The Number System: Rational Numbers
Students will make connections from positive integers to negative integers. Students will connect what they know about addition and subtraction, to add, subtract, multiply, and divide positive and negative numbers. Students will also deepen their understanding of rational numbers.
Essential Outcomes
The Number System
- NY-7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.
Other Standards Addressed in this Unit
The Number System
- NY-7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers. Represent addition and subtraction on a horizontal or vertical number line.
- NY-7.NS.1a Describe situations in which opposite quantities combine to make 0.
- NY-7.NS.1b Understand addition of rational numbers; p + q is the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
- NY-7.NS.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
- NY-7.NS.1d Apply properties of operations as strategies to add and subtract rational numbers.
- NY-7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
- NY-7.NS.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real- world contexts.
- NY-7.NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then – (p/q) = -p/q = p/-q. Interpret quotients of rational numbers by describing real-world contexts.
- NY-7.NS.2c Apply properties of operations as strategies to multiply and divide rational numbers.
- NY-7.NS.2d Convert a fraction to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Essential Questions and Big Ideas
- What are integers?
- Integers are numbers that can be written as whole numbers.
- Integers can be positive or negative.
- How do I add with positive and negative integers?
- To add positive integers, combine their values.
- To add a positive and a negative integer, find the difference between the values. If there is more negative, the answer will be negative. If there is more positive, the answer will be positive.
- To add two negative integers, combine the value of the integers and it remains negative.
- How do I subtract with positive and negative integers?
- To subtract positive integers, find the difference between the two. If you’re taking away a larger number, the difference will be negative. If you’re taking away a smaller number, the difference will be positive
- To subtract a positive number from a negative number, it is the same as adding two negative numbers.
- Subtracting a negative number means taking away a negative, which is the same as adding a positive.
- How do I multiply with positive and negative integers?
- Multiplying a positive number by a negative number creates a negative number, as either you are taking away groups of a positive number or combining groups of a negative number.
- Multiplying a negative number by a negative number creates a positive number, because it represents taking away groups of a negative, and taking away a negative is actually creating a positive.
- How do I divide with positive and negative integers?
- Dividing a positive number by a negative number or a negative number by a positive number, leads to a negative quotient, because it represents splitting up a negative total into groups or splitting a positive number up into negative groups, which would require the negative groups to be subtracted.
- Dividing a negative number by a negative number leads to a positive quotient, because it represents splitting a negative total into groups of a negative and identifying how many groups there are.
- What are rational numbers and how do I complete all four operations with them?
- Rational Numbers are numbers that can be represented as a fraction or a terminating or repeating decimal.
- Fractions can be converted to decimals by dividing the numerator by the denominator.
- The integer rules for addition, subtraction, multiplication, and division apply to all rational numbers.
Download the complete Grade 7 Math Unit 1 framework to customize for your own planning.