Grade 5 Clarification Document Unit 2
5.4 Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to:
5.4A
Identify prime and composite numbers.
Supporting Standard
Including, but not limited to:

Whole numbers o
Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n} o
Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Prime number – a whole number with exactly two factors, 1 and the number itself


Composite number – a whole number with more than two factors
Special numbers o
2 is the only even prime number. o
1 is the only common factor in all of the factor pairs of prime numbers.

Ex: (1, 3), (1, 2), (1, 13), etc.

1 is neither prime nor composite. o
0 is neither prime nor composite.

0 cannot be expressed as a product of primes.
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0 x 0 and/or 0 x any number yields an infinite number of factor pairs that have the product 0.
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Various representations to identify prime and composite numbers o
Arrays
Ex: Tile arrangements

Generalizations from arrays used to determine if a number is prime or composite
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Composite numbers have more than two different rectangular arrays that can be made.

Prime numbers have exactly two different rectangular arrays that can be made. o
Organizational factor lists

Ex: 35
Ex: 13
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Generalizations from organizational factor lists used to determine if a number is prime or composite

Composite numbers have more than two factors.

Prime numbers have exactly two factors.
5.4B
Represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.
Readiness Standard
Including, but not limited to:

Whole numbers o
Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time
{1, 2, 3, ..., n } o
Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n }

Addition o
Sums of whole numbers

Subtraction o
Differences of whole numbers

Multiplication o
Product – the total when two or more factors are multiplied o
Factor – a number multiplied by another number to find a product o
Products of whole numbers up to three-digit factors by two-digit factors

Division o
Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor o
Dividend – the number that is being divided o
Divisor – the number the dividend is being divided by o
Whole numbers with quotients up to four-digit dividends and two-digit divisors

Representations of an unknown quantity in an equation o
Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other o
Any single letter to represent the unknown quantity (e.g., 24 – 8 = y , etc.) o
Equal sign at beginning or end and unknown in any position

Ex: g = 6 + 4; 6 + 4 = g

Ex: x = 10 – 4; 10 – 4 = x

Ex: 10 = x + 4; x + 4 = 10

Ex: r = 6 x 4; 6 x 4 = r

Ex: p
= 24 ÷ 4; 24 ÷ 4 = p

Ex : 24 = 6 • z ; z • 6 = 24

Recognition of addition, subtraction, multiplication, and/or division in mathematical and real-world problem situations

Representation of problem situations with equations o
Relationship between quantities represented and problem situation

Addition and subtraction problem structures o
Join problems

Start unknown

Ex:
Ex:
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Change unknown

Result unknown
Ex:
Ex: o
Separate problems

Start unknown
Ex:

Change unknown

Result unknown

Ex:
Ex: o
Part-part-whole problems
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Part unknown

Whole unknown
Ex:
Ex: o
Compare problems

Difference unknown

Larger part unknown
Ex:

Smaller part unknown
Ex:

Ex:

Multiplicative structures o
Product unknown
Ex: o
Factor unknown
Ex:

Division structures o
Partitive division

Total amount known

Number of groups known

Size or measure of each group unknown o
Quotative division (also known as Measurement division)

Total amount known


Size or measure of each group known
Number of groups unknown

Ex:
Ex:

Multi-step problem situations
Ex:

Ex:
Ex:

Ex:
Ex:

Ex:
5.4E
Describe the meaning of parentheses and brackets in a numeric expression.
Supporting Standard
Including, but not limited to:


Expression – a mathematical phrase, with no equal sign, that may contain a number(s), a unknown(s), and/or an operator(s)
Parentheses and brackets – symbols to show a group of terms and/or expressions within a mathematical expression o
Up to two levels of grouping

Parentheses and brackets

Ex: 3 × [7 + 2 - (8 + 4)] + 2

Double parentheses

Ex: 3 × (7 + 2 - (8 + 4)) + 2

Division bar


Ex:
Generalization about grouping symbols within a numerical expression o
When both parentheses and brackets or a double set of parentheses is used within a numerical expression, the inner most grouping should be evaluated first.

Ex:


Parentheses without an operation symbol may be used to represent multiplication. o Various symbols to represent multiplication include x, •, parentheses, or brackets.
 Ex: 5 x 15, 5 • 15, 5(15), (5)(15), 5[15], [5][15]
Relationship between numbers and operators separated by parentheses and/or brackets o
Ex: 5(15 + 2) is 5 times larger than (15 + 2) o
Ex: [4 x (2 + 5) – 3] ÷ 3 is the quotient of the difference between a value 4 times larger than (2 + 5) and 3, divided by 3
5.4F
Simplify numerical expressions that do not involve exponents, including up to two levels of grouping.
Readiness Standard
Including, but not limited to:

Whole numbers o
Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time
{1, 2, 3, ..., n } o
Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n }

Decimals (less than and greater than one to the tenths, hundredths, and thousandths) o
Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part

Fractions (proper, improper, or mixed numbers with equal or unequal denominators) o
Fraction – a number in the form where a and b are whole numbers and b is not equal to zero . A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division. o
Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero o
Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero o
Mixed number – a number that is composed of a whole number and a fraction o
Unit fraction – a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number

Addition o
Sums of whole numbers o
Sums of decimals up to the thousandths o
Sums of fractions with equal and unequal denominators

Subtraction o
Differences of whole numbers

o
Differences of decimals with values limited to the thousandths o
Differences of fractions with equal and unequal denominators

Multiplication o
Product – the total when two or more factors are multiplied o
Factor – a number multiplied by another number to find a product o
Products of whole numbers up to three-digit factors by two-digit factors o
Products of decimals limited to three-digit factors by two-digit factors with products to the hundredths

Multiply tenths by tenths (e.g., 0.3 x 0.7 = 0.21, 1.2 x 1.2 = 1.44, 14.3 x 1.3 = 18.59, etc.)

Multiply tenths by hundredths or vice versa (e.g., 0.5 x 0.12 = 0.06, 1.4 x 0.15 = 0.21, 21.4 x 0.45 = 9.63, etc.)

Multiply tenths by thousandths or vice versa (e.g., 0.4 x 0.125 = 0.05, 0.125 x 8.4 = 1.05, etc.)

Multiply whole numbers by tenths, hundredths, and thousandths or vice versa (e.g., 3 x 1.3 = 3.9, 42 x 7.45 =
312.9, 7.02 x 78 = 547.56, 6 x 0.125 = 0.75, etc.) o
Products of fractions where factors are limited to a fraction and a whole number

Division o
Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor o
Dividend – the number that is being divided o
Divisor – the number the dividend is being divided by o
Whole numbers with quotients up to four-digit dividends and two-digit divisors o
Quotients of decimals limited to four-digit dividends and two-digit whole number divisors, with quotients to the hundredths

Dividend to the tenths and whole number divisor (e.g., 1.2 ÷ 24 = 0.05, 358.8 ÷ 23 = 15.6, 721.7 ÷ 14 = 51.55, etc.)

Dividend to the hundredths and whole number divisor (e.g., 8.68 ÷ 4 = 2.17, 8.25 ÷ 15 = 0.55, 62.76 ÷ 12 =
5.23, etc.)

Whole number dividends and whole number divisors (e.g., 3 ÷ 4 =.0.75, 10 ÷ 8 = 1.25, 1000 ÷ 16 = 62.5, etc.) o
Quotients of fractions where dividend and divisors are limited to whole numbers by unit fractions and unit fractions by whole numbers

Expression – a mathematical phrase, with no equal sign, that may contain a number(s), a unknown(s), and/or an operator(s)

Numerical expressions without exponents

Parentheses and brackets – symbols to show a group of terms and/or expressions within a mathematical expression o
Up to two levels of grouping

Parentheses and brackets

Ex: 3 × [7 + 2 - (8 - 4) ÷ 3] + 2

Double parentheses

Ex: 3 × (7 + 2 - (8 - 4) ÷ 3) + 2

Division bar

Ex:

Order of operations – the rules of which calculations are performed first when simplifying an expression o
Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right o
Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right

Various symbols to represent multiplication include x, •, parentheses, or brackets.

Ex: 5 x 15, 5 • 15, 5(15), (5)(15), 5[15], [5][15] o
Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right

Ex: