1. Pedagogical Content Knowledgeof Numbers & Operations, Relationship &AlgebraPrepared by: Mdm. Catherine Kiu (Week 1)
Pedagogical ContentKnowledge• PCK allows teachers to transformtheir knowledgefor the benefit of students in their classroom(Shulman, 1986) • Pedagogical content knowledgeexamples: - finding multiple ways to represent ideas- creating developmentally appropriatelessons- adapting material to the needs of specificstudents.
Pedagogical ContentKnowledge
Pedagogical ContentKnowledgeofWholeNumbers &OperationsBasic Concept: Whole Numbers and Operations
Basic Concept of Whole Number • Whole numbers can be represented in various ways • Eg. numerals (5), words (five), and on a number line. Representation Counting• Wholenumbersareusedfor counting. • They represent thenumberof objectsinaset,andthecountingsequencestartsfromzero(0) andextendsinfinitely: 0, 1, 2,3,.PositiveNumbers• always non-negative.• Equal to or greaterthan“0”• without anyfractional ordecimal parts. Basic operations • Can be used in basic mathematical operations (+，-，x, ÷ ) • Fundamental for solving variousmathematical problems. Place Value • Whole numbers have place value, where the position of a digit in a number determines its value. • The places are ones, tens, hundreds, thousands, and so on. Ordering and Comparing • Whole numbers can be ordered and compared using symbols such as greater than (>), less than (<), and equal to (=). Cardinality • Whole numbers are used to express the cardinality or quantity of a set. • E.g. There are five apples in the basket.
1. Addition (+): • Combining two or more numbers to find their total or sum. • E.g. 3+4=7; 3+4=7 2. Subtraction (-): • Finding the difference between two numbers. • E.g. 8−5=3; 8−5=3 3. Multiplication (×): • Repeated addition or combining equal groups. • E.g. 2×6=12; 2×6=12 4. Division (÷): • Sharing or grouping a quantity into equal parts. • E.g. 10÷2=5; 10÷2=5 Basic Concept of Operations
• Choose and explain threebasicconcepts of Whole NumbersandOperations with appropriateexamples. • Multiplication and divisionarerespectively definedasrepeatedaddition and repeatedsubtraction. Justifystatementabove with suitable examples.
Stages of conceptual development of Whole Numbers and Operations.Pedagogical ContentKnowledgeofWholeNumbers &Operations
1. Pre-Counting Stage: Age Range: 0-3 years Characteristics: • Recognition of quantities in everyday situations. • Informal exposure to numbers in daily life. • Basic understanding of one-to-one correspondence(e.g.,matching one object to one count). Stages of conceptual development of WholeNumbers and Operations
2. One-to-One Correspondence:Age Range: 3-4 years Characteristics: • Ability to count with understanding. • Developing the concept of one-to-one correspondence(each object gets one count). • Basic understanding of the order of counting. Stages of conceptual development of WholeNumbers and Operations
3. Counting Objects: Age Range: 4-6 years Characteristics: • Counting objects with understanding upto10andbeyond.• Recognizing and writing numerals. • Developing the concept of cardinality (understandingthelast number counted represents the total). Stages of conceptual development of WholeNumbers and Operations
4. Counting to Solve Problems: Age Range: 6-7 years Characteristics: • Using counting to solve simple additionandsubtractionproblems. • Understanding basic addition and subtractionconcepts.• Developing a sense of quantity relationships. Stages of conceptual development of WholeNumbers and Operations
5. Multiplicative Thinking: Age Range: 9-11 years Characteristics: • Understanding multiplication as repeatedaddition.• Developing an understanding of the inverserelationshipbetween multiplication and division. • Solving problems involving multiplicationanddivision.Stages of conceptual development of WholeNumbers and Operations
6. Developing Fluency andAutomaticity:Age Range: 11+ years Characteristics: • Automatizing basic addition, subtraction, multiplication,anddivision facts. • Applying operations fluently and efficiently. • Developing problem-solving skills in morecomplexcontexts. Stages of conceptual development of WholeNumbers and Operations
7. Abstract Understanding: Age Range: Adolescence and beyond Characteristics: • Abstract thinking and understanding of thepropertiesofwhole numbers. • Applying mathematical reasoning to solvecomplexproblems. • Exploring more advanced mathematical conceptsbeyondbasic operations. Stages of conceptual development of WholeNumbers and Operations
Read more for better understandingofconceptual development of WholeNumbers and Operationsat: https://numeracyguidedet.global2.vic.edu.au/numeracy-focus-areas-developing-number-sense/
Compare andcontrastthestages of conceptual development ofWholeNumbers andOperations
Non-standard units in Whole Numbers and OperationsPedagogical ContentKnowledgeofWholeNumbers &Operations
• refer to using unconventional or non-traditional units of measurementwhen dealing with quantities and performing operations. • Used in measurement (length, weight and volume). • Length : use paper clips, straws, or other objectstomeasurethelength of an item – E.g. weight of an item. – E.g. measurepaperclipsto measure the length of a desk. • Volume: use cups, bottle caps, or small containerstomeasurethevolume of a container- E.g. measure the volume of ajugusingsmallcups • Weight: use marbles, stones, or other small objectstomeasuretheweight of an item. – E.g. measure the weight of a fruit usingmarblesNon-standardunits
Based on the topicsofmeasurements(weight, lengths,and volumes), determinetwonon-standardunitsusedtoteach each of thetopics. Justify your answer.
Example of ExerciseNon-standard units(Length)
Exampleof ExerciseNon-standardunits(Volume)
Example of ExerciseNon-standard units(Weight)
Estimation in Whole Number and OperationPedagogical ContentKnowledgeofWholeNumbers &Operations
• involves making approximate calculationsorguesses using whole numbers. • a useful skill that allows an individual toquicklydetermine reasonable or approximateanswerswithout precise calculations. • Estimation is made by statingthequantitybasedon a reference set and using “approximate”,“lessthan” and “more than” Estimation in WholeNumbers
1. Rounding: • round numbers to the nearest place value. • E.g. - Estimate the sum of 48 and 72 - round them to 50 and 70, respectively- add the rounded values (50 + 70 =120) toget anestimate of the sum. Techniques of Estimation
2. Front-End Estimation: • involves using only the leadingdigitsof thenumbers. • E.g. - estimate 387 + 216 - add the leading digits (300 +200=500)togetanapproximate sumof 500. Techniques of Estimation
3. Compatible Numbers: • numbers that are easy to work with whenperformingmentalor estimated calculations - chosen to simplifycalculations• E.g. - estimate 139 ÷ 7 - use the compatible number 140 (whichisdivisibleby7and get an estimate of 20. Techniques of Estimation
4. Compensation: • adjust one number to make calculations easier. • E.g. - estimate 68 + 37 - compensate: 68 + 2 = 70, 37 - 2 = 35; resultingin70+35.- 70 + 35 = 105 Techniques of Estimation
Select suitable estimationtechniques to solve thequestionsbelow. A technique canbeusedtosolve not more than 2questions. a) 232 + 347 = b) 89 – 78 = c) 58 x 63 = d) 878 ÷ 5 =
Pedagogical ContentKnowledgeof Fractions,Decimals andPercentageMeanings of Fractions, Decimals and Percentage
❑ A fraction - a numerical value that representsapartofawhole. ❑ written as a ratio - the numerator (thenumber abovetheline) representing the measured quantityandthedenominator (the number belowthe line) representingthetotal amount. ❑ E.g. if someone cuts a cake into eight equal pieces,eachpiece would be one-eighth of the cake. What is a Fraction?Numerator(themeasuredquantity)Denominator(totalamount)
Meaningof FractionsPart-wholeQuotient Ratio
Meaningof Fractionsquotient
Meaningof FractionsRatio 3 5
Meaning of FractionsPart-whole Main focus
Models of the Part-wholeMeaning• Attributes such as region, length, set andareaareuseful in modelling the part-wholemeaningofafraction. • Region model is most often used–thesimplest
Models of the Part-wholeMeaning• the region = whole (the unit) - the parts are congruent (same size & shape). • The region may be any shape, such as a circle, rectangle, square or triangle. • should use a variety of shapes when presenting the region model so that the children do not think that a fraction is always a particular shape. Region Model
Models of the Part-wholeMeaningLength Model 4
Models of the Part-wholeMeaningSet Model
Models of the Part-wholeMeaningArea • In the area model fractions are representedaspartsofanarea or region. • General version of the region model - partsmust beequalin area but not necessarily congruent. • Before using this model, must make surestudentshavesome idea of when 2 different shapes haveequal areas.• more appropriate for older children (Year 4&Year5)
Models of the Part-wholeMeaningArea Model • Useful manipulatives include rectangular, or circular fractionsets,patternblocks, geoboards and tangrams. • Rectangular, or circular fraction sets helps to: - develop the understanding that fractions are parts of awhole- to compare fractions - to generate equivalent fractions and to explore operations withfractions.• the rectangular model - easier for students to drawprecisely• the circular model - emphasizes the part-whole concept of fractionsandthe meaning of the relative size of a part to the whole(Cramer, Wyberg& Leavitt, 2008).
Models of the Part-wholeMeaningArea Model
Models of the Part-wholeMeaningArea
Three Main Types of Fractions
Other Types of Fractions
Other Types of Fractions
What is a Decimal
Fractions can bedefinedasanumerical value thatrepresentsapart of a whole, quotientorratio.Explain fractionasapart-whole.with suitable examples.
MeaningofDecimals