Naplan numeracy prep_bright_ideas


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Naplan numeracy prep_bright_ideas

  1. 1. 2010 NAPLAN Numeracy Preparation and Bright Ideas National Assessment Program Literacy and Numeracy 1
  2. 2. ContentsPreparing the students Key Actions 3 Interpret and unpack the language of the test questions 4 ESL support strategies 4 Strategies for answering the test questions 5 Plan of attack 5 Test instruction icons 5 Multiple choice questions 5 Short answer and calculator questions 6 Polya’s 4 step plan 7 Problem solving strategies 7 Finding out where and why students make mistakes 9 Newman’s error analysis 9 Critical questions 9Using the test and practice questions Classroom games and activities 10 Topic areas 12 Hand on learning activities 13Curriculum Ideas Work station activities 17 Number and Algebra 17 Chance and Data 18 Measurement 19 Space 20 Open-ended questions 21 Stem and leaf plots 22 Websites 23Appendix 1 – Chance Cards for the Probability Continuum 24For further information please contactEllen Herden Gay WestManager Assessment and Reporting Project Manager Numeracy AssessmentPhone: 8999 3784 Phone: 8999 3778Email: Email: 8999 4200 Fax: 8999 4200 2
  3. 3. Key ActionsListed below are recommended actions for teachers and administrators to enable a smooth andsuccessful implementation of the testing program.Before Testing Become familiar with the DET web site and place the URL on your staff and parent notice board: Download and use practice tests or other NAPLAN resources from the website. Familiarise students with the different types of NAPLAN questions and create similar questions for units of work being studied in your classroom. Encourage students to work independently on sets of practice questions. For younger students practise ‘All by myself time’, while for older students establish ‘Independent Work time’. Gradually increase the times to prepare students for test conditions. Identify students who may require special provisions or exemptions and discuss their needs with specialist staff and parents. Complete official documentation by end of Term 1. Identify students who may need separate supervision during the tests and/or may need to have the numeracy test questions read to them. Check that you have enough calculators (Years 7 and 9 only), scrap paper and erasers for the tests. Read through the 2010 NAPLAN Information for Schools package and the School administrators handbook. Attend the NAPLAN Information sessions then meet with other staff to check understanding and/or contact Assessment and Reporting with any queries about the NAPLAN testing process (see page 1 for contact details). Ask: ‘What is the best way to administer the test?’ ‘Do I need extra support from either my colleagues, school administration team or regional officers?’During Testing Administer the test in the students’ usual learning environment however charts and specific teaching posters that could be relevant to the tests, e.g. times-tables, calendars, shape or formula charts must be covered, removed or turned over. Numeracy questions can be read to students – however no reading of number digits, elaboration of terminology and/or concepts is allowed. If you are absolutely sure that the test is far too difficult and not accessible for some students ask them to complete the first couple of questions and leave the rest of the test, rather than just guessing and filling in random bubbles. This gives inaccurate data. Have some other test or activity ready for them to do. Walk around during the testing sessions to encourage students and to make sure that they are working effectively and have answered all questions without skipping pages. Record participation details on the Student Test Participation and Summary Report.After Testing No marking of the test is required, but having a quick look through the books to gain an overall impression of the students’ responses is advisable. Blank test booklets and stimulus materials can be kept and these will be useful for test review with students. Discussing the test and asking for student feedback will alert you to the curriculum areas that students found problematical. Digital copies of the tests are on the DET website. 3
  4. 4. Interpret and unpack the language of the test questionsIt is important that teachers explicitly teach the following strategies and skills to adequatelyprepare their students for the language of the NAPLAN tests. Teach the meaning of words or phrases that are critical for students to understand exactlywhat a question is asking. Create a variety of questions that use these words and phasesespecially for the topics you are working on in class. Make a class chart or glossary of these words, phrases and questions; add new entries asthey are ‘discovered’ in the NAPLAN test and practice questions or text book questions. The listthat follows contains some common examples NAPLAN GlossaryLabel, key or legend Refers to Groups ofMultiple choice Means List or tableIdentify Match, matches How many more …Information Circled How many are left?Single or short answer Represents How many are needed?Relates to Number pattern What is the rule?Which statement Time line Which number comes next?Stacked, stacks, fits Number line The next numberShortest Definition What number will make this numberDistance, route, path Review sentence true?Is closest to Nearest to Total area of whole . . .Round off or rounded Closest to Opposite or adjacentMore than Less than Which expression is equivalent toGreater than Value of Complete the tableLongest, highest Same value as About how much…Shared equally Dimensions of How many times larger…Divide …is between Which indicates or showsMost likely Overview Fill in the missing number . . .Least likely Difference Is removed from …Impossible Altogether Sequence or in what orderBest chance Calculate The sum ofLook atHint: Explanations or example questions could be added after each word or phrase to remindstudents of their meaningESL support strategiesESL learners have to master specialised mathematical language and may also have to learnnew ways of organising knowledge. Teachers can encourage ESL learners to keep personal glossaries adding new terminologyas it arises and explaining the new words, phrases and questions in their own language. Classroom charts can be used to display anything that will help the learners e.g. the steps inproblem solving, different meanings of the same word , alternative words that can be used tomean the same thing like take away, subtract, minus, less than, difference etc. As well as using charts, frameworks, picture cues and glossaries, teachers can providescaffolding for ESL learners by modelling appropriate mathematical language to • accompany actions and the use of materials which allow for a hands-on, manipulative approach to learning • talk about Western concepts in the mathematical activity • link to what learners already know 4
  5. 5. • prompt using guided questioning, probing, paraphrasing, clarifying • supply words, phrases and sentences that learners aren’t able to give at the point where they need to use mathematical language. Teachers need to create opportunities for learners to use and practise the language orallybefore being required to read and/or write it. ESL learners need sufficient time to becomefamiliar with the vocabulary and teachers should encourage and expect the learner to use therephrased language to restate what was said, or later to use it in a similar activity. (See Gledhill, Ruth & Morgan, Dale 2000, Risk taking: Giving ESL students an edge in ESLFundameNTals 2004)Strategies for answering the test questions Help the students to establish a ‘plan of attack’ to solve a problem, one that works for them and becomes a habit to be used for every question. • Read the problem carefully first and notice the instruction icons. • Underline/circle/highlight key words or phrases in the question. • Understand what information is required; use scrap paper for working out. • Chose and use the best problem solving strategy (or strategies) to find the answer. • Use specific techniques for answering different types of questions as shown below for ….multiple choice, short answer and calculator questions.Giving students the rights ‘tools’ to use for the job of answering questions and making themfamiliar with the questions types can assist with decreasing student stress and increasingstudent confidence during the test week. Test instruction icons In the Numeracy tests, students will encounter four types of instruction icons. It is important to alert them to these icons and discuss what they mean. multiple choice short answer calculator and non-calculator Multiple choice questions • Go through the ‘plan of attack’ above and solve the problem. • Eliminate the obvious incorrect response/s or distracters. • Refer or check back to the question. • Choose the best response (shade/fill in the bubble). Underline main Read question words/phrases. carefully. e.g. 6 groups of 5 pens is the same number of pens as 3 groups of 10 6 5 3 Strategy: Write two number sentences 6 x 5 = 30 Check back to see Eliminate 3 x _ = 30 that ‘10’ correctly the incorrect to solve the problem. answers the question. responses. 5
  6. 6. Short answer or missing number questions• Go through the ‘plan of attack’ and solve the problem.• Write the answer neatly and clearly in the box.• Check back to make sure that your answer makes sense. Read Underline main question words/phrases. Write an answer carefully. neatly and clearly in the answer box. e.g. $4 is shared equally among 5 girls. 80 cents How much does each girl get? Strategy: Draw a number line or guess Check back to see that and check to solve ’80 cents’ correctly the problem. answers the question.Calculator questions• Students should use calculators that they are familiar with.• Go through the ‘plan of attack’ and solve the problem by estimating.• Estimate an approximate answer, use the calculator to find the exact answer.• Always double check answers on the calculator (make it a habit).• Check back to make sure the answer makes sense. Read question Underline main carefully. words/phrases. e.g. Zoe bought a bike on sale at 15% off the original price. The original price was $420. How much did Zoe pay for the bike? $63 $357 $378 $405 Strategy: Estimate Eliminate the 10% of 420=42 Use calculator to incorrect 5% is half = 21 check back that responses Take 60 from 420 ’$357’ correctly and the correct answers the answer is about question. $360. 6
  7. 7. Polya’s 4 step plan To become a good problem solver, students need to have a plan or method that is easy to follow to determine what needs to be solved. The key is to have a plan which works in any mathematical problem solving situation. Polya’s 4 step plan is a useful starting point to help students develop a ‘plan of attack’ in problem solving. This will set them on the trail to develop their own successful problem solving technique. Once students find a problem solving method that suits them, they can use it in other curriculum areas and in real life problems that they encounter. Polyas four steps to problem solving 1. Understand the Calmly examine all the information in the problem. __problem Decide what information is important and what seems unimportant. Would a FIGURE or DIAGRAM help? Write down or underline all relevant information. 2. Devising a plan Have I seen a similar problem? Can I solve part of the problem? Could I organize the data into a table? Could I find a pattern? [See problem solving strategies below] 3. Carrying out the ---- Can I follow through each step of the plan? -plan Is each step correct? Can I prove it is correct? 4. Looking back Can I check the result? Does the answer make sense? Is there another way I could solve the problem? Could I use the results in a problem I have seen before? From: Polya, G. (1973) How To Solve It Princeton University Press: USA Previous years NAPLAN tests and the other practice test questions on the DET website provide teachers with useful vehicles to model problem solving using the plan outlined above. Many problems can be solved in more than one way and using a variety of problem solving strategies. Problem solving strategiesA critical skill or essential learning for working mathematically, is to be able to solve problems.Teachers need to explicitly teach the steps involved in the problem solving process:• reading or looking carefully at a problem• following a plan to analyse what is required• choosing which strategy, or strategies in combination, will help you solve the problem• checking back to see if the answer is the one required and• finally reflecting on the problem and seeing if it leads to other questions or possibilities. This last stage of the process, the ‘looking back’, may be the most important step as it provides students with an opportunity to learn from the problem.Becoming a successful problem solver is embedded in the Constructive learner and theCreative learner domains of the EsseNTial Learnings:• A Constructive learner – learns mathematics in ways that makes him or her Numerate, i.e. able to apply their mathematical knowledge appropriately in real life.• A Creative Learner – is a creative, strategic and effective problem solver. 7
  8. 8. Make a Problem Solving Strategies chart as shown below and find some other strategies to addto it that work in your classroom. The 2009 NAPLAN Numeracy Analysis and Alignmentdocument gives further ideas on which strategies would be most appropriate to use for selectedproblems.Look at the ideas on setting up a ‘Mathematician’s Strategy Toolbox’ in Working Mathematically:The Process on the Maths300 Website: Problem Solving Strategies Locate key words and phrases Turn the diagram around/ (underline/highlight/circle) look at it from a different angle Note the bolded word/s Guess and check Use lots of scribble paper Estimate Break into smaller Write out a number sentence more manageable parts or equation Draw a sketch, Make an organised list picture or diagram or table Work backwards Construct a graph Act it out or Solve a simpler problem use materials Draw a number line Try all possibilities Draw a clock face Make a model Look for a pattern ……. Whole class modelling activityAs students become familiar with using the test language glossary, ‘plan of attack’ and problemsolving strategies chart, the teacher can direct a whole class lesson as follows:• use an interactive white board, data projector or overhead transparency projector (OHP)• download the 2008 and 2009 NAPLAN test questions or other practice test questions• select and discuss questions with the whole class or a small group, particularly those questions that relate to a unit of work that you are studying e.g. 2D shapes• ‘unpack’ any complex wording or phrasing (add to class chart/glossary)• discuss and understand what information is required• read through the answer choices (called distractors)• ask students to work in small groups to solve and select an answer• encourage students to use different problem solving strategies e.g. use scrap paper to draw diagrams and check calculations, work backwards etc.• ask each group to report back to the class and demonstrate how they solved the problem, chose their answer and why the other choices were incorrect• reinforce the notion that there are many methods/ways to reach an answer, some are more efficient than others depending on the problem type and complexity• support groups or individuals to gradually create their own questions and distractors on the topic that they are working on. 8
  9. 9. Finding out where and why students make mistakes Newman’s error analysis• After students have established a ‘plan of attack’ to solve a problem, are familiar with the language of the problems and have had practice using a number of problem solving strategies, then the teachers need to analyse areas that are still causing difficulties.• To identify errors, teachers need listen to students as they talk through the problem solving process i.e. to ‘think out loud’. By using Newman’s five levels of questioning (Newman’s error analysis) shown in the table below, teachers can pinpoint the step or steps in the process where students’ errors occur. Newman’s five levels of questioning Processes used to Prompts or questions used by the teacher solve a problem to determine where the errors occur 1. Reading the problem Reading Please read the question to me. If you dont know a word, 50% leave it out. to 70% 2. Comprehending what is read Comprehension of Tell me what the question is asking you to do. errors occur 3. Carrying out a transformation from the words of the problem to the selection of an appropriate mathematical before Transformation strategy step 4 Tell me how you are going to find the answer. 4. Applying the process skills demanded by the selected strategy Process skills Show me what to do to get the answer. "Talk aloud" as you do it, so that I can understand how you are thinking. 5. Encoding the answer in an acceptable written form. Encoding Now, write down your answer to the question. Newman, A. (1983). The Newman language of mathematics kit. Sydney, NSW: Harcourt, Brace & JovanovichStudies have shown that from 50% to 70% of errors occur in the first 3 steps of the problemsolving process as shown above. Critical questions to ask before the NAPLAN test.• Are the students having difficulty in any area of numeracy?• Are the students having difficulty with any question type?• Are the students using a ‘plan of attack’?• Is there any unfamiliar language that needs to be ‘unpacked’?• Do some students need to have the numeracy test questions read to them individually or in a small group?• Are the students choosing and using the various problem solving strategies successfully? 9
  10. 10. Classroom games and activities Class quiz sessionsAfter working on specific mathematics concepts play different sorts of quiz games.• Choose three or four contestants, like a TV quiz show, another student can keep the scores. Put the test questions on the Smart Board, data projector or OHT and uncover questions one by one or read questions out loud, show a diagram or graph, multiple choice is good for this. The first student to ring a bell, blow a whistle, click or clap can answer the question.• Play ‘Who wants to be the class champion?’ Class quiz with ‘phone a friend’, ‘3 lifelines’, ‘50/50’, ‘Trust the audience’ etc.• Play Celebrity Heads using concepts like 2 and 3 D shapes. Contestant asks yes/no questions e.g. Do I have 4 sides? Are any of my surfaces curved?• Divide the class in half and alternate questions to each side, have scorers out the front using tally marks.• Organise individuals or small groups to create their own questions about the topic. Computer number patternsStudents learn to:• use Microsoft Word to insert a 10 by 10 grid and make their own 100 chart• create and colour a number pattern to 20, 50 or 100, e.g. 2, 6, 10, 14• print these out and swap with other students to work out the pattern• include subtraction, squaring and cubing of numbers in the patterns. Open-ended activityTo play Here’s the Answer! What’s my Question?, the teacher:• chooses a topic that the class has been working on• reads out an answer e.g. $2.35, 1.30pm etc.• pairs or groups students and asks them to create questions to match this answer• discusses with class the variety and different types of questions created for the answer. Student-constructed test questionsFor this activity teachers help students to:• brainstorm a list of topics that they have been working on during the term• form into groups of 2 or 3, with students choosing a topic e.g. 3D shapes, time etc.• create different types of questions and write/print/draw/trace these onto blank cards – multiple choice, match-ups, sequencing or short answer• use these as part of the term test, together with the teacher’s questions. (The teacher can review and check the questions before they are put into the test).Student-constructed test questions are a powerful Assessment for Learning tool. During theprocess, students either independently or in small discussion groups gain a deeperunderstanding of the topic by having to create their own questions and answers. They alsogain insight into the assessment process. Research shows that student-constructed tests:• foster a positive assessment culture• are a critical component of student-centred classrooms• are a powerful group review strategy• empower students to take responsibility for their own learning• reduce test anxiety/stress. 10
  11. 11. ‘Around the World’Play this game using ideas from the NAPLAN questions. • Students sit in a large circle. • One person stands behind someone in the circle. • Teacher or designated student calls out a question. • If the person standing up correctly answers first, he or she moves to stand behind the next person in the circle. The goal is to work their way around the circle. • If the person sitting down calls out the correct answer first then he or she stands up, while the person who was standing sits in their place. • Then the process begins again. Who can travel right around the entire circle (world)? • Ideas for the questions can come from the 2008 and 2009 NAPLAN test questions and practice questions on the website or real objects around the classroom such as: - large class clock or calendar - numbers and sizes of windows, doors, and chairs. - 4 big dice or 6 apples or 10 teddies etc. - a variety of containers and packages - a selection of 2D shapes and 3D objects. • Some examples of the types of questions are: Number and Algebra - There are 6 apples, I eat 2, how many are left? - How many groups of 2 teddies can I make? - Add up the numbers on the top of each dice. - Multiply the numbers on the top of each dice. - What is the next term in the number pattern shown on the dice? Space - Look at the 3D shapes, which shape has the most edges? - Name a 2D shape with 6 sides. Measurement - Look up at the clock, what was the time 2 hours ago? - The calendar on the wall shows _______ (month) - What date is the second Sunday? - Next month will start on which day of the week? - What date is 2 weeks from today? - What container would hold the least amount of water? - Which container holds closest to 2 litres? - How tall is the door, to the nearest metre? - Find something in the classroom that is taller than the desk but shorter than the door. • Questions involving times tables, multiplication, division, addition, subtraction and any other standard mental maths skills are suitable. • Another strategy is to ask the students (in pairs) to create their own questions about particular topics e.g. time questions about the class calendar or the clock. Students could be rostered on for ‘Make-up NAPLAN Questions’ sessions. 11
  12. 12. Topic areas Individual practice questions can be separated and sorted into groups and used for review and assessment of units of work. • There are a variety of practice test questions (2004-2009) and the 2008 and 2009 NAPLAN test questions on the website to use for this activity. • Photocopy the questions onto coloured paper or card. Each band or year level can be a different colour for quick visual recognition e.g. Band 1 – green, Band 2 – blue etc. • Down load the test papers and cut up into individual questions (laminate if possible) and group them into sections like: Number and Algebra - Patterning - Money - Fractions - Operations - Equations Measurement - Length - Area - Time - Volume/capacity Chance and Data - Graphs/tables - Probability - Statistics Space - Shapes/angles - Mapping and co-ordinates. • Use questions for quiz time during and after a unit of work on these topics. • Write the answers on the back and the children can quiz each other e.g. the student who gets the correct answer can pick up the next card and ask the question. • The question cards can be left in a box or tray for students to work on individually or in small groups. • They can be used as a workstation activity for rotating maths or ‘hot potato’ sessions. • Provide students with questions that cover a range of Bands or year levels so they can find their own level of ability for a topic. Independent Time or ‘All-by-Myself-Time’ • Schedule regular times during the week when it is ‘All by myself time’ or ‘Independent Work Time’. During this period, the students must attempt a numeracy (or literacy) task without help from the teacher or their peers. Use an egg timer or stop watch and increase the period of time gradually. • This becomes an excellent Assessment as Learning opportunity as students reflect on their strengths and weaknesses. 12
  13. 13. Hands on learning activitiesThe following activities enable richer concept development using 2008/2009 NAPLANtest questions or practice questions as a springboard to hands-on learning activities. Patterns• Make patterns with tiles, like building a stair case.• Count the tiles and determine how many are needed for the next row, make different patterns e.g. go up 2 each time.• Create KGP 3 questions like: Draw the next line of tiles in the pattern. Numbers to 10 and numbers to 20• Use counters (a bundle of 10 or 20) and divide them into two different groups, each time record the numbers that add up to the total, e.g. separate 8 and 2 counters and record, 8 and 2 make 10 or 8 + 2 = 10, work through all the combinations.• Use blank ten frames and dice e.g. roll a 4, colour in 4 squares, use a different colour for the other 6 squares to represent the numbers that make up 10.• Use a number strip/line to 20 and put two different lots of coloured counters on the strip to represent different ways for two numbers to make 20, find all the combinations and record them.• Use counters to find the combinations of other numbers e.g. 15, 18, etc.• Create open KGP 3 tasks like: If 10 and 3 make 13 then show me another way to make 13 by writing different numbers in the spaces: ___ and ___ make 13. Grids/references• Use blank grids and draw treasure in one square/cell, follow directions to get to the treasure.• Use blank grids and write the numbers and letters down each side, place a counter in one square/cell and talk about the reference and position of that square/cell.• Create Band 1 questions like: Start at X which is in C3, move two spaces left, next go two spaces down, now go one space to the right, put a tick in this box. What is the new grid reference? Ordering• Roll two or three dice to make a 2 or 3 digit number, make four different numbers and then put the numbers in order from smallest to largest.• Use 2 digit numbers to start with and create Band 1 tasks like: Write these numbers in order from smallest to largest. 638, 386, 863, 683. ________ ________ ________ ________ Smallest Largest Place these numbers on a number line and find the difference between them. 13
  14. 14. Number Patterns, Sequencing and Algebra• Use the 100 grid and create/colour in number sequences/patterns.• Practise whisper or skip counting, emphasise specific numbers and whisper all other numbers, count backwards as well as forwards 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 . . .• Use coloured counters to emphasis patterns.• Start with tally marks or bundles of sticks to count in fives.• Create Band 1 to Band 4 tasks as follows: Band 1 Write the next number in this sequence 15, 20, 25, 30, Band 2 Write the next number in this sequence 9, 8.5, 8, 7.5, Band 3 Write the next number in this sequence 400 000, 8 000, 160, Band 4 Write the next number in this sequence 1.5, 1.0, 0.5, 0, – 0.5,• Make a pattern of triangles with paddle pop sticks. Count the number of triangles and the number of paddle pop sticks needed. Discuss the pattern and create Band 1 questions like There is a pattern in the way these triangles are made. How many sticks would you need to make the next triangle?• Make a list of values – number of triangles with number of paddle pop sticks.• Create Band 3 questions like s=2t+1 Fill in the missing values. How many sticks are needed to make 10 triangles, . . . . . 100 triangles?• Create Band 4 questions like Which rule describes how many sticks s are needed for t triangles? Brackets and Order of Operations• Use small cards with brackets, equals and algorithm signs such as ( ) = + – x ÷ < > 2 and digit cards from 0 to 9 to create a variety of equations. Review the BODMAS rules: B Brackets first O Orders (i.e. Powers, exponents, indices and square roots, etc.) DM Division and Multiplication (left to right) AS Addition and Subtraction (left to right)• Show how equations with the same numbers can have different answers: 2 + 5 x 3 = 17 (2 + 5) x 3 = 21 30 ÷ 5 x 3 = 18 30 ÷ (5 x 3) = 2 5 x 2 2 = 20 (5 x 2) 2 = 100 14
  15. 15. Calendars• Use a current calendar to read dates and days regularly.• Students draw the current month from scratch, without looking at the calendar or use a blank calendar and fill it in.• Teacher asks questions like Band 1 What is the name of this month? How many days in a week? How many days/ weeks in this month? Band 2 Write the date of the second Tuesday in May. Write the date that is 2 weeks after this date. The month is _ _ _ _ _ _ _ _ _ _ _ _ _ Use horizontal Monday and vertical Tuesday calendars Wednesday Thursday Friday Saturday Sunday 2D Shape Patterns• Cut out a 2D shape and trace around it, move it, trace around it again and make a continuous pattern.• Cut out several copies of the one shape and stick them onto paper to make a pattern.• Trace or stick the shape in different ways in a continuous pattern, by flipping it, sliding it, or turning it.• Talk about the language of flipping, sliding, or turning the shape and how the shape looks the same or different. Use the words reflect, translate and rotate for older students.• Create activities like Band 2 Redraw the shape below to show that it has been flipped to the right. Band 3 Rotate this shape by 180 degrees. 3D Objects – nets and cross sections• Draw a net of a 3D shape, cut it out, fold it to see if it works.• Draw a variety of nets and see if your buddy can work out what 3D object it is.• Make as many different nets of a cube as possible – there are 11, can you find them all?• Make some 3D objects with plasticine or play dough. Use fishing line to cut through the objects and make as many different cross sectional shapes as possible. These shapes can be used to make interesting patterns and designs. 15
  16. 16. Money• Make different combinations of an amount of money, say $3.85 or $42.• Talk about how many different ways you can make these amounts.• Use real or play money to add and take away amounts.• Use real or play money to work out change from $2, $5, $10, $20 etc.• Make use of junk mail e.g. calculate the cost of a loaf of bread, a carton of milk and a dozen eggs, then work out the change from $10 ($20 or $50).• Role play shopping to practice counting out amounts of money, adding and subtracting amounts, and giving change out from $10, $20 or $50.• Create questions and tasks such as Band 1 What is the total value of these coins? Band 2 Show two different ways of making $4.25. The cost of a CD was $15.85. What was the change from $20?• Using money to help with the understanding of percentage – interest, discounts and price increases. Junk mail and advertisements are useful for these activities.• Partitioning amounts of money into percentage parts with questions like Band 4 What is $10 as a percentage of $40? A skate board is normally $85. Work out the new price after a discount of 20%. Spinners• Talk and ask questions about the chance of events occurring What number are we most likely/least likely to spin up? Why? Why not?• Make an easy spinner (a sharp pencil held vertically with its point in the centre of the spinner surface, flick a paper clip around the lead end).• Make spinners with coloured parts of the circle e.g. ½ red, ¼ blue, 1/8 yellow, and 1/8 green. Discuss what could happen, where the spinner might land and how often. What colour/number are we most likely/least likely to spin? Why? Why not?• Record outcomes of spins using tally marks, graph results and create questions like Band 2 Write the number that is most likely to come up on this spinner. Band 4 What is the chance that the spinner will land on …? Chance and Data• Use a variety of different objects e.g. cards, stickers, stamps, marbles, to set up groups in bags and containers. Work out the probability of selecting items from these sets.• Use the probability cards in Appendix 1 to teach the specific language of chance.• Place 12 counters in a sock or bag – 3 different colours. Next shake the sock and take a counter out without looking. The pupils tally the colour and the counter is replaced in the sock. Continue sampling the counters until you have a representative/fair sample. Pupils are told there are 12 counters in the sock, from that and the results, they have to estimate how many of each colour there are. 16
  17. 17. Work station activitiesThe following activities can be used with the 2008 and 2009 NAPLAN test questions or theother practice questions available on the web. They can be used on the interactive whiteboardor photocopied onto coloured card, cut up into individual questions and divided into differentconcept areas like money, time, patterns, maps, graphs, etc.Teachers can add questions while students can be challenged to create their own questions,then write these onto some blank cards, with or without the answer/s on the back. This is agreat way to familiarise students with different question types like multiple choice, sequencing,and short answer.Number & Algebra Activities and TasksTest questions cards for • Use a variety of manipulatives to investigate addition, subtraction, grouping, multiplication, division, patterning and algebra.Number & Algebra • Model numbers using MABs, 10 frames, unifixMaterials: cubes, bundles of pencils, paddle pop sticks,Blocks and beans match sticks, straws and use place value charts.Cups and counters • Decimals are modelled with straws (cut up for tenths) and globs of plasticine (rolled and divided up for tenths10 frames and hundredths)Plasticine • Manipulatives can be used for algebra conceptsMatchsticks e.g. create repeating patterns, record the details using a list, working out the pattern, find theStraws formula and draw the graph.Paddle pop sticks • Cards, dice, dominoes, racetrack activities to enhance number skillsCards and dice (Domino Deductions, Dice Dazzlers, Card Capers by Paul Swan).Dominoes • Hundred grid activities – patterning, jigsaws, change with money, addition, subtraction, multiples, use for decimal and percentageJunk mail concepts.Calculators • My answer is 18, what could my question be? Use a variety of answers like 180, 18x, 18 hours, 18 metres, etc.Play money • Large grid tablecloth to use as a 100 board for counting,Rope and pegs • addition, subtraction, multiples and number pattern work.Laminated 100 grids • Calculator activities enhance number concepts.(numbers 1-100, 101- Good habits like estimating first and then double checking should be encouraged.200, 0.01-1.00 or blank) • Play money, plastic or cut-out coins and notes, largeLarge grid tablecloth and and small, different combinations for same totals,number cards (100 grid) racetrack game to give change (out of $10, $20 etc). Money Wall games and jigsaws for making amounts ofColoured card circles money in different ways.Paper plates • Use junk mail to work out multiple buys and $ change, also for percentage discount or price increases on items. • Fraction activities using tape, rope and pegs, paper folding, paper plate drawing and cutting, fraction games (commercial or home-made) or interactive computer estimation games e.g. Maths300 site. • Fraction estimation using coloured card circles, fraction wall games. Fraction concepts using real objects like fruit, pizza, cakes, sandwiches etc. • Ratio situations can be modelled using food and liquid (recipes, dried fruit, lollies and cordial mixtures work well). Integrate with measurement – mass, volume and capacity. 17
  18. 18. Chance and Data Activities and TasksTest question cards for • Coloured pegs or other objects in a bag, students ‘guess’ which colour will beChance & Data withdrawn from the bag, record with tally marks.Materials: • Dice games where the beetle body parts are numbered, roll the dice to put the beetle together, or use other animals, objects orSpinner template football players, see MCTP Chance and Data Investigation Vol 1Spinners made from by Charles Lovitt and Ian Lowe. • Ideas for tasks and activities using coloured spinners and pegspencils and paper clips are in MCTP Chance and Data Investigation Vol 1 by CharlesVariety of dice Lovitt and Ian Lowe.Coloured pencils • Play ‘Dice Football’ is a Maths300 computer activity.Coloured Pegs • Use football results from newspapers, magazinesBags for objects or AFL/NFL web sites to graph real results.Tape or rope Start a class footy tips competition. • Play ‘Dice Cricket’ is in MCTP Activity Bank Vol 2Card to cut up by Charles Lovitt and Doug Clarke.Wall height chart • Investigate and record ‘Ice-cream Combinations’ and ‘CrazyTape measures Animals’ activities in Math300 and MCTP books.Quad paper • Make an easy spinner with a sharp pencil held vertically with its point in the centreRulers of the spinner template. Flick a paper clipComputers around the sharp lead end. • Analyse the results of various body measurements, like footNewspapers length, arm span, hand span, height etc. early years classes canPaddle pop sticks use lengths of tape and informal units. Upper classes can list data using stem and leaf plots. Find lowest, highest, range, mean, median, mode, (this real data will interest and motivate the students). • Compare and make statements about relationships between data sets e.g. height and arm spans, head diameter and arm length etc. • Paddle Pop Stick Drop gives data to order, graph and discuss (Count Me In Too activity). • Calculate averages using the students’ families e.g. the average age of 3 children in one family is 12. How old could the three siblings be? • Collect and discuss data about birth months of class members. • Discuss graphs and data lists found in newspapers and magazines about real world situations. • Use data from sport’s day or athletic events (Potato Olympics) to create a variety of graphs – pie, histogram, bar, stem and leaf. • Using the language of probability, discuss and peg up a probability continuum with words like ‘impossible’ to ‘certain’ pegged onto a clothes line or chart, find the students own language for words like possible, impossible, even chance etc. (See Appendix 1) • For older children, match up the words with arithmetic values – 0, ½, 1, 0.5, 50% etc. 18
  19. 19. Measurement Activities and TasksTest questions cards for • Estimating, ordering, weighing and measuring everyday objectsMeasurement – length, area, like cups, cans, packages, lollies, fruit, pencils, erasers, multi- link blocks, MAB’s, paperclips etc. using beam balances, scales.volume, capacity • Capacity and volume measurement activities using water, sand, blocks, rice, pasta, marbles to order/estimate/measure /compareMaterials: different containers or objects using formal and informal units.Tape measures (Volume with different block constructions.)Paper tape or string • See body measurements activity above in Chance & Data, younger children can useRulers paper tape and ‘measure’ using informalBlocks or MABs units like multi-link blocks or felt pens.Grid and dot paper • Perimeter of our bodies, hands, feet by placing string along the outline of the body or part which has been chalked on the carpetBalance beam or concrete area, or drawn on grid paper, then lengthen stringWeighing scales out and order, measure and compare results.Kitchen scales • Area of body, find out how many notebooks or paper plates can be placed inside the body outline. • Feet and hand areas on grid paper, or use other informal, uniform units e.g. blocks. • Perimeter, length and area of classroom items and different rooms. What is taller and wider than . . .? • Geoboards and coloured rubber bands are useful to consolidate and investigate the concepts of area and perimeter, redraw ontoTest questions cards for grid or dot paper and write about what you discovered. Area = 2 24 m , Perimeter could be?Time and Temperature • Routine calendar work – look at it everyday and talk aboutMaterials: events that have been and are coming up e.g. 3 weeks till sport’s day. Seasonal calendar – bush foods, animal life,Calendars weather changes etc.Thermometer and rain gauge • Note daily temperatures and rainfall by reading the thermometer at different times and the rain gauge. Graph the results andPaper plates and spit pin relate to seasons and months.paper fasteners for the hands • Draw from scratch, fill in empty calendars,Circular table cloth or ‘construct’ the month using a calendar jigsaw by cutting up old calendars.Working analogue and digital • Investigate the number patterns on theclocks in classroom completed calendars e.g. 3 by 3 grid, diagonals etc.Newspapers – TV and movie • Construct a class set of paper plate clocks with moveable hands fastened with split pins for time activities.guides • Use a large circular table cloth and paper plate numbers to construct a large clock face, use legs and arms as the hour and minute hands to make the times, or a metre ruler and a 30 cm ruler for the 2 hands. Use the clock for number counting and counting by 5’s – forwards and backwards. • Refer to the clock throughout the day with questions like – How many minutes until recess? What was the time 30 minutes ago? What’s the time in 24 hour time? Nominate students to make up and ask some time questions. • Use TV Guides to work out elapsed time of shows & movies. • Investigate Australian time zones and world time zones. 19
  20. 20. Space Activities and TasksTest question cards on • Look at everyday objects in the classroom and the outsideSpace environment (desks, rooms, buildings, packages, nature) and describe the 2D shapes and 3D objects that occur.Materials: • Construct 2D shapes and 3D objects from a variety of realColoured paper and card materials, glue to make mobiles and repeating patterns. • Visualise and create nets of 3D objects fromColoured cardboard scratch rather than from BLM nets and compareprotractors/angle wheels variations. Discuss number of faces (surfaces), edges and vertices (corners).Paddle pop sticks • Use coloured paper to cut and fold 2D shapes, observe,Matchsticks compare and discuss properties like symmetry, tessellations,Blocks and MABs transformations (flip, slide, turn), and design. Tell me everything you can about this shape?Plasticine and fishing line • Use cardboard angle wheels, protractors, compasses, setStraws squares and rulers. (‘Geopaperpolygons by J. Burnett, C.Pipe cleaners Irons, Origo Publishers).Rolled up newspapers • Manipulate elastic bands on a geoboard to make shapes then draw these onto dot, grid or isometric paper. Find as manyGeo-boards different types of hexagon shapes that you can.coloured rubber bands • Use a length of rope, tape or large elastic loop to make andAssorted cans and packages manipulate the outlines of 2D shapes. • Create repeating patterns with shapes, copy designs andBoxes from small to large colour them, draw in the sand or use chalk on the cement,Street directories students need to describe and compare what they have created.Tourist maps and atlases • Paint various sides of 3D objects and use to print repeatingGoogle Earth patterns on paper or cloth (flip, slide, turn).Large circular table cloth • Use plasticine to ‘create’ 3D solids and use fishing line to cutN, S, E, W cards through the object and investigate different cross-sections.angle cards • Use 4 MABs or other blocks to create as many different structures as possible. Visualise and draw the front, side and overhead views of the structures into grid paper, then check these. Draw these onto isometric paper. Try 12 MABs. • Location activities – use ‘Google Earth’, street directories, atlas or tourist maps of Alice Springs, Darwin, Katherine, Tennant Creek and the Northern Territory to ask and create questions using co-ordinates and compass directions (integrate with SOSE Unit on mapping). • Draw local mud maps of the classroom, school yard layout, community from a bird’s eye view and use directions – left, right, up, down – to describe how to get to certain places. • Work out the real North, South, East and West directions in the playground or local area using the different positions of the sun during the day. • Use a large circular tablecloth and N S E W cards to place compass directions and model turning in a specified direction and for an certain 0 number of degrees e.g. turn 45 W. • Use the above for angle and bearings work. 20
  21. 21. Open-ended questionsMany of the questions we traditionally ask students call for one finite answer – a number, anamount of money, a percentage, a shape or some mathematical object. For example, askingthe question ‘What is 5 + 6?’ requires students to answer ‘11’. Similarly, asking students whichtriangle in a set of triangles is isosceles requires them to identify a specific object. These kindsof questions are closed because the expected answers are predetermined and specific.In contrast, open-ended questions allow a variety of correct responses, elicit a different kind ofstudent thinking and allow students to demonstrate their own ways of solving the problem.Both closed-ended and open-ended questions are appropriate for assessing studentsmathematical thinking. A test consisting solely of open-ended questions would take aninordinate amount of time to grade and might not cover the curriculum adequately. Closed-ended questions are a reasonable way to sample students understanding of a broad range oftopics. But closed-ended questions do not allow students to reveal their thinking processes aswell as open-ended questions.Some suggestions for opening up the closed NAPLAN questions are included in the NTCFcurriculum support documents for the 2008 and 2009 numeracy tests. Each question has theanswer, the NTCF link, the outcome descriptor and includes open/extension activities for eachquestion. How to make questions open-ended.Method 1:• Identify a topic.• Think of a closed question and write down the answer.• Make up a new question that includes (or addresses) the answer. How many chairs are in the room? Four This can become . . . I counted something in our room. There were exactly four. What might I have counted?Method 2:• Identify a topic.• Think of a standard question. ?• Adapt it to make an open-ended question. What is the time shown on this clock? This can become . . . My friend was sitting in class and she looked up at the clock. What time might it have shown? Show this time on a clock.References:Clarke, D (1992) MCTP – Assessment Alternatives in Mathematics Curriculum Corp: Vic.Downton, A. Knight, R. Clarke, D. Lewis, G. (2006) Mathematics Assessment for Learning: RichTasks and Work Samples Mathematics Teaching & Learning Centre ACU: Melbourne.Sullivan, P. & Lilburn, P. (2004) Open-ended maths activities: Using ‘good’ questions toenhance learning in mathematics South Melbourne: Oxford. 21
  22. 22. Stem and leaf plotsThis is a very effective way of listing and ordering data. An example of this would be whencollecting data about the heights of students in the class. The teacher writes the ‘stem’ on thewhiteboard – the expected range of results – see column 1. below.The students help each other to measure their heights and then they write it in cm, e.g. 9students call out their heights:163cm, 138cm, 140cm, 155cm, 155cm, 149cm, 144cm, 135cm, 153cm.The teacher points out the lowest number,135, and the highest number, 163.The hundreds and tens will make up the stem, 13, 14, 15, 16, as in figure 1 below.Now add the digits in the ones place, they will be the leaves, as in figure 2 below.Finally, re-order the heights from smallest to largest on each line, as in figure 3 below. 1.Blank Stem 2.Heights entered 3.Heights re-ordered 13 13 8, 5 13 5, 8, 14 14 0, 9, 4 14 0, 4, 9 15 15 5, 5, 3 15 3, 5, 5 16 16 3 16 3Looking at the stem and leaf plot, it is easy to see the spread of the data, in fact it is now lookslike a horizontal bar graph: • lowest height = 135cm • greatest height = 163cm • range is 163 –135 = 28 cm • middle height, median = 149cm • most common height, mode = 155cm.It is much easier to estimate the mean or averageof the scores, say somewhere between 144 and 155.Add all the heights(135+138+140+144+149+153+155+155+163)divide by 9 (total number of scores)and the mean or average = 148cm.Other good examples to use for stem and leaf teaching, learning and assessment:‘Greedy Pig’ in Maths No Fear! by Curriculum Corp and NT DET, and in Maths300.‘The Beetle Game’ in MCTP Maths Investigations by Curriculum Corp, and in Maths300.Websites for Stem and Leaf Information: (excellent video explanation) 22
  23. 23. WebsitesThe national NAPLAN website details, samples, parent information, FAQS etc: NT site has a variety of administration details, forms, updates, 2008 and 2009 NAPLANtest and other practice tests as well as useful support materials, NTCF alignment and extensiondocuments, reporting guides and RAAD information: Vic site has a variety of practice AIM tests from Year 3 to Year 9: SA also has a selection of State Assessments from Year 3 to Year 9: educational websites with a variety of teaching, learning and assessment ideas:Curriculum Corporation, Assessment for Learning: Training Institute: http://www.assessmentinst.comAssessment is for Learning (AifL), Scotland of Education, QLD of Education, NSW ideas for your classroom can be found on the following sites:Australian Association of Mathematics Teachers Corporation Maths300 investigative tasks, UK, Maths Enrichment http://nrich.maths.orgMath Forum http://mathforum.orgUtah State University, National Library of Virtual Manipulatives Archive site for specific topics and themes Mathematical Society Council of Teachers of Mathematics Tell a friend,Rubric Construction add to favourites!For any sort or size graph paper interactive levels maths games site for lesson plans, scope and sequence DET website has a variety of curriculum units of work and ideas for your classroom:[Please contact Gay West on p: 8999 3778 or e: for further details about theactivities in this booklet or with any feedback.] 23
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