# Numbers in Grade 4

- Whole Numbers within 10 000
- Part-to-Whole Relationships as Fractions & Decimals
- Refined Additive Thinking Strategies
- Refined Multiplicative Thinking Strategies

Grade 4 | Whole Numbers Within 10 000 |
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Procedural Knowledge | Skip 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 Knowledge | Each 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 Skills | Compose 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 Misconceptions | 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 its place value, and explain the connections. | |

Key Vocabulary | Base ten system. |

Positional system. | |

Place value. | |

Ones, tens, hundreds, thousands. |

Grade 4 | Part-to-whole Relationships as Fractions and Decimals |
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Procedural Knowledge | Relate 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 Knowledge | Fractions 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 Skills | Manipulate equivalent fractions. |

Represent fractions as decimals and decimals as fractions. | |

Compare fractions and decimals. | |

Common Misconception | Fractions 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 Strategies | Communicate 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 Vocabulary | Fractions. |

Numerator and denominator. | |

Decimals. | |

Tenths. |

Grade 4 | Refined Additive Thinking Strategies |
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Procedural Knowledge | Refine 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 Knowledge | Additive 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 Skills | Addition 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 Strategies | Use 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 Vocabulary | Addition, sum. |

Subtraction, difference. | |

Place value. | |

Tenths, ones, tens, hundreds, thousands. | |

Regrouping. |

Grade 4 | Refined Multiplicative Thinking Strategies |
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Procedural Knowledge | Refine 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 Knowledge | Multiplicative 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 Skills | Multiplication 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 | |

#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 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 Vocabulary | Multiplication, product. |

Division, divisor, dividend, quotient. |