Getting Started Getting Started Introduction The purpose of this booklet is to provide guidelines for organising your assessment information and developing your classroom programme. NumPA provides a wealth of diagnostic assessment information about students. There needs to be an appropriate link between this data and the learning experiences you provide for your students through the classroom programme. Your planning should also meet the achievement aims and objectives of Mathematics in the New Zealand Curriculum (MiNZC). This book consists of three main parts: • Organising your NumPA information • Developing the number component of your classroom programme • Choosing from useful formats to simplify the task of planning and ongoing assessment. Successful numeracy teachers emphasise connections. These connections can be between the ideas in mathematics, between mathematics and other learning areas, and between mathematics and the real-life experiences of students. Book 1: The Number Framework and the NumPA data give a direct window into students’ number knowledge and strategy application. This provides clear direction for teaching and allows you to choose appropriate learning experiences for your students. Organising Test Information There are three clear steps to organising your NumPA information: 1. Analysis of knowledge hot spots 2. Assigning strategy stages to students 3. Grouping for instruction. Analysis of Hot Spots Examining the Knowledge section of the NumPA will give you a collective picture of your students’ number knowledge as well as highlighting areas of strength and weakness for individual students. From this data, you will be able to identify hot spots, which are problematic areas of knowledge that are common to many students in your class. These hot spots should form the core of your whole class knowledge teaching at the warm-up phase of each lesson. You should choose suitable Knowledge Learning Activities (from Book 4: Teaching Number Knowledge) to target these hot spots during your warm-up or group teaching time (refer to the models for daily numeracy lessons on pages 4–5 of this book). Pages 18 and 19 provide possible formats for recording knowledge hot spots in your class grouping sheet for NumPA. It is quite common to see a mismatch between a student’s strategy stage and their knowledge. This suggests that either the student has more knowledge than they are able to use or the student has powerful strategies but lacks the knowledge to apply them to different numbers. This is why the identification of knowledge hot spots is crucial for quality teaching. 1 Getting Started Assigning a Strategy Stage Your initial grouping of students should be by their dominant strategy stage. Stages 0–6 describe the transition from counting to part-whole addition and subtraction strategies. Responses to the Operational Strategy Windows of NumPA will dictate which strategy stage to assign to your students. Addition and subtraction form the basis for identifying strategy stages 0–6 because students need to be able to add and subtract fluently before they can derive multiplication and division facts independently. For example, a student might be stage 5 for addition and subtraction and stage 6 for multiplication and division. You should assign the student as stage 5 and focus initially on addition and subtraction. This student understands how to derive multiplication facts but lacks the addition and subtraction strategies to do so efficiently. Strategy stage 7 is assigned to students who have reached stage 6 for addition and subtraction and have demonstrated high-level strategies for multiplication or division (Advanced Multiplicative stage). Strategy stage 8 is assigned to students who have reached stage 6 for addition and subtraction, stage 7 for multiplication and division, and have demonstrated high-level strategies for solving problems with fractions, ratios, and proportions (Advanced Proportional stage). Grouping for Instruction Group your students for instruction by their assigned strategy stages. You can record this information onto a class grouping sheet for NumPA. Pages 18 and 19 provide a format for this purpose and a hypothetical example is provided on page 20. Choose the page that has the relevant strategy stages for your class and write each student’s name in the relevant column. The progress of a student can be shown by highlighting their name through to their new stage. This constitutes a useful record of achievement. In situations where there is a wide range of strategy stages within your class, you need to consider either cross-grouping between classes or compromising by putting together students from close strategy stages to reduce the number of teaching groups. For example, you may have groups at these strategy stages: Stage 2 Counting from One on Materials Stage 3 Counting from One by Imaging Stage 4 Advanced Counting Stage 5 Early Additive Part-Whole 2 Getting Started Students at stages 3 and 4 could be combined, provided that the advanced counting students are only counting on for simple addition problems. Such a combined group could be termed “early advanced counters”. Students from stages 4 and 5 could be combined provided that the advanced counting students are imaging counting on to solve addition and subtraction problems. The main obstacle to these students doing partitioning is likely to be a lack of knowledge, for example, of the “teen” code. Another grouping scenario may be that most of the students are at one strategy stage. In this case, it may be necessary to form two or more groups, according to both the knowledge and the strategies they demonstrate. For example, you have 20 students grouped at stage 4, advanced counting. Eight students in the group solve problems by using counting on with their fingers, and 12 students solve problems of counting on and back by imaging in their heads. There is a clear division between these students, and this enables you to confidently divide them into two groups. The students using counting on by imaging are more ready for the transition to part-whole thinking than their counting-on-with-fingers counterparts. Finger counters are likely to need work on imaging before the part-whole transition is successful. Developing Your Number Programme Number lessons can have many structures. These structures include: whole-class knowledge lessons, a combination of both knowledge and strategy teaching, and pure strategy teaching. Over time, a balanced programme should contain both knowledge and strategy teaching. Whole-class instruction has the benefit of involving all students. It is very effective in simplifying management and preparation. It is therefore most suited to situations where there is a common goal for instruction. This approach is typically used to develop students’ thinking in the knowledge section of the Number Framework. Students can learn to count together, recognise numerals, and practise relevant basic facts effectively in wholeclass situations. Your identification of knowledge hot spots provides good mathematical content for whole-class instruction. For example, you may have identified and recorded on the class grouping sheet for NumPA (on page 18 or 19) that most of your students are lacking in a similar concept of number knowledge, like backwards counting. Using the knowledge-only lesson model enables in-depth teaching of the knowledge needed by most students in the class. This will enhance further strategy development for all students. 3 Getting Started Knowledge-only Lesson Warm-up 10 minutes Practice 30 minutes Warm-down 5 minutes • Whole-class knowledge activities • Hot spot focus is shared with students as learning outcomes. • One or two short activities related to numeral recognition, counting (including both sequencing and ordering), grouping, or basic facts are shared with the whole class. • Activity-based practice in pairs or small groups or as a whole class • Activities may be organised as stations for students to visit. • Students share their thinking with the class. • The teacher summarises learning outcomes from today’s lesson, making connections to previous lessons and existing knowledge. • The students reflect on their learning through teacher questioning. Strategy-only Lesson Whole-class instruction has both strengths and weaknesses as an approach to developing students’ number strategies. When problem-solving strategies are shared, the students provide models for others to adopt and adapt. They can encounter strategies they were previously unaware of. A disadvantage of whole-class teaching is that the students at different strategy stages solve operational problems at greatly differing speeds and levels of sophistication. This can make management very difficult unless open-ended problems are used. Optimum strategy development is more likely to occur when the students are working in groups with others who are functioning at strategy stages close to their own. Below, in table form, is a model for managing strategy teaching group rotation. In this model, each group gets two lessons on consecutive days, and three lessons during the week. Mon Group 1 Group 2 Group 3 Teacher Teacher Tues Teacher Teacher Wed Thurs Class Together Teacher Teacher Teacher Teacher Class Together Fri A combination of: - Assessment - Individual help - Activity introduction Teacher A model for a lesson based on this mixed knowledge and strategy teaching is shown on the opposite page. 4 Getting Started Knowledge and Strategy Mixed Lesson Warm-up 10 minutes Knowledge hot spot focus identified through analysis • • Direction of Groups • Group Teaching 1 • 2 minutes 15 minutes Based on strategy groups identified on class grouping sheet for NumPA (see pages 18 and 19) • • • • • Redirection of Groups • Group Teaching 2 • 3 minutes 15 minutes Warm-down 5 minutes • • Whole-class knowledge activities (hot spot focus), for example, counting, place value modeling, fraction recognition Short activities related to numeral recognition, counting – sequencing and ordering, grouping, or basic facts The teacher ensures that students are ready to move to their first activity. A task board (see page 11) may be used to guide students to the activities. The teacher provides in-depth teaching for one group of students. Group teaching is aimed at key knowledge or transition from one strategy stage to another. Learning outcomes are shared with the students. The teaching group moves to a suitable reinforcement activity at the completion of this session. The other students are engaged in independent activities. The teacher ensures that the students are ready to move onto their next activity. A task board (see page 11) may be used to guide students to the activities. As for Group Teaching 1, but with a new group of students The teaching group will be set up to begin tomorrow’s lesson with a suitable reinforcement activity. Possible activities include: • Sharing examples of students’ work; • Discussing a key idea from the unit; • Demonstrating independent activities; • Checking work; • Students reflecting on their learning through teacher questioning; • Students write up their work for that session. 5 Getting Started A Model for Strategy Teaching Book 4 of the Numeracy series, Teaching Number Knowledge, contains knowledge activities. Knowledge is defined as those things that students need to recall automatically. Conventional teaching methods like playing games, individual practice, and rehearsing number sequences can help students to increase their set of automatically recalled facts. In books 5, 6, and 7, Teaching Addition, Subtraction, and Place Value; Teaching Multiplication and Division; and Teaching Fractions, Decimals, and Percentages respectively, the activities are designed to develop mental strategies. As an example, to distinguish between knowledge and strategy, consider how students may choose to work out 8 + 9. Knowing that 8 + 8 is 16, students use a strategy to work out that the answer to 8 + 9 is one more than 16. Compared with learning knowledge like counting and basic facts, learning to strategise is more complex. To help Existing you to plan for student learning, the Knowledge & Strategies strategy booklets incorporate a teaching model. The model used for strategy teaching activities is shown in this diagram. Using Materials Existing knowledge and strategies are prerequisites for developing more advanced strategies. In the Using Materials phase, students are presented with problems to be solved with the support of materials. Using Imaging Using Number Properties New Knowledge & Strategies In the Using Imaging phase, the materials are shielded from the students, and they are encouraged to image what actions they would take when using these materials. For students who are struggling with this imaging, the backwards direction of the arrow indicates that you should “fold back” to allow access to materials. This working between Using Materials and Using Imaging may occur for some time until students have developed the ability to image the problems. Using Number Properties focuses the students’ attention on solving problems by discarding imaging in favour of working with the numbers and operations as ideas. This is generally achieved by using numbers that are too large or complex to image. Once again, for students who are not making the abstractions, the backwards direction of the arrow indicates that you should “fold back” to posing problems that can be imaged. In the process of developing more advanced strategies, new knowledge and strategies have been created. 6 Getting Started This model applies to all strategy teaching. It should always be your aim to introduce important mathematical ideas by using materials and to progress to Using Imaging then Using Number Properties as soon as your students are able. When applying the model, you should listen and observe to see whether individual students are making the connections at each phase. If these connections are not being made, it is suggested you should not attempt to push on to the next phase. Either drop back to the previous phase or end the lesson. Similar activities can be revisited at a future time. An example of a strategy lesson being applied is provided on pages 13–17. Planning Numeracy Lessons When filling in your mathematics long-term plan, remember that in the first four years of schooling, the emphasis should be on the number strand. In the middle and upper primary years, the emphasis may change depending on the needs of individual students identified by NumPA. Number will also be integrated into the teaching of other strands. Theme units are an excellent way to work across strands and to connect mathematics to other learning areas and to daily life experiences. Your specific coverage of number operations should be clearly documented on a long-term plan. This plan may cover one term or a full year. It should be seen as a statement of intent that may legitimately vary in response to student needs and progress. A variety of unit structures can be used. Number units may be generic and deal with the achievement objectives from Mathematics in the New Zealand Curriculum in an integrated way, or units may target a specific operational domain, like addition and subtraction or fractions. Theme units might be used to develop a collection of key number ideas. Unit Planning You need to identify (highlight) the following aspects in planning number units: • Key knowledge outcomes • Strategy outcomes • Activities for learning these outcomes. It is important that the knowledge and strategy outcomes are aligned. Choosing a small number of outcomes is preferable to inadequate coverage of many outcomes. Pages 21 to 37 provide possible unit planning formats for number. These formats are organised by strategy stages and operation, consistent with the layout of Book 1: The Number Framework. They provide links to the relevant achievement objectives from the curriculum statement. Each planning form has a reference bar that spans two stages. This suggests that the outcomes on the form are suitable for students who need to make the transition between these stages. The outcomes are also suitable for broadening students who are already working at the higher stage. 7 Getting Started For example, if the bar highlights AC and EA, the outcomes are suitable for transition of Advanced Counting students to Early Additive Part-Whole, or for broadening Early Additive Part-Whole students. AC EA You can use a single planning format for more than one unit of teaching. An existing plan can be highlighted using a different colour to indicate a new teaching period. Once the unit plans are complete, you can use a weekly plan format or a planning diary to record the day-to-day details. There is a risk of over-planning. Some teachers wisely fill in weekly plans to Wednesday or Day 3. This allows for flexibility for the rest of the week, particularly when the students have not completed activities or new learning needs are identified. Pages 38 and 39 contain possible weekly planning formats. You might find the Numeracy Planning Assistant on the NZmaths website useful. You can access it at www.nzmaths.co.nz/numeracy 8 Getting Started Finding Your Way around Unit Plan Templates Relevant strategy stages and operational domain Links to MiNZC Mathematical Processes and Achievement Objectives Key Knowledge Required Knowledge needed by the students to meet the strategy outcomes Knowledge Being Developed Knowledge taught alongside the strategy outcomes Strategy Learning Outcomes to explain what the students are learning within the operational domain Knowledge Activities Referenced activities from Book 4: Teaching Number Knowledge and Enriching the Number Framework with Beginning School Mathematics Strategy Activities Referenced activities from Book 5: Teaching Addition, Subtraction, and Place Value; Book 6: Teaching Multiplication and Division; Book 7: Teaching Fractions, Decimals, and Percentages; Enriching the Number Framework with Beginning School Mathematics and the Figure It Out series 9 Getting Started Ongoing Assessment Give priority to assessing students’ progress through strategy stages and in the key knowledge required. You may wish to use checklists of the key knowledge required to help you to identify potential blocks to students making strategy transitions. Progress across strategy stages can be documented easily on the class grouping sheet for NumPA (see pages 18–19) by highlighting a student’s name through to the next stage. Where possible, you should use non-obtrusive methods of assessment. Group teaching provides many opportunities for discussions with students that can form the basis of judgments about achievement. Snapshots can be used to assess students’ progress. Such snapshots consist of mini-interviews, where you ask a problem, such as those suggested in NumPA, that might produce a range of responses from the students. Possible assessment portfolios can be developed. See pages 40–48 for a sample of these portfolios. They list learning outcomes for both strategy and knowledge and are organised by stage. They may be used in different ways: • As individual portfolio assessments • As group assessment sheets • As student self or peer assessment sheets • As goal-setting sheets. You may also use these assessment sheets as a teaching coverage guideline. Managing Independent Activities The success of small-group strategy teaching depends on the ability of the rest of the class to engage productively in independent activities. Routines need to be established for: • introducing “new” activities; • resolving student uncertainty without involving the teacher; • indicating to the students which activities to attempt and monitoring their completion (for example, name tags, check sheets); • caring for and maintaining materials (for example, class monitors); • checking answers and written recording. You need to have some system for informing your students about their independent work for the lesson. This can take the form of a task board (see page 11) or a group box, which is a box that is assigned to a particular group that contains instructions and activities. Working in pairs during independent activity time encourages students to engage intensively with others. Students tend to sustain their effort longer when they are working in pairs than they do individually. There is tremendous potential for students to learn by interacting with each other, provided that the abilities of the students in each pair are similar. 10 Getting Started Individual work is useful in establishing whether the student can solve problems independently. The types of activities that are appropriate for independent work are: • practice that is related to the most recent strategy lesson; • games and puzzles that are suitable for the student’s strategy stage; • knowledge-practice activities; • problem-solving or knowledge-based worksheets; • students creating their own problems for classmates; • exploring activities or equipment related to a new unit of work. A Model for Informing Students about Independent Group Work Maths Task Board Group names and or shapes may be shifted along to show each daily programme. The icons for activities may be on labels that can be moved around the task board. Teachers may use pictorial representations or writing to further explain the activities. T Pr KA T Pr G T = Teaching Pr = Practice G = Games KA = Knowledge Activity KA before T allows students to revise key knowledge or knowledge being developed before a strategy teaching session. The key to recommended activities during a number lesson The task board may be drawn up on a whiteboard or made out of cardboard and laminated. You may choose to use other existing task boards (for example, a reading task board) and change the symbols or pictures when it is mathematics time. 11 Getting Started Classroom Resources Materials Masters Material Masters are photocopiable resources from which you can make key pieces of cardware that are referred to in the teaching books of this folder. They can be downloaded as PDF files from the NZ Maths website (nzmaths.co.nz). Several commercial firms also produce this cardware. Purchasing it is often a more cost effective option than making it yourself. If a teaching activity requires a Material Master, it will be referenced using a code number. For example, Material Master 5-2 is first introduced in Book 5: Teaching Addition, Subtraction, and Place Value. The suffix -2 indicates that it is the second material master referred to in that book. You will require these Material Masters in order to teach many of the strategy development and knowledge lessons effectively. Ministry Publications Available in Schools • Figure It Out series. Currently available: two number books at level 2, nine books at levels 2–3, nine books at level 3, eight books at levels 3–4, six number books at years 7–8. Under development: two number books at level 3, two number books at levels 3–4, one theme book at levels 3–4, 14 books at years 7–8 (including two on number sense and four on algebra). The available books are referenced throughout the planning forms on pages 21–37. • www.nzmaths.co.nz This website is funded by the Ministry of Education. It contains units of work on all strands of MiNZC, links to useful websites throughout the world, has a “Bright Sparks” section for extending able students, and includes a Numeracy Planning Assistant for planning units of work for students working at stages one to six of the Number Framework. • Connected series • Problem Solving CD • Beginning School Mathematics (BSM) • Development Band Mathematics Useful Hardware The list below provides a basic set of equipment for numeracy. This hardware should be available in every numeracy project class. • Dice (dotted and numbered 1–6) and/or blank cubes (for making your own dice) • Slavonic abacus • Tens frames • Number lines and number strips, used with counters and pegs • Strings of 100 beads in groups of five beads of each colour and supermarket tags • Transparent counters and other types of novelty counters (for example, teddies, fruit) • Iceblock sticks and pipe cleaners (or rubber bands) • Beans, film canisters, and plastic containers (to package hundreds) • Play money • Interlocking cubes, preferably Unifix or similar • Numeral cards • Ice-cream containers • Hundreds board and thousands book • Calculators, overhead projector, and student models • Counting flip boards • Dominoes • Fraction kits, circular and strip models. 12 Getting Started Applying the Strategy Teaching Model The lesson illustrated below is “Subtraction in Parts” from Book 5: Teaching Addition and Subtraction and Place Value, pages 26–27. This lesson is not a script to be imitated rigorously, but simply one possible lesson using the strategy teaching model detailed in this book, on page 6. Using the same activity with different groups of students will require you to take different instructional paths. The lesson agenda should be determined by the responses of students. An important point is not to focus on lesson differences but rather the central place the teaching model occupies in strategy lessons. Student’s Objective for Lesson I am learning to subtract by splitting numbers into parts instead of counting down. Prior Knowledge and Strategies Before attempting to develop students’ advanced part-whole ideas, the teacher knows or establishes that the students can: • identify any number from 1 to 10 on tens frames instantly; • instantly recall addition of single-digit numbers up to a total of 10 and know the related subtraction facts; • identify tens and ones in any two-digit number. Materials Required • A metal plate with two tens frames drawn on it and magnetized counters (or use two blank tens frames [Material Master 4-6] and counters), bundled sticks (ones and tens) or beans in film canisters (ones and tens). Lesson Transcript Actions/Words Commentary Knowledge Check Phase The teacher asks sufficient questions Teacher (T): “I am going to ask you some like these to assure herself that the questions. Cross your arms when you know the students know the basic facts adding answer. to 10 and their related subtraction Six plus what makes 10? Seven plus what makes facts. John’s knowledge/instant recall 10? 10 minus seven gives what?” is not good enough. The teacher notes The teacher continues with similar problems. this for future teaching. All the The teacher opens one film canister of beans students can produce 36 in tens and and shows the students that there are 10 beans in ones. The teacher decides to proceed it. with the lesson. T: “I am going to write a number on the board. I want each of you to make this number of beans using the canisters.” The teacher writes 36 on the whiteboard. The students attempt to show three canisters and six loose beans. Using Materials Phase The teacher organises the students into groups of four. T: “Our problem to solve is: Brian has 14 oranges, and he eats six of them. How many are left? Firstly how could we write this on the board as a take-away? Andrew?” A: “14 minus six”. The teacher begins the teaching with the Using Materials part of the teaching model. She is conscious that she will not supply methods of solution but rather lets the students discuss how they might solve the problem. When getting the students to explain their solutions, she asks questions and links the answers to the materials. 13 Getting Started The teacher writes 14 – 6 on the board. T: “On this tens board, I am putting 10 green magnetic counters. To make 14 altogether, how many yellow counters do I need? Sarah?” S: “Four.” T: “Good. So these 14 counters stand for oranges. Look carefully at the counters and work out how many are left when you take away six counters. Discuss this in your groups. No, Sarah, I don’t want anyone to touch the counters. Just look at them and imagine taking away six. There could be more than one way of doing this.” Group discussions follow. T: “Who thinks they have got the answer? Moana?” M: “You need to take off two.” T: “Why do I need to take off two more?” M: “Because four and two makes six.” T: “I see. Well, Moana, when you take off two more, how many green counters will be left?” M: “Eight, I think.” T: “How do you know there will be eight?” M: “Because if you take two away that leaves eight.” T: “Can you come out the front and show everyone how you did it?” Moana comes to the front. T: “What colour did you take away first?” M: “Yellow.” T: “Why did you choose the yellow counters?” M: “Because there would be 10 left.” T: “Okay, take the yellow ones off the board.” Moana removes the four yellow counters. T: (asking the other students) “How many more counters does Moana need to take?” Z: “Two more … that would leave eight.” T: “Well done, Zoe. Does everyone else understand that? Moana, take the two counters off to show us.” Moana removes two of the green counters. T: “Did anyone do it a different way? Charles, okay. Can you come out to the front? Can you show us which six oranges you ate?” Charles takes six green counters off the left-hand tens frame. T: “So how do you know the answer is eight?” C: “Because four and four is eight.” T: “Where are your four and four, Charles?” Charles points at the four green counters and the four yellow counters.” A set of similar problems follows, with the students having access to materials. 14 The teacher encourages multiple solutions because she is trying to help the students construct the fact that part/whole thinking is superior to counting on and is not concerned about any particular part/whole reasoning at this stage. The teacher observes which students are making the connections. She notices that Helen insists on using counting by ones methods and is unlikely to benefit from the next Using Materials problems. She decides to leave Helen in the group for today. Getting Started T: “Let’s try another one. In your groups get out 34 beans.” The students get three canisters of 10 and four loose beans. The teacher writes 34 – 5 on the whiteboard. T: “Tara has 34 beans, and she eats five. How many beans does she have left?” All groups solve the problem by opening one canister and discuss the answer. T: “Do you all agree on the answer? Could any of your group explain how you worked out the answer? Explain your method to each other.” More discussion occurs within the groups. T: “Who can tell me the answer? Natalie?” N: “It is 29.” T: “Yes. Can you show everyone using the canisters?” Natalie removes the four loose beans, opens a canister, and removes one more. The teacher repeats the previous teaching using the beans with bigger numbers. She notices in the group discussion that Helen continues to use her fingers to count back. She notes that after three examples everyone else appears to be making sense of the part/whole reasoning so she then moves on to the Using Imaging phase of the teaching model. T: “How many are left in the canister?” N: “Nine.” T: “How do you know that, Natalie? Did you count them?” N: “No, I know 10 take away one is nine.” T: “Does everyone understand that? Has anyone got another way?” There is no reply. The students do more examples using materials. Using Imaging Phase T: “Let’s do another one. This time I am going to hide the beans, and you are going to imagine how to solve the problem.” The teacher writes 43 – 5 on the board and shields 43 beans from the students’ view using an icecream container. T: “There are 43 beans under here. What do they look like?” A: “Four canisters and three more beans.” T: “Yes (lifting the ice-cream container and replacing it), you’re right. I’m going to take away five. Imagine what I will do. What is 43 – 5? Discuss this in your groups.” Discussion occurs within the groups. T: “Okay, what is the answer?” S: “38.” In the Using Imaging phase, the teacher deliberately shields the materials from the students to encourage them to think about the groups of beans (and hence numbers) that can be rearranged mentally to solve subtraction problems. Once she has asked for the answer, the teacher folds back to materials to allow any students who could not do the imaging to connect with the materials. After three examples, the teacher notices that seven of the eight students seem to be able to reason correctly with imaging. She infers they have a chance of making connections at the Using Number Properties phase. Helen is not coping even at the Using Materials level. 15 Getting Started T: “Yes, that’s right. Show everyone how you worked it out. Use the beans.” Sarah removes three beans, opens a canister, and removes two more. T: “So why is the answer 38?” S: “I took two out of the canister. That leaves eight.” T: “Where is the 30 part of 38?” S: “There are three canisters and eight.” T: “I see, so three tens is 30, and eight is 38. Good.” The students attempt several more problems, using imaging of the materials. Using Number Properties T: “Now this time we are not going to use any beans.” The teacher writes 75 – 8 on the whiteboard. T: “I have 75 lollies, and I eat eight of them. How many are left? Discuss this in your groups. Remember if your classmates can’t work out the answer, your job is to teach them.” Discussion follows. T: “Okay, who thinks they have the answer?” N: “67.” J: “66.” T: “Hmmm … you can’t both be right. John, how did your group work it out?” J: “Um … First we took off five.” T: “Why did you take off five?” J: “Because that made 70.” T: “But the problem is to take off eight. How many more did you have to take off?” J: “Three … Oh, the answer is 67.” T: “How do you know that?” J: “Well, 70 take off three is 67.” T: “Great, so you all agree now?” M: “Yes, we got 67.” S: “But I got 77?” The teacher notes the number size has been increased to the point where imaging is difficult, and she encourages the students to reason directly on the numbers. So 75 is seen as seven tens and five ones, and the method used in Using Imaging is abstracted to act on numbers as abstract ideas. The teacher collects various answers. She concludes from John’s answer of 66 that either John has made an arithmetic slip or he is having difficulty applying his recall of basic facts to larger numbers. She suspects that Sarah’s answer of 77 is basically from correct reasoning but she has lost track of the fact that the step 10 – 3 decomposes a 10 so there are only 6 tens left. The teacher encourages the students to fold back to imaging to understand why the answer is 67. The teacher writes all the answers 66, 67, and 77 on the board. T: “Well, let’s have a look. Imagine how many tens and ones we would need to make 75.” The teacher points at 75 – 8 on the whiteboard. M: “Seven tens and five ones.” T: “So we have seven tens and five ones, right. How can we remove eight?” S: “Take away five ones first.” T: “Stop for a moment. How many are left?” S: “70.” T: “How many more to go? 16 Getting Started S: “Three.” The teacher writes 70 – 3 on the whiteboard. T: “What is 70 take away three?” M: “67.” T: “Because?” S: “You take three out of one of the tens.” T: “Leaving how many tens?” S: “Six ... I know what I did now.” T: “Well done Sarah, good thinking. Now, let’s try this one.” The teacher writes 85 – 9 on the whiteboard, and the lesson continues. The teacher observes that Andrew, Sarah, Moana, and Charles have understood Using Number Properties, at least for today. John would be able to cope but doesn’t know or apply his basic facts of subtraction well enough. Zoe and Natalie can cope with imaging but have not made the transition to number properties. Helen cannot understand part/whole at any level. The teacher decides to put the Helen with the Advanced Counting group. With the rest she plans to use other activities on part/whole thinking and repeat the use of the teaching model. 17 Getting Started 18 Class Grouping Sheet for NumPA Teacher School Operation Domain Knowledge Hot Spots Emergent One-to-One Counting Counting from One Materials Counting from One Imaging Advanced Counting Early Additive Class Grouping Sheet for NumPA Teacher School Operation Domain Knowledge Hot Spots Counting from One Advanced Counting Early Additive Advanced Additive Advanced Multiplicative Advanced Proportional Getting Started 19 Teacher Knowledge Hot Spots Ms Print School Tapworth Primary Operation Domain Addition/Subtraction One more/one less than the number, up to 1 000 Recognition of fractions symbols and diagrams Groupings with fives and tens, tens/hundreds in whole numbers to 1 000 Counting from One Advanced Counting Early Additive Advanced Additive Skye Tania Leon Allan Zoe Jane Bruce Vey-un Tui Carla Leyton Rawiri Damion Toru Kahu Maria Advanced Multiplicative Advanced Proportional Getting Started 20 Class Grouping Sheet for NumPA Getting Started Transition Emergent to One to One Counting Addition and Subtraction Mathematical Processes (Level 1) • Problem Solving: Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 1) • Make up, tell, and record number stories, up to nine, about given objects and sequence pictures. • Form a set of up to 20 objects. • Read and write any two-digit whole number. • Rote count to at least 50. Knowledge Being Developed The students are learning to: • Identify all of the numbers in the range 0–10. • Say the forwards and backwards number word sequences in the range 0–10. • Say the number before and after a given number in the range 0–10. • Order the numbers in the range 0–10. • Instantly recognise patterns to five, including finger patterns. Exploring Computation and Estimation (Level 1) • Make sensible estimates and check the reasonableness of answers. • Model and explain addition calculations with a sum of up to 20. • Using up to 20 objects, model and explain subtraction calculations. E CA AC EA AA AM Knowledge Activities Teaching Number Knowledge Counting Based Number Mat and Lily Pads: page 2 Tens Frames: page 2 Pipe Cleaner Numbers: page 4 Number Fans: page 4 Counting: page 12 Number Line Flips: page 16 Grouping Based Fabulous Fives: page 23 AP BSM (for Emergent to One-to-one Counting): 2-1-1, 2-1-2, 2-1-4, 2-1-21, 2-1-84, 2-3-7, 2-3-8, 2-3-24, 2-3-55, 3-1-2, 3-1-3, 3-1-4, 3-1-7, 3-1-21, 3-1-22, 3-1-23, 3-1-46, 3-1-47, 3-3-9, 3-3-22, 3-3-48. BSM (for One-to-one Counting): 2-1-5, 3-1-43, 4-1-5, 4-1-22, 4-1-23, 4-1-44, 4-1-45, 4-1-48, 4-1-83, 4-3-5, 4-3-6, 4-3-21, 4-3-44, 4-3-84, 4-3-46, 4-3-24, 4-3-25, 4-3-51, 5-1-3, 5-1-6, 5-1-7, 5-1-8, 5-1-21, 5-1-22, 5-1-46, 5-1-83, 5-3-5, 5-3-22, 5-3-45, 5-3-46, 5-3-82, 5-3-6, 5-3-48, 5-3-7, 5-3-23, 5-3-49, 5-3-83, 6-1-3, 6-1-21, 6-1-4, 6-1-43, 6-1-5, 6-1-6, 6-1-44, 6-1-7, 6-1-45, 6-1-46, 6-1-81, 6-1-82, 6-1-9, 6-2-21, 6-2-22, 6-2-9, 6-2-48, 6-3-3, 7-1-1, 7-1-41, 7-1-42, 7-1-81, 7-1-2, 7-1-43, 7-1-3, 7-1-44, 7-1-46, 7-2-7, 7-2-48, 7-2-49, 7-2-82, 7-3-4, 8-2-8, 8-2-47, 8-2-48, 8-2-83, 9-2-18, 9-2-61, 9-2-85. Strategy Learning Outcomes The students are learning to: • Count a set of objects in the range 1–10. • Form a set of objects in the range 1–10. Strategy Activities Teaching Addition, Subtraction, and Place Value Lucky Dip: page 1 Match It Up: page 1 Counting as We Go: page 2 How Many Now?: page 2 How Many …?: page 2 Loud and Soft: page 2 Tick Tock: page 3 Before and After: page 3 Clapping: page 3 Walk the Bridge: page 3 Up or Down: page 4 Ordering Numerals: page 4 Dice Groups: page 4 How Many Cubes?: page 4 Caterpillar Legs: page 5 Petals and Flower Centres: page 5 Feed the Elephant: page 5 Birthday Cake: page 5 Turtles 5 and …: page 5 Facts to 10: page 6 BSM (for One-to-one Counting) 2-1-5, 4-1-23, 4-3-21, 4-3-44, 4-3-46, 5-1-7. The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 21 Getting Started Transition One-to-One Counting to Counting from One on Materials Addition and Subtraction Mathematical Processes (Level 1) • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 1) • Make up, tell, and record number stories, up to nine, about given objects and sequence pictures. • Form a set of up to 20 objects. • Read and write any two-digit whole number. • Rote count to at least 50. Exploring Computation and Estimation (Level 1) • Make sensible estimates and check the reasonableness of answers. • Model and explain addition calculations with a sum of up to 20. • Using up to 20 objects, model and explain subtraction calculations. E CA AC EA AA AM Key Knowledge Required for Addition and Subtraction Knowledge Activities The students can: • Rote count to 10, at least. • Order all of the numbers in the range 0–10. • Say the forwards and backwards number word sequences in the range 0–10. • Say the number before and after a given number in the range 0–10. • Instantly recognise patterns to five, including finger patterns. Teaching Number Knowledge Counting Based Number Mat and Lily Pads: page 2 Tens Frames: page 2 Pipe Cleaner Numbers: page 4 Number Fans: page 4 Counting: page 12 Number Line Flips: page 16 Grouping Based Fabulous Fives: page 23 BSM (for One-to-one Counting): 2-1-5, 3-1-43, 4-1-5, 4-1-22, 4-1-23, 4-1-44, 4-1-45, 4-1-48, 4-1-83, 4-3-5, 4-3-6, 4-3-21, 4-3-44, 4-3-84, 4-3-46, 4-3-24, 4-3-25, 4-3-51, 5-1-3, 5-1-6, 5-1-7, 5-1-8, 5-1-21, 5-1-22, 5-1-46, 5-1-83, 5-3-5, 5-3-22, 5-3-45, 5-3-46, 5-3-82, 5-3-6, 5-3-48, 5-3-7, 5-3-23, 5-3-49, 5-3-83, 6-1-3, 6-1-21, 6-1-4, 6-1-43, 6-1-5, 6-1-6, 6-1-44, 6-1-7, 6-1-45, 6-1-46, 6-1-81, 6-1-82, 6-1-9, 6-2-21, 6-2-22, 6-2-9, 6-2-48, 6-3-3, 7-1-1, 7-1-41, 7-1-42, 7-1-81, 7-1-2, 7-1-43, 7-1-3, 7-1-44, 7-1-46, 7-2-7, 7-2-48, 7-2-49, 7-2-82, 7-3-4, 8-2-8, 8-2-47, 8-2-48, 8-2-83, 9-2-18, 9-2-61, 9-2-85. Knowledge Being Developed for Addition and Subtraction Knowledge Activities Teaching Number Knowledge Counting Based Number Mat and Lily Pads: page 2 Pipe Cleaner Numbers: page 4 Counting: page 12 Card Ordering: page 13 Number Line Flips: page 16 The students are learning to: Counting Based • Identify numbers in the range 0–20. • Say the forwards and backwards number word sequences in the range 0–20. • Say the number before and after a given number in the range 0–20. • Order numbers in the range 0–20. • Record the results of counting and operations using symbols, pictures, and diagrams. BSM (for Counting from One on Materials): 2-1-5, 6-1-3, 6-1-4, 6-1-7, 6-1-9, 6-1-21, 6-1-46, 6-3-22, 6-3-3, 6-3-4, 6-3-5, 6-3-7, 6-3-49, 7-1-1, 7-1-2, 7-1-41, 7-1-42, 7-1-43, 7-3-9, 8-1-4, 8-1-53. Strategy Learning Outcomes The students are learning to: • Solve simple addition problems to 20 by counting all the objects. • Solve simple subtraction problems from 20 by counting all the objects. Strategy Activities Teaching Addition, Subtraction, and Place Value Adding and Subtracting with Counters: page 7 Adding and Subtracting with One Hand: page 7 Fly Flip: page 8 Using Fives: page 8 Challenging Hand Problems: page 9 Teens and Fingers: page 9 Ones and Tens: page 10 More Ones and Tens: page 11 BSM (for Counting from One on Materials): 6-1-3, 6-1-9, 6-1-46, 6-1-47, 6-1-81, 6-1-82, 7-1-6, 7-1-52, 7-1-53, 7-1-84, 7-1-85, 7-3-4, 7-3-7, 7-3-50, 7-3-51, 7-3-52, 7-3-82, 8-1-5, 8-1-8, 8-1-46, 8-1-51, 8-1-52, 8-3-7, 8-3-8, 8-3-9, 8-3-50, 8-3-51. The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 22 AP Getting Started Transition Counting from One on Materials to Counting from One by Imaging Addition and Subtraction Mathematical Processes (Level 1) • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 1) • Make up, tell, and record number stories, up to nine, about given objects and sequence pictures. • Form a set of up to 20 objects. • Read and write any two-digit whole number. • Rote count to at least 50. Exploring Computation and Estimation (Level 1) • Make sensible estimates and check the reasonableness of answers. • Model and explain addition calculations with a sum of up to 20. • Using up to 20 objects, model and explain subtraction calculations. AC EA AA AP Knowledge Activities The students can: • Identify numbers in the range 0–20. • Say the forwards and backwards number word sequences in the range 0–20. • Say the number before and after a given number in the range 0–20. • Order numbers in the range 0–20. Teaching Number Knowledge Counting Based Pipe Cleaner Numbers: page 4 Knowledge Being Developed for Addition and Subtraction Knowledge Activities Teaching Number Knowledge Counting Based Number Mat and Lily Pads: page 2 Number Fans: page 4 Counting: page 12 Card Ordering: page 13 Lucky Dip: page 15 Number Line Flips: page 16 Grouping Based Fabulous Fives: page 23 Using the Slavonic Abacus to Reinforce Five Grouping : page 35 Patterns to 10: page 36 BSM (for Counting from One by Imaging): 2-1-5, 6-1-3, 6-1-4, 6-1-7, 6-1-9, 6-1-21, 6-1-46, 6-3-22, 6-3-3, 6-3-4, 6-3-5, 6-3-7, 6-3-49, 7-1-1, 7-1-2, 7-1-41, 7-1-42, 7-1-43, 7-3-9, 8-1-4, 8-1-53. Strategy Learning Outcomes The students are learning to: • Solve simple addition problems by counting all the objects in their head (by imaging). • Solve simple subtraction problems by counting all the objects in their head (by imaging). CA AM Key Knowledge Required for Addition and Subtraction The students are learning to: Counting Based • Identify numbers in the range 0–20, at least. • Say the forwards and backwards number word sequences in the range 0–20, at least. • Say the number before and after a given number in the range 0–20, at least. • Order numbers in the range 0–20, at least. Grouping Based • Recall groupings with five. • Recall groupings within five and 10. • Instantly recognise patterns to 10 (doubles and five based), including finger patterns. • Recall doubles to 10. • Record the results of counting and operations using symbols, pictures, and diagrams. E Strategy Activities Teaching Addition, Subtraction, and Place Value Using One Hand: page 11 Using Tens Frames: page 12 Both Hands: page 12 Imaging with Tens Frames: pages 13 What’s Hidden?: page 14 Imaging Many Hands: page 14 Making Tens: page 14 Crossing the Five Barrier: page 15 Fingers Again: page 16 Ten Sweets per Packet: page 16 BSM (for Counting from One by Imaging): 6-1-3, 6-1-9, 6-1-46, 6-1-47, 6-1-81, 6-1-82, 7-1-6, 7-1-52, 7-1-53, 7-1-84, 7-1-85, 7-3-4, 7-3-7, 7-3-50, 7-3-51, 7-3-52, 7-3-82, 8-1-5, 8-1-8, 8-1-46, 8-1-51, 8-1-52, 8-3-7, 8-3-8, 8-3-9, 8-3-50, 8-3-51. The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 23 Getting Started Transition Counting from One on Materials to Counting from One by Imaging Multiplication and Division/Fractions Mathematical Processes (Level 1) • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 1) • Make up, tell, and record number stories, up to nine, about given objects and sequence pictures. • Form a set of up to 20 objects. • Read and write any two-digit whole number. • Rote count to at least 50. Exploring Computation and Estimation (Level 1) • Make sensible estimates and check the reasonableness of answers. • Find by practical means, one half and one quarter of a shape, and a half of a set of objects. Knowledge Being Developed for Mult/Div, Fractions Knowledge Activities Teaching Number Knowledge Counting Based Counting: page 12 Skip-counting on the Number Line: page 12 Beep: page 13 Grouping Based Fabulous Fives: page 23 Using the Slavonic Abacus to Reinforce Five Grouping : page 35 Patterns to 10: page 36 Teaching Number Knowledge Counting Based Number Mat and Lily Pads: page 2 Number Fans: page 4 Card Ordering: page 13 Lucky Dip: page 15 Number Line Flips: page 16 Teaching Fractions, Decimals and Percentages Refer to Tasks for Key Knowledge on page 1. Strategy Learning Outcomes Strategy Activities Teaching Fractions, Decimals, and Percentages Fair Shares: page 2 BSM (for Counting from One by Imaging): 8-3-6, 8-3-47, 8-3-48, 8-3-49, 8-3-82. The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 24 AC EA AM Knowledge Activities The students are learning to: • Solve simple multiplication and division problems by counting all the objects. • Find halves and quarters of shapes or sets of objects to 20 by equal sharing of the objects. • Find halves, and quarters of shapes and objects, e.g., half a glass of water. CA AA Key Knowledge Required for Multiplication/Division and Fractions The students can: Counting Based • Identify numbers in the range 0–20, at least. • Say the forwards and backwards number word sequences in the range 0–20, at least. • Say the number before and after a given number in the range 0–20, at least. • Order numbers in the range 0–20, at least. Grouping Based • Recall doubles to 10. The students are learning to: Counting Based • Say the forwards and backwards skip-counting sequences in the range 0–20 for twos and fives. Grouping Based • Instantly recognise patterns to 10 (doubles and five-based), including finger patterns. • Recall doubles to 10. • Record the results operations using symbols, pictures, and diagrams. E AP Getting Started Transition Counting from One by Imaging to Advanced Counting Addition and Subtraction Mathematical Processes • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 2) • Read any three-digit whole number. • Explain the meaning of the digits in two- or three-digit whole numbers. • Order any set of three or more whole numbers (up to 99). • Write and solve comparison problems. Exploring Computation and Estimation (Level 2) • Make sensible estimates and check the reasonableness of answers. • Recall the basic addition and subtraction facts. • Mentally perform calculations involving addition and subtraction. • Write and solve story problems that involve whole numbers using addition and subtraction. Knowledge Activities The students can: Counting Based • Identify numbers in the range 0–20, at least. • Say the forwards and backwards number word sequences in the range 0–20, at least. • Say the number before and after a given number in the range 0–20, at least. • Order numbers in the range 0–20, at least. Grouping Based • Instantly recognise patterns to 10 (doubles and five based), including finger patterns. Teaching Number Knowledge Counting Based Number Mat and Lily Pads: page 2 Counting: page 12 Grouping Based Fabulous Fives: page 23 Using the Slavonic Abacus to Reinforce Five Grouping : page 35 Patterns to 10: page 36 BSM (for Counting from One by Imaging): 2-1-5, 6-1-3, 6-1-4, 6-1-7, 6-1-9, 6-1-21, 6-1-46, 6-3-22, 6-3-3, 6-3-4, 6-3-5, 6-3-7, 6-3-49, 7-1-1, 7-1-2, 7-1-41, 7-1-42, 7-1-43, 7-3-9, 8-1-4, 8-1-53. Knowledge Being Developed for Addition and Subtraction The students are learning to: Counting Based • Say the forwards and backwards number word sequences in the range 0–100, at least. • Say the number before and after a given number in the range 0–100, at least. • Identify all of the numbers in the range 1–100, at least. Grouping Based • Recall groupings with 10 and the patterns of teens. • Recall groupings within 20. • Recall the number of tens within decades. • Recall the names for 10, e.g., 4 + 6 and 5 + 5, and the teen facts, e.g. 16 = 10 + 6. • Recall the doubles to 20, and the corresponding halves. • Record the results of mental addition and subtraction using equations. Knowledge Activities Teaching Number Knowledge Counting Based “Teen” and “Ty” Numbers: page 3 Number Fans: page 4 Counting: page 12 Card Ordering: page 13 Lucky Dip: page 15 Number Line Flips: page 16 Squeeze - Guess My Number: page 17 Hundreds Boards and Thousands Book: page 17 Bead Strings: page 18 Grouping Based Slavonic Abacus: page 24 Tens and Ones: page 25 Up to Ten: page 33 Double Trouble: page 34 Number Boggle: page 35 Tens Frames Again: page 36 BSM (for Advanced Counting): 9-1-4, 9-1-6, 9-1-12, 9-1-42, 9-1-83, 9-3-9, 9-3-48, 9-3-84, 10-1-3, 10-1-4, 10-1-42, 10-1-47, 10-3-6, 10-3-9, 10-3-11, 10-3-56, 11-1-6, 11-1-47, 11-1-48, 11-1-84, 11-3-4, 11-3-8, 11-3-9, 11-3-13, 12-1-1, 12-1-5, 12-1-44, 12-1-45, 12-1-84. Figure It Out Number, Level 2, Book 1, pages 4–5 The students are learning to: • Solve addition problems by counting on in their head from the largest number, using supporting materials, if necessary, but progressing to imaging the count on number. • Solve subtraction problems by counting back from the largest number in their head, using supporting materials, if necessary, but progressing to imaging the count back number. • Solve addition and subtraction problems by counting on in ones and tens, e.g., 43 + 32 as 43, 53, 63, 73, 74, 75. CA AC EA AA AM Key Knowledge Required for Addition and Subtraction Strategy Learning Outcomes E Strategy Activities Teaching Addition, Subtraction, and Place Value Number Tiles: page 17 The Number Strip: page 18 The Bears’ Picnic: page 19 Frog Jumps: page 19 The Bigger Number First: page 20 Change Unknown: page 20 Counting Back: page 21 Adding Tens: page 21 Subtracting Tens: page 22 Adding Ones and Tens: page 23 Subtracting Ones and Tens: page 23 The Missing Ones and Tens: page 24 The Thousands Book: page 24 BSM (for Advanced Counting): 9-1-8, 9-1-12, 9-1-13, 9-1-51, 9-1-52, 9-1-53, 9-3-11, 9-3-12, 9-3-13, 9-3-48, 10-1-6, 10-1-41, 10-1-42, 10-1-47, 10-1-49, 10-3-8, 10-3-11, 10-3-53, 10-3-56, 10-3-86, 11-1-3, 11-1-7, 11-1-47, 11-1-48, 11-1-83, 11-3-4, 11-3-10, 12-1-1, 12-1-5, 12-1-44. Figure It Out Basic Facts Levels 2–3: pages 1, 3, 5, 21 Number Levels 2–3: page 1 Basic Facts Level 3: page 1 The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 25 AP Getting Started Transition Counting from One by Imaging to Advanced Counting Multiplication and Division/Fractions Mathematical Processes • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 2) • Write and solve story problems that involve halves, quarters, thirds, and fifths. Exploring Computation and Estimation (Level 2) • Make sensible estimates and check the reasonableness of answers. • Write and solve story problems that involve whole numbers, using addition, subtraction, multiplication, or division. CA AC EA AA AM Key Knowledge Required for Multiplication/Division and Fractions Knowledge Activities The students can: Counting Based • Say the forwards and backwards skip-counting sequences in the range 0–20 for twos and fives. Grouping Based • Instantly recognise patterns to 10 (doubles and five-based), including finger patterns. • Recall doubles to 10. Teaching Number Knowledge Grouping Based Fabulous Fives: page 23 Using the Slavonic Abacus to Reinforce Five Grouping: page 35 Patterns to 10: page 36 Teaching Multiplication and Division Refer to Tasks for Key Knowledge on page 1 Teaching Fractions, Decimals, and Percentages Refer to Tasks for Key Knowledge on page 1. Knowledge Being Developed for Multiplication/Division and Fractions The students are learning to: Counting Based • Say the forwards and backwards skip-counting sequences in the range 0–100 for twos, fives, and tens. • Identify the symbols for halves, quarters, thirds, and fifths. Grouping Based • Recall the number of tens in decades, e.g., four tens in 40. Knowledge Activities Teaching Number Knowledge Fraction Pieces: page 6 Counting: page 12 Skip-counting on the Number Line: page 12 Beep: page 13 Using Calculators: page 15 Strategy Learning Outcomes The students are learning to: • Solve multiplication problems by skip-counting in twos, fives, and tens, for example, 4 x 5 as 5, 10, 15, 20, by tracking the count with their fingers, if necessary, but progressing to tracking the count by imaging. • Find simple fractions of shapes and lengths, starting with halves and quarters. • Find a fraction of a number by sharing out the objects equally, moving towards anticipating the sharing by imaging or skip-counting, for example, 1 of 12 by 4 sharing 12 objects into four sets or by trial skip-counting 3, 6, 9, 12. Strategy Activities Teaching Multiplication and Division Introduction: page 1 Number Strips: page 2 Teaching Fractions, Decimals, and Percentages Fair Shares: page 2 BSM (for Advanced Counting Multiplication): 9-1-45, 9-1-46, 9-1-47, 9-1-83, 9-1-84, 9-3-10, 9-3-53, 10-1-4, 11-1-6, 11-3-12, 11-3-13, 11-3-54. BSM (for Advanced Counting Fractions): 9-2-18, 9-2-61, 9-2-85, 9-3-10, 9-3-53, 11-2-18, 11-2-59. Figure It Out Multiplication: Basic Facts, Levels 2–3: pages 7, 9 Fractions: Number, Levels 2–3: pages 17, 18 The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 26 E AP Getting Started Transition Counting from Advanced Counting to Early Additive Addition and Subtraction Mathematical Processes • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 2) • Read any three-digit whole number. • Explain the meaning of the digits in two- or three-digit whole numbers. • Order any set of three or more whole numbers (up to 99). Exploring Computation and Estimation (Level 2) • Make sensible estimates and check the reasonableness of answers. • Recall the basic addition and subtraction facts. • Mentally perform calculations involving addition and subtraction. • Write and solve story problems that involve whole numbers, using addition, subtraction, multiplication, or division. Knowledge Activities The students can: Grouping Based • Recall groupings with 10 and the patterns of teens. • Recall groupings within 20. • Recall the number of tens within decades. • Recall the names for 10, e.g., 4 + 6 and 5 + 5, and the teen facts, e.g., 16 = 10 + 6. • Recall the doubles to 20, and the corresponding halves. Teaching Number Knowledge Counting Based “Teen” and “Ty” Numbers: page 3 Grouping Based Up to Ten: page 33 Double Trouble: page 34 Number Boggle: page 35 Tens Frames Again: page 36 BSM (for Advanced Counting): 9-1-4, 9-1-6, 9-1-12, 9-1-42, 9-1-83, 9-3-9, 9-3-48, 9-3-84, 10-1-3, 10-1-4, 10-1-42, 10-1-47, 10-3-6, 10-3-9, 10-3-11, 10-3-56, 11-1-6, 11-1-47, 11-1-48, 11-1-84, 11-3-4, 11-3-8, 11-3-9, 11-3-13, 12-1-1, 12-1-5, 12-1-44, 12-1-45, 12-1-84. Figure It Out Number, Level 2, Book 1, pages 4–5 Knowledge Being Developed for Addition and Subtraction The students are learning to: Counting Based • Identify all of the numbers in the range 0–1 000. • Say the forwards and backwards number word sequences by ones, tens, and hundreds in the range 0–1 000. • Say the number 1, 10, or 100 more or less than a given number • Order numbers in the range 0–1 000. Grouping Based • Recall the number of groupings of tens that can be made from a three-digit number. • Recall the number of tens and hundreds in centuries and thousands. • Recall addition and subtraction facts to 20. • Round three-digit whole numbers to the nearest 10, or hundred. • Identify the multiples of 10 that add to 100. Knowledge Activities Teaching Number Knowledge Counting Based Number Fans: page 4 Place Value Houses: page 5 Number Hangman: page 5 Counting: page 12 Card Ordering: page 13 Arrow Cards: page 14 Lucky Dip: page 15 Number Line Flips: page 16 Squeeze – Guess My Number: page 17 Hundreds Boards and Thousands Book: page 17 Bead Strings: page 18 Who is the Richest?: page 19 Grouping Based Slavonic Abacus: page 24 Tens and Ones: page 25 Close to 100: page 25 Nudge: page 26 Estimating: page 27 Traffic Lights: page 27 Zap: page 28 Number Mats and Number Fans: page 36 Loopy: page 39 Addition Flash Cards: page 39 BSM (for Early Additive Part-Whole): 11-3-13, 11-3-54, 11-3-55, 12-1-3, 12-1-4, 12-1-7, 12-1-9, 12-1-46, 12-1-43, 12-1-48, 12-1-50, 12-1-86, 12-1-87, 12-3-6, 12-3-7, 12-3-45, 12-3-46, 12-3-49, 12-3-65, 12-3-81, 12-3-82, 12-3-83. Figure It Out Number, Level 2, Book 1, page 1 Book 2, pages 2, 4–6 The students are learning to: • Solve adding and subtracting problems in their head by working out the answer from basic facts they know, e.g., 8 + 7 as 8 + 8 – 1 (using doubles), or (5 + 3) + (5 + 2) = 5 + 5 + 5 (using fives), or 10 + 5 (making tens). • Solve adding and subtracting problems with two- and three-digit numbers using groupings of 10 and 100, for example, 43 + 25 as (40 +20) + (3 +5), or (43 + 20) + 5 (standard place value partitioning), or 39 + 26 as 40 +25 ( tidy numbers with compensation) CA AC EA AA AM Key Knowledge Required for Addition and Subtraction Strategy Learning Outcomes E Strategy Activities Teaching Addition, Subtraction, and Place Value Make Ten: page 25 Grouping then Subtracting: page 26 Subtraction in Parts: page 26 Up over the Tens: page 27 Adding in Parts: page 28 Comparisons: page 29 More Comparisons: page 29 How Many Ten-dollar Notes?: page 30 BSM (for Early Additive Part-Whole) 11-3-12, 11-3-54, 11-3-55, 12-1-6, 12-1-9, 12-1-46, 12-1-47, 12-1-52, 12-1-55, 12-1-56, 12-1-85, 12-1-89, 12-3-8, 12-3-13, 12-3-47, 12-3-52, 12-3-53, 12-3-81. Figure It Out Number, Level 2, Book 1, pages 2–3, 6–11, 16, 22–24 Number, Level 2, Book 2, pages 1, 3, 7–15 Basic Facts, Levels 2–3: pages 12, 15, 19 Number, Levels 2–3: page 2 Basic Facts, Level 3: pages 4, 5, 6, 7, 8 Number, Level 3: page 19 Number, Years 7–8, Link, Book One, pages 1–2, 6–8, 15–17 The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 27 AP Getting Started Transition Advanced Counting to Early Additive Multiplication and Division/Fractions Mathematical Processes • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 2) • Write and solve story problems that involve halves, quarters, thirds, and fifths. Knowledge Activities Knowledge Being Developed for Multiplication/Division and Fractions The students are learning to: Counting Based • Say the forwards and backwards skip-counting sequences in the range 0–100 for twos, threes, fives, and tens at least. • Identify the symbols for halves, quarters, thirds, fifths, and tenths including fractions greater than 1. Knowledge Activities Teaching Multiplication and Division Refer to Tasks for Key Knowledge on page 4. Teaching Fractions, Decimals, and Percentages Refer to Tasks for Key Knowledge on page 4. Teaching Number Knowledge Creating Fractions: page 6 Counting: page 12 Fraction Pieces: page 6 Skip-counting on the Number Line: page 12 Beep: page 13 Using Calculators: page 15 Order fractions with like denominators, e.g., 1 and 2 . 4 4 Grouping Based • Recall groupings of two in numbers to 20, groupings of five in numbers to 50, and groupings of 10 in numbers to 1 000. • Automatically recall the multiplication and division facts for the multiples of 2, 5, and 10. • Record the results of mental multiplication calculations using equations and diagrams. Strategy Learning Outcomes The students are learning to: • Use repeated addition and adding and subtracting from known facts to solve multiplication problems with twos, threes, fours, fives, and tens at least. • Use repeated addition and adding and subtracting from known addition or subtraction facts to solve simple division problems of these types: sharing , e.g., 16 lollies among four friends sets of, e.g., 16 lollies in packets of four • Find fractions of shapes and lengths, including fractions greater than one, e.g., 5 of a circle. • • 2 Put fractions in order from smallest to largest. Find a fraction of a set by skip-counting, by using repeated addition, or by adding and subtracting from known addition or subtraction facts, for example, 4 + 4 + 4 = 12, so 1 of 12 is 4. 3 Strategy Activities Teaching Multiplication and Division Introduction: pages 4–5 Animal Arrays: page 5 Pirate Crews: page 7 Biscuit Boxes: page 8 Twos, Fives, and Tens: page 10 Teaching Fractions, Decimals, and Percentages Wafers: page 5 Animals: page 7 Fraction Circles: page 8 Hungry Birds: page 11 BSM (for Early Additive Part-Whole Multiplication): 11-3-12, 11-3-55, 12-3-13. BSM (for Early Additive Part-Whole Fractions): 12-3-7, 12-3-49, 12-3-50, 12-3-83, 12-3-84. Figure It Out Multiplication: Number, Level 2, Book 1, pages 12–15, 19 Number, Level 2, Book 2, pages 16–19 Basic Facts, Levels 2–3 pages: 10, 14, 20, 22 Number, Levels 2–3 page 13 Fractions: Number, Level 2, Book 1, pages 17–18, 21 Number, Level 2, Book 2, pages 20–21, 24 Number, Levels 2–3, page 19 Number, Level 3, page 9 Number, Years 7–8, Link, Book One, pages 20–24 The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 28 E CA AC EA AA Key Knowledge Required for Multiplication/Division and Fractions The students can: Counting Based • Say the forwards and backwards skip-counting sequences in the range 0–100 for twos, fives, and tens. • Identify the symbols for halves, quarters, thirds, and fifths. Grouping Based • Recall the number of tens in decades, e.g., four tens in 40. • Exploring Computation and Estimation (Level 2) • Make sensible estimates and check the reasonableness of answers. • Write and solve story problems that involve whole numbers using addition, subtraction, multiplication, or division. AM AP Getting Started Transition Early Additive to Advanced Additive Addition and Subtraction Mathematical Processes • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 3) • Explain the meaning of the digits in any whole number. Exploring Computation and Estimation (Levels 2 and 3) • Make sensible estimates and check the reasonableness of answers. • Mentally perform calculations that involve addition and subtraction. • Write and solve story problems that involve whole numbers using addition, subtraction, multiplication, or division. E CA AC EA AA Key Knowledge Required for Addition and Subtraction The students can: Counting Based • Identify all of the numbers in the range 0–1 000. • Say the forwards and backwards number word sequences by ones, tens, and hundreds in the range 0–1 000. • Say the number 1, 10, or 100 more or less than a given number • Order numbers in the range 0–1 000. Grouping Based • Recall the number of groupings of tens that can be made from a three-digit number. • Recall the number of tens and hundreds in centuries and thousands. • Recall addition and subtraction facts to 20. • Round three-digit whole numbers to the nearest ten or hundred. Knowledge Activities Teaching Number Knowledge AM Counting Based Counting: page 12 Grouping Based AP Slavonic Abacus: page 24 Tens and Ones: page 25 Number Mats and Number Fans: page 36 Loopy: page 39 Addition Flash Cards: page 39 BSM (for Early Additive Part-Whole): 11-3-13, 11-3-54, 11-3-55, 12-1-3, 12-1-4, 12-1-7, 12-1-9, 12-1-46, 12-1-43, 12-1-48, 12-1-50, 12-1-86, 12-1-87, 12-3-6, 12-3-7, 12-3-45, 12-3-46, 12-3-49, 12-3-65, 12-3-81, 12-3-82, 12-3-83. Figure It Out Number, Level 2, Book 1, page 1 Book 2, page 2, 4-6 Knowledge Being Developed for Addition and Subtraction The students are learning to: Counting Based • Identify all of the numbers in the range 0–1 000 000. • Say the forwards and backwards whole-number word sequences by ones, tens, hundreds, and thousands in the range 0–1 000 000, including finding numbers that are 10, 100, and 1 000 more or less than a given number. • Order whole numbers in the range 0–1 000 000. Grouping Based • Recall addition and subtraction facts to 20. • Recall groupings within 1000, e.g., 240 + 760. • Record the results of mental calculation using addition and subtraction equations and diagrams. • Recall how many tens and hundreds there are in four-digit numbers. • Round whole numbers to the nearest ten, hundred, or thousand. • Carry out column addition and subtraction with whole numbers of up to four digits. Knowledge Activities Teaching Number Knowledge Counting Based Number Fans: page 4 Place Value Houses: page 5 Number Hangman: page 5 Card Ordering: page 13 Arrow Cards: page 14 Lucky Dip: page 15 Rocket-Where Will It Fit?: page 16 Number Line Flips: page 16 Squeeze - Guess My Number: page 17 Hundreds Boards and Thousands Book: page 17 Bead Strings: page 18 Who is the Richest?: page 19 Grouping Based Close to 100: page 25 Nudge: page 26 Estimating: page 27 Traffic Lights: page 27 Zap: page 28 Swedish Rounding: page 30 Bridges: page 37 Bowl a Fact: page 37 Figure It Out Number, Years 7–8, Link, Book Two, pages 4, 16 The students are learning to: • Choose appropriately from a full range of strategies to solve addition and subtraction problems mentally, including: Compensation, e.g., 632 – 179 as 632 – 180 + 1 Place Value, e.g., 273 – 106 as (273 – 100) - 6 Compatible Numbers, e.g., 47 + 86 + 53 as (47 + 53) + 86 Reversibility, e.g., 903 – 798 as 798 + = 903. Equal Additions, e.g., 754 – 529 as 755 – 530. Decomposition, e.g., 82 – 48 as 712 - 48. Strategy Activities Teaching Addition, Subtraction, and Place Value Saving Hundreds: page 31 Jumping the Number Line: page 32 Don’t Subtract – Add!: page 33 Problems like … page 33 How Many Tens and Hundreds?: page 34 Problems like … page 35 When One Number Is Near a Hundred: page 36 Problems like … page 36 Equal Additions: page 37 People’s Ages: page 38 A Balancing Act: page 38 Near Doubles: page 39 Three or More at a Time: page 40 Problems like … page 40 A Standard Written Form for Addition: page 41 Large Numbers Roll Over: page 41 Decomposition: page 42 Mixing the Methods: page 43 Mental or Written?: page 44 Estimation as a Check: page 44 Figure It Out Basic Facts, Levels 2–3: page 4 Number, Levels 2–3: pages 6, 10, 11, 12 Basic Facts, Level 3: page 9 Number, Level 3: pages 6, 18, 20 Number, Levels 3–4: pages 13, 14 Number, Years 7–8, Link, Book One, pages 9–10, 14, 18–19 Number, Years 7–8, Link, Book Two, pages 8, 10–12, 15 Number Sense, Years 7–8, Link, Book One, page 15 Strategy Learning Outcomes • Use pencil and paper or calculators to work out addition and subtraction answers where the numbers are large or untidy. The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 29 Getting Started Transition Early Additive to Advanced Additive Multiplication and Division Mathematical Processes • Problem Solving: Devise and use problem solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 3) • Explore number patterns showing multiples. Exploring Computation and Estimation (Levels 2 and 3) • Make sensible estimates and check the reasonableness of answers. • Recall the basic multiplication facts. • Write and solve problems that involve whole numbers, using addition, subtraction, multiplication, or division. • Write and solve problems that involve whole numbers and decimals and that require a choice of one or more of the four arithmetic operations. Key Knowledge Required for Multiplication and Division The students can: Counting Based • Say the forwards and backwards skip-counting sequences in the range 0–100 for twos, threes, fives, and tens at least. Grouping Based • Recall groupings of two in numbers to 20, groupings of five in numbers to 50, and groupings of 10 in numbers to 1 000. • Automatically recall the multiplication and division facts for the multiples of two, five, and 10. • Recall the number of groupings of tens that can be made from a three-digit number. • Recall the number of tens and hundreds in centuries and thousands. Knowledge Activities Knowledge Being Developed for Multiplication and Division The students are learning to: Grouping Based • Recall groupings of twos, threes, fives, and tens that are in numbers to 100 and the resulting remainders. • Recall groupings of 10 and 100 that can be made from a four- digit number. • Recall all the basic multiplication and division facts. • Recall multiplication facts for squares to 100. • Record the results of mental calculation using multiplication and division equations and diagrams. Knowledge Activities Teaching Number Knowledge Skip-counting on the Number Line: page 12 Beep: page 13 Using Calculators: page 15 Zap: page 28 Tens in Hundreds and More: page 29 Number Mats and Number Fans: page 36 Bowl a Fact: page 37 In and Out: page 38 Multiplication Madness: page 39 Loopy: page 39 Strategy Learning Outcomes The students are learning to: • Solve multiplication and division problems from other facts I know, using a variety of strategies, including: Doubling, e.g., 4 x 6 as 2 x 6 x 2 Adding and subtracting, e.g., 9 x 7 as 10 x 7 – 7 Reversing, e.g., 56 ÷ 8 as 8 x = 56 Rounding, e.g., 19 x 4 as 20 x 4 – 4 • Multiply by tens, hundreds, thousands, and other multiples of ten. • Change the order of the factors to make a multiplication problem easier, e.g., 26 x 3 as 3 x 26. • Solve division problems of the types: equal sharing finding the number of equivalent sets Teaching Number Knowledge Counting: page 12 Teaching Multiplication and Division Refer to Tasks for Key Knowledge on page 12. Strategy Activities Teaching Multiplication and Division Introduction: page 12 Fun with Fives: page 13 Multiplying Tens: page 14 A Little Bit More/A Little Bit Less: page 16 Turn Abouts: page 18 Long Jumps: page 19 Goesintas: pages 21 Figure It Out Basic Facts, Levels 2–3: pages 11, 13, 15, 17, 18, 23, 24 Number, Levels 2–3: pages 14, 15, 16 Basic Facts, Level 3: pages 10, 11, 12, 15, 16, 17, 18, 19, 21, 22, 23, 24 Number, Level 3: page 7 Basic Facts, Levels 3–4: page 20 Number, Years 7–8, Link, Book One, pages 3–5, 11–13 Number, Years 7–8, Link, Book Two, pages 2–3 Number Sense, Years 7–8, Link, Book One, pages 7–9, 13 The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 30 E CA AC EA AA AM AP Getting Started Transition Early Additive to Advanced Additive Fractions and Decimals Mathematical Processes • Problem Solving: Devise and use problem solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 3) • Explain the meaning of the digits in decimal numbers with up to three decimal places. • Order decimals with up to three decimal places. Exploring Computation and Estimation (Level 3) • Make sensible estimates and check the reasonableness of answers. • Write and solve problems that involve whole numbers and decimals that require a choice of one or more of the four arithmetic operations. • Solve practical problems that require finding fractions of whole number and decimal amounts. Key Knowledge Required for Fractions and Decimals Knowledge Activities The students can: Counting Based • Identify the symbols for halves, quarters, thirds, fifths, and tenths including fractions greater than 1. Teaching Number Knowledge Creating Fractions: page 6 Counting: page 12 Fraction Pieces: page 6 Teaching Fractions, Decimals, and Percentages Refer to Tasks for Key Knowledge on pages 13–14. • Order fractions with like denominators, e.g., 1 and 2 . 4 4 Knowledge Activities Teaching Number Knowledge Number Fans: page 4 Place Value Houses: page 5 Fraction Pieces: page 6 Creating Fractions: page 6 More Geoboard Fractions: page 7 Non-Unit Fractions: page 7 Packets of Lollies: page 8 Reading Decimal Fractions: page 9 Skip-counting on the Number Line: page 12 Beep: page 13 Card Ordering: page 13 Arrow Cards: page 14 Lucky Dip: page 15 Using Calculators: page 15 Rocket – Where Will It Fit?: page 16 Number Line Flips: page 16 Squeeze – Guess My Number: page 17 Bead Strings: page 18 Super Liquorice: page 21 Zap: page 28 Strategy Learning Outcomes The students are learning to: • Use a variety of multiplication and division strategies to solve problems that involve finding a fraction of a whole number amount, for example, 1 of 27 from 27 ÷ 3 = 9. • 3 Use multiplication and division to compare the size of fractions with whole numbers, especially fractions greater than one, e.g., Position of 17 on a numbers line showing 0–10? • • 3 Use multiplication and division to create equivalent ratios, e.g., 2:3 as 10:15. Use symmetry, area, and volume to find simple fractions of shapes, lengths, and objects. CA AC EA AA AM AP Grouping Based • Recall groupings of two in numbers to 20, groupings of five in numbers to 50, and groupings of 10 in numbers to 1 000. • Automatically recall the multiplication and division facts for the multiples of two, five, and 10. Knowledge being Developed for Fractions and Decimals The students are learning to: Counting Based • Identify decimals to three places. • Identify symbols for any fraction, including tenths, hundredths, thousandths, and those greater than 1. • Say the forwards and backwards word sequences for halves, quarters, thirds, fifths, and tenths. • Order unit fractions for halves, quarters, thirds, fifths, and tenths. Grouping Based • Recall the number of tenths and hundredths in decimals to two places. • Round decimals with up to two places to the nearest whole number. E Strategy Activities Teaching Fractions, Decimals, and Percentages Birthday Cakes: page 14 Fractional Blocks: page 16 Seed Packets: page 18 Trains: page 20 Figure It Out Number, Level 2, Book 2, pages 22–23 Number, Levels 2–3: pages 20, 21, 22, 23, 24 Number, Level 3: pages 10, 11 Number, Levels 3–4: page 3 Number, Years 7–8, Link, Book Two, pages 1, 22–23 The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful. 31 Getting Started Transition Advanced Additive to Advanced Multiplicative Addition and Subtraction Mathematical Processes • Problem Solving: Devise and use problem solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 3) • Explain the meaning of the digits in any whole number. • Explain the meaning of the digits in decimal numbers with up to three decimal places. • Order decimals with up to three decimal places. Exploring Computation and Estimation (Levels 2 and 3) • Make sensible estimates and check the reasonableness of answers. • Mentally perform calculations that involve addition and subtraction. • Write and solve problems that involve whole numbers and decimals that require a choice of one or more of the four arithmetic operations. Key Knowledge Required for Addition and Subtraction Knowledge Activities The students can: Counting Based • Identify decimals to three places. • Identify symbols for any fraction, including tenths, hundredths, thousandths, and those greater than 1. Grouping Based • Recall the number of tenths and hundredths in decimals to two places. • Round decimals with up to two places to the nearest whole number. Teaching Number Knowledge Counting Based Lucky Dip: page 15 Number Line Flips: page 16 Hundreds Boards and Thousands Book: page 17 Bead Strings: page 18 Who is the Richest?: page 19 Grouping Based Estimating: page 27 Traffic Lights: page 27 Bridges: page 37 Bowl a Fact: page 37 Figure It Out Number, Years 7–8, Link, Book Two, pages 4, 16 Knowledge Being Developed for Addition and Subtraction The students are learning to: Counting Based • Say the forwards and backwards decimal word sequences by thousandths, hundredths, tenths, ones, tens, etc., starting at any whole number in common usage. • Say the number one-thousandth, one-hundredth, onetenth, one, and ten, etc., before and after any given whole number in common usage. • Order decimals to three places. Grouping Based • Recall the number of groupings of tens, hundreds, and thousands that can be made from a number of up to seven digits. • Round whole numbers and decimals, with up to two places, to the nearest whole number, or tenth. • Record the results of mental calculations using equations, and diagrams, for example, empty number line. • Carry out column addition and subtraction for whole numbers. Knowledge Activities Teaching Number Knowledge Counting Based Number Fans: page 4 Place Value Houses: page 5 Number Hangman: page 5 Card Ordering: page 13 Arrow Cards: page 14 Rocket-Where Will It Fit?: page 16 Squeeze - Guess My Number: page 17 Grouping Based Close to 100: page 25 Nudge: page 26 Zap: page 28 Swedish Rounding: page 30 Figure It Out Number, Years 7–8, Link, Book Two, pages 17–19 Number, Years 7–8, Level 4, Book Three, pages 13, 15, 22, 24 Number, Years 7–8, Level 4, Book Four, page 3 Number, Years 7–8, Level 4, Book Four, pages 12–13, 24 Strategy Learning Outcomes The students are learning to: • Use a broad range of mental strategies to solve addition and subtraction problems with whole numbers and decimals, including: Tidy numbers, e.g., 3.1 – 2.79 as (3.1 – 2.8) + 0.01 Place value, e.g., 3.06 + 2.7 as (3.06 + 2) + 0.7 Reversibility and commutativity, e.g., 6.7 – 4.9 as 4.9 + = 6.7, and 2.3 + 4.8 as 4.8 + 2.3. Exploring equal addition, e.g., 4.2 – 2.8 as 4.4 – 3.0 Exploring decomposition, e.g., 7.1 – 2.8 as 6.11 – 2.8. Using negative numbers, e.g., 64 – 38 as 4 – 8 = - 4, 60 – 30 = 30, 30 – 4 = 26. • Use a pencil and paper or a calculator to add and subtract decimals and whole numbers where the numbers are difficult or untidy. 32 Strategy Activities Teaching Fractions, Decimals, and Percentages Pipe Music with Decimals: page 24 Candy Bars: page 30 Figure It Out Number, Level 3: page 5 Number, Levels 3–4: pages 6, 7 Basic Facts, Levels 3–4: page 19 Number, Years 7–8, Level 4, Book Three, page 12 Number, Years 7–8, Level 4+, Book Six, page 18 Number Sense, Years 7–8, Link, Book One, pages 15, 21 Number Sense, Years 7–8, Book Two, pages 1, 7–8, 11 E CA AC EA AA AM AP Getting Started Transition Advanced Additive to Advanced Multiplicative Multiplication and Division Mathematical Processes • Problem Solving: Devise and use problem solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 3) • Explain the meaning of the digits in any whole number. Exploring Computation and Estimation (Level 3) • Make sensible estimates and check the reasonableness of answers. • Recall the basic multiplication facts. • Write and solve problems that involve whole numbers and decimals and require a choice of one or more of the four arithmetic operations. Knowledge Activities The students can: Grouping Based • Recall groupings of twos, threes, fives, and tens that are in numbers to 100 and the resulting remainders. • Recall groupings of 10 and 100 that can be made from a four- digit number. • Recall all the basic multiplication and division facts. • Recall multiplication facts for squares to 100. • Record the results of mental calculation using multiplication and division equations and diagrams. Teaching Number Knowledge Number Mats and Number Fans: page 36 Bowl a Fact: page 37 In and Out: page 38 Multiplication Madness: page 39 Loopy: page 39 Teaching Multiplication and Division Refer to Tasks for Key Knowledge on page 23. Knowledge Being Developed for Multiplication and Division The students are learning to: Grouping Based • Recall the number of groupings of tens, hundreds, and thousands that can be made from a number of up to seven digits. • Recall multiplication and division facts to 10 x 10, and the corresponding division facts. • Record the results of mental calculations using equations and diagrams, for example, empty number line. • Carry out a short written algorithm for multiplication and division of a three-digit whole number by a single-digit number. Knowledge Activities Teaching Number Knowledge Skip-counting on the Number Line: page 12 Beep: page 13 Using Calculators: page 15 Zap: page 28 Tens in Hundreds and More: page 29 Dividing? Think About Multiplying First: page 39 Multiplication Flash Cards: page 40 Figure It Out Number, Years 7–8, Level 4, Book Three, pages 8–9 Number, Years 7–8, Level 4, Book Five, pages 6–7 The students are learning to: • Use a diverse range of strategies to solve problems involving multiplication and division with whole numbers including: Compensating from tidy numbers Place value Reversibility and commutativity, e.g., 84 ÷ 7 as 7 x = 84 , or 2.37 x 6 as 6 x 2.37. Proportional adjustment, for example, doubling and halving. Changing the starting number, the dividing number, or both numbers, e.g. 201 ÷ 3 as (99 ÷ 3) + (99 ÷ 3) + (3 ÷ 3). • Solve division problems that have remainders and express the answer in fraction, decimal, or whole number form, e.g., 76 • ÷ 5 as 15 15 , or 15.2, or 15 r 1. Use written working forms or calculators where the numbers are difficult and/or untidy. CA AC EA AA AM Key Knowledge Required for Multiplication and Division Strategy Learning Outcomes E AP Strategy Activities Teaching Multiplication and Division Introduction: pages 23–25 Cut and Paste: page 25 Multiplication Smorgasboard: page 27 Proportional Packets: page 29 Royal Cooking Lessons: page 31 Remainders: page 33 Paper Power: page 35 Cross Products: page 38 Figure It Out Basic Facts, Level 3: pages 13, 14, 20 Number, Level 3: pages 4, 21, 22, 23, 24 Number, Levels 3–4: pages 15, 16, 17, 18, 20 Basic Facts, Levels 3–4: pages 3, 12, 13, 15, 16, 17 Number, Years 7–8, Link, Book Two, pages 6–7, 13–14 Number, Years 7–8, Level 4, Book Three, pages 7–10, 16, 23 Number, Years 7–8, Level 4, Book Five, pages 1, 2 , 5, 17 Number Sense, Years 7–8, Link, Book One, pages 1–3, 10–12, 14, 17, 20, 23–24 Number Sense, Years 7–8, Book Two, pages 2, 16, 24 33 Getting Started Transition Advanced Additive to Advanced Multiplicative Fractions, Decimals, Proportions, and Ratios Mathematical Processes • Problem Solving: Devise and use problem solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 3) • Explain the meaning of the digits in decimal numbers with up to three decimal places. • Order decimals with up to three decimal places. Key Knowledge Required for Fractions, Decimals, Proportions, and Ratios The students can: Counting Based • Identify decimals to three places. • Identify symbols for any fraction, including tenths, hundredths, thousandths, and those greater than 1. • Say the forwards and backwards word sequences for halves, quarters, thirds, fifths, and tenths. • Order unit fractions for halves, quarters, thirds, fifths, and tenths. Grouping Based • Recall the number of tenths and hundredths in decimals to two places. • Round decimals with up to two places to the nearest whole number. Knowledge Activities Knowledge Being Developed for Fractions, Decimals, Proportions, and Ratios The students are learning to: Grouping Based • Order decimals to three places, for example, 6.25 and 6.3. • Order fractions, including halves, quarters, thirds, fifths, and tenths. • Recall equivalent fractions for halves, thirds, quarters, fifths, and tenths with numbers to 100 and with 1 000. • Recall fraction ↔ decimal ↔ percentage conversions for halves, thirds, quarters, fifths, and tenths. • Record the results of mental calculations using equations and diagrams, for example, empty number line. • Round whole numbers and decimals with up to two places to the nearest whole number or tenth. Knowledge Activities Strategy Learning Outcomes The students are learning to: • Use mental strategies based on multiplying and dividing to solve problems with fractions, decimals, proportions, and ratios, including: Finding equivalent fractions, decimals, and percentages, e.g., 3 = 0.75 = 75% - 4 Using unit fractions, e.g., 5 of 72 by 1 of 72 = 9, 5 x 9 = 45. - 34 8 8 Place value, e.g., 6 x 3.4 = (6 x 3) + (6 x 0.4). Compensating from tidy numbers or fractions, e.g., 2.9 x 6.3 as 3 x 6.3 = 18.9, 0.1 x 6.3 = 0.63, 18.9 – 0.63 = 18.27. Exploring Computation and Estimation (Level 3) • Make sensible estimates and check the reasonableness of answers. • Recall the basic multiplication facts. • Write and solve problems that involve whole numbers and decimals that require a choice of one or more of the four operations. • Solve practical problems that require finding fractions of whole number and decimal amounts. E CA AC EA AA Teaching Number Knowledge Fraction Pieces: page 6 Rocket – Where Will It Fit?: page 16 Number Line Flips: page 16 Squeeze – Guess My Number: page 17 Bead Strings: page 18 Zap: page 28 Figure It Out Number, Years 7–8, Link, Book Two, pages 9, 20 Number, Years 7–8, Level 4, Book Three, pages 2-3 Number, Years 7–8, Level 4, Book Five, pages 16 Teaching Fractions, Decimals, and Percentages Refer to Tasks for Key Knowledge on pages 22–23. Teaching Number Knowledge Number Fans: page 4 Place Value Houses: page 5 Creating Fractions: page 6 More Geoboard Fractions: page 7 Non-Unit Fractions: page 7 Packets of Lollies: page 8 Reading Decimal Fractions: page 9 More Reading of Decimal Fractions: page 10 Linking Money and Decimal Fractions: page 10 Card Ordering: page 13 Arrow Cards: page 14 Using Calculators: page 15 Who Has More Cake?: page 20 Super Liquorice: page 21 Who Wins?: page 21 Nudge: page 26 Sensible Rounding: page 30 Locating Decimal Fractions: page 31 Digits on the Move: page 32 The Same But Different: page 33 Figure It Out Number, Years 7–8, Link, Book Two, pages 21, 24 Number, Years 7–8, Level 4, Book Three, pages 1, 4, 17–19 Number, Years 7–8, Level 4, Book Five, pages 11, 19, 23 Strategy Activities Teaching Fractions, Decimals, and Percentages Introduction: pages 22–24 Deci-mats: page 27 Candy Bars: page 30 Figure It Out Number, Level 3: pages 12, 13, 14 Number, Levels 3–4: pages 4,–7, 9–12, 22–24 Basic Facts, Levels 3–4: pages 22, 23 Number Sense, Years 7–8, Link, Book One, pages 6, 16, 19, 22 Number Sense, Years 7–8, Book Two, pages 4, 6, 10, 18-19, 22-23 AM AP Getting Started Transition Advanced Additive to Advanced Multiplicative Number Patterns and Relationships Mathematical Processes • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 4) • Explain the meaning of negative numbers. • Explain the meaning and evaluate powers of whole numbers. Exploring Computation and Estimation (Level 4) • Make sensible estimates and check the reasonableness of answers. • Explain satisfactory algorithms for addition, subtraction, and multiplication. • Demonstrate knowledge of the conventions for order of operations. CA AC EA AA AM AP Key Knowledge Required for Number Patterns Knowledge Activities The students can: • Recall all the basic multiplication and division facts. • Recall multiplication facts for squares to 100. • Recall groupings of 10 and 100 that can be made from a four-digit number. Teaching Number Knowledge Skip-counting on the Number Line: page 12 Beep: page 13 Using Calculators: page 15 Zap: page 28 Tens in Hundreds and More: page 29 Dividing? Think About Multiplying First: page 39 Multiplication Flash Cards: page 40 Knowledge Being Developed for Number Patterns The students are learning to: • Recall the number of groupings of tens, hundreds, and thousands that can be made from a number of up to seven digits. • Recall multiplication and division facts to 10 x 10. • Record the results of mental calculations using equations and diagrams, for example, empty number line. • Carry out short multiplication and division of a three-digit whole number by a single-digit number. Knowledge Activities Teaching Number Knowledge Arrow Cards: page 14 Strategy Learning Outcomes The students are learning to: • Find out whether a whole number is prime or non-prime. • Solve problems that involve the ordering, and addition and subtraction of integers. • Find powers and square roots of whole numbers, particularly squares and cubes. • Use factorials to solve problems. • Find general rules for finding unknown members of a repeating sequence of numbers or objects, for example, - recursion rules, e.g., 6, 10, 14, 18, … (+ 4 to the previous number) - function rules, e.g., 1 2 3 4 … 4 7 10 13 … Bottom number equals (top number x 3) + 1 • Show and interpret relationships using equations, tables, and graphs. • Solve number problems that have one or more “unknowns”. E Strategy Activities Figure It Out Number, Years 7–8, Level 4, Book Four, pages 1–7 (Prime Nos) Number, Years 7–8, Level 4, Book Four, pages 8–13, 15, 21 (Integers) Number, Years 7–8, Level 4, Book Four, pages 14, 16-20, 22–24 Number, Years 7–8, Level 4, Book Five, page 10 (Powers and square roots) Number, Years 7–8, Level 4, Book Five, pages 4, 21 (Factorials) Number Sense, Years 7–8, Book Two, pages 5, 9, 12 Algebra, Levels 3–4, pages 1–24 Algebra, Years 7–8, Link, Book One, pages 1–24 Algebra, Years 7–8, Book Two, pages 1–9, 11–13, 15, 20–21 Algebra, Years 7–8, Book Three, pages 1–7, 10–17, 20–21 Algebra, Years 7–8, Level 4+, Book Four, pages 6–7 35 Getting Started Transition Advanced Multiplicative to Advanced Proportional Fractions, Decimals, Proportions, and Ratios Mathematical Processes • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 4) • Find fractions equivalent to the one given. • Express a fraction as a decimal and vice versa. • Express a decimal as a percentage and vice versa. • Express quantities as fractions or percentages of a whole. Exploring Computation and Estimation (Level 4) • Make sensible estimates and check the reasonableness of answers. • Find a given fraction or percentage of a quantity. • Write and solve problems involving decimal multiplication and division. • Explain satisfactory algorithms for addition, subtraction, and multiplication. Key Knowledge Required for Fractions and Decimals Knowledge Activities The students can: Grouping Based • Order decimals to three places, for example, 6.25 and 6.3. • Order fractions, including halves, quarters, thirds, fifths, and tenths. • Recall equivalent fractions for halves, thirds, quarters, fifths, and tenths with numbers to 100 and with 1 000. • Recall fraction ↔ decimal ↔ percentage conversions for halves, thirds, quarters, fifths, and tenths. • Record the results of mental calculations using equations and diagrams, for example, empty number line. • Round whole numbers and decimals with up to two places to the nearest whole number or tenth. Teaching Number Knowledge Place Value Houses: page 5 More Geoboard Fractions: page 7 Non-Unit Fractions: page 7 Packets of Lollies: page 8 Reading Decimal Fractions: page 9 Card Ordering: page 13 Arrow Cards: page 14 Using Calculators: page 15 Super Liquorice: page 21 Rocket – Where Will It Fit?: page 16 Squeeze – Guess My Number: page 17 Figure It Out Number, Years 7–8, Link, Book Two, pages 21, 24 Number, Years 7–8, Level 4, Book Three, pages 1, 4, 17–19 Number, Years 7–8, Level 4, Book Five, pages 11, 19, 23 Teaching Fractions, Decimals and Percentages Refer to Tasks for Key Knowledge on pages 32-33. Knowledge Being Developed for Fractions and Decimals The students are learning to: Counting Based • Say the forwards and backwards decimal word sequences by thousandths, hundredths, tenths, ones, and tens, starting at any decimal number. • Order fractions, decimals, and percentages. Grouping Based • Recall the number of tenths, hundredths, and onethousandths in numbers of up to three decimal places. • Recall what happens when a whole number or decimal is multiplied or divided by the power of 10. • Recall fraction ↔ decimal ↔ percentage conversions for fractions in common use, e.g., eighths, tenths, twentieths. • Record the results of mental calculations using equations. • Carry out column addition and subtraction for whole numbers and decimals to three places. • Carry out short multiplication and division of whole numbers and decimals by a single-digit number. • Carry out multiplication of a three- or four-digit whole number by a two-digit whole number. Knowledge Activities Teaching Number Knowledge More Geoboard Fractions: page 7 Non-Unit Fractions: page 7 Packets of Lollies: page 8 Reading Decimal Fractions: page 9 More Reading of Decimal Fractions: page 10 Linking Money and Decimal Fractions: page 10 Measurement and Zeros: page 11 Who Has More Cake?: page 20 Little Halves and Big Quarters: page 20 Who Wins?: page 21 Who Gets More?: page 22 Equivalent Fractions, Decimals, and Percentages: page 22 Difficult Fractions to Percentages: page 23 The students are learning to: Choose appropriately from a broad range of mental strategies to solve problems with fractions and decimals, ratios, and proportions, including: Applying reversibility to find fractions, decimals and percentages, e.g., 13.6 ÷ 0.4 = , as 0.4 x = 13.6 Applying equivalent ratios and proportions by finding common factors, Strategy Activities Teaching Fractions, Decimals, and Percentages Introduction: pages 32–34 Hot Shots: page 34 Mixing Colours: page 37 Folding Fractions and Decimals: page 39 Figure It Out Number, Levels 3–4: pages 19, 21 Basic Facts, Levels 3–4: pages 8, 9 Number, Years 7–8, Level 4, Book Three, pages 5–6, 11, 14, 20–21 Number, Years 7–8, Level 4, Book Five, pages 8–9, 14–15, 20–22 Number, Years 7–8, Level 4, Book Six, pages 10–11, 17, 21 Number Sense, Years 7–8, Book Two, pages 13–15, 20–21 Strategy Learning Outcomes e.g., 28 ÷ 42 = - 42 - 6 3 Finding proportional relationships between measures, e.g., 12 → 15 as - % as 28 = 4 = 2 = 66.6% → 25 by 12 is 4 of 15, 20 is 4 of 25 5 5 Using proportional adjustment, e.g., 7.2 ÷ 0.3 as 8 x 0.9 = 7.2 so 24 x 0.3 = 7.2 Converting from ratios to proportions and vice versa, e.g., 3 : 5 out of a total of 96 as 3 : 5 is 3 , 3 of 96 is 36 so 36 : 60 36 8 8 E CA AC EA AA AM AP Getting Started Transition Advanced Multiplicative to Advanced Proportional Number Patterns and Relationships Mathematical Processes • Problem Solving: Devise and use problem-solving strategies to explore situations mathematically. • Use equipment appropriately when exploring mathematical ideas. • Logic and Reasoning: Classify objects. • Communicating Mathematical Ideas: Devise and follow a set of instructions to carry out a mathematical activity. • Record and talk about the results of mathematical exploration. Exploring Numbers (Level 4) • Explain the meaning of negative numbers. • Explain the meaning and evaluate powers of whole numbers. Exploring Computation and Estimation (Level 4) • Make sensible estimates and check the reasonableness of answers. • Explain satisfactory algorithms for addition, subtraction, and multiplication. • Demonstrate knowledge of the conventions for order of operations. E CA AC EA AA AM AP Key Knowledge Required for Number Patterns Knowledge Activities The students can: • Order integers. • Recall square numbers to 100. • Recall multiplication facts to 10 x 10, and the corresponding division facts. • Record the results of calculations using equations. Teaching Number Knowledge Arrow Cards: page 14 Knowledge Being Developed for Number Patterns The students are learning to: • Order integers and positive and negative decimals • Recall square numbers to 100, and the corresponding square roots. • Recall the prime numbers to 20, at least. • Express whole numbers in standard form. Knowledge Activities The students are learning to: • Express whole numbers as the product of prime numbers (prime factorisation). • Solve problems involving the addition, subtraction, and multiplication of integers. • Solve problems with powers, including exponential growth, multiplication of powers with the same base (e.g., 16 x 8), estimating and finding square roots of whole numbers. • Work in number bases other than 10. • Find general rules for finding unknown members of a repeating sequence of numbers or objects, including nonlinear repeating patterns, for example, - recursion rules, e.g., 1 4 9 16 25… +3 +5 +7 +9 - function rules, e.g., 1 2 3 4 … 2 4 8 16 … Bottom number equals (2 to the power of top number, 2n) • Show and interpret relationships using algebraic expressions and equations, tables, and graphs. • Solve problems that involve one or more unknowns, including equations with letters, e.g., r = (4 x t) + 3, if r = 51 what is t? Strategy Activities Figure It Out Number, Years 7–8, Level 4+, Book Six, pages 2–3, 7 (Prime Nos) Number, Years 7–8, Level 4+, Book Six, pages 14–16, 22–24 (Integers) Number, Years 7–8, Level 4+, Book Six, pages 4–5, 8, 18–20 (Powers and square roots) Number, Years 7–8, Level 4+, Book Six, pages 12–13 (Bases) Strategy Learning Outcomes Number, Years 7–8, Level 4+, Book Six, pages 6, 9 (General rules) Algebra, Years 7–8, Level 4, Book Two, pages 10, 14, 16–19, 22–24 Algebra, Years 7–8, Level 4, Book Three, pages 8–9, 18–19, 22–24 Algebra, Years 7–8, Level 4+, Book Four, pages 1–5, 8–24 37 Week: Room: Hot Spot Focus: SLO: Monday G Tuesday Pr Wednesday T Thursday G Friday Pr T KA Pr T KA T G Pr T G Pr T KA Pr T Pr T G Pr T KA Pr T KA Pr Warm-down: SLO = Strategy Learning Outcome, T = Teaching, Pr = Practice, G = Games, KA = Knowledge Activity Getting Started 38 Weekly Number Plan Getting Started Weekly Number Plan Week: Room: Hot Spot Focus SLO: SLO: SLO: Day 1 G T Pr T Pr KA Warm-down: Day 2 Pr G T KA T Pr Warm-down: Day 3 T Pr G Pr KA T Warm-down: Day 4 G T Pr T Pr KA Warm-down: Day 5 Pr G T KA T Pr Warm-down: Name: SLO = Strategy Learning Outcome, T = Teaching, Pr = Practice, G = Games, KA = Knowledge Activity 39 Getting Started Name: E Date achieved I can … Emergent I am learning to … Knowledge • Read 0 1 2 3 4 5, first, and then … 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5, first, and then … 1 2 3 4 5 6 7 8 9 10 5 4 3 2 1 0, first, and then … 10 9 8 7 6 5 4 3 2 1 0 The number after a number • Count • Count • Say 3, 4, 5, ___ • Say The number before a number ___2, 3, 4 • Order 5 3 1 2 4 0, first, and then … 5 3 8 1 2 9 7 10 4 6 0 Strategy • Count a set of objects 1 • Get 3 4 5 a set of objects, like seven teddies 1 40 2 2 3 4 5 6 7 Getting Started E Name: CA Date achieved I can … One-to-one Counting I am learning to … Knowledge • Read • Count 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 • Count 10 9 8 7 6 5 4 3 2 1 0 • Say The number after a number • Say 4, 5, 6 ___ The number before a number • Order ___3, 4, 5 5 3 8 1 2 9 7 10 4 6 0 • Know Patterns to five Strategy • Join Groups of objects together and • Split A number of objects into groups 41 Getting Started Name: CA Date achieved I can … Counting from One on Materials I am learning to … Knowledge • Read • Count • Count • Say • Order • Read 0 1 2 11 12 13 3 4 5 6 7 14 15 16 17 1 2 3 4 5 6 7 8 9 … 20 8 9 10 18 19 20 10 11 20 19 18 17 16 15 14 13 12 11 10 …0 The number after a number between 0 and 20 4, 5, 6 _____ … 11, 12, 13 ____ The number before a number between 0 and 20 ____ 4, 5, 6 … ____ 11, 12, 13 Numbers to 20 8 17 3 14 11 as 3 8 11 14 17 1 1 2 4 • Know Groups within five + + Groups with five Strategy • Solve + and – problems up to 10 by 42 12 Counting all the objects and 3 + 2 Getting Started CA Name: Counting from One by Imaging I am learning to … Knowledge 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 • Read • Count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 • Count 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 • Say The number after a number in the range 1-20 Say 14, 15, 16, ______ The number before a number in the range 1-20 • • Order • Know • Know • Know • Skipcount forwards Strategy • Solve + and problems to 10 by • Share objects equally to make 1 ’s and 41 ’s 2 Date achieved I can … _____14, 15, 16 7, 15, 12, 6, 18, 13, 5, 19, 10, 1, 8, 16, 9, 3, 11, 2, 14, 17, 20, 0, 4 1 1 2 4 Groupings within 10 5+5 6+4 3+7 Doubles to 10 2+2 3+3 4+4 5+5 In twos 2 4 6 8 10 12 14 16 18 20 In fives 5 10 15 20 25 30 35 40 45 50 In tens 10 20 30 40 50 60 70 80 90 100 Counting all the objects in my head 4+3= Share 12 muffins amongst four 1, 2, 3, 4, 5, 6, 7 1 4 of 12 43 Getting Started CA Name: AC Advanced Counting I am learning to … Knowledge • Read Any number up to 100, for example, 17, 26, 38, 47, 53, 74, 86, 99 • Count Forwards from any number up to 100, for example, 34, 35, 36 … • Count Backwards from any number from 100, for example, 47, 46, 45 … • Say The number after a number in the range 1-100 54, 55, 56, ______ The number before a number in the range 1-100 • Say • Order • Count • Read _____54, 55, 56 Numbers to 100, for example, 37, 26, 55, 73, 22, 34, 45 Forwards and backwards in twos, fives, and tens Unit fractions 1 2 • Know 10 + 5 = 15 • Know • Know Strategy • Solve + and problems by • Solve x problems by • Find 1 2 and 41 of sets 44 1 4 1 3 1 5 1 10 Teen numbers 10 + 8 = 18 10 + = 16 Tens in decades, for example, “How many tens in 60?” Tens that add to 100, for example, 30 + 70 Doubles to 20 and halves 1 1 3 + 3 6 - 3 2 of 6 7 + 7 14 - 7 of 14 2 Counting on or back from the largest number, in my head. 16 + 3 17, 18, 19 Skip-counting in twos, fives, or tens 4x5 Share 12 muffins among four 5, 10, 15, 20 1 4 of 12 Date achieved I can … Getting Started AC Name: Date achieved I can … Early Additive Part-Whole I am learning to … Knowledge • Read • Count Numbers to 1 000 333 479 983 Forwards by ones, tens, and hundreds up to 1 000 Backwards by ones, tens, and hundreds from 1 000 • Say The number one more, 10 more, 100 more than numbers to 1 000 The number one less, 10 less, 100 less than numbers to 1 000 • Order • Know • Order Numbers from 0 – 1 000 58 376 1 1 2 1 4 1 3 1 5 837 1 10 Fractions with the same denominators 1 4 > • Skipcount Forwards and backwards in threes 3 6 9 12 15 30 • Round Three-digit numbers to the nearest 10 or 100 246 250 (nearest 10) Addition and subtraction facts to 20 12 + 8 = 20 11 + 9 = 20 20 – 12 = 8 20 – 9 = 11 20 – 8 = 12 20 – 11 = 9 • Know Strategy • Solve + and – problems in my head by • Use repeated addition to solve x problems by • Find a fraction of a number by 3 4 Using doubles, for example, 8 + 7 as 8 + 8 – 1 Using fives, for example, 8 + 7 as 5 + 3 + 5 + 2 Using making tens, for example, 8 + 7 as 10 + 5 Using making tens, for example, 19 + 6 as 20 + 5, 29 + 8 as 30 + 7 Using place value, for example, 33 + 16 as 30 + 10 + 3 + 6 Twos 2 + 2 + 2 + 2 = 4 x 2 Threes 3 + 3 + 3 + 3 + 3 = 5 x 3 Fours 4 + 4 + 4 = 3 x 4 Fives 5 + 5 + 5 + 5 + 5 = 5 x 5 Tens 10 + 10 = 2 x 10 Using repeated addition or subtraction, for example, 1 3 of 12 as 4 + 4 + 4 for example, 12 – 2 – 2 – 2 = 6, 6 – 2 – 2 – 2 = 0, 1 3 of 12 is 2 + 2 45 EA Getting Started EA Name: Advanced Additive Part-Whole I am learning to … Knowledge Numbers to 1 000 000, for example, 1 374; 98 765; 763 104 • Read and order Decimals up to three places, for example, 0.764; 0.14; 0.8 • Read Symbols for any fraction, for example, 1 1 • Say • Know • Order • Know • Know • Round • Know • Record Strategy • Solve + and – problems by • Solve x and ÷ problems by • Solve problems with fractions 46 3 4 5 13 3 The number 1 000 more/1 000 less How many tens and hundreds are in four-digit numbers Fractions with the different denominators, for example, 1 4 2 3 4 5 Groups within 1 000, for example, 240 + 760 How many twos, threes, fives, and tens in numbers to 100 and any remainders, for example, threes in 17 Whole numbers to the nearest 10, 100, and 1 000, for example, 5 508 6 000 Decimals to the nearest whole number, for example, 3.49 3 Multiplication facts for squares to 100, for example, 5 x 5, 9 x 9 Results of calculations using empty number lines and written algorithms Using compensation from tidy numbers 725 - 389 as 725 - 400 + 11 = 336 Using place value 376 + 431 as 300 + 400 + 70 + 30 + 6 + 1 = 807 Using compatible numbers 35 + 37 + 65 as (35 + 65) + 37 = 100 + 37 = 137 Using reversibility 814 - 789 = as 789 + = 814 Using equal additions 72 - 37 as 75 - 40 (add three to both numbers) Using decomposition 83 - 28 as renaming 83 so 87 13 - 28 Using doubling, for example, 2 × 6 = 12 so 4 × 6 = 24 Deriving facts, for example, 2 × 6 = 12 so 3 × 6 = 12 + 6 = 18 Using reversibility, for example, 7 × 4 = 28 so 28 ÷ 4 = 7 Using proportional adjustment, for example, 3 × 12 is the same as, 6 × 6 = 36 (doubling and halving), or 24 ÷ 4 = 6 so 24 ÷ 8 = 3 Mentally, using known multiplication and division facts, for example, 1 3 of 36 as, 3 × 12 = 36 so, 1 3 of 36 = 12 AA Date achieved I can… Getting Started AA Name: Advanced Multiplicative Part-Whole Date achieved I can … I am learning to … Knowledge • • • • • Count Say Order Order Know • Know Forwards and backwards in 0.001s, 0.01s, 0.1s, ones, tens Number 0.001, 0.01, 0.1, 1, 10 before/after any whole number Decimals up to three places, for example, 6.25 and 6.3 Fractions including halves, thirds, quarters, fifths, tenths Groupings of 10, 100, 1 000, made from a number of up to seven digits, for example, tens in 47 562 Equivalent fractions and proportions for 1 s, 1 s, 1 s, 1 s, 1 s 2 3 4 with denominators of 10, 100, 1 000, for example, • Round • Recall • Recall Whole numbers and decimals to the nearest 1 or x and ÷ facts to 10 x 10, 100 ÷ 10 Fraction decimal percentage conversions for and Strategy • Solve + and problems by using • Solve x and ÷ problems using • Solve problems with fractions, decimals, proportions, and ratios, using 1 4 1 10 s, for example, 3 4 1 2 s, = 0.75 = 75% 5 = 1 10 1 s, 1 3 4 10 25 100 s, 1 5 s, Compensation from tidy numbers, e.g., 3.2 + 1.95 as 3.2 + 2 – 0.05 Place value, e.g., 8.65 – 4.2 = (8 – 4) + (0.6 – 0.2) + 0.05 or 8.65 – 4 = 4.65 then 4.65 – 0.2 = 4.45 Reversibility and commutativity, e.g., 6.03 – 5.8 = as 5.8 + = 6.03 (reversibility) or + 3.98 = 7.04 as 3.98 + = 7.04 (commutativity) Equal additions, e.g., 7.24 – 3.8 as 7.44 – 4.0 = 3.44 Using negatives, e.g., 6.4 – 2.5 as 0.4 – 0.5 is -0.1; 6.0 – 2.0 = 4.0; 4.0 – 0.1 = 3.9 Decomposition, e.g., 9.25 – 6.83 as 8. 125 – 6.83 Compensation from tidy numbers, e.g., 6 x 998 as, (6 x 1 000) – (6 x 2) or 56 ÷ 4 using (60 ÷ 4) – 1 Place value, e.g., 28 x 7 as (20 x 7) + (8 x 7) or 72 ÷ 4 as (40 ÷ 4) + (32 ÷ 4) Reversibility, e.g., 96 ÷ 6 as 6 x = 96 and commutativity, e.g., 17 x 6 as 6 x 17 Proportional adjustment, e.g., 4 x 18 as 8 x 9 or 81 ÷ 3 as (81 ÷ 9) x 3 Written working forms or calculators where the numbers are difficult and/or untidy Unit fractions, e.g., 4 x 18 as ( 1 x 18) x 4 9 9 Place value, e.g., 3.4 x 8 as (3 x 8) + (0.4 x 8) = 24 + 3.2 = 27.2 Compensating from tidy numbers or fractions, e.g., 83 x 28 as 1 2 of 3 4 x 28 or 1.9 x 3.4 as (2 x 3.4) – (0.1 x 3.4) Using equivalent fractions and ratios, e.g., 40% of 35 as 2 of 35 = 14 5 47 AM Getting Started AM Name: Advanced Proportional Part-Whole I am learning to … Knowledge • Count Forwards and backwards in 0.001s, 0.01s, 0.1s, ones, tens, etc. • Say The number 0.001, 0.01, 0.1, 1, 10 before/after decimal numbers • Order Fractions, decimals, and percentages, e.g., 40%, 3 , 0.5 • Know • Know • Round • Recall Strategy • Solve x and ÷ problems with fractions and decimals by • Find • Solve problems with ratios and proportions by 48 5 Number of tenths, hundredths, and thousandths that are in numbers up to three decimal places, e.g., tenths in 45.6. What happens when a whole number or decimal is x or ÷ by a power of 10, e.g., 4.5 x 100; 67.3 ÷ 10 Decimals to the nearest 100, 10, 1, 1 10 , 1 100 Fractional, decimal, and percentage conversions for commonly used fractions, e.g., 81 = 0.125 = 12.5% Conversion between fractions and decimals, e.g., 0.75 x 2.4 as 43 x 2.4 Place value, e.g., 0.15 x 3.6, as (0.1 x 3.6) + (0.05 x 3.6) Doubling and halving, etc., e.g., 7.2 ÷ 0.4 as (7.2 ÷ 0.8) x 2 Commutativity, e.g., 48 x 0.125 as 0.125 x 8 = 81 of 8 = 1 Multiplying numerators and denominators, e.g., 43 x 25 as 43 xx 25 Fractions, decimals, and percentages of given amounts, e.g., 65% of 24 as 50% of 24 is 12, 10% of 24 is 2.4, and 5% is 1.2 so 12 + 2.4 + 1.2 = 15.6 Finding equivalent ratios with a common factor, e.g., 21:28 as :8 as 21:28 is 3:4 so 6:8 18 e.g., 27 = 23 so 23 = 10 15 Finding a multiplier between the units, e.g., 18 out of 27 as 10 out of 15 by multiplying 15 by 2 3 Date achieved I can … AP Book 3 Numeracy Professional Development Projects 2003 (Draft) NUMERACY AND THE MATHEMATICS CURRICULUM Numeracy arises out of effective mathematics teaching. All the strands within Mathematics in the New Zealand Curriculum are important in the pathway to numeracy. Number is central to this pathway, although the relative emphasis on this strand changes with the stages of schooling: The general aims of mathematics education in New Zealand define the features of school programmes that contribute to the development of numerate people. Such programmes: • help students to develop a belief in the value of mathematics and its usefulness to them, to nurture confidence in their own mathematical ability, to foster a sense of personal achievement, and to encourage a continuing and creative interest in mathematics; • in the first four years of schooling, the main emphasis should be on the number strand; • in the middle and upper primary years of schooling, the emphasis is spread across the strands of the curriculum; • towards the end of compulsory schooling, number sense becomes a tool for use across the other strands. develop in students the skills, concepts, understandings, and attitudes which will enable them to cope confidently with everyday life; • help students to develop a variety of approaches to solving problems involving mathematics, and to develop the ability to think and reason logically; • At all stages, students should: • develop an understanding of numbers, the ways they are represented, and the quantities for which they stand; • help students to achieve the mathematical and statistical literacy needed in a society which is technologically oriented and information rich; • develop accuracy, efficiency, and confidence in calculating – mentally, on paper, and with a calculator; • provide students with the mathematical tools, skills, understandings, and attitudes they will require in the world of work; • develop the ability to estimate and to make approximations, and to be alert to the reasonableness of results and measurements. • provide a foundation for those students who may continue studies in mathematics or other learning areas where mathematical concepts are central; • help to foster and develop mathematical talent. (Mathematics in the New Zealand Curriculum, page 31) (Mathematics in the New Zealand Curriculum, page 8) These achievement aims enable students to develop the ability and inclination to use mathematics to solve problems in a range of contexts. Numeracy Professional Development Projects 2003 (Draft) Published by the Ministry of Education. PO Box 1666, Wellington, New Zealand. Copyright © Crown 2003. All rights reserved. Enquiries should be made to the publisher. ISBN 0 478 27243 X Dewey number 372.7 Topic Dewey number 510 Item number 27243 Although the groundwork is laid in mathematics, other curriculum areas also provide opportunities for numeracy learning. In addition, the home, early childhood settings, and the community assist in the development of numeracy. ACKNOWLEDGMENTS The Ministry of Education wishes to acknowledge the following people and organisations for their contribution towards the development of this draft handbook. Holmes (Dunedin College of Education), Errolyn Taane (Dunedin College of Education), Malcolm Hyland (Ministry of Education), Ro Parsons (Ministry of Education). THE PARTICIPANTS: The New Zealand numeracy project personnel – facilitators and principals, teachers and children from more than eight hundred New Zealand schools who contributed to this handbook through their participation in the numeracy development projects in 2000, 2001, and 2002. THE NUMERACY REFERENCE GROUP: Professor Derek Holton, convenor (The University of Otago), Professor Megan Clark (Victoria University), Dr Joanna Higgins (Wellington College of Education), Dr Kay Irwin (Auckland University), Dr Gill Thomas (Dunedin College of Education), Dr Jenny Young Loveridge (The University of Waikato), Dr Glenda Anthony (Massey University), Tony Trinick (Auckland College of Education), Garry Nathan (Auckland College of Education), Graham Cochrane (Education Review Office), Eleanor Burt (Christchurch College of Education), Dr Joanna Wood (New Zealand Association of Mathematics Teachers), Peter Hughes (Auckland College of Education), Vince Wright (The University of Waikato School Support Services), Geoff Woolford (Parallel Services), Kevin Hannah (Christchurch College of Education), Chris France (School Trustees' Association), Julie Hepburn (NZPF), Jo Jenks (Early Childhood Division, Wellington College of Education), Gary Sweeney (New Zealand Association of Intermediate and Middle Schools), Diane Leggatt (NZEI Te Riu Roa), Sului Mamea (Pacific Island Advisory Group, Palmerston North), Sally Peters (The University of Waikato School of Education), Andrew Kear (PPTA), Ro Parsons (Ministry of Education), Malcolm Hyland (Ministry of Education). PUBLISHING: Kathy Campbell, Jocelyn Cranefield (Learning Media Limited), Kirsty Farquharson (Learning Media Limited), Jan Kokason (Learning Media Limited), Dr Gill Thomas (Dunedin College of Education), Joe Morrison (Maths Technology Limited), Bronwen Wall (Learning Media Limited). In addition, the Ministry of Education wishes to acknowledge Professor Bob Wright (Southern Cross University, Lismore, NSW), Dr Noel Thomas (Charles Sturt University, Bathurst, NSW), Dr Keono Gravemeier (Freudenthaal Institute, Utrecht, Netherlands), Jim Martland (The University of Manchester, UK). The Ministry of Education also wishes to acknowledge The New South Wales Department of Education and Training for permission to trial Count Me In Too in 2000 through a one-year arrangement. The findings from the use of this pilot project informed the development of the numeracy policy in New Zealand. Count Me In Too is the registered Trade Mark of the Crown in the Right of the State of New South Wales (Department of Education and Training). Copyright of the Count Me In Too Professional Development Package materials (1997–2002), including the Learning Framework in Number and the Schedule for Early Number Assessment, is also held by the Crown in the Right of the State of New South Wales (Department of Education and Training) 2002. The cover design is by Dave Maunder (Learning Media Limited) and Base2 Communication Design Ltd. All other illustrations are by Noel Eley and James Rae. THE WRITERS AND REVIEWERS: Vince Wright (The University of Waikato), Peter Hughes (Auckland College of Education), Lynn Tozer (Dunedin College of Education), Sarah Vokes (Bayview School), Gaynor Terrill (The University of Waikato School of Education), Carla McNeill (The University of Waikato School of Education), Professor Derek Holton (The University of Otago), Dr Gill Thomas (Dunedin College of Education), Bruce Moody (mathematics consultant), Lynne Petersen (Dominion Road School), Marilyn All illustrations copyright © Crown 2003.

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