Teaching & Learning Plans Introduction to Patterns Junior Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes to achieve. Prior Knowledge points to relevant knowledge students may already have and also to knowledge which may be necessary in order to support them in accessing this new topic. Learning Outcomes outline what a student will be able to do, know and understand having completed the topic. Relationship to Syllabus refers to the relevant section of either the Junior and/or Leaving Certificate Syllabus. Resources Required lists the resources which will be needed in the teaching and learning of a particular topic. Introducing the topic (in some plans only) outlines an approach to introducing the topic. Lesson Interaction is set out under four sub-headings: i. Student Learning Tasks – Teacher Input: This section focuses on teacher input and gives details of the key student tasks and teacher questions which move the lesson forward. ii. Student Activities – Possible and Expected Responses: Gives details of possible student reactions and responses and possible misconceptions students may have. iii. Teacher’s Support and Actions: Gives details of teacher actions designed to support and scaffold student learning. iv. Assessing the Learning: Suggests questions a teacher might ask to evaluate whether the goals/learning outcomes are being/have been achieved. This evaluation will inform and direct the teaching and learning activities of the next class(es). Student Activities linked to the lesson(s) are provided at the end of each plan. Teaching & Learning Plan: Introduction to Patterns Aims • Recognise a repeating pattern • Represent patterns with tables, diagrams and graphs • Generate arithmetic expressions from repeating patterns Prior Knowledge Students should have some prior knowledge of drawing basic linear graphs on the Cartesian plane. Learning Outcomes As a result of studying this topic, students will be able to: • use tables, graphs, diagrams and manipulatives to represent a repeatingpattern situation • generalise and explain patterns and relationships in words and numbers • write arithmetic expressions for particular terms in a sequence • use tables, diagrams and graphs as tools for representing and analysing patterns and relations • develop and use their own generalising strategies and ideas and consider those of others • present and interpret solutions, explaining and justifying methods, inferences and reasoning Catering for Learner Diversity In class, the needs of all students whatever their level of ability are equally important. In daily classroom teaching, teachers can cater for different abilities by providing students with different activities and assignments graded according to levels of difficulty so that students can work on exercises that match their progress in learning. For less able students, activities may only engage them in a relatively straightforward way and more able students can engage in more open–ended and challenging activities. This will cultivate and sustain their interest in learning. In this T & L Plan for example teachers can provide students with the same activities but with variations on the theme e.g. draw a picture, put it in words, write a multiplication sentence, apply the algorithm. Teachers can give students various amounts and different styles of support during the class for example, providing more clues. In interacting with the whole class, teachers can make adjustments to suit the needs of students. For example, they can ask less able students simple questions (e.g. Student Activity 1A and 1B) and reserve more challenging questions for the more able students (e.g. Student Activity 4A and 4B). © Project Maths Development Team 2011 www.projectmaths.ie 1 Teaching & Learning Plan: Introduction to Patterns Besides whole-class teaching, teachers can consider different grouping strategies to cater for the needs of students and encourage peer interaction. Students are also encouraged in this T & L Plan to verbalise their mathematics openly and share their work in groups to build self-confidence and mathematical knowledge. Relationship to Junior Certificate Syllabus Topic Number Description of topic Students learn about Learning outcomes Students should be able to 4.1 Generating arithmetic expressions from repeating patterns Patterns and the rules that govern them and so construct an understanding of a relationship as that which involves a set of inputs, a set of outputs and a correspondence from each input to each output.ms involving fractional amounts. • use tables to represent a repeating-pattern situation 4.2 Representing situations with tables, diagrams and graphs. • generalise and explain patterns and relationships in words and numbers • write arithmetic expressions for particular terms in a sequence Relations derived from some • use tables, diagrams and graphs as tools for kind of context – familiar, representing and analysing everyday situations, imaginary contexts or arrangements of patterns and relations tiles or blocks. They look at • develop and use their own various patterns and make generalising strategies and predictions about what comes ideas and consider those next. of others • present and interpret solutions, explaining and justifying methods, inferences and reasoning Resources Required Unifix cubes (optional), coloured paper, graph paper Introducing the Topic Ask students to give you examples of patterns e.g. • 5, 10, 15, 20 • 50, 100, 150, 200 • Apple, orange, apple, orange • Sue, Bob, Tom, Sue, Bob, Tom, Sue © Project Maths Development Team 2011 www.projectmaths.ie 2 Teaching & Learning Plan: Introduction to Patterns Real Life Context The following examples could be used to explore real life contexts. • The pattern that traffic lights follow • The Fibonacci Sequence in nature, e.g. leaves on trees • The pattern of the tides in the oceans • The pattern an average school day follows © Project Maths Development Team 2011 www.projectmaths.ie 3 Teaching & Learning Plan: Introduction to Patterns Lesson Interaction Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses Teacher’s Support and Actions Assessing the Learning • 5,10, 15» »» Write student examples of patterns on the board. »» Are students able to provide examples of patterns? Teacher Reflections »» Today we are going to discuss patterns in mathematics; e.g. 2, 4, 6, 8 10, 20, 30, 40 »» Can anyone give me some examples of number patterns? • 3, 6, 9, 12,» • 2, 4, 6, 8, 10 »» Could the numbers 1, 5, 12, 13, 61 be a pattern? Give me a reason for your answer. • No, because you can’t predict the next number.» »» Can students give a reason why this sequence of numbers could not be a pattern? • There isn’t any logical sequence in the list of numbers. »» What do you think the properties or characteristics of a pattern could be? • You have to be able to predict what number comes next.» • If it’s a pattern then it will have a logical sequence.» »» Begin to make a list of the “properties” patterns usually have on the board. »» Are students able to list the properties which define a pattern? • Most patterns have some kind of rule that helps you figure out what the next number could be. »» Do you think patterns can only occur in numbers? What about colours? e.g. Traffic lights: red, amber, green, or letters e.g. A, B, C, A, B, C, A., could these be called patterns? © Project Maths Development Team 2011 • Once there is a logical sequence and you can predict what comes next, then yes, colours or letters or anything could be a pattern. www.projectmaths.ie »» Can students explain why patterns are not limited to “lists” of numbers? »» Can students explain why almost anything could be a pattern if it has a logical predictable sequence? KEY: » next step • student answer/response 4 Teaching & Learning Plan: Introduction to Patterns Student Learning Tasks: Student Activities: Possible Teacher’s Support and Actions Teacher Input and Expected Responses Assessing the Learning »» Can you give me an example of a pattern that doesn’t contain numbers? »» Are students able to give examples of patterns which don’t contain numbers? • Yellow, Green, Blue, Yellow, Green, Blue.» • Toffee, Chocolate, Toffee, Chocolate. »» Working in pairs, complete Student Activity 1A • Students report back to the rest of the class on Student Activity 1A. © Project Maths Development Team 2011 Teacher Reflections »» Can students represent the given pattern, using diagrams, or unifix cubes, »» Distribute resources which may be required, e.g. unifix cubes, coloured or a table?» markers.» »» Are students interacting and cooperating with »» Circulate around the room to each other to find the see what students are doing and solutions on Student provide help where necessary.» Activity 1A? »» Get a selection of students to report back on the various solutions »» Can students verbalise they found when completing their thinking when Student Activity 1A. reporting back to the class? »» Distribute Student Activity 1A. »» We are now going to look at an activity about patterns.» »» What do you think the key to the pattern in this activity was? »» Write these new examples of patterns without numbers on the board. • The red blocks were always odd numbers, and the black blocks were always even numbers. www.projectmaths.ie »» Were students able to identify and verbalise what the key to the pattern was? KEY: » next step • student answer/response 5 Teaching & Learning Plan: Introduction to Patterns Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Support and Actions and Expected Responses »» We are now going to look at a slightly more complicated pattern.» »» Distribute Student Activity 1B »» Walk around the room and listen to what students are saying. Offer help, guidance where it is required.» »» Working in pairs, complete Student Activity 1B. • Students report back to the rest of the class on Student Activity 1B. »» How does this represent a pattern?» »» Can students represent the pattern?» Teacher Reflections »» Are students interacting and cooperating with each other?» »» Get a selection of students »» Have students identified to report back on the various the key to this pattern solutions they found when and can they verbalise completing Student Activity 1B. their thinking? »» Write the following on the board: “Starting from today, I am going to save 5 euro every day, for the next 7 days” »» We are now going to look at a sentence which might describe a number pattern. Let me give you an example. »» Do you think that this sentence could represent a pattern?» Assessing the Learning • Yes» • Because you can predict the next number every time.» »» Can students work out the answer?» »» Do students know if the word sentence represents a pattern?» • It has a logical sequence» »» So, looking at this, how much will I have saved in 7 days? © Project Maths Development Team 2011 • 35 euro www.projectmaths.ie »» Can students give a reason why the word sentence is a pattern? KEY: » next step • student answer/response 6 Teaching & Learning Plan: Introduction to Patterns Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Support and and Expected Responses Actions »» We are now going to look at another type of pattern in Student Activity 2A. • Students report back to rest of class on Student Activity 2A. • Yes» »» Sometimes, we can understand a pattern more easily if we can draw • No a graph of it.» Assessing the Learning »» Distribute Student Activity 2A to students. »» Can students represent the pattern?» »» Walk around the room and listen to what students are saying. Offer help, guidance where it is required.» »» Are students interacting and cooperating with each other?» »» Get a selection of students to report back on the various solutions they found when completing Student Activity 2A. »» If students are confused on how to plot points on coordinate plane, a quick recap will be required. Teacher Reflections »» Have students identified the key to this pattern and can they verbalise their thinking?» »» Did students realise that the start amount of money could be included in the answer to question 7? »» Are all students familiar with how to plot points on the coordinate plane? »» Does everyone remember how to plot points on the coordinate plane? »» We are now going to look at Student Activity 2B. © Project Maths Development Team 2011 »» Distribute Student Activity 2B. www.projectmaths.ie KEY: » next step • student answer/response 7 Teaching & Learning Plan: Introduction to Patterns Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Support and and Expected Responses Actions Assessing the Learning • Students report back to rest of class on Student Activity 2B. »» Can students represent the pattern?» »» Walk around the room and listen to what students are saying. Offer help, guidance where it is required.» »» Get a selection of students to report back on the various solutions they found when completing Student Activity 2B. Teacher Reflections »» Are students interacting and cooperating with each other?» »» Have students identified the key to this pattern and can they verbalise their thinking?» »» Were students able to correctly plot the points on the coordinate plane? »» Do you think that drawing a graph to show the relationship between the number of days gone by and the amount of money in the money box has any advantages? • Yes, because you can see how it increases all the time. » »» What things did you notice about this graph? • It goes up» »» Can students explain the advantages of drawing a graph to represent a pattern? »» Write comments about the graph on the board. Introduce the phrase “linear” for a straight line graph. »» Could students describe the properties of the graph? • Yes, because you can extend the graph and find out the value for any day, without having to build a table • It’s a straight line» • It doesn’t start at zero © Project Maths Development Team 2011 »» Write the benefits of drawing a graph on the board. www.projectmaths.ie KEY: » next step • student answer/response 8 Teaching & Learning Plan: Introduction to Patterns Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Support and Actions and Expected Responses »» To ensure that everyone is clear on how to draw a graph of a pattern, can you all complete Student Activity 2C. »» Distribute Student Activity 2C • Students report back to rest of class on Student Activity 2C. Assessing the Learning »» Can the students complete the activity?» »» Walk around the room and listen to what students are saying. Offer help, guidance where it is required.» »» Are students able to choose a relevant scale for drawing a coordinate graph?» »» Get a selection of students to report back on the various solutions they found when completing Student Activity 2C. »» Have students identified the key to this pattern and can they verbalise their thinking? Teacher Reflections »» Can students represent both patterns on the same coordinate plane?» »» Were students able to correctly plot the points on the coordinate plane? »» Graphs can be a very useful way of comparing two things.» »» Distribute Student Activity 3A. »» For example, let’s just look at the question on Student Activity 3A • Probably Bill, because he »» Before we begin the starts with 30 texts and question, can you predict gets an extra 3 free every who might have the day. better deal for the most texts in a month? © Project Maths Development Team 2011 www.projectmaths.ie KEY: » next step • student answer/response 9 Teaching & Learning Plan: Introduction to Patterns Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» The next activity we • Students report back to rest are going to look at of class on Student Activity requires you to draw 3A. a graph of 2 different patterns on the same graph paper, and compare them. When we have done this, we will know for sure which person has the better deal from the mobile phone company »» What do you think was the benefit of representing the information graphically in Student Activity 3A? © Project Maths Development Team 2011 • It was much easier to figure out who had the better deal» Teacher’s Support and Actions Assessing the Learning »» Walk around the room and listen to what students are saying. Offer help, guidance where it is required.» »» Were students able to correctly complete the table?» »» Get a selection of students to report back on the various solutions they found when completing Student Activity 3A. »» Could students correctly answer the questions in the activity?» Teacher Reflections »» Were students able to represent both sets of data on the same graph?» »» Were students able to give a reason why Amy had the better deal for free texts each month? »» Were students able to give the benefits of drawing a graph for this question? • You can just extend your graph and see who gets the most texts, you don’t have to keep filling in the table. www.projectmaths.ie KEY: » next step • student answer/response 10 Teaching & Learning Plan: Introduction to Patterns Student Learning Tasks: Teacher Input Student Activities: Possible and Expected Responses »» Now we are going to • Students report back try Student Activity 3B. to the rest of class on This time you will have Student Activity 3B. to figure out how you are going to represent the information and find the answers to the questions asked. Teacher’s Support and Actions Assessing the Learning »» Distribute Student Activity 3B »» Were students able to construct a table to represent the data?» »» Walk around the room and listen to what students are saying. Offer help, guidance where it is required.» »» Were students able to choose an appropriate scale for the graph?» »» Get a selection of students to report back on the various solutions they found when completing Student Activity 3B. • They have to have a »» So, what have we logical sequence» learned about patterns so far? • They don’t always have to be numbers» »» Write student answers on the board. Teacher Reflections »» Could students draw both sets of data on the same axes?» Were students able to answer the questions based on the graph correctly?» »» Could students verbalise their mathematical thinking? »» Are students able to state what they have discovered about patterns to date? • You can represent them in lots of ways, e.g. tables, graphs, diagrams» • Sometimes a graph is a good way to represent a pattern, because you can see what will happen over a period of time. © Project Maths Development Team 2011 www.projectmaths.ie KEY: » next step • student answer/response 11 Teaching & Learning Plan: Introduction to Patterns Student Learning Tasks: Teacher Input Student Activities: Possible Teacher’s Support and Actions and Expected Responses Assessing the Learning HIGHER LEVEL MATERIAL: Student Activity 4 and Student Activity 5 contain optional activities for students taking higher level. »» Now we are going to look at a pattern containing some blocks. • Selection of students report back to rest of class on Student Activity 4A. »» Distribute Student Activity 4A »» Distribute materials which may be required for task e.g. coloured blocks, or coloured paper etc » »» Walk around the room and listen to what students are saying. Offer help, guidance where it is required.» Teacher Reflections »» Were students able to model the pattern using coloured blocks or paper?» »» Did students successfully create a table / diagram to represent the pattern?» »» Were students able to verbalise their mathematical thinking to you and their colleagues? »» Get a selection of students to report back on the various solutions they found when completing Student Activity 4A. »» May have to explain what the term “algebraic formula” means. »» The final step we are going to do on patterns is to take a look at how we might represent a pattern using an algebraic formula.» »» Does everyone understand what the term “algebraic formula” means? © Project Maths Development Team 2011 • Is it when we write the formula using x’s and y’s?» • Is it when we try to explain a pattern using algebra? www.projectmaths.ie KEY: » next step • student answer/response 12 Teaching & Learning Plan: Introduction to Patterns Student Learning Tasks: Student Activities: Possible Teacher’s Support and Actions Teacher Input and Expected Responses Assessing the Learning »» We are now going to look at Student Activity 4B »» Are students interacting and cooperating with each other to find the solutions on Student Activity 4B? »» Distribute Student Activity 4B »» Walk around the room and listen to what students are saying. Offer help, guidance where it is required. Teacher Reflections »» Can students accurately model the problem using the resources supplied »» Get a selection of students to report back on the various solutions they found when completing Student Activity 4B. »» Were students able to develop a formula which would describe the pattern?» »» Did students verify their equation for the pattern?» »» Are students able to verbalise their mathematical thinking and defend their answers? © Project Maths Development Team 2011 www.projectmaths.ie KEY: » next step • student answer/response 13 Teaching & Learning Plan: Introduction to Patterns Student Activity 1A 1.Represent this repeating pattern - red, black, red, black, red, black, – by building it with blocks or colouring it in on the number strip below: 1 2 3 4 5 6 7 8 9 10 2.Complete the following table based on your diagram above: Block position number on strip 1 2 3 4 5 6 7 8 9 10 Colour Red Black 3.Answer the following questions: 1. List the position numbers of the first 5 red blocks:_______________________________ 2. What do you notice about these numbers?______________________________________ 3. List the position numbers of the first 5 black blocks:_ ____________________________ 4. What do you notice about these numbers?______________________________________ 5. What colour is the 100th block?_ _______________________________________________ 6. What colour is the 101st block?_________________________________________________ 7. What position number on the strip has the 100th black block?____________________ 8. What position number on the strip has the 100th red block?______________________ 9. What colour will the 1000th block be?_ _________________________________________ Explain how you found your answer for question 8:_________________________________ ______________________________________________________________________________ © Project Maths Development Team 2011 www.projectmaths.ie 14 Teaching & Learning Plan: Introduction to Patterns Student Activity 1B 1.Represent this repeating pattern – yellow, black, green, yellow, black, green – by building it with blocks or colouring it in on a number strip or drawing a table or in any other suitable way. 1 2 3 4 5 6 7 8 9 10 1. List the numbers of the first 3 yellow blocks. Is there a pattern in these numbers? ______________________________________________________________________________ 2. List the numbers of the first 3 black blocks. Is there a pattern in these numbers? ______________________________________________________________________________ 3. List the numbers of the first 3 green blocks. Is there a pattern in these numbers? ______________________________________________________________________________ 4. What colour is the 6th block?___________________________________________________ 5. What colour is the 18th block?_ ________________________________________________ 6. What colour is the 25th block?_ ________________________________________________ 7. What colour is the 13th block?_ ________________________________________________ 8. What colour will the 100th block be in the sequence?____________________________ 9. What colour will the 500th block be in this sequence?____________________________ 10. Explain how you found your answers to questions 8 and 9, ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 11. What rule could you use to work out the position number of any of the (i) yellow blocks, (ii) black blocks, (iii) green blocks? ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ © Project Maths Development Team 2011 www.projectmaths.ie 15 Teaching & Learning Plan: Introduction to Patterns Student Activity 2A Money box problem John receives a gift of a money box with €4 in it for his birthday. This is his starting amount. John decides he will save €2 a day. Represent this pattern by drawing a table, or a diagram, or by building it with blocks. 1. How much money will John have in his money box on the following days? Day 5: ___________ Day: 10: ___________ Day 14: ___________ Day 25: ___________ 2. By looking at the pattern in this question, can you explain why the amount of money John has on day 10 is not twice the amount he has on day 5?______________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 3. How much money does John have in his money box on day 100?_________________ 4. Explain how you found your answer for the amount for day 100:_________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 5. How much money has John actually put in his money box after 10 days? Explain how you arrived at this amount.________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ 6. John wants to buy a new computer game. The game costs €39.99. What is the minimum number of days John will have to save so that he has enough money to buy the computer game?_ _____________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ © Project Maths Development Team 2011 www.projectmaths.ie 16 Teaching & Learning Plan: Introduction to Patterns Student Activity 2B Annie has a money box; she starts with €2 and adds €4 each day. Create a table showing the amount of money Annie has each day over a period of 10 days. 1. Using the axes below, draw a graph to show how much money Annie has saved over 6 days 32 30 28 26 24 Number of days 22 20 18 16 14 12 10 8 • 6 4 2 • 0 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10 Start Amount (Euros) 2. List two things that you notice about this graph a. ______________________________________________________________________________ b. ______________________________________________________________________________ 3. Could you extend the line on this graph to find out how much money Annie has in her money box on day 10? a. Amount on day 10 =___________________________________________________________ © Project Maths Development Team 2011 www.projectmaths.ie 17 Teaching & Learning Plan: Introduction to Patterns Student Activity 2C Owen has a money box; he starts with €1 and adds €3 each day. 1. Draw a table showing the amount of money Owen has each day. 2. Draw a graph to show the amount of money Owen has saved over 10 days. Hint: Think carefully about the following before you draw your graph: a._ Where will you put “Number of days” and “Amount of money” on your graph? b._What scale will you use for the amount of money? (Will you use 1, 2, 3,... or will you decide to use 5, 10, 15, 20..... or perhaps a different scale?) © Project Maths Development Team 2011 www.projectmaths.ie 18 Teaching & Learning Plan: Introduction to Patterns Student Activity 3A Using Graphs to represent information Amy and Bill are discussing phone network offers. Bill says that on his network he begins each month with 30 free texts and receives 3 additional free texts each night. Amy says that she gets no free texts at the beginning of the month but that she receives 5 free texts each night. To see how many texts each person has over a period of time, complete the tables below. Amy’s texts Time Start amount Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10 Bill’s texts Time Start amount Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10 Number of free texts 0 5 10 Number of free texts 30 33 36 1. Who has the most free texts after 10 days?______________________________________ Number of Texts 2. Using the graph paper provided, draw a graph showing the number of texts Amy has, and using the same axes draw a graph of the texts Bill has, for 10 days. 60 55 50 45 40 35 30 25 ■■ ■ Graph Key: Amy = Bill = ■ 20 15 10 5 •• 0 1 • • 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time (Days) 3. What do you notice about each graph?_ ________________________________________ ______________________________________________________________________________ 4. Extend the lines of each graph to “Day 20”. In your opinion, who do you think has the better deal on free texts Bill or Amy? Why?__________________________________ ______________________________________________________________________________ 5. Will Amy and Bill ever have the same number of texts on a particular day? If so, which day? If not, why?_ ______________________________________________________ ______________________________________________________________________________ © Project Maths Development Team 2011 www.projectmaths.ie 19 Teaching & Learning Plan: Introduction to Patterns Student Activity 3B Examine the situation below Liam begins the month with 20 free texts and receives 2 additional free texts each night. Jessie does not have any free texts at the beginning of the month but receives 3 free texts each night. 1. Draw a table showing the following information. a. The number of free texts Liam has after 10 days b. The number of free texts Jessie has after 10 days 2. Represent this information on a graph, (note: show Liam’s and Jessie’s number of free texts on the same graph) 3. Will Liam and Jessie ever have the same number of free texts on a certain day? If so, which day? If not, why not?_______________________________________________ ______________________________________________________________________________ 4. Who in your opinion has the better deal for free texts each month? Give a reason for your answer._ _____________________________________________________________ ______________________________________________________________________________ © Project Maths Development Team 2011 www.projectmaths.ie 20 Teaching & Learning Plan: Introduction to Patterns Student Activity 4A (Higher Level Material) Finding Formulae The figures below are made up of white and red square tiles. The white squares are in the middle row and have a border of red tiles around them. For 1 white tile, 8 red tiles are needed; for 2 white tiles, 10 red tiles are needed, and so on. Using the information above, create a table or diagram which will show how the number of red tiles increases as the number of white tiles increases. (Hint: Look at the way the number of red tiles change each time a white tile is added, can you see a pattern?) © Project Maths Development Team 2011 www.projectmaths.ie 21 Teaching & Learning Plan: Introduction to Patterns Student Activity 4B (Higher Level Material) Breaking down the pattern and developing your own formula Let’s look at how each shape is built. Complete the blank spaces below To make the next shape in the sequence, I have to add ____ white tile(s) to the middle and _____ extra red tiles to the outside So each time I make a new figure the number of white tiles increases by ______ and the number of red tiles increases by _______; complete the pattern in the table below. +1 Number of white tiles 1 Number of red tiles 8 2 10 © Project Maths Development Team 2011 www.projectmaths.ie +2 22 Teaching & Learning Plan: Introduction to Patterns Student Activity 4B (continued) Let the number of white tiles = n. We know from our first shape that the white tile is surrounded by 8 red tiles, and each time we add a white tile we must add two red tiles. +1 White If n is the number of white tiles and there are the same number of red tiles above and below it (i.e. a total of 2n) and two lots of 3 at either side (i.e. +6). Then the general formula (or expression) for the number of red tiles must be 2n + 6. (Remember we are calculating the total number of RED tiles, not the total number of tiles in the shape) Using the logic above, can you develop another formula or expression based on the information below; (let the number of white tiles = n) Hint: The number of red tiles above and below is 2 more than the number of white tiles and 2 extra at the sides. 1 How many red tiles will there be if there are 100 white tiles? (Check your answer using both equations, i.e. 2n + 6 AND the one you have found above.)____________ _ ____________________________________________________________________________ _ ____________________________________________________________________________ © Project Maths Development Team 2011 www.projectmaths.ie 23 Teaching & Learning Plan: Introduction to Patterns Student Activity 5 (Higher Level Material) Group Work: Graphing Functions Set 1. 1. I have €36 to buy a gift. The amount of money I have left depends on the price of the purchase. (x-axis, price of purchase; y-axis, amount of money I have left). 2. A straight line, 36cm long, is divided into 2 parts A and B. The length of part B depends on the length of part A. (x-axis, length in cm of part A; y-axis, length in cm of part B ). For each of the sets assigned to your “expert” group: • Find some data points that fit the given problem, e.g. (1, 35) or (10, 26) etc • Organise the data into a table. Use the quantity identified as the x-axis for the left column and the quantity identified as the y-axis for the right column. • Organise the data into a graph. Use the quantity identified as the x-axis for the horizontal axis and the quantity identified as the y-axis for the vertical axis. IMPORTANT: Each member of the group will need a copy of each graph to share with the next group. • Share and compare your graphs with other groups • Identify similarities (things that are the same) and differences among the various graphs. © Project Maths Development Team 2011 www.projectmaths.ie 24 Teaching & Learning Plan: Introduction to Patterns Student Activity 5 (continued) Set 2. 3. The amount of money collected for a particular show is dependent on the number of tickets sold. Tickets cost €8 each. (x-axis, number of tickets sold; y-axis, amount of money collected). 4. The number of empty seats in a cinema that seats 350 people depends on the number of seats that are sold (x-axis, number of seats that are sold; y-axis, number of seats that are empty). For each of the sets assigned to your “expert” group: • Find some data points that fit the given ticket problem, e.g. (1,8) or (2,16) etc • Find some data points that fit the cinema probelm, e.g. (0,350) or (20,330) etc • Organise the data into a table. Use the quantity identified as the x-axis for the left column and the quantity identified as the y-axis for the right column. • Organise the data into a graph. Use the quantity identified as the x-axis for the horizontal axis and the quantity identified as the y-axis for the vertical axis. IMPORTANT: Each member of the group will need a copy of each graph to share with the next group. • Share and compare your graphs with other groups • Identify similarities (things that are the same) and differences among the various graphs. Set 3. 5. The perimeter of a square depends on the length of a side (x-axis, length of side; y-axis, perimeter.) 6. If I fold a paper in half once I have 2 sections. If I fold it in half again I have 4 sections. What happens if I continue to fold the paper? (x-axis, number of folds; y-axis, number of sections ) 7. The area of a square depends on the length of a side. (x-axis, length of side; y-axis, area of the square). For each of the sets assigned to your “expert” group: • Find some data points that fit the given problem. • Organise the data into a table. Use the quantity identified as the x-axis for the left column and the quantity identified as the y-axis for the right column. • Organise the data into a graph. Use the quantity identified as the x-axis for the horizontal axis and the quantity identified as the y-axis for the vertical axis. IMPORTANT: Each member of the group will need a copy of each graph to share with the next group. • Share and compare your graphs with other groups • Identify similarities (things that are the same) and differences among the various graphs. © Project Maths Development Team 2011 www.projectmaths.ie 25

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