Africa meets Africa Making a Living through the Mathematics of Zulu Design Lesson pack: 1 A note to the educator: This booklet contains a set of Arts and Culture and a set of Mathematics learning materials at Grade 9 level, which are linked conceptually. Your learners develop beadwork skills from the Zulu cultural heritage, and in doing so they work with mathematical ideas prescribed by the Revised National Curriculum Statements (RNCS). These activities, worksheets and research tasks for learners and the RNCS linked educator's guidelines included, serve as an example for developing further innovative materials. The book and film Africa meets Africa: Making a Living through the Mathematics of Zulu Design serves as primary resource material. Both the film and the book's full colour images have been designed to help you illustrate your lessons in class. Have a look at our website at www.africameetsafrica.co.za, and share your teaching ideas with educators from all over the country! Below: Ubeschwana (beaded aprons) from the Standard Bank African Art Collection (Wits Art Galleries) SBAAC Cover image: Isithwalo (headband) from SBAAC 1 When we are faced with new problems, do we try to solve each one on its own, or is there value in looking for solutions in what is already around us? Problems seldom exist in isolation. Art and Mathematics are both about problem solving. Cultural expression and mathematical problem solving help us to make sense of the world. Of course, understanding Mathematics well and making beautiful things seem to be very different kinds of skills. A well made piece of beadwork can speak to us in many different ways. It can speak of the past and it can relate to our lives, here and now. The patterns and colours in a young Nongoma girl's ubeschwana (apron) on the inside cover, or a Maphumulo bride's beaded headband, like the one on the cover of this book, tell us about the women who wore them, about where and how they lived. And they speak of the social values that helped them solve problems and make up the daily patterns of their lives. The beaded Zulu love letter (above right), understood only by the person it was made for, solved problems of the heart! Nowadays the beaded HIV/AIDS symbol gives new meaning to the traditional love letter, sold in shops all over South Africa and worn as a pin. People's needs and customs have generated many forms and patterns in the things they have made for everyday use. The shapes and colours interlinked in beaded and woven patterns can also help us understand Mathematical ideas. We can see some interesting properties of numbers demonstrated in beadwork, for example. This lesson pack will show you how to recognize them. Implementing the concept of Indigenous Knowledge Systems (IKS) One of the RNCS Critical Outcomes tells us that problem solving contexts seldom exist in isolation, because the world is made up of related systems. We live within a rich heritage of southern African knowledge systems. The Zulu cultural tradition is one of these. We see evidence of it in things around us every day; like the grass baskets we use or buy just for their beauty, or the beaded jewelry we wear, which comes from a long tradition of beaded adornment. Some carefully crafted objects reveal new solutions to problems of making - such as weaving with contemporary materials like wire and plastic, instead of grass. Can we learn how to recognise Mathematical ideas in the things we see in the world around us? And can we learn how to use what we see to help us understand more abstract Mathematical ideas? What are Indigenous Knowledge Systems exactly? Join our online debate on this question at www.africameetsafrica.co.za. 2 Part 1: Educator's guide: Arts and Culture: Zulu Beadwork Design Our methodology: The key activity in this lesson pack guides learners step by step as they make a Zulu love letter with a triangular design. The same design is used in a series of Mathematics lessons and worksheets in the second half of this lesson pack. When you show your learners how to do beadwork, try to bring some Zulu baskets, pots and beadwork to class and let them handle and enjoy the rich textures, colours and complex patterns in the objects. Introduce the idea that the things we use every day can tell us much about who we are. Of course, inviting a beadwork maker from your community to show her work, or someone who sells her wares in your city to your classroom, would be even better! LESSON PLAN AND OVERVIEW FOR ARTS AND CULTURE, GRADE 9. Critical and Developmental Learning Outcomes and Assessment Standards Critical and Developmental Outcomes: Learners will: CO 1 Identify and solve problems and make decisions using critical and creative thinking. CO 7 Demonstrate an understanding of the world as a set of related systems by recognising that problem solving contexts do not exist in isolation. DO 3 Be culturally and aesthetically sensitive across a range of social contexts DO 5 Develop entrepreneurial opportunities. Arts and Culture Learning Outcomes and Assessment Standards LO 1: Creating, interpreting and presenting Learners will: Be able to create, interpret and present artwork: AS: The learner creates art/craftwork that translates ideas into concepts. ? Demonstrates the confident use of elements of design. ? Develops entrepreneurial awareness of how to market art products in terms Integration of the art market and with an awareness of tourism. recommended with ? Additional: Exploration and representation of specific patterns and design Learning motifs which feature in South African history. Outcomes from LO 2: Reflecting. The learner will be able to reflect critically and creatively on artistic HSS (History) and cultural processes, products and styles in past and present contexts. EMS and AS The learner identifies the constituent parts of an integrated African art form. Technology Teaching and assessment contexts Focus of Research LO 4: Expressing and Communicating: .The learner will be able to use multiple forms of communication and expression in Arts and Culture AS The learner is able to explain how art reflects cultures, lifestyles, beliefs and fashion Learners are introduced to the cultural context of Zulu beadwork. (30 to 45 minutes) Learners make a piece of beadwork. Instruction sheet, materials provided. They complete a worksheet as guided research task (homework assessment task) The learner worksheet guides research in beadwork being produced locally, as well as investigating what marketing opportunities could be created in the community Forms of assessment Class discussion on the cultural context of Zulu beadwork, the names of pieces and their meaning - a group assessment by the educator. The finished beadwork pieces and the completed worksheets assessed individually by the educator. See assessment grid provided. Exhibitions of learner beadwork can be held at Teachers' Centres. Prizes will be awarded, sponsored by the Shuttleworth Foundation, for excellence in beadwork and also to educators for designing innovative learning materials. Recommended references 1.The resource books and films Africa meets Africa: the Power to Speak and Africa meets Africa: Making a living though the Mathematics of Zulu Design. 2. Museums with beadwork collections and commercial outlets listed in the back of the Africa meets Africa books and on the Africa meets Africa website 3. Speaking with Beads: Eleanor Preston-White and Jean Morris. 3 Lesson 1: Introducing the lessons This lesson relates to Learning Outcomes 2 and 4, as set out in the lesson plan and overview. There are several ways to introduce Zulu beadwork to your learners. Try to use as many of the following as you can to introduce your learners to how, as Learning Outcome 2 states, art reflects cultures, lifestyles, beliefs and fashion. o Invite a local member of your community if you live in a rural environment, or someone who makes and sells beadwork in your city, perhaps through the Durban African Arts Centre or the Victoria street market, to the classroom. Ask her to demonstrate her work and to talk about who first showed her how to make beadwork. o Show all of Part 1, or the section on beadwork in the Africa meets Africa film titled Things to Use that you received with this lesson pack. o Look at the range of beaded adornments illustrated on the back cover of this lesson pack and in the Africa meets Africa resource book with your learners, explaining where in KwaZulu Natal particular designs and colours are used (see page 5 to 13 of the book resource). o Allow enough time for learners to explore the actual and documented pieces of beadwork you bring to class and to share the knowledge and experience of beadwork they have. Spend at least 30 minutes or more on this introduction. You will be dealing with Learning Outcomes 2 and 4 and preparing learners for LO 1 - making a piece of beadwork. Based on discussion generated in your class, do a group assessment on the prior learning about beadwork represented in your class and write down the questions raised, so that these might be sought out in the research task. Lesson 2 in summary: The learners make their own piece of beadwork with a triangular design, as explained step by step on the next page. This lesson shows how to use the brick stitch and the netting stitch. The brick stitch is often used to make love letters, durable aprons and anklets, as its thorough threading technique reinforces the garment. The netting stitch, used to make the fringe of triangles at the bottom of the love letter, is more delicate and decorative. Make a copy of the next page for each learner to work from as you demonstrate the technique. Encourage the learners to finish their love letter within the time you set aside. An hour is suggested, depending on how familiar the learners are with beadwork. Some might need more or less time than this! 4 Learner activity sheet: Making a love letter in the brick stitch and the netting stitch Step 1: How to start To start the brick stitch, one threads beads in pairs of two on to a safety pin. Thread nylon through a very thin beading needle and make a double knot at the end. Now thread two white beads, and then pull the thread over one side of the pin (as shown above) and thread again, inserting your needle through the top end of the same two beads again and pulling down through to the bottom. Thread the next two beads and loop the thread around the pin and down through the two beads again. Continue like this until you have a double row of 14 white beads. Step 2: Now create your design, following this code: W = white beads, B = blue beads and R = red beads. Row 3 –Turn your pin around with the beads facing up. The thread will now be on the left. Start row 3, working from left to right. It is very important always to start with two beads only on the edge, and then to work with one bead at a time. Stitch as follows: Thread two beads for the edge, and then put the thread through the loop of the bead directly underneath the first bead, from the outer side towards you. Put the thread through the second bead again, making sure that it goes around the loop below, or else your bead will fall out. Pull the thread tight, creating a loop for the next row. Carry on with one bead at a time each time, going through the loop underneath and then again through the bead itself, until you reach the edge. Now keep threading the following sequence of coloured beads Row 3 - from left to right-7W,1B, 6W Row 4 - now from right to left-6W,2B, 6W Row 5 - left to right - 6W,1B, 1R,1B, 5W Row 6 - right to left - 5W,1B, 2R, 1B, 5W Row 7 - left to right - 5W,1B,1R,1B,1R,1B,4W Row 8 - right to left - 4W,1B,1R, 2 B, 1R,1B,4W Row 9 - left to right - 4W,1B,1R,1B, 1R,1B,1R,1B, 3W Row 10 - right to left - 3W,1B,1R,1B, 2R,1B,1R,1B, 3W Row 11 - left to right - 3W,1B,1R,1B,1R,1B,1R,1B,1R,1B,2W Row 12 - right to left - 2W,1B,1R,1B,1R,2B,1R,1B,1R,1B,2W Row 13 - left to right - 2W,1B,1R,1B,1R,1B,1R,1B,1R,1B,1R,1B,1W Row 14 - right to left - 1W,1B,1R,1B,1R,1B,2R,1B,1R,1B,1R,1B,1W Rows 15-16 - white beads further throughout Step 3: Making the bottom fringe in the netting stitch The fringe is made with the netting stitch, which leaves spaces between strings of beads. Thread 5 W, 1B, 1R, 1B, then thread the red and blue through W no.5 and pull tight to create a small triangle. Now thread 4W and put it through bead no. 5 on the edge to create a larger triangle The next one will stretch from no. 6-9 on the edge, and the last one from 10-15. Finish off with a knot. 5 Lesson 2: Educators guide: This activity relates to all the assessment standards mentioned in the overview for Learning Outcome 1. Introducing the activity: Refer to the book resource Africa meets Africa: Making a Living through the Mathematics of Zulu Design from page 6 to 13. Some background information follows for you to talk about while the learners are working, and to lead their thinking towards aspects of Mathematics (note the two phrases in bold). Use the assessment grid below to evaluate the individual love letters produced An introduction: Glass beads were first made in Egypt and Iran. They arrived in Africa with the Arab trade of the 9 th century and were used as a form of currency for trading African gold, ivory and slaves. So counting has always been important as far as beads are concerned, and in more ways than one! The brick stitch has been used in rural KwaZulu Natal at least since the 19th century, when European colonial traders brought glass beads to the royal courts of King Shaka and Dingaan. At that time the Zulu king still controlled the economy and all trade on his land, as the early colours and patterns worn by his subjects show. The distinctive beaded colours and patterns worn by people in his kraal displayed the king's wealth. When the gold mines of Johannesburg and the diamond fields of Kimberley needed a large labour force, the young women whose lovers left to become migrant workers on the mines would bead love letters for them to wear. Each beaded design carried a secret message which would be understood by the young man it was meant for. So beadwork was a coded visual language long before the Zulu language was written down! The Zulu love letter has remained popular, though how and what it communicates has changed - now it also helps break the silence caused by fear of HIV/AIDS and communicates knowledge and awareness. Extending the activity: Once the basic technique is mastered, encourage learners to make more love letters or any other beaded items such as toys, bracelets and pins based on their own designs. These could relate to issues that are meaningful to them, and social issues in their school community. Encourage the learners to think about marketing what they make. At a school near Durban for example, beads are sold cheaply by the spoon full, and once learners have made things with them, they sell their products to their educators and fellow learners at a profit! Keep the beadwork made and show the pieces to your subject advisor. The Africa meets Africa Project will choose the best made pieces and most original designs for an exhibition at the district Teachers' Centre, and prizes will be awarded, also to educators for the most innovative teaching ideas inspired by this project. Arts and Culture: Grade 9 Assessment sheet for beadwork activity Learner's name: School: District: Educator: Assessment Rubric: 1- Totally inadequate. 2- Acceptable. 3- Good with limitations. 4- Outstanding. 1 2 3 4 Overall mastering of technique Neatness of threading Good use of time Economic use of materials and presentation Cultural awareness displayed in participation in and contribution to class discussion Additional comments Note additional pieces made by learner Lesson 3: A research task: Educator's guide The following worksheet has been designed as a guided research task for your learners. It serves to make learners aware of the Zulu beadwork heritage, but also of cultural change and how closely this relates to problem solving in everyday life. One of the problems young adults are most aware of is the problem of making a living after school. The worksheet stimulates entrepreneurship and HIV/AIDS awareness. Introduce the worksheet by watching Part 3 of the film Africa meets Africa: Making a Living through the Mathematics of Zulu Design with the learners. Make copies of the worksheet. Researched answers might require more paper! 6 Bead there, done that: Zulu beadwork design then and now An Arts and Culture research task for Grade 9 learners Name: School: 1. Culture and its expression is dynamic. Who would have worn this love letter, the imitamatama (bandolier) above and the elibhantohi (waistcoat) and more or less when would they have worn them? Who would have made them? Research your answer in the Africa meets Africa resource book your teacher has in class Answer: ………………………………………………………………………….. ……………………………………………………………………………………… …………………………………………………………………………………….. 2. Is one of these more traditional than another? Explain your answer. …………………….…………………………...……………..…………………. ...……………………………………………………………..………………….. 3. What is the relationship between cultural expression and economics? First research your answer in the Africa meets Africa resource, then by talking to someone who makes and sells beadwork. In the city you could speak to beadwork makers at the Durban African Arts Centre, on the street or at Victoria street market near the station. Some questions to ask could be: ? Who taught you to make beadwork? How old were you then? ? Why do you sell your work here? ? What made you decide on this price? Materials, customers? ? Does anyone else sell your work? For what price? ? Do you ever change your product, its patterns and colours? Why? 7 4. Can a traditional form of adornment, a symbol or a technique of making take on a new use and a new meaning in the present? Give at least one example. ……………………………………………………………………………………………………………………. ……………………………………………………………………………………………………………………. ……………………………………………………………………………………………………………………. 5. Who gives an object meaning? The person who makes it or the one who uses it? Or both? Give reasons for your answer. …………………………………………………………………………………………………………………… …………………………………………………………………………………………………………………… 6. Beadwork was a coded language long before Zulu was written down. Research this statement and explain it. ……………………………………………………………………………. 7. Design a beaded product that would sell well in your community (at school or at home). Are there particular issues that are meaningful to your community, that you could somehow link your design to? Write down what you could make and how you would go about marketing it. Could beaded items be made and sold for special events at school? What would tourists want to buy? Could you involve your friends? In some schools beads are bought in large quantities and sold very cheaply by the large spoon full. Learners buy beads, make beaded love letters as pins and sell them to teachers and other learners. Lebolile Ximba made this crucifix.of a woman Photos on this page: Kate Wells .………………………………………………………………………………………. ............................................................................................................. ............................................................................................................. 8. Work out a profitable retail price for your product and explain how you arrived at it. Think of the cost of materials and the time it takes to make a product. Also think about what your market can afford. Could you ask for a deposit when an order comes in? Think about: What your product is. Is it something small, people want to buy often, or something people will buy less often, but at a higher price? How many do you need to make? How long will it take to make each one and what will the materials cost? What will it cost to hire helpers? My cost price is ………………….My retail price is………………My profit is…………. 8 Part 2: Educator's guide: Beadwork and Mathematics: Patterns, functions and algebra The following lessons link into the Mathematics section of Africa meets Africa: Making a Living through the Mathematics of Zulu Design. The book offers you support learning material on four areas of Mathematics: Numbers (including natural numbers, integers, irrational numbers and series), Polygons, Tilings (Tesselations) and Symmetry. Part two of the film offers you ten minutes of viewing that you can play to the learners in class on each one of the four sections, as Dr Chonat Getz and narrator Wandile Molebatsi look at examples of beadwork and weaving together. Dr Getz explores abstract mathematical ideas in a concrete way. You will find source material in the book and film for teaching Mathematics from Grade 7 to Grade 12 level LESSON PLAN AND OVERVIEW FOR MATHEMATICS, GRADE 9. Learning Outcomes and Assessment Standards. Related Learning Outcomes and Assessment Standards Teaching, Learning and Assessment Contexts LO2: PATTERNS, FUNCTIONS AND ALGEBRA The learner will be able to recognise, describe and represent patterns and relationships, as well as to solve problems using algebraic language and skills. We know this when the Grade 9 learner ?Investigates, in different ways, a variety of numeric and geometric patterns and relationships by representing and generalising them, and by explaining and justifying the rules that generate them (including patterns found in natural and cultural forms and patterns of the learner's own creation). ?Represents and uses relationships between variables in order to determine input and/or output values in a variety of ways using: ? Verbal descriptions ? Flow diagrams ? Tables ? Formulae and equations LO3: SPACE AND SHAPE ( to be explored further in Lesson Pack 2) The learner will be able to describe and represent characteristics and relationships between two-dimensional shapes and three-dimensional objects in a variety of orientations and positions. We know this when the Grade 9 learner ? Recognises, visualises and names geometric figures and solids in natural and cultural forms and geometric settings, including: ? Regular and irregular polygons and polyhedra ? Spheres ? Cylinders The material consists of five worksheets which link patterns to beadwork. Each worksheet focuses on a different beadwork design, with corresponding differences in the general formulae. The focus of each worksheet is as follows: WORKSHEET 1 Growing Triangles WORKSHEET 2 AIDS Ribbons WORKSHEET 3 Diamond Shapes WORKSHEET 4 Triangles WORKSHEET 5 Square and Triangular Numbers Answers to each worksheet are given at the end of the lesson pack. 9 Focus of the Worksheets ? The focus of LO2 in the Senior Phase is to formalise the rules generating patterns. The learners need to investigate numerical and geometric patterns in order to establish the relationships between variables; and also to express rules governing patterns in algebraic language or symbols. The five worksheets provide material with which to achieve these aims. ? The worksheets also provide opportunities for the learners to represent and describe relationships either in words, using flow diagrams, or using equations (or general formulae). Learners discover that all of these are equivalent. ? Note that there is a sheet of blank bead grids on the last page of this booklet. This sheet can be photocopied and handed out to the learners. Structure of the Lessons INTRODUCTION ? Spend about 5 minutes introducing the subject to the learners. If possible, bring pieces of beadwork to class. In the Arts and Culture section of these materials learners see images of bead work and they recognize different patterns and styles of beadwork from different areas. Now show the learners the first five minutes of Part 2 of the film. BODY OF THE LESSON ? Encourage the learners to work together in their groups on each activity. They should discuss the tasks with all the members of their group and reach consensus on the answers. ? They could either write their answers on the worksheets or in their Mathematics books. CONCLUSION ? It is important for you to bring the class to a conclusion at the end of the lesson. ?Ask two or three groups to report back on their answers. Encourage the rest of the class to discuss the answers. (After each lesson, ask different groups to report back.) ? Encourage the learners to discover the equivalence between the different ways of representing relationships ? Learners work together on the tasks with the members of their group. For this reason peer assessment is used. ?Groups report back on their findings to the rest of the class. The class then discusses each group's contributions. ? Where learners do not finish tasks in class, they should be asked to complete them for homework. Forms of Assessment References ? Africa meets Africa Making a Living through the Mathematics of Zulu design ISBN 0-620-32178-4 10 WORKSHEET1 Triangular patterns are often found in beadwork, baskets and pottery. 1) Study the beaded love letter above. Use two different colours to copy the pattern onto the circles on the right. 2) How many red beads are there in Pattern 1? 3) How many black beads are there in Pattern 2? Pattern 1 Pattern 2 4) Use your drawing to work out the number of red beads there would be in Pattern 3. Write the total on the table below. 5) How many black beads would there be in Pattern 4? Fill in the total on the table. 6) Use your table to work out how many beads there would be in Pattern 5 and Pattern 6. Check if you are right by drawing each pattern. Pattern Number Number of new beads 1 3 2 7 3 4 5 6 11 7) The learners in a Grade 9 class were asked to write down what they noticed about the patterns, and this is what they said: A: MAKHOSI: This is what I noticed about the pattern: Pattern 1: Top bead + 1 bead on the left + 1 bead on the right Pattern 2: Top bead + 3 beads on the left + 3 beads on the right Pattern 3: Top bead + 5 beads on the left + 5 beads on the right Pattern 4: Top bead + 7 beads on the left + 7 beads on the right B: MARY: This is what I noticed about the pattern: Pattern 1: 2 beads on the left + 1 bead on the right Pattern 2: 4 beads on the left + 3 beads on the right Pattern 3: 6 beads on the left + 5 beads on the right Pattern 4: 8 beads on the left + 7 beads on the right C: TSHIDI I worked out that the nibmer of beads in each shape was equla to 4times the pattern number minus 1. I showed this using a flow diagram: 1 3 4 times the Pattern Number minus 1 2 3 7 11 4 15 INPUT OUTPUT D: KHWEZI I used algebra and found the general formula. I used N for the Number of Beads and P for the Pattern Number. I could then say: N=4P-1 Work with a group of 4. Each member of your group should use one of the above methods to work out the number of beads in: a) Pattern 7 b) Pattern 8 c) Pattern 9 d) Pattern n 8) Compare your methods. Which method is quickest to use? Give a reason for your answer. 12 13 3) Use your table to work out the number of beads needed to make Pattern 7 and Pattern 8. Check if you are right by drawing the patterns. Pattern Number 1 2 3 4 5 6 7 8 4) Number of beads Andile looked at the drawings of the different sized AIDS ribbons and discovered the following: Pattern 1: Number of beads = 5 x 2 + 1 = 11 Pattern 2: Number of beads = 5 x 3 + 1 = 16 Pattern 3: Number of beads = 5 x 4 + 1 = 21 a) Investigate whether Andile's formula holds for Patterns 4, 5, 6, 7, and 8 b) Use Andile's formula to find the number of beads in i) Pattern 9 ii) Pattern 10 iii) Pattern 20 iv) Pattern x c) Write a general formula for finding the number of beads as an equation using N for Number of beads and P for the Pattern Number. 14 WORKSHEET 3 Some bead patterns use the rhombus (or diamond shape) as a basis for the design. Notice that all four sides of a rhombus are equal in length, and that opposite sides are parallel to one another Circles Diagram 1) Use two colours to complete the pattern on the Circles Diagram. 2) Count the number of beads added on each time and fill in the answers on the table. Pattern Number 1 2 3 4 5 6 3) Number of New Beads 4 12 The learners in a Grade 9 class were asked to look at the Circles Diagram and to write down any patterns they noticed. Mpume wrote Pattern 1: Pattern 2: Pattern 3: Pattern 4: Number of beads = top bead + bottom bead + 1 bead to the left + 1 bead to the right =1+1+1+1=4 Number of beads = top bead + bottom bead + 5 beads to the left + 5 beads to the right = 1 + 1 + 5 + 5 = 12 Number of beads = top bead + bottom bead + 9 beads to the left + 9 beads to the right = 1 + 1 + 9 + 9 = 20 Number of beads = top bead + bottom bead + 13 beads to the left + 13 beads to the right = 1 + 1 + 13 + 13 = 28 Use Mpume's pattern to work out the number of beads in a) Pattern 5 b) Pattern 6 c) Pattern 7 d) Pattern 8 15 4) This is what Reuben wrote about the patterns he noticed: Pattern 1: Number of beads = 1 bead at each corner + 0 beads on each of the four sides = 4 + (4 x 0) = 4 Pattern 2: Number of beads = 1 bead at each corner + 2 beads on each of the four sides = 4 + (4 x 2) = 12 Pattern 3: Number of beads = 1 bead at each corner + 4 beads on each of the four sides = 4 + (4 x 4) = 20 Pattern 4: Number of beads = 1 bead at each corner + 6 beads on each of the four sides = 4 + (4 x 6) = 28 Use Reuben's pattern to work out the number of beads in: a) Pattern 5 b) Pattern 6 c) Pattern 7 d) Pattern 8 5) Compare your answers for each Pattern in 2), 3) and 4). If you don't get the same number of beads for each Pattern, investigate where you went wrong. 6) Use Mpume's pattern or Reuben's pattern or any ideas of your own to write down a general formula for finding the Number of Beads in each pattern. Use N for Number of Beads and P for the Pattern Number. 7) Draw a flow diagram to illustrate the formula using 1, 2, 3 and 4 as Input values and the corresponding number of beads as Output values. 16 WORKSHEET 4 Nomsa is making bead designs and builds patterns like this: Pattern 1 1) Pattern 2 Pattern 3 Pattern 4 Pattern 5 Use beads (or counters, stones, mealies, coins or small pieces of paper if you don't have beads) to make the same five patterns as Nomsa. Count the number of beads that you use for Pattern 1, Pattern 2, Pattern 3, Pattern 4 and Pattern 5, and fill in the number of beads you used on the table. 2) Study the table. Use your answers to work out the number of beads needed to make Patterns 6, 7, 8, 9 and 10. Fill in the numbers on the table. Pattern Number 1 2 Number of beads 1 4 3 4 5 6 7 8 9 10 3) Write down a formula that can be used to work out the Number of beads (N) in any Pattern Number (P) 4) Draw a flow diagram to show your general formula from question 3. Use 1, 2, 3 and 4 as input values. Calculate the output values and fill them in. 5) Use your formula to work out the number of beads needed for: a) Pattern 50………………………………………………………………………………. b) Pattern 100………………………………………………………………………………. c) Pattern 2 500………………………………………………………………………………. d) Pattern n………………………………………………………………………………. 17 WORKSHEET 5 This headband or Isithwalo, from the Standard Bank African Art Collection (SBAAC), Wits Art Galleries, was probably made in the mid 19th Century, that is, around 1850. It was made with glass beads and is 23,5cm long and 3,3 cm wide. Show your class the first five minutes of Part 2 of the film. A lot of counting is done in beadwork. If we count the number of beads in one triangle, we see that the number of beads = 1 + 2 + 3 + 4 = 10. 10 is called a triangular number because you can arrange 10 beads in the shape of a triangle. 17 is NOT a triangular number because you cannot arrange 17 beads in the shape of a triangle. Other triangular numbers are 3; 6; 10; 15; … NOTE: When you write a list of numbers like 3; 6; 10; 15; … we call the list a sequence of numbers. Each number in the sequence is called a term. The three dots at the end of the sequence indicate that the sequence of numbers continues. 1) a) Complete the following table: Method 1 Method 2 1st triangular number 1 1 2nd triangular number 1+2=3 1+2=3 3rd triangular number 1+2+3=6 3+3=6 4th triangular number 1 + 2 + 3 + 4 = .... 6 + 4 = 10 5th triangular number 1+2+3+4+5= 10 + 5 = 6th triangular number 7th triangular number 8th triangular number 9th triangular number 10th triangular number b) Write down the first eight triangular numbers ………………………………………….. 18 Look at one of the square patterns on this front apron, also from the SBAAC collection Of course it is not exactly a square as the beads used to weave it are cylindrical rather than spherical in shape. The red beads on the bottom left form a triangle, and the green beads on the top right form a triangle. Counting the number of beads in the whole square, we see that we have 10 x 10 = 100 altogether. If we add together the beads on the top right, we get the triangular number 1 + 2 + 3 + … + 9 = 45 19 If we add together the beads on the bottom left, we get the triangular number 1 + 2 + 3 + … + 10 = 55 And 45 + 55 = 100. So the sum of the two triangular numbers is a square number. 2) Let's call the first triangular number ? 1 , the second triangular number ? 2 , the third triangular number ? 3, and so on. Complete the following table: ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1=1 2=3 2=3 3=6 3=6 4 = 10 4= 5= 5= 6= 6= 7= 7= 8= ? 1+? 2=1+3=4 4=2x2 ? 2+? 3=3+6=9 9=3x3 ? 3 + ? 4 = 6 + 10 = 16 ? 4+? 5= ? 5+? 6= ? 6+? 7= ? 7+? 8= 3) What type of number do you get when you add together the values of the two triangular numbers? 4) Without calculating the triangular numbers, write down the answers to the following: a) ? 8 + ? 9 …………………………………………………………………………………….. b) ? 9 + ? 10 …………………………………………………………………………………… 5) Write down a formula for ? n + ? (n + 1), where n is any natural number. 20 ANSWERS TO WORKSHEET 1 1) 2) 3) 4) 5) 6) Learners colour in the circles using two different colours Pattern 1: 3 beads Pattern 2: 7 beads Pattern 3: 11 beads Pattern 4: 15 beads Pattern Number Number of New Beads 1 2 3 4 5 6 3 7 11 15 19 23 NOTE TO THE EDUCATOR: 7=3+4 11 = 7 + 4 15 = 11 + 4 In other words, each time 4 is added on to get the new number. 7) The solutions for each method is as follows A: MAKHOSI Pattern 1: Top bead + 1 bead on the left + 1 bead on the right = 1 + (2 x 1 - 1) + (2 x 1 - 1) = 1 + 1 + 1 = 3 Pattern 2: Top bead + 3 beads on the left + 3 beads on the right = 1 + (2 x 2 - 1) + (2 x 2 - 1) = 1 + 3 + 3 = 7 Pattern 3: Top bead + 5 beads on the left + 5 beads on the right = 1 + (2 x 3 - 1) + (2 x 3 - 1) = 1 + 5 + 5 = 11 Pattern 4: Top bead + 7 beads on the left + 7 beads on the right = 1 + (2 x 4 - 1) + (2 x 4 - 1) = 1 + 7 + 7 = 15 Pattern 5: Top bead + 9 beads on the left + 9 beads on the right = 1 + (2 x 5 - 1) + (2 x 5 - 1) = 1 + 9 + 9 = 19 Pattern 6: Top bead + 11 beads on the left + 11 beads on the right = 1 + (2 x 6 - 1) + (2 x 6 - 1) = 1 + 11 + 11 = 23 a) Pattern 7: Top bead + 13 beads on the left + 13 beads on the right = 1 + (2 x 7 - 1) + (2 x 7 -1) = 1 + 13 + 13 = 27 b) Pattern 8: Top bead + 15 beads on the left + 15 beads on the right = 1 + (2 x 8 - 1) + (2 x 8 - 1) = 1 + 15 + 15 = 31 c) Pattern 9: Top bead + 17 beads on the left + 17 beads on the right = 1 + (2 x 9 - 1) + (2 x 9 - 1) = 1 + 17 + 17 = 35 d) Pattern n: 1 + (2 x n - 1) + (2 x n - 1) = 1 + 2n - 1 + 2n - 1 = 4n - 1 B: MARY Pattern 1: Pattern 2: Pattern 3: Pattern 4: Pattern 5: Pattern 6: a) Pattern 7: b) Pattern 8: c) Pattern 9: d) Pattern n: 2 beads on the left + 1 bead on the right = (2 x 1) + (2 x 1 - 1) = 2 + 1 = 3 4 beads on the left + 3 beads on the right = (2 x 2) + (2 x 2 - 1) = 4 + 3 = 7 6 beads on the left + 5 beads on the right = (2 x 3) + (2 x 3 - 1) = 6 + 5 = 11 8 beads on the left + 7 beads on the right = (2 x 4) + (2 x 4 - 1) = 8 + 7 = 15 10 beads on the left + 9 beads on the right = (2 x 5) + (2 x 5 - 1) = 10 + 9 = 19 12 beads on the left + 11 beads on the right = (2 x 6) + (2 x 6 -1) = 12 + 11 = 23 14 beads on the left + 13 beads on the right = (2 x 7) + (2 x 7 - 1) = 14 + 13 = 27 16 beads on the left + 15 beads on the right = (2 x 8) + (2 x 8 -1) = 16 + 15 = 31 18 beads on the left + 17 beads on the right = (2 x 9) + (2 x 9 - 1) = 18 + 17 = 35 2n + 2n - 1 = 4n - 1 C: TSHIDI 8 9 n u 4 times the u Pattern Number u minus 1 t 7 27 t 31 t 35 u t 4n - 1 INPUT OUTPUT 21 D: KHWEZI Pattern 7: Pattern 8: Pattern 9: Pattern n: 8) N = 4P - 1 = 4 x 7 - 1 = 28 -1 = 27 N = 4P - 1 = 4 x 8 - 1 = 32 - 1 = 31 N = 4P - 1 = 4 x 9 - 1 = 36 - 1 = 35 N = 4P - 1 = 4n - 1 Accept any answer as long as the reason is given for the answer. Generally, it is quicker to find the answer using a formula in other words using Khwezi's method. ANSWERS TO WORKSHEET 2 1) Learners colour in the circles. 2) See the table below. 3) Pattern number Number of beads 11 1 16 2 21 3 26 4 31 5 36 6 41 7 46 8 NOTE TO THE EDUCATOR: 16 = 11 + 5 21 = 15 + 5 26 = 21 + 5 In other words, each time 5 is added on to get the new number. 4) a) Pattern 1: Pattern 2: Pattern 3: Pattern 4: Pattern 5: Pattern 6: Pattern 7: Pattern 8: Number of beads = 5 x 2 + 1 = 11 Number of beads = 5 x 3 + 1 = 16 Number of beads = 5 x 4 + 1 = 21 Number of beads = 5 x 5 + 1 = 26 Number of beads = 5 x 6 + 1 = 31 Number of beads = 5 x 7 + 1 = 36 Number of beads = 5 x 8 + 1 = 41 Number of beads = 5 x 9 + 1 = 46 b) i)Pattern 9: Number of beads = 5 x 10 + 1 = 51 ii)Pattern 10: Number of beads = 5 x 11 + 1 = 56 iii)Pattern 20: Number of beads = 5 x 21 + 1 = 105 + 1 = 106 iv)Pattern x: Number of beads = 5 x (x + 1) + 1 = 5x + 5 + 1 = 5x + 6 c) N=5x(P+1)+1 N=5P+5+1 N=5P+6 ANSWERS TO WORKSHEET 3 1) The learners use two colours to complete the pattern on the Circles Diagram. 2) Pattern number Number of new beads 1 2 3 4 5 4 12 20 28 36 6 44 22 3) Mpume’s pattern: Pattern 1: Pattern 2: Pattern 3: Pattern 4: Pattern 5: Pattern 6: Pattern 7: Pattern 8: No of beads = top bead + bottom bead + 1 bead to the left + 1 bead to the right = 1 + 1 + 1 + 1 = 4 No of beads = top bead + bottom bead + 5 beads to the left + 5 beads to the right = 1 + 1 + 5 + 5 = 12 No of beads = top bead + bottom bead + 9 beads to the left + 9 beads to the right = 1 + 1 + 9 + 9 = 20 No of beads = top bead + bottom bead + 13 beads to the left + 13 beads to the right = 1 + 1 + 13 + 13 = 28 No of beads = top bead + bottom bead + 17 beads to the left + 17 beads to the right = 1 + 1 + 17 + 17 = 36 No of beads = top bead + bottom bead + 21 beads to the left + 21 beads to the right = 1 + 1 + 21 + 21 = 44 No of beads = top bead + bottom bead + 25 beads to the left + 25 beads to the right = 1 + 1 + 25 + 25 = 52 No of beads = top bead + bottom bead + 29 beads to the left + 29 beads to the right = 1 + 1 + 29 + 29 = 60 4) Reuben’s pattern: Pattern 1: Pattern 2: Pattern 3: Pattern 4: Pattern 5: Pattern 6: Pattern 7: Pattern 8: Number of beads = 1 bead at each corner + 0 beads on each of the four sides = 4 + (4 x 0) = 4 Number of beads = 1 bead at each corner + 2 beads on each of the four sides = 4 + (4 x 2) = 12 Number of beads = 1 bead at each corner + 4 beads on each of the four sides = 4 + (4 x 4) = 20 Number of beads = 1 bead at each corner + 6 beads on each of the four sides = 4 + (4 x 6) = 28 Number of beads = 1 bead at each corner + 8 beads on each of the four sides = 4 + (4 x 8) = 36 Number of beads = 1 bead at each corner + 10 beads on each of the four sides = 4 + (4 x 10) = 44 Number of beads = 1 bead at each corner + 12 beads on each of the four sides = 4 + (4 x 12) = 52 Number of beads = 1 bead at each corner + 14 beads on each of the four sides = 4 + (4 x 14) = 60 5) Learners compare their answers for each Pattern in 2), 3) and 4). If they don't get the same number of beads for each Pattern, they should investigate where they went wrong. 6) Using Mpume's Pattern: Using Reuben's Pattern N = 1 + 1+ (4P - 3) + (4P - 3) N = 4 + [4 x (2P - 2)] N = 2 + 4P - 3 + 4P - 3 N = 4 + 8P - 8 N = 8P - 4 N = 8P - 4 7) 1 u 1 u 2 u u 4 8P - 4 3 u 4 u 9 u 16 u INPUT ANSWERS TO WORKSHEET 4 1) The learners make the five patterns. 2) Pattern number 1 2 Number of beads 3) N = P x P = P 4) 1 4 OUTPUT 3 4 5 6 7 8 9 10 9 16 25 36 49 64 81 100 2 1 u 2 u 3 u 4 u P 1 u 2 9 u 16 u INPUT OUTPUT 5) a) b) c) d) u 4 Pattern 50: Number of beads = 502 = 2 500 Pattern 100: Number of beads = 1002 = 10 000 Pattern 2 500: Number of beads = 2 5002 = 6 250 000 Pattern n: Number of beads = n2 23 ANSWERS TO WORKSHEET 5 1) a) Method 1 1st triangular number Method 2 1 1 2nd triangular number 1 + 2 = 3 1+2=3 3rd triangular number 1+2+3=6 3+3=6 4th triangular number 1 + 2 + 3 + 4 = 10 6 + 4 = 10 5th triangular number 1 + 2 + 3 + 4 + 5 =15 10 + 5 = 15 6th triangular number 1 + 2 + 3 + 4 + 5 + 6 = 21 15 + 6 = 21 7th triangular number 1 + 2 + 3 + 4 + 5 + 6 + 7 =28 21 + 7 = 28 8th triangular number 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 =36 28 + 8 = 36 9th triangular number 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 36 + 9 = 45 10th triangular number 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10=55 45 +10 = 55 b) 2) The first eight triangular numbers are 1; 3; 6; 10; 15; 21; 28; and 36 ? 1=1 ? 2=3 ? 1+? 2=1+3=4 4=2x2 ? 2=3 ? 3=6 ? 2+? 3=3+6=9 9=3x3 ? 3=6 ? 4 = 10 ? 3 + ? 4 = 6 + 10 = 16 16 = 4 x 4 ? 4 = 10 ? 5 = 15 ? 4 + ? 5 = 10 + 15 = 25 25 = 5 x 5 ? 5 = 15 ? 6 = 21 ? 5 + ? 6 =15 + 21 = 36 36 = 6 x 6 ? 6 = 21 ? 7 = 28 ? 6 + ? 7 = 21 + 28 = 49 49 = 7 x 7 ? 7 = 28 ? 8 = 36 ? 7 + ? 8 = 28 + 36 = 64 64 = 8 x 8 3) When you add together two consecutive triangular numbers, you end up with a square number. 4) a) B) 5) ? 8 +? 9 = 8th triangular number + 9th triangular number = 92 = 81 ? 9 + ? 10 = 9th triangular number + 10th triangular number = 102 = 100 ? n + ? (n+1) = nth triangular number + (n + 1)th triangular number = (n + 1)2 24 25 Back cover: Top row, left to right: Crucifix by Lebolile Ximba of the Siyazama Project, Elibhantohi(waistcoat) made in Ulundi,1950's, Love letter with HIV/AIDS symbol , Siyazama Project Middle: Bridal cape (Isikoti) Estcourt area, Beaded jewelry entrepreneur Alexia Mkhize wearing one of her beaded necklaces Bottom middle and right : Legpieces (zingusha) from Msinga, approximately 1910, Girdle (sigege) and neck piece (Iqabane) from Eshowe The Africa meets Africa Project Registration number 2004/0076692/08 Public Benefit Organisation (NPO 037-722-NPO) This lesson pack was developed and published in 2006 by the Africa meets Africa Project. to accompany the educator's resource pack . Africa meets Africa: Making a Living though the Mathematics of Zulu Design ISBN 0-620-32178-4. Arts and Culture learning material written by Helene Smuts. Beadwork activity designed by Jannie van Heerden Mathematics learning material written by Jackie Scheiber. Edited by David Andrew, Ruth Sack, Annebel Klopper, Chonat Getz. Thanks to Jannie van Heerden, Kate Wells and the Standard Bank African Art Collection (Wits Art Galleries) SBAAC, for photographic images used. © Copyright: Africa meets Africa: Making a Living though the Mathematics of Zulu Design Lesson pack 1 and 2: Helene Smuts Arts Education Consultants cc and the Shuttleworth Foundation Email: [email protected] Tel/fax +27 (011) 622 7871. Postal address: 32 Nottingham Rd. Kensington, Johannesburg, South Africa 2094 Available on www.africameetsafrica.co.za All rights reserved Africa meets Africa: Making a Living though the Mathematics of Zulu Design Lesson Packs 1 and 2 and the 2006/2007 implementation programme in partnership with the KwaZulu Natal Department of Education was generously sponsored by the Shuttleworth Foundation 26

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