Maths Exam Prep PDF

***[Maths exam prep]*** ***[Maths pedagogical practices:]*** Using cognitively challenging tasks Fostering productive disposition- Promoting maths talks Encouraging playfulness Emphasising maths modelling ***[Maths proficiencies:]*** Adaptive reasoning- using logic to explain/understand thin...

***[Maths exam prep]*** ***[Maths pedagogical practices:]*** Using cognitively challenging tasks Fostering productive disposition- Promoting maths talks Encouraging playfulness Emphasising maths modelling ***[Maths proficiencies:]*** Adaptive reasoning- using logic to explain/understand thinking Procedural fluency- using maths procedures in effective way Productive disposition- seeing maths as useful/beneficial Strategic competence -- skills to solve maths problems- Conceptual understanding- comprehension of maths concepts ***[Types of picture books:]*** Explicit- teaches maths concepts Embedded- Entertains and portrays maths ideas Perceived- Entertains and maths concepts are unintentional/ incidental ***[Number strand units:]*** Uses of number Numeration and Counting Place value and Base ten Sets and operations- problems -- Change (Join/separate, book- 3 tigers and here comes 2 more) part part whole and compare Fractions ***[Experiences required by children to count effectively:]*** (**C**an **S**arah **M**cCarthy **R**un **R**agged **C**atching **S**weets) Classifying/ Sorting- Common properties, similarities/differences (book- Grandmas button box) Matching- relationships and similarities/differences or equal/not equal (book- Fox socks, 3 little pigs) Rote- reciting number names in correct order from memory (Experiences- counting forwards/backwards, songs- 10 little fingers/1,2,3,4,5/ 5 little monkeys, games- pass the teddy- I am 1, I pass it to 2, I am 2, I pass it to 3, or start at 3 and count onwards) Rational- Object of a set and number names (resources- counters, cubes, shapes, 5/10 frame, number fans-) Context -- Suzie's party (no. of sweets in bowl, no. of party hats, knocks on the door for secret code) Combining- objects close or spread out the quantity in set stays the same Subitising- Recognising number of items in a group without counting (Activity- Display picture dots for a second, students' close eyes and then guess without counting, or getting Suzie home, throw the die and can only move a space if they roll a 5) ***[The Zero set:]*** Show empty box of crayons, tin of sweets Book- Handa's surprise ***[Experiences required by children to develop number sense:]*** Games to develop number sense- sneaky snake- roll 2 dice and add/take away. Cover all number on the snake. Similar to bingo (**R**achel **C**arroll **M**isses **O**rdering **C**hinese **P**rawns) Rational counting Comparing- size, weight, height, colour, properties (No. of choc chips on cookies, no. of sweet in a tube, making towers from cubes) Matching sets and numerals- (number fan and numeral card matching, bingo, suzies party- ingredients for the buns, 3 eggs, 2 flour, 4 sugar spoons) Ordering- Making stair, add 1 block per step, number cards and line up in order, make harder by starting on diff number such as 3, number line- put counter on number after 4, between 3 and 5 (Ordering cookie monster t shirt, building stairs for him to reach clothesline) Combining -- story problems Partitioning- diff ways to make a number, number bonds, we can make 6 loads of ways 1+5,2+4,3+3,4+2,5+1,6+0 and 0+6. (Lola ladybirds' outfit, book- monster maths picnic, cubes, lollipop sticks, clothes hanger, rhymes, number problems) ***[Recognising numerals:]*** Straight/curved lines- rhymes (Guess my number, guess the shape/describe the number, feel in the bag, bingo, show me 245, number line and counter, I spy) ***[Forming numerals:]*** Familiar with shape, describe numeral, songs and rhymes Trace, sand basin, partner back, playdough, water paint, dotted line, air trace Error- reverse numbers ***[Ordinal number:]*** Cardinal number- 1,2,3,4,5 Ordinal number- 1^st^, 2^nd^, 3^rd^, 4^th^ Games- hide and see with counter under cups, close eyes and move comes Stories/ books- Hare and the tortoise/ hungry caterpillar ***[Stages of counting:]*** - Count all- combine all groups of objects and count total - Count on - Count on from 1^st^ number- Adding 2+3, I have 2 so 3,4,5. Adding 3+4, I have 3 so 4,5,6,7 - Counting on from larger number- awareness of **commutative property-** 2+3=3+2. Order doesn't matter. Maths language- and, plus, in addition, combine, together ***[Addition:]*** Language- sum of, add, combine, together, total, increase, sum of Symbols - teach symbols last ***[Properties of zero:]*** Number remains unaltered when 0 is added ***Commutative property***- order doesn't matter 3+4=4+3 ***[Basic facts for addition:]*** 1 more than/ 2 more than Facts with zero Doubles/ near doubles Make 5 Make 10 Doubles +2 ***[Addition strategies:]*** Split method- 46+38= 40+30= 70 6+8=14 70+14=84 Jump method- 76+8= 76+4=80 80+4=84 76+8=84 Short cut method- 46+38= 44+40= 84 take 2 from 46 and give 2 to 38 Or 46+40-2= 86-2=84 Mixed method- mixture of all of them ***[Estimation strategies:]*** Rounding Front end- 154+635+99+251= 100+600+200=900 Clustering Compatible numbers ***[Renaming:]*** Addition situations where new groups of 10/100 are added ***[Place value mat vs transition board]*** Place value mat- shows place values Transition boards- shows operations Use transition board first, then algorithm with symbol Book- Circumference and all the king tens, a fair bear share ***[Subtraction:]*** Types of problems: Reduction- Change (separate) problem and book- elevator magic Complementing- Part part whole problem- and book- Subtracting with Sebastian pig and friends on a camping trip Differences- Comparison problems Language of reduction- take away, minus, subtract, less Language of complementing- I have 5, how many more do I need? Language of differences- Difference between 5 and 7 Book- Shark swimathon ***[Subtraction facts:]*** Think as addition 8-3 or think of addition 3 +? =8 All numbers -- 0 are the same number 5-0, 6-0, 7-0 Numbers who's difference is 0 1-1, 2-2, 3-3 Numbers -- 1or 2 5-1, 3-2 ***[Subtraction strategies:]*** Jump methods- 85-58 58+20=78 78+2=80 80+5=85 20+2+5=27 85-58 58+30=88 88-3=85 30-3=27 85-58 58+2=60 60+10+10+5= 85 2+10+10+5=27 85-58 85-50=35 Split method- 80-50=30 5-8= - 3 30-3=27 Shortcut method- 85-60=25+2=27 ***[Estimation methods:]*** Round all up= 419-65 goes to 420-70 Round only one number: 6724-1863 goes to 6724 -- 2000 Round all down (front end) 51-22 goes to 50- 20 Round 1 up 1 down (both close to the middle) 65-24 goes to 70-20 Round to (compatible numbers) 72-18 goes to 75-15 ***[Multiplication:]*** Repeated addition Equal group contexts- group of 2 (feet, eyes) 4 (wheels, animals' legs) 6 (eggs) Books- What comes in 2s, 3s and 4s/ How many feet in the bed/ Spunky monkeys on parade ***[Multiplicaton problems: ]*** Comparison problem- Book Minnies diner ***[Multiplication models:]*** Sets/ collections Arrays Number line Hundred squares ***[Properties of multiplication:]*** Commutative- order doesn't matter 4x3=3x4 Distributive- 7x3= 5 x 3 + 2 x 3= 15 + 6= 21 Zero - Any number x 0 will always be 0= 4x0, 5x0, 6x0 = 0 Identity- Any number x 1 is the number = 3 x 1, 4 x 1, 8 x 1 = 8 Associative- 3 numbers multiplied together, order in which may be performed may be varied according to convenience (3x4) x5 = 3 x (4x5) ***[Basic facts of multiplication:]*** Repeated addition- 7 x 3= 3+3+3+3+3+3+3= 21 Commutative property= 7 x 3 = 3 x 7 Skip counting= 2, 5, 10 Zero property= 2 x 0, 8 x 0, 6 x 0= 0 Nifty nines- Finger nines Book- The best of times ***[Strategies for multiplication:]*** Repeated addition= 29 x 4 - 29 + 29 + 29+ 29 Partitioning- 29 x 4 -- (20 x 4) + (9 x 4) = 80 + 36 = 116 Short cut method -- 29 x 4= (30 x 4) -- (1 x 4) = 120-4 = 116 ***[Estimate first and then introduce algorithm:]*** Regular multiplication- 15 x 4 = (10 x 4) + (5 x 4) = 40 + 20 = 60 Long multiplication- 36 x 24 = (36 x 4) + (36 x 20) = 36 x 24 = (30 x 4) + (6 x 4) = 120 + 24= 144 36 x 24 = (36 x 20) or (36 x 2) x 10= 72 x 10= 720 720 + 144= 864 ***[Multiplication and division books:]*** 1 grain of rice 2 ways to count to 10 A remainder of 1 Divide and ride ***[Division:]*** Emphasis sharing or grouping ***[Types of problems:]*** Type 1- Equal groups (groups of 2, 4,6) (eyes, feet, wheels, eggs) a. Partitioning- fair share (size of set is unknown) 24 apples to share evenly amongst 4 friends. How many apples do they get. Book- clean sweep campers b. Measurement (size of set is known) 24 apples, each bag has 6 apples. How many bags Book- divide and ride, a remainder of 1 Type 2- Multiplicative comparison (set size unknown) Jill scored 24 goals in league, 6 more than Alan. How many did Alan score. 583 cookies produced every day. Divide between 4 factories. Resources- paper plates ***[Fractions:]*** ***Models to develop understanding of fractions:*** 1: Area model: paper folding, geoboards, pattern blocks, dot paper (book- Give me half or full house) 2 Length model: Measurement model- number line, fraction strips/ walls 3: Set model: Collection of objects- cubes, counters (book- Jump, Kangaroo, Jump, the doorbell rang) ***[Finding fraction of a unit:]*** Partitioning a unit into equal sized pieces (Geoboards, pattern blocks) ***[Impact of unit size:]*** Is a 1/3 bigger than a ½ Use paper folding to determine ¼, ½ etc ***[Comparing fractions:]*** Use benchmarks to compare fractions (similar to a fraction number line) 0, ¼, ½, ¾, 1, 1¼, 1½, 1¾ 2 Is the sum between 0 and1 or 1 and 2? Books- Apple fractions, Full house, Give me half ***[Why use fraction models:]*** Demonstrate the nature of fractions- geoboards, dot paper and elastics (Different ways of making quarters) Compare fractions- is 7/12 bigger than 9/10s Find a fraction of a unit- 6 pizzas between 4 kids. How many slices each? Find the unit when given the fraction Demonstrate the impact of unit size- Would you rather 1/3 or ½ a pizza ***[Children's understanding of:]*** Equivalence- To find the equivalent fraction, we multiply/ divide the numerator and denominator by the same number. 2/2=1 5/5=1 Operations- ***[To simplify a fraction:]*** To simplify a fraction, you must divide the numerator and the denominator by the same number -- 6/9 divide both numbers by 3 is 2/3. ***[Adding fractions:]*** Create story problem of fractions with the same denominator- 3/8 + 2/8 ***[Same fraction family:]*** ¼ + 3/8 3/12 + 4/12= 7/12 ***[Different fraction families:]*** ¼ + 2/3 3/8 + 7/9 = 9/24 + 14/24 = 23/24 ***[Subtracting fractions:]*** ¾ - 2/3 (Paper folding, fraction wall) ***[Mixed fractions:]*** 3 1/5 -- 1 3/10 ***[Multiplying fractions:]*** Repeated addition -- 4 x 5= 20 4+ 4+ 4+ 4+4=20 4 x ½ ½ + ½ + ½ + ½ = 2 Commutative property- 4 x5 = 5 x 4 3/8 x 2/8 = 6/ 8 ***[Fraction division:]*** Grouping- how many 4s in 12? Sharing- Divide 12 between 4 people ***[Decimals:]*** Introduce to tenths = 1/10 -- 0.1 1/100- 0.01 1/1000- 0.001 Fractions and decimals are two different systems of representing the same thing 4 ½ = 4.5 ***[Role of the decimal:]*** Decimals separate whole numbers from fractions 1/10= 0 ones and 1 ten= 0.1 ***[Models to promote understanding:]*** Fraction wall Decimal walls- represent tenths and equivalent tenths e.g halves/ fifths Hundred squares- tenths, hundredths and equivalents Tenths- 100 square= 1 unit Each row is 1/10 = 0.1 Hundreths- Each square is a hundredth = 0.01 Dienes blocks- concrete materials and represent tenths, hundredths, thousandths [Introducing tenths:] Large cube is 1 unit/ whole 1 tenth -- 1 flat [Introducing hundredths:] 1 hundreth -- 1 long = 0.01 [Introducing thousandths:] Small cube -- thousandth = 0.001 162/1000= 1 flat (tenth), 6 longs (hundredths) and 2 cubes (thousandths) Number lines ***[Ordering decimals:]*** Start with comparison between 2 decimals From smallest to largest- largest to smallest ***[Converting fractions to decimals and decimals to fractions:]*** To change halves, quarters, fifths and tenths into decimals, we can change them into tenths or hundreths. ½ = 5/10 = 0.5 ¼ = 25/100 = 0.25 or we can divide 2 by 1.0= 0.5 ***[Adding and subtracting decimals:]*** Use real life examples e.g weight of shopping Start adding without renaming then start renaming ***[Multiplying decimals:]*** Answer will always be bigger Repeated addition- 5 x 1.1= 1.1 + 1.1 + 1.1 + 1.1 + 1.1= 5.5 x 10 -- decimal moves 1 place to the right x 100 -- decimal moves 2 places to the right 1 decimal place x 1 decimal place= 2 decimal place answer- 0,1 x 0.1 = 0.01 2 decimal place x 1 decimal place= 3 decimal place answer= 0.25 x 0.1 = 0.025 ***[Dividing decimals:]*** Answer will always be smaller ***[Percentages:]*** Percent means per hundred 100/100= 100% 30/100= 30% 75/100= 0.75= 75% Book- piece= part= portion