, (I) 3. Determine the lengths in m of Persons 5 and 6, the values of (E) and (F). 4. Determine the mass in kg, the value of (G) and (H) of Persons 8 and 9. 5. An American exchange student said that they use the following formula: BMI = 703 × Use the information of Person 10 to calculate the BMI to show that the formula is correct. Page 7

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 2. Mr and Mrs Bush’s domestic worker Phelo and her daughter Thuli, stay with them. Thuli and her friend Busi are worried about their weight. Study the growth chart on THE NEXT PAGE and answer the following questions. 2.1 Determine Thuli’s BMI (Body Mass Index). She is 17 years old and her BMI places her in the 75th percentile. 2.2 Busi is 18 years old and she is 1,74 m tall. She has a weight of 96 000 g. TABLE 1 shows the relationship between BMI and weight status. TABLE 1: WEIGHT STATUS TABEL BMI for age Percentile range Weight status < 5th percentile Underweight 5th to < 85th percentile Healthy 85th to < 95th percentile Risk of overweight ≥95th percentile Overweight 2.2.1 Determine her weight status using TABLE 1 and the given formula: BMI = 2.2.2 What advice should Mrs Bush give Busi’s parents? Page 8

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 Page 9

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 TERMINOLOGY Term Definition Breadth How wide something is. From the word “broad”. A straight line passing through the centre of a circle and Diameter touching the circle at both ends, thus dividing the circle into two equal halves. The measurement between two points, in a straight line, Length e.g. the length of a room. The distance from the centre of the circle to any point on Radius the circumference of the circle. Page 10

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 SESSION 2 | MODELS Packing a rectangular box in a rectangular container/large box: SUMMARY PACKAGING METHOD 1: Length - wise: The LENGTH of the small box is packed along the LENGTH WHAT YOU SHOULD KNOW of the LARGE CONTAINER / BOX. CALCULATION: Prior knowledge of the 3 - dimensional figures especially the The number of small boxes that can be packed along length = identification of length, width and Length (ℓ) of large container / box ÷ length (ℓ1) of small box height; rounding; ratio between radius and diameter, calculator skills and logical reasoning are knowledge and skills that come in handy for the topic that actually contains a strong practical component. Width -wise: The WIDTH of the small box is packed along the WIDTH / BREADTH of the LARGE CONTAINER / BOX. CALCULATION: The number of small boxes that can be packed along WIDTH = Width (w) of large container / box ÷ width (w1) of the small box Height - wise: How many boxes can be packed along the HEIGHT of the BIG BOX / CONTAINER? CALCULATION: The number of small boxes that can be packed along the HEIGHT = Height (h) of large container / box ÷ Height (h1) of the box Page 11

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 What aspects of PACKAGING PACKAGING METHOD 2: are important to us? What is packed? Rectangular Length-wise: The width of the small box is packed along the LENGTH boxes and cylindrical cans/tins of the large container / box. What is it packed into? CALCULATION: Rectangular Containers / Boxes The number of small boxes packed along the length = How is it packed? There are Length (ℓ) of large container / box ÷ width (b1) of the small box different packaging ways e.g., packaging methods 1 and 2 but optimal use of space and cost-effectiveness must also be taken into account. OPTIMAL / MAXIMUM USE OF SPACE: The boxes or cans should be packed in such a way that it Width - wise: The LENGTH of the small box is packed along the fits the largest number of WIDTH of the large container / box. boxes/cans/tins in a CALCULATION: container/box. The number of small boxes that can be packed along the WIDTH = Width (w) of large container / box ÷ length (ℓ1) of the small box COST EFFICIENCY: Packaging materials are expensive. The smallest container may need to be used for packaging of a certain number of boxes /tins /cans. Height - wise: How many boxes can be packed along the height? CALCULATION: The number of small boxes that can be packed next to HEIGHT = Height (h) of large container / box ÷ Height (h1) of the small box Page 12

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 EXAMPLE 1: PROBLEM: The small boxes must be packed in the big box. Verify which of the following packaging options will be the most cost effective: Option 1: The length of the small box along the length of the large box/container Option 2: The width of the small box along the length of the large box/container Show all calculations to justify your answer. OPTION 1: The length of the small box along the length of the large box / container Length - wise: The number of small boxes that can be packed along the LENGTH = length of large box ÷ length of small box = 56 ÷ 7 = 8 boxes Width - wise: The number of small boxes that can be packed along the WIDTH = width of large box ÷ width of small box = 40 ÷ 4 = 10 boxes Height - wise: The number of small boxes that can be packed along the HEIGHT = Height of big box ÷ Height of small box = 24 ÷ 6 = 4 boxes TOTAL BOXES PACKED = Number at length × Number at width × Number at height = 8 × 10 × 4 = 320 boxes Page 13

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 OPTION 2: The width of the small box along the length of the large box / container. Length-wise: The number of small boxes packed along the LENGTH = Length of large box ÷ width of small box = 56 ÷ 4 = 14 boxes Width - wise: The number of small boxes that can be packed along the WIDTH = Width of large box ÷ length of small box = 40 ÷ 7 = 5.7 ≈ 5 Height: The number of small boxes that can be packed along the HEIGHT = Height of big box ÷ Height of small box = 24 ÷ 6 = 4 boxes TOTAL BOXES PACKED = Number on length × Number on width × Number on height = 14 × 5 × 4 = 280 boxes Conclusion: Option 1 (More boxes can be packed, and it will be more compact) ANSWERING TECHNIQUES: Make sure all units of measurement are exactly the same Use headings e.g., Length - wise etc. Write down your formula, which indicates exactly which values you need to use. Use only the Natural Numbers; ignore decimals. Use your calculator correctly for calculations. Page 14

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 DETERMINING THE NUMBER OF CYLINDRICAL TINS/CANS IN A RECTANGULAR CONTAINER / LARGE BOX: Length - wise: The DIAMETER of the tin is packed along the LENGTH of the large container / box. CALCULATION: The number of tins that can be packed along the LENGTH = Length (ℓ) of large container / box ÷ diameter (d1) of the tin Width - wise: The DIAMETER of the tin packed along the WIDTH of the large container / box. CALCULATION: The number of tins that can be packed along the WIDTH = Width (w) of large container / box ÷ diameter (d1) of the tin Height-wise: How many tins can be packed along the height? CALCULATION: The number of tins that can be packed along the HEIGHT = Height (h) of large container / box ÷ Height (h1) of the tin TOTAL TINS/CANS PACKED = number at length × number at width × number at height NUMBER OF TINS/CANS PACKED ON BASIS = number at length × number at width NB: Diameter = 2 × radius Page 15

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 EXAMPLE 2: PROBLEM Determine how many cans/tins can be packed in the box. Show all calculations. SOLUTION: Diameter = 3.5 × 2 = 7 cm Conversion: Height = 180 mm ÷ 10 = 18 cm Length - wise: The number of cans/tins that can be packed along the LENGTH = Length of large box ÷ diameter of the tin = 85 ÷ 7 = 12,14 ≈ 12 cans/tins Width - wise: The number of cans/tins that can be packed along the WIDTH = Width of large container / box ÷ diameter of the tin = 43 ÷ 7 = 6,14 ≈ 6 cans/tins Page 16

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 Height - wise: The number of cans/tins that can be packed next to HEIGHT = Height of large container / box ÷ Height of the tin = 18 ÷ 8.57 = 2,1 ≈ 2 (cans/tins/layers) TOTAL CANS/TINS PACKED = number at length × number at width × number at height = 12 × 6 × 2 = 144 tins TOTAL TINS/CANS PACKED = number at length × number at width × number at height NUMBER OF TINS/CANS PACKED ON BASIS = number at length × number at width 1. Calculate the number of small boxes that can be packed in the ACTIVITY ON PACKAGING: large container/box if the length (ℓ) of the small box is packed along the width(w) of the container / box. 2. A box of 24 cans of Diet Coke is displayed. The radius of a can of Diet Coke is 3.5 cm. The cans are packed as in the picture. Calculate the minimum length and minimum width of the box to contain the 24 cans of Diet Coke. Page 17

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 3. Use the bookshelf and a book's dimensions to determine how many books can be packed into the TWO shelves if the books are packed in its width (w) along the length of the bookshelf. 4. Use the following diagrams to answer the question that follows: Verify which of the following packaging options will be the most cost-effective: Option 1: The length (ℓ) of the box along the length of the trailer Option 2: The width (w) of the box along the length of the trailer NB: Show all calculations to justify your answer. ANSWERS: ACTIVITY ON PACKAGING 1. The length of the box is packed along the width of the container/box: Conversion of box width = 87 mm ÷ 10 = 8,7 cm Width - wise: The number of boxes that can be packed along the WIDTH = Width of large box ÷ length of small box = 42.1 ÷ 10.2 = 4,127 ≈ 4 boxes Page 18

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 Length - wise: The number of small boxes packed along the LENGTH = Length of large box ÷ width of small box = 54.8 ÷ 8.7 = 6,298 ≈ 6 boxes Height - wise: The number of small boxes that can be packed along the HEIGHT = Height of big box ÷ Height of small box = 33.5 ÷ 6.5 = 5,153 ≈ 5 boxes / layers TOTAL BOXES PACKED = number at height × number at width × number at height =4×6×5 = 120 boxes 2. Diameter = 2 × 3,5 = 7 Minimum length = 7 × 6 = 42 cm Minimum width = 7 × 4 = 28 cm 3. Conversion of width of book = 36 mm ÷ 10 = 3.6 cm Number of books on 1 shelf = 90 ÷ 3.6 = 25 books Number of books on 2 shelves = 25 × 2 = 50 books 4. OPTION 1: The length of the box along the length of the trailer. Length - wise: The number that can be packed along the LENGTH = length of trailer ÷ length of the box = 10 ÷ 0.31 = 32.25 ≈ 32 boxes Page 19

2022 WORKBOOK | Grade MATHEMATICAL LITERACY 12 Width - wise: The number of boxes that can be packed along the WIDTH = width of trailer ÷ width of the box = 2.8 ÷ 0.25 = 11.2 ≈ 11 boxes Height - wise: The number of boxes that can be packed along the HEIGHT = Height of trailer ÷ height of the box = 2.4 ÷ 0.2 = 12 boxes TOTAL BOXES PACKED = number at height × number at width × number at height = 32 × 11 × 12 = 4 224 boxes OPTION 2: The width of the boxes along the length of the trailer Length - wise: The number of boxes packed along the length = Length of trailer ÷ width of the box = 10 ÷ 0.25 = 40 boxes Width - wise: The number of boxes that can be packed along the WIDTH = Width of trailer ÷ length of the box = 2.8 ÷ 0.31 = 9.03 ≈ 9 boxes Height - wise: The number of boxes that can be packed along the HEIGHT = Height of trailer ÷ height of the box = 2.4 ÷ 0.2 = 12 boxes TOTAL BOXES PACKED = number at height × number at width × number at height = 40 × 9 × 12 = 4 320 boxes Conclusion: Option 2 (More boxes can be packed, and it will be more compact) Page 20