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Procedural KnowledgeSkip count forward and backward by 2, 5, 10, and 100, starting at any number.
Recognize patterns created by skip counting.
Count and record the monetary value of collections of coins or bills (limited to either cents or dollars) of varying denominations.
Estimate quantities using referents.
Recognize and represent numbers.
Order numbers using benchmarks on a visual or spatial representation.
Conceptual KnowledgePlace value and unitizing applies to larger numbers.
Place value is the basis for the base-ten number system.
Estimation can be applied to larger numbers.
There are patterns in how numbers are named and represented symbolically.
A visual or spatial representation of quantity can be extended to include larger numbers, up to 1000, and does not have to start at 0.
Essential SkillsNumber counting by 2’s, 5’s, 10’s, and 100’s, forward and backward.
Determining ones, tens, hundreds, and thousands in terms of place value.
Common MisconceptionsDue 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.
Students 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 the number of ones, tens, hundreds, or thousands that the digit represents, and explain the connections.
Create a numerical system and compare it to the base ten positional system. Describe and explain differences and similarities.
Key VocabularyBase ten system.
Positional system.
Place value.
Ones, tens, hundreds, thousands.
Procedural KnowledgePartition a set, a length, and an area to create halves, thirds, quarters, fifths, and tenths.
Represent fractions symbolically.
Compare different unit fractions from the same set, length, and area.
Count by unit fractions to build one whole, limited to 1/2, 1/3, 1/4, 1/5, and 1/10.
Determine the location of a unit fraction on a linear representation of a whole.
Conceptual KnowledgeFractions are numbers used to represent part-to-whole relationships.
Fraction notation shows the relationship between the whole (denominator) and the number of parts (numerator).
Fractions occupy space in a visual or spatial representation of quantity.
A fraction can be associated with a specific point on a linear representation of quantity.
Essential SkillsCount unit fractions of the same whole.
Compare unit fractions of the same whole.
Locate unit fractions on a linear representation.
Common MisconceptionsFractions are not related to a whole.
Fractions are comparable if they do not refer to the same whole.
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.
#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.
Key StrategiesUse concrete, pictorial and symbolic representations to explain what a unit fraction is.
Communicate what a unit fraction is.
Locate unit fractions on linear representations of a whole.
Use and explain strategies to compare unit fractions.
Experience unit fractions in different contexts.
Explain how to figure out how many unit fractions are needed to build one whole.
Key VocabularyUnit fraction.
Parts and whole.
Procedural KnowledgeApply concrete, pictorial, symbolic, or mental math strategies.
Add and subtract in joining, separating, and comparing situations.
Recognize reliability of a chosen strategy.
Recall single-digit addition number facts to a sum of 18 and related subtraction number facts.
Add and subtract numbers within 1000, including 0, without a calculator.
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 compose and decompose larger numbers.
Unitizing is used for the purpose of regrouping in addition and subtraction.
When subtracting, the order of numbers is important.
Problems can be solved in different ways.
Strategies can be chosen based on the nature of the problem.
Estimation can be used in problem-solving situations, including when an exact value is not needed or to verify a solution.
Knowledge of single-digit addition and subtraction number facts is used to add and subtract larger numbers.
Subtraction strategies.
Addition number facts to 18 and related subtraction number facts.
Common Misconceptions“Carrying” or “trading” 1 in addition and subtraction are arbitrary processes. Students do not connect to regrouping.
When adding a number to zero, the result will be zero.
When subtracting zero from a number, the result will be zero.
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 add and subtract.
Connect concrete, pictorial and symbolic representations of addition and subtraction.
Communicate and explain addition and subtraction strategies based on place value.
Role play and explain what regrouping means in addition and subtraction.
Create and solve problems with addition and subtraction.
Estimate sums and differences in problem solving.
Find, explain and correct errors in problems involving addition and subtraction.
Subtraction, minuend, subtrahend, difference.
Ones, tens, hundreds, thousands.
Regrouping.
Procedural KnowledgeRepresent multiplication and division strategies concretely, pictorially, or symbolically.
Explore patterns in multiplication and division.
Multiply and divide in sharing, grouping, array, and combination situations, with or without remainders.
Apply strategies to single-digit multiplication number facts for products to 81 and related division number facts.
Relate place value to multiplication by 10.
Multiply by 0.
Conceptual KnowledgeMultiplication and division are operations used when applying multiplicative thinking strategies.
Multiplication and division involve a whole, a number of groups, and a quantity in each group.
Multiplication and division are sharing and grouping situations that can be represented symbolically (×, ÷, =).
A multiplication situation can be represented as a division situation (multiplication and division are inverse operations).
Two numbers can be multiplied in any order (commutative property).
Essential SkillsMultiplication strategies.
Division strategies.
Multiplication number facts to 81 and related division number facts.
Common MisconceptionsWhen multiplying a number by zero, the result will be the number.
When dividing a number by zero, the result will be zero.
When dividing zero by a number, the result will be the number.
Adding a zero when multiplying by 10 is an arbitrary process. Students might not connect to place value.
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.
Key StrategiesUse concrete, pictorial and symbolic strategies to multiply and divide, with and without remainders.
Connect concrete, pictorial and symbolic representations of multiplication and division, with and without remainders.
Communicate and explain different strategies for multiplication and division, with and without remainders.
Role play and explain what the commutative property mean in multiplication.
Find, explain and correct errors in multiplications and divisions, with and without remainders.
Key VocabularyMultiplication.
Division.
Ones, tens.
Remainder