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Welcome to Dingo State School Numeracy Afternoon Our Aim • Help children feel confident when working mathematically … How is mathematics education different for this generation? Previously we were taught mainly facts and procedures. Lessons were about practise and recall Lessons ensure students have deep and connected understandings. Students are expected to explain, reason and justify. In the Australian Curriculum, we expect students to have: • Understanding • (connecting, representing, identifying, describing, interpreting, sorting, …) • Fluency • (calculating, recognising, choosing, recalling, manipulating, …) • Problem solving • (applying, designing, planning, checking, imagining, …) • Reasoning • (explaining, justifying, comparing and contrasting, inferring, deducing, proving, …) “But, I wasn’t good at Maths” When we focus too much on facts and procedures, students may develop misconceptions and inefficient methods. This quite often leads to maths anxiety, disengagement and poorer performance in all subjects that require mathematics. Going mental first • Mental computation and estimation account for approximately 80% of the calculations we do in real life • Using a calculator 15-18% of the time • Paper and pencil methods 3-5% of the time Most adults were not taught mental computation methods. Discussion • 30% of Australian adults are unable to use the procedure that was drilled into us. • Most adults were shown one way to add and subtract, multiply and divide. • Often, these adults have ‘ANXIETY’ in situations that involve calculation Counting and Subitising The counting principles •Each object must be counted only once •Numbers must be said once •Arrangement of objects does not affect how many there are in the collection •The last number said tells how many in the collection Key Terms •Rainbow Facts or friends of 10 •Doubles •Near doubles •Subitising •Partitioning •Compensating •Jumping Key Terms •Rainbow Facts or friends of 10 Key Terms •Doubles •Near doubles 2+2 3+3 22+22 46+46 6+7 8+9 22+23 46+47 Counting and Subitising Subitising •Students recognise numbers at a glance •Familiar with collections or arrangements such as dice patterns •Seeing smaller parts within collections Subitising assists students to quantify collections and build whole-part relationships and an understanding of decimal numbers. Partitioning •Breaking a number up into smaller numbers, usually hundreds, tens and ones. 156 •100 + 50 + 6 •50 + 50 + 50 + 6 9731 •9000 + 700 + 30 + 1 Builds fluency and flexibility of thinking [ ]=7+5+3+5+6 How would you add these numbers? [ ]=7+5+3+5+6 Child A adds in order: 7,5,3,5 and then 6 (counting all on her fingers) Child B says ‘7 and 3 make 10’, ‘5 and 5 make 10’ and ‘6 more is 26’ Child C says ‘5+5 = 10’ , then ‘6 + 3 = 9’, ‘so that’s 19’. He then counts on 7 more Child D takes 2 from 7 and adds it the 3 to make 4 lots of 5. She says, ‘Four fives are 20. 20 and 6 more is 26.’ Insight into misconceptions 7+4=[ ]+5 16 11 6 11 & 16 55% 20% 15% 10% Here are the four most common student responses (Years 2-7) What is a typical response rate in Year 4? 6 is the correct answer. 201-198 201 - 198 Jumped straight to a procedure and got it wrong Jumped straight to a procedure and got it correct Reasoned mentally Trusted Numbers 2/3 14% 25% 61% 4/5 25% 35% 40% 6/7 28% 41% 31% Give your child a number fact. 7 8 3 4 7 4 7 4 3 4 7 7 3 3 3 4 Can they create the related number facts? 12 4 To 12 ensure they 8 4 UNDERSTAND 12 4 8 Ask your child to say or 12 a problem draw 8 4that (USES) one of these 12 facts 4 8 24 8 8 3 24 24 3 3 3 8 3 8 12 24 24 8 3 LINK this to FRACTIONS Eg ⅓ of 12 = 4 3 4 12 12 4 4 3 3 4 12 12 4 3 FOUR SQUARE NUMBER FACTS 8 + 2 = 10 2 + 8 = 10 RELATED FACTS 10 - 8 = 2 10 – 2 = 8 STORY ( drawing or oral ) A chef made 10 cakes and I ate 2 of them. There are 8 cakes left. EXTENDED FACTS If I know 8 + 2 = 10 then I also know: 80 + 20 = 100 100 – 20 = 80 What methods do we use as adults? 38 + 57 72 - 28 JUMP- addition & subtraction SPLIT: addition & subtraction Students must be confident and flexible with Place Value concepts to assist with mental computation. Left to right adding and subtracting • Students and adults naturally use left to right thinking for mental calculations. • Left to right written methods connect to students’ mental strategies. Compensating 38 + 57 72 - 28 What does Australian Curriculum expect? • Addition and subtraction facts by Year 3 • Multiplication and division facts by Year 3/4 • A greater focus on mental methods to prepare students for real world situations • A range of written methods (not just the one method) • Students to choose methods and strategies to suit individual problems How can you help at home? •Talk about ways that you use to add and subtract mentally. •Practise number facts in game situations. •Don’t expect your children to naturally add and subtract like you do. •Let your children discuss their ways for adding and subtracting with you. Resist the temptation “ This is how you should do it” Ask your child to explain their strategies for adding and subtracting Ask your child for an estimate before they calculate the exact answer Ask “Is that answer reasonable?” Adding number plates Other Ideas • Play board games with dice, spinners, dominoes and cards. • Cook with the kids, scales, measuring cups and timers etc. • Smarties or any sweets are great for counting large numbers, sharing….. • Family bar of chocolate is great for fractions • Shopping or travel challenges. How long, How much If students are shown the links they are less likely to forget them 1 8 2 3 16 24 4 32 5 6 7 8 9 40 48 56 64 72 The x1, x2, x5 and x10 facts can provide a foundation for all multiplication and division facts. 10 80 10x10 How many multiplication facts are there? 10x9 9x9 10x8 9x8 8x8 8x7 7x10 7x9 7x7 10x6 7x6 9x6 8x6 6x6 9x5 5x6 7x5 5x10 8x5 5x5 4x10 9x4 4x7 4x8 5x4 4x6 4x4 3x5 4x3 3x10 3x6 8x3 9x3 7x3 3x3 4x2 2x5 9x2 2x6 10x2 2x7 8x2 2x2 45 2x3 What do we do if my child doesn’t know a table fact? I have forgotten 8 x 7? I know 7x2=14 Double 14 is 28 Double 28 is 56 I know that 5 x 8 = 40 2 x 8 = 16 40+16 = 56 10 x 7 = 70 Take two 7s (14) from 70 That’s 56 Teaching Number Facts What is more important. To know your number facts or to understand strategies to solve number facts? 37 You need to know the basics 1’s 2’ 10s 1x9 1 x 29 1 x 379 2x9 doubling 2 x 29 2 x 279 10 x 9 9 groups of 10 NOT add zero 10 x 29 279 x 10 Term 1 2007 38 Number Facts 3X What strategies do you use? 3x6 27 x 3 Term 1 2007 231 x 3 39 Number Facts 3X (3x) double plus 1 3x6 Double 6 + 6 12 + 6 = 18 3x8 Double 8 + 8 16 + 8 = 24 What other strategies? 27 x 3 Double 27 + 27 54 + 27 = 81 231 x 3 Double 462 + 231 462+ 231 = 693 3 x 200 = 600 3 x30 = 90 3x1= 3 Term 1 2007 40 Number Facts 3X Scan Think Do See Plan Do Check 3 x 275 Term 1 2007 41 Number Facts 4X What strategies do you use? 4x9 47 x 4 Term 1 2007 135 x 4 42 Number Facts 4X (x4) Double Double Double the other number, then double again. 4x6 Double 6 + Double 6 12 + 12 = 24 4x9 Double 9 + Double 9 18 + 18 = 36 What other strategies? 37 x 4 Double 37 + Double 37 74 + 74 = 148 135 x 4 Double 135 + Double 135 270 + 270 = 540 4 x 100 = 400 4 x 30 = 120 4x5= 20 Term 1 2007 43 Number Facts 4X Scan Think Do See Plan Do Check 4 x 128 Term 1 2007 44 Number Facts 5X What strategies do you use? 5x6 37 x 5 Term 1 2007 135 x 3 45 Number Facts 5X (x5) Multiply by 10 then half 5x6 10 x 6 60 ÷ 2 = 30 5x9 10 x 9 then divide 2 90 ÷ 2 = 45 What other strategies? 37 x 5 10 x 37 then divide 2 370 ÷ 2 = 185 135 x 5 10 x 135 then divide 2 1350 ÷ 2 = 675 5 x 100 = 500 5 x 30 = 150 5x5= 25 Term 1 2007 46 Number Facts 5X Scan Think Do See Plan Do Check 5 x 246 Term 1 2007 47 Number Facts 6X What strategies do you use? 6x7 34 x 6 Term 1 2007 267 x 6 48 Number Facts 6X (x6) Multiply by 10 then half plus the number OR Double the #, then add it one more time then double again 6x7 ((Double 7) + 7) x 2 (14 + 7) x 2 21 x 2 = 42 6 x7 10 x 7 then divide 2 +6 70 ÷ 2 + 6 = 42 What other strategies? 34 x 6 ((Double 34) + 34) x 2 (68 + 34) x 2 102 x 2 = 204 34 x 6 10 x 34 then divide 2 + 34 340 ÷ 2 + 34 = 170 + 34 = 204 267 x 6 200 x 6 = 1200 60 x 6 = 360 7x6= 42 1602 Term 1 2007 49 Number Facts 6X Scan Think Do See Plan Do Check 6 x 262 Term 1 2007 50 Number Facts 7X What strategies do you use? 7x7 24 x 7 Term 1 2007 379 x 7 51 Number Facts 7X (x7) Multiply Fives strategies plus double strategies 7x7 7 x 10 ÷ 2 + (7 x 2) 70 ÷ 2 + 14 35 + 14 = 49 8’s -1? 9’s – double? Does anyone know a better mental strategy? 24 x 7 24 x 10 ÷ 2 + (24 x 2) 240 ÷ 2 + 48 168 138 x 7 138 x 10 ÷ 2 + (138 x 2) 1380 ÷ 2 + 276 690 + 276 966 379 x 7 300 x 7 = 2100 70 x 7 = 490 9x7= 63 2653 52 Number Facts 7X Scan Think Do See Plan Do Check 7 x 425 Term 1 2007 53 Number Facts 8X What strategies do you use? 7x8 34 x 8 Term 1 2007 154 x 8 54 Number Facts 8X (x8) Double + Double + Double When multiplying by 8 double the other #, double the result, and then double again. 6x8 Double 6 + Double 12 + Double 24 6x2x2x2 48 7X8 Double 7 + Double 14 + Double 28 7x2x2x2 56 Does anyone know a better mental strategy? 34 x 8 Double 34 + Double 68 + Double 136 34 x 2 x 2 x 2 = 272 154 x 8 Double 154 + Double 308 + Double 616 154 x 2 x 2 x 2 1232 278 x 8 200 x 8 = 1600 70 x 8 = 560 8x 8 = 64 2224 55 Number Facts 8X Scan Think Do See Plan Do Check 8 x 256 Term 1 2007 56 Number Facts 9X What strategies do you use? 3x9 27 x 9 Term 1 2007 231 x 9 57 Number Facts 9X (x9) Multiply by 10 then subtract the number 6x9 6 x 10 then take 6 60 – 6 = 54 8x9 8 x 10 then take 8 80 – 8 = 72 ROTE 34 x 9 34 x 10 then take 34 340 – 34 = 306 156 x 9 156 x 10 then take 156 1560 – 156 = 1404 456 x 9 400 x 9 = 3600 50 x 9 = 450 6x9 = 54 4104 58 Number Facts 9X Scan Think Do See Plan Do Check 9 x 455 Term 1 2007 59 Double Digit Multiplication What aspects of learning multiplication are the most important? 60 By the end of primary 27 x 33 = Trusting the numbers In order for our students to do this they must TRUST the numbers. Thank you 62 Cheers