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}
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

Prime number – a whole number with exactly two factors, 1 and the number itself

Various representations to identify prime and composite numbers
o Arrays
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.
 0 x 0 and/or 0 x any number yields an infinite number of factor pairs that have the product 0.
Ex: Tile arrangements

o
Generalizations from arrays used to determine if a number is prime or composite
 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.
Organizational factor lists
Ex: 35
Ex: 13

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:

Change unknown

Result unknown
Ex:
Ex:
o
Separate problems
 Start unknown
Ex:

Change unknown

Result unknown
Ex:
Ex:
o
Part-part-whole problems
 Part unknown
Ex:

Whole unknown
Ex:
o
Compare problems
 Difference unknown
Ex:

Larger part unknown

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
Ex:
o
Quotative division (also known as Measurement division)
 Total amount known
 Size or measure of each group known
 Number of groups unknown
Ex:

Ex:
Ex:
Multi-step problem situations
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
o
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
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
where a and b are whole numbers and a < b where b is not equal to zero
Addition
o
o
o
Sums of whole numbers
Sums of decimals up to the thousandths
Sums of fractions with equal and unequal denominators
Subtraction
o Differences of whole numbers
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o
o
Differences of decimals with values limited to the thousandths
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: