Numbers in Grade 4

Grade 4Whole Numbers Within 10 000
Procedural KnowledgeSkip count by place value.
Estimate quantities using referents.
Recognize and represent quantities with numbers.
Order numbers using benchmarks on a visual or spatial representation.
Conceptual KnowledgeEach place value is 10 times the value of the place to its right.
Estimation can be applied to larger numbers.
There are patterns in how numbers are named and represented symbolically (International System of Units (SI) representation).
A visual or spatial representation of quantity can be extended to include large numbers and does not have to start at 0.
Essential SkillsCompose and decompose numbers by place value.
Due to the lack of analysis and investigation of a base ten positional system, students might create their own misunderstandings of what this sort of system is about.
Common MisconceptionsStudents might not realize that:
☆ In a base ten system, we count in groups of ten.
☆ In a base ten system, we only need digits up to 9.
☆ In a positional system, the position of the digit matters.
It is relevant to listen to students’ explanations to address misconceptions.
Big Ideas#1 – The set of whole numbers is infinite, and each whole number can be associated with a unique point on the number line.
#2 – The base ten numeration system is a scheme for recording numbers using digits 0-9, groups of ten, and place value.
#3 – Any number can be represented in an infinite number of ways that have the same value.
#4 – Numbers can be compared by their relative values.
#8 – Numerical calculations can be approximated by replacing numbers with other numbers that are close and easy to compute with mentally.
Key StrategiesExperience counting in different ways, and in different orders.
Represent numbers (concretely, pictorially and symbolically) based on the base ten positional system, and explain the representations.
Connect a digit in a determined position to its place value, and explain the connections.
Key VocabularyBase ten system.
Positional system.
Place value.
Ones, tens, hundreds, thousands.
Grade 4Part-to-whole Relationships as Fractions and Decimals
Procedural KnowledgeRelate fractions and decimals, limited to tenths.
Represent decimals concretely, pictorially, or symbolically, limited to tenths.
Make meaning of equivalent fractions concretely or pictorially, limited to denominators of 10 or less.
Count beyond 1 using improper fractions, limited to the same denominator.
Count beyond 1 using decimals, limited to tenths.
Compare fractions and decimals to the benchmarks of 0, 1/2, and 1.
Determine the location of fractions and decimals on a linear representation of a whole.
Conceptual KnowledgeFractions are numbers used to represent part-to-whole relationships.
Decimals are numbers used to represent part-to-whole relationships.
The same part-to-whole relationship can be represented with fractions with different denominators (equivalent fractions).
The same part-to-whole relationship can be represented with a fraction and a decimal.
Place value patterns extend to decimals.
Fractions and decimals occupy space in a visual or spatial representation of quantity.
A fraction or a decimal can be associated with a specific point on a linear representation of quantity.
Essential SkillsManipulate equivalent fractions.
Represent fractions as decimals and decimals as fractions.
Compare fractions and decimals.
Common MisconceptionFractions and decimals are not related to a whole.
Fractions are comparable even if they do not refer to the same whole.
Students might not realize that:
☆ 0.1 is one tenth (1/10) of 1 one.
☆ 1 one is ten groups of 0.1
Big Ideas#1 – The set of rational numbers is infinite, and each rational number can be associated with a unique point on the number line.
#2 – The base ten numeration system is a scheme for recording numbers using digits 0-9, groups of ten, and place value.
#3 – Any number can be represented in an infinite number of ways that have the same value.
#4 – Numbers can be compared by their relative values.
#5 – The same number sentence can be associated with different concrete or real-world situations, and different number sentences can be associated with the same concrete or real-world situation.
Key StrategiesCommunicate and explain how and why equivalent fractions have the same value.
Role play why identical fractions of different wholes might not represent the same quantity.
Use concrete, pictorial and symbolic representations to explain what tenths are.
Communicate and explain how fractions relate to decimals.
Represent fractions as decimals and decimals as fractions and explain the strategies used.
Use and explain strategies to compare fractions and decimals.
Locate fractions and decimals on linear representations of a whole.
Experience fractions and decimals beyond 1 (concretely, pictorially and symbolically).
Key VocabularyFractions.
Numerator and denominator.
Decimals.
Tenths.
Grade 4Refined Additive Thinking Strategies
Procedural KnowledgeRefine concrete, pictorial, symbolic, or mental math strategies.
Add and subtract in joining, separating, and comparing situations.
Refine a chosen strategy.
Add and subtract whole numbers within 10 000, including dollars, without a calculator.
Add and subtract whole numbers to calculate totals within 100 cents without a calculator.
Express a preferred strategy for addition and subtraction of whole numbers in algorithmic form.
Add and subtract decimals, limited to tenths.
Create and solve problems that involve addition and subtraction.
Estimate sums and differences as part of a problem-solving process.
Conceptual KnowledgeAdditive thinking strategies can be applied to whole numbers and decimals.
Problems can be solved in different ways.
Strategies can be chosen based on the nature of the problem.
Some strategies lend themselves to mental math.
Strategies can be refined over time.
Additive thinking strategies can be represented with step-by-step procedures (algorithms).
Essential SkillsAddition strategies for whole numbers and decimals.
Subtraction strategies for whole numbers and decimals.
Common Misconceptions“Carrying” or “trading” 1 in addition and subtraction are arbitrary processes. Students do not connect to regrouping.
In addition or subtraction strategies, decimals should be aligned from the first number on the right, and not based on place value.
The number of decimal places in a sum or difference is equal to the sum of decimal places in the parcels.
Addition and subtraction algorithms are arbitrary.
Big Ideas#2 – The base ten numeration system is a scheme for recording numbers using digits 0-9, groups of ten, and place value.
#3 – Any number or numerical expression can be represented in an infinite number of ways that have the same value.
#5 – The same number sentence can be associated with different concrete or real-world situations, and different number sentences can be associated with the same concrete or real-world situation.
#7 – Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones.
#8 – Numerical calculations can be approximated by replacing numbers with other numbers that are close and easy to compute with mentally.
Key StrategiesUse concrete, pictorial and symbolic strategies to add and subtract whole numbers and decimals.
Connect concrete, pictorial and symbolic representations of addition and subtraction of whole numbers and decimals.
Communicate and explain different strategies for adding and subtracting whole numbers and decimals.
Communicate and explain addition and subtraction of whole numbers and decimals based on place value.
Role play and explain what regrouping means in addition and subtraction of whole numbers and decimals.
Create and solve problems with addition and subtraction of whole numbers and decimals.
Estimate additions and subtractions of whole numbers and decimals in problem solving.
Find, explain and correct errors in problems involving addition and subtraction of whole numbers and decimals.
Key VocabularyAddition, sum.
Subtraction, difference.
Place value.
Tenths, ones, tens, hundreds, thousands.
Regrouping.
Grade 4Refined Multiplicative Thinking Strategies
Procedural KnowledgeRefine concrete, pictorial, symbolic, or mental math strategies.
Recognize patterns in multiplication and division.
Multiply and divide in sharing, grouping, array, combination, area, and comparison (rate) situations, with or without remainders in context.
Refine a chosen strategy.
Recall single-digit multiplication number facts for products to 81 and related division number facts.
Multiply a 2- or 3-digit number by a 1-digit number, limited to whole numbers, concretely, pictorially, or symbolically, without a calculator.
Divide a 2-digit number by a 1-digit number, limited to whole numbers, concretely, pictorially, or symbolically, without a calculator.
Multiply or divide in parts (distributive property).
Estimate products and quotients as part of a problem-solving process.
Conceptual KnowledgeMultiplicative thinking strategies can be applied to larger numbers.
Numbers can be multiplied in any order (commutative and associative properties).
When dividing, the order of numbers is important.
Problems can be solved in different ways.
Strategies can be chosen based on the nature of the problem.
Some strategies lend themselves to mental math.
Strategies can be refined over time.
Estimation can be used in problem-solving situations, including when an exact value is not needed or to verify a solution.
Division situations may or may not have remainders.
Essential SkillsMultiplication number facts to 81 and related division number facts.
Multiplication strategies.
Division strategies.
Distributive property.
Common Misconceptions“Carrying” 1 in multiplication is an arbitrary process. Students might not connect to regrouping.
Adding zeros when multiplying by 10 (or its multiples) is an arbitrary process. Students might not connect to place value.
Multiplication and division algorithms are arbitrary.
a(b + c) = ab + c
Big Ideas#2 – The base ten numeration system is a scheme for recording numbers using digits 0-9, groups of ten, and place value.
#3 – Any number or numerical expression can be represented in an infinite number of ways that have the same value.
#5 – The same number sentence can be associated with different concrete or real-world situations, and different number sentences can be associated with the same concrete or real-world situation.
#6 – For a given set of numbers there are relationships that are always true, and these are the rules that govern arithmetic and algebra.
#7 – Basic facts and algorithms for operations with whole numbers use notions of equivalence to transform calculations into simpler ones.
#8 – Numerical calculations can be approximated by replacing numbers with other numbers that are close and easy to compute with mentally.
Key StrategiesUse concrete, pictorial and symbolic strategies to multiply and divide.
Connect concrete, pictorial and symbolic representations of multiplication and division.
Communicate and explain multiplication and division strategies based on place value.
Create and solve problems with multiplication and division.
Estimate products and quotients in problem solving.
Find, explain and correct errors in problems involving multiplication and division.
Key VocabularyMultiplication, product.
Division, divisor, dividend, quotient.