Mathematics 4. Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.
a. Identify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses.
b. Interpret the sum of two or more rational numbers, by using a number line and in real-world contexts.
c. Explain subtraction of rational numbers as addition of additive inverses.
d. Use a number line to demonstrate 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.
e. Extend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.
f. Divide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number.
g. Convert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats.
Unpacked Content
Evidence Of Student Attainment:
Students:
Explain situations where opposite quantities combine to make zero, known as additive inverses.
Apply their knowledge of addition and subtraction of rational numbers to describe real-world contexts.
Add and subtract rational numbers using number lines to show connection to distance
Explain the connection between subtraction and addition of additive inverses.
Model multiplication and division of rational numbers (number horizontal and vertical number lines, integer chips, bar models).
Use properties of operations to multiply signed numbers.
Convert rational numbers to a decimal using long division and determine if the result is terminating or repeating. Teacher Vocabulary:
Integers Rational numbers Additive inverses opposite quantities Absolute value Terminating decimals Repeating decimals Knowledge:
Students know:
a number and its opposite have a sum of 0.
A number and its opposite are called additive inverses.
Strategies for adding and subtracting two or more numbers.
Absolute value represents distance on a number line, therefore it is always non-negative.
Strategies for multiplying signed numbers.
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).
The decimal form of a rational number terminates or eventually repeats. Skills:
Students are able to:
add rational numbers.
Subtract rational numbers.
Represent addition and subtraction on a number line diagram.
Describe situations in which opposite quantities combine to make 0.
Find the opposite of a number.
Interpret sums of rational numbers by describing real-world contexts.
Show that the distance between two rational numbers on the number line is the absolute value of their difference.
Use absolute value in real-world contexts involving distances.
Multiply and divide rational numbers.
Convert a rational number to a decimal using long division. Understanding:
Students understand that:
finding sums and differences of rational numbers (negative and positive) involves determining direction and distance on the number line.
Subtraction of rational numbers is the same as adding the additive inverse, p - q = p + (-q).
If a factor is multiplied by a number greater than one, the answer is larger than that factor.
If a factor is multiplied by a number between 0 and 1, the answer is smaller than that factor.
Multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers.
Integers can be divided, provided that the divisor is not zero. Diverse Learning Needs:
Essential Skills:
Learning Objectives: M.7.4.1: Define rational numbers, horizontal, and vertical.
M.7.4.2: Recall how to extend a horizontal number line.
M.7.4.3: Recall how to extend a vertical number line.
M.7.4.4: Demonstrate addition and subtraction of whole numbers using a horizontal or vertical number line.
M.7.4.5: Give examples of rational numbers.
M.7.4.6: Define absolute value and additive inverse.
M.7.4.7: Explain that the sum of a number and its opposite is zero.
M.7.4.8: Locate positive, negative, and zero numbers on a number line.
M.7.4.9: Recall properties of addition and subtraction.
M.7.4.10: Model addition and subtraction using manipulatives.
M.7.4.11: Show addition and subtraction of 2 or more rational numbers using a number line within real-world context.
M.7.4.12: Define absolute value and additive inverse.
M.7.4.13: Show subtraction as the additive inverse.
M.7.4.14: Give examples of the opposite of a given number.
M.7.4.15: Show addition and subtraction using a number line.
M.7.4.16: Discuss various strategies for solving real-world and mathematical problems.
M.7.4.17: Identify properties of operations for addition and subtraction.
M.7.4.18: Recall the steps for solving addition and subtraction of rational numbers.
M.7.4.19: Identify the difference between two rational numbers on a number line.
M.7.4.20: Recall the steps of solving multiplication of rational numbers.
M.7.4.21: Identify the pattern for multiplying signed numbers.
M.7.4.22: Recall the steps of solving division of rational numbers.
M.7.4.23: Explain that dividing a rational number zero is undefined.
M.7.4.24: Recall that a fraction can be written as a division problem.
M.7.4.25: Recall the steps to divide two rational numbers.
M.7.4.26: Identify whether a decimal is terminating or repeating.
Prior Knowledge Skills:
Define parentheses, braces, and brackets.
Recall addition and subtraction of fractions as joining and separating parts referring to the same whole.
Identify two fractions as equivalent (equal) if they are the same size or the same point on a number line.
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
Generate equivalent fractions.
Show on a number line that numbers that are equal distance from 0 and on opposite sides of 0 have opposite signs.
Define rational number.
Arrange integers and /or rational numbers on a horizontal or vertical number line.
Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
Identify a rational number as a point on the number line.
Alabama Alternate Achievement Standards
AAS Standard:
Mathematics 8. Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.
a. Identify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses.
b. Interpret the sum of two or more rational numbers, by using a number line and in real-world contexts.
c. Explain subtraction of rational numbers as addition of additive inverses.
d. Use a number line to demonstrate 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.
e. Extend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.
f. Divide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number.
g. Convert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats. [Grade 7, 4]
Unpacked Content
Evidence Of Student Attainment:
Students:
Apply their knowledge of addition and subtraction of rational numbers to describe real-world contexts.
Use physical and visual models to add and subtract integers.
Add and subtract rational numbers.
Model multiplication and division of rational numbers.
Apply the distributive property to rational numbers.
Convert rational numbers to a decimal using long division to determine if the result is terminating or repeating. Teacher Vocabulary:
Integers Rational numbers Additive inverses opposite quantities Absolute value Terminating decimals Repeating decimals Knowledge:
Students know:
a number and its opposite have a sum of 0.
A number and its opposite are called additive inverses.
properties of operations.
Absolute value represents distance on a number line, therefore it is always non-negative.
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).
The decimal form of a rational number terminates in 0s or eventually repeats. Skills:
Students are able to:
add rational numbers.
Subtract rational numbers.
Represent addition and subtraction on a number line diagram.
Describe situations in which opposite quantities combine to make 0.
Find the opposite of a number.
Interpret sums of rational numbers by describing real-world contexts.
Show that the distance between two rational numbers on the number line is the absolute value of their difference.
Use absolute value in real-world contexts involving distances.
Multiply and divide rational numbers.
Convert a rational number to a decimal using long division. Understanding:
Students understand that:
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.
Subtraction of rational numbers is the same as adding the additive inverse, p - q = p + (-q).
If a factor is multiplied by a number greater than one, the answer is larger than that factor.
If a factor is multiplied by a number between 0 and 1, the answer is smaller than that factor.
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.
Integers can be divided, provided that the divisor is not zero. Diverse Learning Needs:
