ALEX Classroom Resource

Students:
Given a division problem involving a fraction divided by a fraction,

• Create an appropriate story context.
• Solve the problem using visual fraction models and an equation.
• Explain the relationship between the model and the problem.
• Interpret the solution.
• Use the inverse relationship between multiplication and division, or concept of division as repeated subtraction, to explain and justify the solution.

Students know:

• Strategies for representing fractions and operations on fractions using visual models,
• The inverse relationship between multiplication and division (a ÷ b = c implies that a = b x c).
• Strategies to solve mathematical and conceptual problems involving quotients of fractions.

Students are able to:

• Represent fractions and operations on fractions using visual models.
• Interpret quotients resulting from the division of a fraction by a fraction.
• Accurately determine quotients of fractions by fractions using visual models/equations.
• Justify solutions to division problems involving fractions using the inverse relationship between multiplication and division.

Students understand that:

• The operation of division is interpreted the same with fractions as with whole numbers.
• The inverse relationship between the operations of multiplication and division that was true for whole numbers continues to be true for fractions.
• The relationships between operations can be used to solve problems and justify solutions and solution paths.

Learning Objectives:
M.6.4.1: Define fraction (including numerator and denominator), reciprocal, equation, and quotient.
M.6.4.2: Construct an equation from a given word problem.
M.6.4.3: Discuss the process of multiplying by the reciprocal.
M.6.4.4: Interpret division of fractions by multiplying by the reciprocal.
M.6.4.5: Demonstrate division of fractions using a visual fraction model.
M.6.4.6: Demonstrate multiplication of fractions using a visual fraction model.

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.

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.

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.

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.

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.