Numbers in Grade 3
- Whole Numbers within 1000
- Part-to-Whole Relationships as Fractions
- Additive Thinking Strategies
- Multiplicative Thinking Strategies
Grade 3 | Whole Numbers Within 1000 |
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Procedural Knowledge | Skip 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 Knowledge | Place 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 Skills | Number 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 Misconceptions | 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. |
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 Strategies | Experience 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 Vocabulary | Base ten system. |
Positional system. | |
Place value. | |
Ones, tens, hundreds, thousands. |
Grade 3 | Part-to-whole Relationships as Fractions |
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Procedural Knowledge | Partition 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 Knowledge | Fractions 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 Skills | Count unit fractions of the same whole. |
Compare unit fractions of the same whole. | |
Locate unit fractions on a linear representation. | |
Common Misconceptions | Fractions 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 Strategies | Use 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 Vocabulary | Unit fraction. |
Parts and whole. |
Grade 3 | Additive Thinking Strategies |
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Procedural Knowledge | Apply 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 Knowledge | Additive 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. | |
Essential Skills | Addition strategies. |
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 Strategies | Use 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. | |
Key Vocabulary | Addition, sum. |
Subtraction, minuend, subtrahend, difference. | |
Ones, tens, hundreds, thousands. | |
Regrouping. |
Grade 3 | Multiplicative Thinking Strategies |
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Procedural Knowledge | Represent 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 Knowledge | Multiplication 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 Skills | Multiplication strategies. |
Division strategies. | |
Multiplication number facts to 81 and related division number facts. | |
Common Misconceptions | When 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 Strategies | Use 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 Vocabulary | Multiplication. |
Division. | |
Ones, tens. | |
Remainder |