# TFI Knowledge Base/Maths/Numbers and Operations

## Grade 1

### Std 1.1

1.1 Students understand how to name, represent and use numbers to 100. Using concrete objects, they appreciate the sizes of these numbers and build their mental images.

- SWBAT write the symbols for numbers up to 10 (EI K.2)
- SWBAT represent a given number through objects (EI K.2)
- SWBAT identify 'how many' up to 10 (EI K.2)
- SWBAT identify numbers bigger and smaller than a given number
- SWBAT identify the largest and smallest number in a set (up to 100)
- SWBAT identify quantities up to 100 based on given representation

### Std 1.2

They count and group objects in ones and tens and understand that a 2-digit number has a tens value and a ones value and build an understanding of the concept of place value (in base ten number system) i.e. the place of a digit gives its value. They are able to represent numbers on an abacus i.e. decompose and compose numbers using place-value. They use place value to order numbers to 100 in ascending and descending order (excluding symbols).

### Std 1.6

Using concrete objects and pictures, they understand the meaning and use the terms addition/subtraction. They understand the commutative nature of addition (without using the term) and its use to calculate efficiently. They understand the effect of zero in addition and subtraction and know the symbols for the two operations and the symbol ‘=’.

- SWBAT combine quantities in 2 separate groups up to 10 (EI K.5)
- SWBAT use pictures to represent an addition problem and use an addition problem to represent given pictures (make the connection between the two)
- SWBAT remove quantities in 2 groups up to 10 (EI K.5)
- SWBAT use pictures to represent a subtraction problem use a subtraction problem to represent given pictures (make the connection between the two)
- SWBAT demonstrate understanding of the commutative property of addition by writing two addition problems to (ex. 3 +5 and 5+3) to represent the same situation

### Std 1.7

They are able to add/subtract using concrete objects and skeletal number line (with divisions marked).

- SWBAT add 1-digit numbers up to a sum of 10 (EI K.5)
- SWBAT subtract 1 digit numbers less than 10 (EI K.5)
- SWBAT add 1 digit numbers up to a sum of 20

### Std 1.10

They know the cases in which addition and subtraction are applied and that they cannot subtract a larger number from a smaller number (as of now). They can identify the operation, represent the problems by mathematical expressions, interpret and solve them.

### Std 1.11

They understand the inverse relation between addition and subtraction and can give subtraction corresponding to an addition and vice-versa in number sentences using symbols +,-,= .

### Std 1.12

They understand that sets of objects, pictures, numbers representing them or expressions like 2+3 are all equivalent in the sense of their number.

## Grade 2

### Std 2.1

Students understand and learn how to name and represent numbers to 1000 and note the pattern in their representation. They appreciate relative sizes of the numbers by regarding 10 as a unit or hundred as a unit (in concrete manipulation or mental images).

- SWBAT read and write numbers up to 100*
- SWBAT identify numbers n greater/less than a given number (n=1,2, 10, 100, etc.)

### Std 2.2

They compose and decompose (expand) numbers using place value. They recognize representations using ones blocks, tens blocks, hundreds blocks and write numbers for these representations.

- SWBAT identify place values of numbers up to 100
- SWBAT write numbers, up to 100, in expanded form
- SWBAT identify 2 digit numbers by the numbers of tens and the number of units.
- SWBAT identify and represent a number using block/visual representation.
- SWBAT convert between tens, and units

### Std 2.3

They compare and order the numbers on a partial number line and understand their relative position. They order them in ascending and descending order using symbols >, <, = .

- SWBAT compare numbers, up to 100, and identify largest and smallest numbers
- SWBAT place numbers, up to 100, in ascending and descending order
- SWBAT create largest/smallest numbers from given digits

### Std 2.4

They count backward and forward starting from any number to 1000. They know ordinal numbers to 20 and use them in different contexts.

### Std 2.5

They are able to add and subtract numbers up to three digits without and with composing/decomposing in expanded form (so that they can understand the concept of composing and decomposing well) as well as short form.

- SWBAT add two single digit numbers(EI 1.8)
- SWBAT identify and create visual representations of addition
- SWBAT identify that additing 0 results in the same numer
- SWBAT add 2 2-digit numbers without carryover
- SWBAT identify and create visual representations of subtraction (counting forwards, counting backwards, crossing out)
- SWBAT subtract a 1 digit number from a 1 digit number
- SWBAT identify that subtracting 0 results in the same number
- SWBAT subtract 2 digit numbers from each other without borrowing
- SWBAT add numbers involving carrover
- SWBAT subtract up to 2-digit numbers involving borrowing

### Std 2.6

They know, understand and use commutative and associative properties of addition (without using the terms) and inverse relationship between addition and subtraction.

- SWBAT use the inverse relation between addition and subtraction to fill in missing quantities in equations (in arithmetic problems)
- SWBAT use the inverse relation between addition and subtraction to determine missing quantities in situations (in word problems)

### Std 2.7

They are able to understand the meaning of multiplication (repeated addition) and division (equal sharing, grouping) and their effect using concrete objects, pictures. They understand inverse relation between multiplication and division using concrete objects.

- SWBAT identify and create visual representations of multiplication
- SWBAT identify repeated addition expressions that correspond with multiplication problems (and vice versa)
- SWBAT apply the commutative property of multiplication

### Std 2.9

They understand that multiplication and division are based on multiplication tables and understand, construct and learn multiplication tables of 2 to 5 and 10.

### Std 2.10

They understand and correctly use vertical algorithm for multiplication and division for up to three digit numbers by one-digit numbers without and with composing/decomposing including those in which remainder is left behind. They know the terms related to division (dividend, divisor, quotient and remainder).

### Std 2.12

They summarize the cases for all four basic operations. They identify operations, write expressions, solve and interpret the results for a mixed bag of problems in meaningful contexts.

- SWBAT solve simple word problems using addition
- SWBAT solve simple word problems using subtraction
- SWBAT solve word problems using addition or subtraction and involving carry-over/borrowing.
- SWBAT solve simple word problems using multiplication
- SWBAT solve application questions based on addition/subtraction (for example, questions based on buying items at a store, etc.)
- SWBAT solve application questions based on multiplication (for example, questions based on buying items at a store, etc.)

### Std 2.13

They know and recognize odd and even numbers.

### Std 2.14

They understand using concrete material and pictures- the meaning of fraction as part of a whole or of a collection where whole or collection is divided into equal parts. They recognize and name the following fractions of a whole or a small set of objects - half, one-third, one fourth etc. and two-third, three-fourth using concrete objects and pictures. They know their symbolical representation and the terms numerator and denominator. They compare and order these fractions using concrete objects and pictures. They are explained the representation of these fractions on a number line. They understand concretely, pictorially that when all fractional parts are included such as two halves or four one-fourths, the result is the whole or the complete set.

### Std 2.15

Estimation - estimate number of objects in a set of upto 50 objects, count and compare the estimate with the actual number - check reasonableness of addition/subtraction answers

## Grade 3

### Std 3.1

Students extend naming and representation of numbers to 10000. They understand the cyclical pattern in which the digits change as the numbers increase and realize that counting can go on endlessly in this fashion. They deepen their understanding of whole numbers, their representation and their relative sizes.

- SWBAT read or write numbers up to to 1,000
- SWBAT place numbers in ascending and descending order
- SWBAT identify the number 10 or 100 greater or less than a given number

### Std 3.2

They understand the place-value structure of the base-ten number system and are able to efficiently decompose (expand) and compose numbers using place value. They recognize equivalent expanded representations for the same number and compare and order the numbers.

- SWBAT identify place value of a digit in a number up to 3 digits
- SWBAT write numbers, up to 1,000, in expanded form
- SWBAT compare numbers using greater and lesser than signs ( > < )
- SWBAT identify the largest and smallest number of a set
- SWBAT create the largest or smaller number from given digits

### Std 3.3

They recognize representations using ones blocks, tens blocks, hundreds blocks and thousands blocks and write numbers for the same correctly.

### Std 3.5

They are able to fluently carry out addition and subtraction of numbers up to 4-digits.

- SWBAT add numbers up to 3 digits without carryover
- SWBAT add numbers up to 3 digits with carryover
- SWBAT subtract numbers up to 3 digits without borrowing
- SWBAT subtract numbers up to 3 digits with borrowing
- SWBAT add more than 2 numbers, up to 3 digits each
- SWBAT create (multiple) addition or subtraction expressions that result in a given quantity.
- SWBAT use the inverse relation between addition and subtraction to fill in missing quantities in equations (in arithmetic problems)

### Std 3.6

They understand the meaning and effect of multiplication and division and the cases in which they are applied. They understand commutative, associative and distributive properties of multiplication (without using the terms) and are able to use them.

- SWBAT show and identify visual representations of multiplication
- SWBAT show and identify visual representations of division
- SWBAT apply the commutative and associative property of multiplication
- SWBAT identify repeated addition expressions that correspond with multiplication problems (and vice versa)

### Std 3.7

They construct and learn multiplication tables of 6, 7, 8, 11 and 12.

### Std 3.8

They understand the effect of multiplication by 1, 10, 100 and division by 1 and also by 10, 100 if the number ends in zeroes i.e. the answer is a whole number. They use this and distributive property of multiplication over addition to understand and use vertical multiplication and division algorithm to multiply and divide by numbers of 2 or more digits without and then with composing or decomposing. They understand the relation, dividend = divisor x quotient + remainder.

- SWBAT multiply numbers up to 3 digits by 1 digit
- SWBAT identify the divisor, divident, quotient, and remainder (EI 2.10)
- SWBAT divide numbers up to 2 digits by 1 digit (based on simple division facts, not long division)
- SWBAT multiply a number by 10
- SWBAT divide a number by 10 (no decimal in the answer).

### Std 3.9

They understand the effect of 0 and 1 in multiplication and division.

### Std 3.10

They understand inverse relation between multiplication and division and use it to check results.

- SWBAT solve multiplication problems with a missing number (ex. 5 x ___ = 45 )
- SWBAT solve division problems with a missing number (ex. ___ divide by 8 = 7 )
- SWBAT identify and apply multiplication and division as inverse operations

### Std 3.11

They appreciate the usefulness and purpose of multiplication and division besides addition and subtraction. They develop the ability to correctly choose suitable operations for a mixed bag of problems (a problem may involve more than one operation), write expressions for the problems, solve, check reasonableness of answers and interpret the results.

- SWBAT solve word problems involving addition or subtraction up to 3 digits
- SWBAT solve word problems involving multiplication
- SWBAT solve word problems involving division.
- SWBAT solve word problems involving finding missing values (based on addition or subtraction)
- SWBAT solve application based questions based on addition or subtraction (for example, questions based on buying items at a store, etc.)
- SWBAT solve application based questions based on multiplication or division (for example, questions based on buying items at a store or based on pictures, etc.)

### Std 3.12

They understand properties of sum and product of odd and even numbers.

### Std 3.14

They understand the meaning and concept of fractions as part of a whole or a collection where the whole or collection is divided into equal parts using concrete objects and pictures. They understand naming and notation of fractions and know the terms- numerator and denominator.

- SWBAT identify (visually) and write simple fractions (with denominator up to 15)
- SWBAT identify the numerator and denominator of a fraction

### Std 3.17

They understand the relative sizes of unit fractions (1 or 2 to 1 or 12) and other simple fractions (2 or 3, ¾) using one long object (e.g. sugarcane), a collection of say 12 objects or their pictures. They are explained the representation of these fractions on a number line

### Std 3.20

They thus understand that decimals are fractions and know decimal and fraction equivalents of tenth, hundredth etc /]]

### Std 3.25

Estimation - estimate number of objects in a set of upto 100 objects, count and compare the estimate with the actual number - use properties of operations and their mutual relation to check reasonableness of results

## Grade 4

### Std 4.1

Students deepen their understanding of whole numbers by knowing number representation to 100000 and may be, beyond. They summarize the decimal numeration system and understand the relative sizes of the numbers.

- SWBAT read/write numbers up to to 100,000
- SWBAT create largest/smallest numbers using a given set of digits
- SWBAT identify numbers n greater/less than another number (where n=10,100,1000).

### Std 4.2

They decompose and compose numbers using place value and are able to efficiently compare and order.

- SWBAT identify place value of numbers. SWBAT decompose numbers into expanded form SWBAT compare numbers using greater than/less than signs up to 100,000
- SWBAT place numbers in ascending/descending order

### Std 4.3

They are able to efficiently apply the appropriate operation and fluently carry out addition and subtraction of whole numbers upto five digits and beyond.

- SWBAT add numbers up to 5 digits without carryover SWBAT add numbers up to 5 digits with carryover SWBAT subtract numbers up to 5 digits without borrowing
- SWBAT subtract numbers up to 5 digits with borrowing.
- SWBAT use apply inverse operations to solve addition/subtraction problems with a missing number

### Std 4.4

They are able to apply and more accurately compute multiplication and division of whole numbers (with whole number quotients). They use the relation, dividend = divisor x quotient + remainder.

- SWBAT multiply numbers up to 4 digits (several steps to build up to)
- SWBAT multiply numbers by 10/100/1000
- SWBAT divide numbers up to 4 digits by 2 digits (several steps to build up to)
- SWBAT divide numbers by 10/100/1000
- SWBAT solve word problems involving multiplication/division.
- SWBAT identify and apply the relation dividend = divisor x quotient + remainder to check their answers.
- SWBAT apply inverse operations to find missing numbers in multiplication/division problems

### Std 4.8

They efficiently apply the appropriate operations for problems (with multiple operations and unitary method) from a mixed bag and fluently and accurately carry out the computations, check reasonableness of results and interpret the results.

### Std 4.10

They deepen understanding of whole numbers by knowing about factors, multiples, prime and composite numbers. They know prime factors to 50 and express a number as a product of prime factors (not using exponents).

### Std 4.11

They know about common factors and common multiples and Least Common Multiple (LCM) and its relevance in solving addition/subtraction of fractions and other simple problems.

### Std 4.12

They understand the meaning of proper/improper fractions and mixed numbers concretely, pictorially and symbolically. They convert improper fractions to mixed numbers and vice-versa and understand their equivalence.

- SWBAT identify proper and improper fractions (including showing them visually)
- SWBAT identify and represent mixed numbers
- SWBAT convert between improper fractions and mixed numbers.

### Std 4.14

Using LCM, they are also able to compare unlike fractions with denominators less than 12 or 20 (for the sake of simplicity) and add/subtract them. They compare or add/subtract mixed numbers by comparing or adding/subtracting whole parts and then fractional parts (like or unlike).

- SWBAT identify like and unlike fractions (EI 3.16)
- SWBAT compare fractions with the same denominator (EI 3.17)
- SWBAT arrange fractions with the same denominators in ascending/descending order (EI 3.17)

### Std 4.15

They summarize methods of comparing and adding/subtracting fractions and mixed numbers.

- SWBAT add/subtract fractions with like denominators (both arithmetically and visually)
- SWBAT add/subtract mixed numbers with equal denominators
- SWBAT to identify 1/2, 1/3, and 1/4 of a given number (resulting in a whole number)

### Std 4.16

They compare like/unlike decimals to thousandth place and beyond. They understand and know addition/subtraction of like/unlike decimals to thousandth place and beyond.

- SWBAT represent parts of a whole as decimals (only to hundredths) and convert between fractions and decimals
- SWBAT identify place value of decimals (up to hundredths place)
- SWBAT add/subtract decimals (up to hundredths place)

### Std 4.18

Estimation - estimate number of objects in a set of upto 500 objects and count and compare the estimate with the actual number - understand meaning and purpose of round numbers i.e. why rounded solutions may be appropriate and when they are called for - know how to round 2-digit number to the nearest 10, 3-digit number to the nearest 100 and 4-digit number to the nearest 1000

- SWBAT round numbers to the nearest 10/100/1000/10000
- SWBAT estimate multiplication products by rounding
- SWBAT estimate lengths using a scale

## Grade 5

### Std 5.1

Students know number representation of very large numbers as crores and beyond in the Indian system. They also know the International system of naming numbers. They understand the relative sizes of the numbers. They decompose and compose numbers using place value and are able to efficiently compare and order.

- SWBAT identify the place value of digits in decimal numbers.
- SWBAT read numbers and identify place values of numbers up to tens of crores (using Indian system and international system)
- SWBAT write numbers in expanded form
- SWBAT compare numbers using greater than/less than signs up to any number of digits
- SWBAT write the biggest and smallest number from a given set of digits
- SWBAT round numbers to a given place value
- SWBAT identify numbers n greater/less than another number (where n=10, 100, 1000, etc.)

### Std 5.2

They efficiently apply the appropriate operation and carry out addition and subtraction and also multiplication and division with large whole numbers.

- SWBAT add 3 numbers including large numbers.
- SWBAT subtract large numbers
- SWBAT solve word problems involving addition/subtraction including those with large numbers
- SWBAT multiply numbers up to a 5-digit by a 3-digit number
- SWBAT divide numbers up to a 5-digit by a 2-digit number
- SWBAT solve word problems involving multiplication/division.

### Std 5.4

They know applications of LCM (other than adding/subtracting fractions) through simple problems e.g 3 bells ring at intervals of 7, 10, 14 minutes respectively, they begin by ringing together. After how long will they ring together again?

### Std 5.6

Besides parts of a whole or a collection when whole or collection is divided in equal parts, they understand fraction from another point of view also as – division of two whole numbers. They understand equivalence of these interpretations and associate a proper fraction to a number between 0 and 1 on the number line.

- SWBAT order fractions on a number line (natural extension of comparing fractions)
- SWBAT identify equivalent fractions (3rd EI)
- SWBAT create equivalent fractions with a missing number. (3rd EI)
- SWBAT reduce fractions to their lowest form (3rd EI, 4th IGCSE)
- SWBAT compare fractions with unlike denominators (4th EI, 5th 4th IGCSE) (by creating equivalent fractions with like denominators)

### Std 5.7

They know representation of a whole number as a fraction and a decimal and understand that division can be regarded as multiplication by reciprocal.

### Std 5.8

They understand the meaning and effect of multiplication and division of fractions by a whole number, by a fraction or a mixed number/improper fraction and know how to carry out multiplication and division of fractions by these.

- SWBAT multiply a fraction by a whole number
- SWBAT multiply fractions
- SWBAT identify recipricols of fractions
- SWBAT divide fractions
- SWBAT divide fractions by whole numbers

### Std 5.9

They understand the meaning and effect of movement of decimal when number is multiplied or divided by 10, 100, 1000. They understand the meaning and effect of multiplication and division of decimals by a whole number, by a decimal alone and by a number with both whole and decimal components. They know how to manipulate the decimal point while multiplying and dividing a decimal number by a whole number, a decimal, and a number with both components.

### Std 5.10

They master unitary method to solve multiplication and division problems involving any of whole numbers, fractions or decimals.

- SWBAT solve proportion problems by finding the unit rate
- SWBAT solve proportion problems by multiplying/dividing (scaling)

### Std 5.11

They summarize the four operations and their effects with whole numbers, decimals and fractions

- SWBAT test for divisibility of a number by 2, 3, 5, 10. (4th EI, 5th IGCSE)
- SWBAT add/subtract fractions with unlike denominators
- SWBAT add/subtract mixed numbers with unlike denominators
- SWBAT add/subtract with decimals to the same decimal place
- SWBAT add/subtract with decimals to different decimal places
- SWBAT multiply a decimal by an integer
- SWBAT multiply a decimal by a decimal

### Std 5.13

They understand the equivalence of the concepts of fractions and decimals. They know that all decimals can be represented in fractional form and all fractions in decimal form by dividing the numerator by the denominator including terminating and repeating decimals.

- SWBAT identify decimal fractions and their decimal conversions for fractions up to denominator of 1000 (EI 3.19 and 3.20)
- SWBAT convert a decimal into a fraction to the thousandths place.
- SWBAT identify and create equivalent decimal fractions (ex. 6/10, 60/100, 600/1000)

### Extra Topics

Profit/Loss, Recipricols, prime/composite numbers