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 |
|---|---|
| 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 |
|---|---|
| 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 |
|---|---|
| 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 |
|---|---|
| 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 |