# Book 5 Teaching Addition, Subtraction, and Place Value

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1 Book 5 Teaching Addition, Subtraction, and Place Value Numeracy Professional Development Projects Revised 2012

3 Book 5 Teaching Addition, Subtraction, and Place Value

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8 For example, for the number 17, students need to be able to read, interpret, and show the following. Read: 17 as seventeen seventeen as 17. Say: seventeen or 17 as seventeen seventeen or 17 as one ten and seven ones. Do/model: seventeen or 17 as seventeen singles seventeen or 17 as one ten and seven ones. Choosing wisely is the aim Students learn to solve number problems mentally and to choose critically from the numeracy strategies at their disposal. They learn to discriminate between strategies according to the numbers involved in a particular problem. Group or class discussions in which students are required to justify their choice are ideal for promoting the idea of efficiency. An efficient strategy uses the minimum number of steps to solve a problem. In applying these strategies, students use their knowledge of grouping and place value. Basic fact knowledge develops as they move up the stages of the Number Framework this knowledge is essential for all more sophisticated strategies. Communicating mathematical thinking The ability to articulate mathematical processes and strategies is crucial to the development of confident and capable mathematicians. Students learn to understand and use mathematical language, texts, symbols, and diagrams to express their mathematical thinking in a range of contexts. Students who use counting strategies will do most of their communicating orally. As they move through the stages of the Number Framework, however, they need to be given opportunities to communicate in a variety of ways. Teachers have a key role in developing students ability to communicate orally and in writing. Listening to students, and knowing when to step in and out of discussions and when to press for further understanding, are vital if students are to develop the skills needed to communicate and justify their mathematical thinking. Written recording can be used to reduce the mental load, to communicate ideas to others, and to provide a window into student thinking. Teachers need to model a variety of ways of recording. Students may use words, diagrams, and symbols and then, as they progress to the higher stages, empty number lines, diagrams, calculators, dotty arrays, or ratio tables. Suggestions on the types of recording appropriate for each stage can be found at There are two aspects to communicating by using symbols: being able to use a symbol to appropriately represent a real-world situation, and understanding what a symbol requires you to do in a particular context. Most problems in Book 5 start with a word problem that gets transformed into an expression and is then solved. Word problems require students to choose which operation to use and how to record the process correctly. Students need multiple opportunities to explore problem-solving processes. The teacher can ask questions instead of simply writing the relevant expression on the board or in the modelling book without discussion. 6 Teaching Addition, Subtraction, and Place Value

11 Features of this book The following diagram shows how the information in this book is organised. Moving students from advanced counting to early additive The important ideas are listed at the start of each section. Introduction Students who have mastered Ideas being developed Addition and subtraction problems can be solved by Numbers can be partitioned... Learning experiences that will advance each key teaching idea are listed here. Some students may not need all the learning experiences. Others may need to repeat them with different equipment before they fully understand the key ideas. Learning experiences Key teaching ideas Learning experiences 1. Our number system is based on ten. More ones 2. Basic fact knowledge... Adding ones and tens Key idea 1: Our number system is based on ten Each Page key idea is indicated by a curriculum nautilus symbol. The key ideas replace learning intentions. They ensure clarity about the focus of the lesson. The learning intention can be developed with the students from the key idea as the lesson progresses. Mathematical knowledge required Students need: instant recall of basic facts to 10 This is the knowledge the student needs in order to work successfully with this key idea. knowledge of 10 and facts (e.g., ). Diagnostic snapshot How many ten-dollar notes would I need to make \$68? Students who answer this with confidence Learning experience More ones and tens Equipment: \$1, \$10, and \$100 notes (Material Master 4-9)... In pairs, one student shows two-digit numbers on the arrow cards. The other student reads that number, then models it in place-value money. Example: \$93, \$47, \$82, \$99... The snapshot is used to find out what students already understand. The teacher can use this information to either move on to the next learning experience or build knowledge further as required. Lists the materials needed for this learning experience. Note that student student talk is encouraged at every opportunity. Teaching Addition, Subtraction, and Place Value 9

12 Learning experiences for emergent and one-to-one counting Introduction The key concept for one-to-one counting is that counting determines the number of objects in a set. Students are learning to count, with understanding, in order to identify how many in any set of objects. Learning experiences at this stage are designed to connect the counting words with the counting sequence (1 10) to correctly produce the number of a set. The students need to be given multiple opportunities to read, count, and model numbers up to 10. The learning experiences listed in the table below will enable students to reliably count the number of objects, one to one, in structured and unstructured collections. Counting principles Counting the number of objects in a set requires students to say all the counting words in the correct order, starting from one (one-to-one principle). Counting the number of objects in a set requires students to assign one word to each object in the set (stable-order principle). Counting the number of objects in a set requires students to recognise that the last number counted defines the number of the set (cardinal principle). The order in which objects are counted does not affect the outcome (order-relevance principle). 1 Any collection of objects can be counted, whether tangible or not (abstraction principle). Recognising patterns to five instantly (by subitising) enables students to recognise small numbers quickly without explicitly counting them. Learning experiences Key teaching ideas Learning experiences Page * 1. Symbols/words for numbers in the range 1 10 are identified. 2. The number word sequence for numbers in the range 1 10 is said accurately. 3. The symbols/words for numbers in the range 1 10 are matched to the number of objects in the set. 4. The sequence of numbers in the range 1 10 is ordered correctly. Lucky dip Number mat Lily pads Pipe cleaner numbers Counting as we go How many now? Loud and soft Clapping Walk the bridge Counting together Counting movements to nine Match it up Magic caterpillar legs Petals and flower centres Feed the elephant Birthday cake Before and after Ordering numerals Up or down How many beans? Making a series to nine BSM BSM BSM 5. Patterns for numbers 1 5 are recognised instantly. * Page numbers in these tables refer to pages in this book. Patterns to five, then ten How many different sets of five can you make? Fabulous fives 15 BSM 16 Beginning School Mathematics 1 Gelman, R., and Gallistel, C. (1978). The Child s Understanding of Number. Cambridge, MA: Harvard University Press. 10 Teaching Addition, Subtraction, and Place Value

13 Although the learning experiences have been matched to specific key ideas, they are all closely linked and can be used to complement each other. Because knowledge and strategy about one-to-one counting are developed simultaneously, no specific knowledge is identified for each of the suggested learning experiences. Key idea 1: Symbols/words for numbers in the range 1 10 are identified Learning experiences Lucky dip Equipment: Numeral cards 1 to 10 (Material Master 4-1); a container. Show the students a card and ask them what number it is. Encourage them to draw the number in the air and/or draw the number on the mat with their fingers. Repeat with further cards. Number mat Equipment: A large piece of cloth or PVC with the numbers 0 to 9 arranged randomly over it, or an A4 number mat for the students to use in pairs (Material Master 4-13). Lay the large number mat out in the middle of a group of students. Ask a student to jump on a number you have chosen for them. Get the students seated around the outside of the mat to draw the number on the floor in front of them with their fingers so that you can quickly see which students can identify the number correctly. The students can play Twister by having three or four numbers called out consecutively. They must use their hands, feet, or head to touch each new number while leaving the other numbers untouched. Two or three students can be on the mat at the same time and can be given different sets of numbers to touch. Lily pads Equipment: Large numeral cards showing 0 to 9 (Material Master 4-3). Line up the numeral cards in order to create lily pads. The students act as frogs and jump on specific numbers until Stop is called. Get the students who are seated around the outside of the lily pads to write the number on the floor in front of them with their fingers so that you can quickly see which students can identify the number correctly. Pipe cleaner numbers Equipment: Two pipe cleaners for each student; large numeral cards showing 0 to 9 (Material Master 4-3). Say a number to the students. (The size of the number will depend on the ability of the students.) Have the students make that number with their pipe cleaner(s). If they have trouble doing this, they can refer to the numeral cards and make their pipe cleaner number either beside or on top of the numeral card. This learning experience can be repeated with play dough. Teaching Addition, Subtraction, and Place Value 11

14 Key idea 2: The number word sequence for numbers in the range 1 10 is said accurately Learning experiences Counting as we go Equipment: Classroom objects to pass around. Have the students form groups and arrange themselves in circles. Nominate a student from each group to start and provide them with one object. The object is passed around the circle from student to student, and the number of passes is counted. The students count as far as they can or until the teacher calls Stop. Repeat the above activity, but count forwards or backwards from a number other than one. Give all the groups the same starting number. Have all the groups count forwards. Play some music. When you stop the music, each student draws the group s current number in the air. Record the number that each group stopped at on the board or in the modelling book so the students can practise ordering the numbers from smallest to biggest. How many now? Equipment: Marbles or heavy counters or pebbles; a container. Have the students close their eyes and listen as you drop objects, one at a time, into the container. The students count aloud as they hear each object being dropped. At the end, ask how many objects are in the container and record the students responses on the board or in the modelling book. The students can check their counts by emptying the container and counting the number of objects. Repeat with students in pairs. One student does the dropping while the other student counts. Then get the students to swap roles. Loud and soft Equipment: Two hand puppets. Put one puppet on each hand. The first puppet speaks loudly, and the second speaks softly. Counting from one, make the puppets say the numbers alternately. Have the students count simultaneously with the puppets loudly and then softly. If you have a puppet that can squeak, get the students to close their eyes and count the squeaks silently or aloud as the puppet squeaks. Discuss how many squeaks the puppet made and record the count on the board or in the modelling book. Repeat by counting forwards or backwards to or from different numbers. Clapping Equipment: None. Get the students to count from one, clapping their hands in time with each count. Repeat by counting forwards or backwards to and from different numbers. Change the learning experience by getting the students to count from one as they alternately clap their hands and slap their knees. The students then say the clapping numbers aloud and the slapping numbers silently (or vice versa). Repeat by varying the parts of the body used. 12 Teaching Addition, Subtraction, and Place Value

15 Walk the bridge Equipment: Large numeral cards 1 to 10 (Material Master 4-3). Place numeral cards on the ground, in order from 1 to 10, to form a bridge. The students count aloud as one student steps across the numbers (like a bridge). The student who is walking the bridge can decide whether to walk forwards or backwards. The other students watch closely and produce the forwardsor backwards-counting sequence. When Stop is called, all the students who are off the bridge have to identify the number on which the walker has stopped and do that many star jumps (or another movement). Repeat by starting from different numbers. Counting together BSM: (page 18) Equipment: Class counting rhymes. Counting movements to nine BSM: (page 19) Equipment: Rope; drum; chairs; dandelions (or other real or artificial flowers). Key idea 3: The symbols/words for numbers in the range 1 10 are matched to the number of objects in the set Learning experiences Match it up Equipment: Numeral cards 1 to 10 (Material Master 4 1) and dot cards 1 to 10 (Material Master 5 11). Have the students place the dot cards face down in one row, and the numeral cards face down in a parallel row. The students take turns to turn over a card from each row, trying to match a numeral card with a dot card. While some students may need to count all the dots from one, others may instantly recognise some of the number patterns. If there is a match, the student keeps the pair. The game continues until all the pairs have been matched. The game can be repeated by having all the cards mixed and face down in the middle of the group, rather than being in rows. Magic caterpillar legs Equipment: Clothes pegs; numeral cards 1 to 10 (Material Master 4-1); caterpillars (Material Master 5-6). Give each student blank caterpillars to which numeral cards have been added. Students with little number recognition should initially be presented with the 1, 2, and 3 numeral cards only. Gradually increase the number size as the students knowledge of the numbers increases. Have the students add the correct number of legs (i.e., pegs) to each of their caterpillars. They can then practise writing in the air, or on the floor, the number of legs of their caterpillars. The students can also order their caterpillars according to the number of legs each caterpillar has. Teaching Addition, Subtraction, and Place Value 13

16 Petals and flower centres Equipment: Numeral cards 1 to 10 (Material Master 4-1); counters or ice-block sticks. Give each student a flower (a numeral card). Have the students surround the centre of a flower with the correct number of petals (counters or ice-block sticks). They can then order their flowers according to which flower has the greatest or least number of petals. This learning experience can be repeated with spots on dogs, spots on butterflies, and/or lollies on an ice cream. Feed the elephant Equipment: Paper cups; counters or beans or Checkout Rapua material in the School Entry Assessment kit; laminated elephants (Material Master 5-7). Add single-digit numbers to the speech bubbles on the elephants and clip each elephant to a paper cup. In pairs, students take turns feeding their elephant the correct amount of food (counters, beans, etc.) while their partner checks the answer. Students can then order the elephants according to how much each elephant ate. This experience can be used to develop five and thinking. Using numbers from six to nine only, the students feed the elephant five yellow bananas (yellow counters or beans) and green coloured vegetables (green counters or beans) for the rest of the numbers. Birthday cake Equipment: Ice-block sticks; numeral cards 1 to 10 (Material Master 4-1); laminated birthday cake (Material Master 5-8). In pairs, have the students select numeral cards that tell the age of the student who is having the party. They then put the matching number of candles (ice-block sticks) on the birthday cake. This learning experience can be used to develop simple addition and subtraction of one. Repeat the activity, but this time ask questions such as: How many more candles will be on the cake next year when [name] is one year older? or How many fewer candles would have been on the cake last year when [name] was one year younger? Discuss, and record the story problems on the board or in the modelling book. Key idea 4: The sequence of numbers in the range 1 10 is ordered correctly Learning experiences Before and after Equipment: Counters; numeral cards 1 to 10 (Material Master 4-1); game boards (Material Master 5-4). Have the students take turns to pull a card from the stack of cards. One at a time, they say the number that comes directly before or after the number they have pulled out and then cover this number on their game board. If a student cannot cover a square, the next student has their turn. The pulled-out card is replaced in the stack so that it can be pulled out again. The game ends when a student covers all their numbers. 14 Teaching Addition, Subtraction, and Place Value

17 Ordering numerals Equipment: Numeral cards 1 to 10 (Material Master 4-1). Have the students order the numeral cards from 1 to 10, forwards or backwards. They can then practise saying the sequence as they point at the cards. For example, have them point to a number and then ask, What number comes before that number? or What number comes after that number? Up or down Equipment: Pegs; vertical number lines (Material Master 5-3). Ensure each student has a vertical number line from 1 down to 20. Have each student place a peg on 1 and a peg on 20 on their number line. Choose a student to think of a mystery number between 1 and 20. The rest of the students take turns to guess the number and ask the chooser if their number is up (less than) or down (more than) from the mystery number. The students move their pegs to narrow the range of answers until the mystery number is found. Repeat by choosing another student to select a mystery number. How many beans? Equipment: Plastic beans. In pairs, have each student pick up a handful of beans and, before counting them, say how many they think they have. Each student then counts their partner s beans. They compare who has got more and who has got less by lining up the beans side by side. Record the students comparison stories on the board or in the modelling book for example, Brian had eight beans, and Margaret had six beans. Brian had more beans than Margaret. Making a series to nine BSM: (page 8) Equipment: Cards showing sets of one to nine objects; blank cards and stamps; numeral cards 1 to 9 (Material Master 4 1); an egg timer. Key idea 5: Patterns for numbers 1 5 are recognised instantly Learning experiences Patterns to five, then ten Equipment: Tens frames (Material Master 4-6); playing boards (Material Master 5-10); counters; dominoes and dice. Have a pile of tens frames with five or fewer dots on each. Flash a tens frame briefly. The students say aloud how many dots they see and model that pattern on one hand. Once you are sure the students successfully know all their patterns to five, explore the numbers between six and ten. Use a pile of tens frames with five or more dots on each. Show one of the students a tens frame briefly. That player says how many empty spaces there are on the frame. If correct, the player moves their counter forwards that number of spaces on the playing board. Alternate turns. The first person to reach the end of their playing board is the winner. Teaching Addition, Subtraction, and Place Value 15

18 Dominoes are also excellent for getting students to learn to instantly recognise patterns for the numbers 0 6 (standard set) and 0 9 (extended set). By playing the game, students learn to match without counting. Dice are similarly useful for the numbers 1 6. How many different sets of five can you make? BSM: (page 34). Equipment: Cubes; nursery sticks; beans; three dice one showing cubes, one showing nursery sticks, and one showing beans. Fabulous fives Equipment: Ten-bead string. Students practise making finger patterns for the numbers one to ten. The focus is on grouping, not counting. Begin with finger patterns the students will know, like one, two, five, and ten. Say, Show me five fingers Now, show me six. What did you do to make six? Show me ten fingers How do you know it is ten? (five and five). Now, show me nine. What did you do to make nine? Using imaging Students image the count when they are able to count physical objects in their minds. Repeat this activity with the students hiding their fingers behind their backs. As they pretend that their fingers are tall soldiers standing at the castle gate, get them to image the finger patterns from one to five, and then from six to ten. 16 Teaching Addition, Subtraction, and Place Value

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