Getting Started

Type of Unit: Introduction

Prior Knowledge

Students should be able to:

Understand ratio concepts and use ratios.
Use ratio and rate reasoning to solve real-world problems.
Identify and use the multiplication property of equality.

Lesson Flow

This unit introduces students to the routines that build a successful classroom math community, and it introduces the basic features of the digital course that students will use throughout the year.

An introductory card sort activity matches students with their partner for the week. Then over the course of the week, students learn about the routines of Opening, Work Time, Ways of Thinking, Apply the Learning (some lessons), Summary of the Math, Reflection, and Exercises. Students learn how to present their work to the class, the importance of students’ taking responsibility for their own learning, and how to effectively participate in the classroom math community.

Students then work on Gallery problems, to further explore the resources and tools and to learn how to organize their work.

The mathematical work of the unit focuses on ratios and rates, including card sort activities in which students identify equivalent ratios and match different representations of an equivalent ratio. Students use the multiplication property of equality to justify solutions to real-world ratio problems.

Students use ratio cards to find equivalencies and form partnerships for the week. As a class, students discuss and decide on classroom norms.Give each student a ratio card. Instruct students to find a classmate whose card has a ratio that is equivalent to theirs. Classmates with equivalent ratios are now partners for the week. With the class, discuss and decide on classroom norms, or rules. Tell students how to access the application they will use this year.Key ConceptsStudents understand that ratio relationships are multiplicative. They use ratio tables to show ratio relationships.Goals and Learning ObjectivesDistinguish between ratio tables and tables that do no show equivalent ratios.Understand how ratio tables are used to solve ratio problems.Use the basic features of the application.Create and understand the classroom norms.Use mathematical reasoning to justify an answer.

Working With Rational Numbers

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Compare and order positive and negative numbers and place them on a number line.
Understand the concepts of opposites absolute value.

Lesson Flow

The unit begins with students using a balloon model to informally explore adding and subtracting integers. With the model, adding or removing heat represents adding or subtracting positive integers, and adding or removing weight represents adding or subtracting negative integers.

Students then move from the balloon model to a number line model for adding and subtracting integers, eventually extending the addition and subtraction rules from integers to all rational numbers. Number lines and multiplication patterns are used to find products of rational numbers. The relationship between multiplication and division is used to understand how to divide rational numbers. Properties of addition are briefly reviewed, then used to prove rules for addition, subtraction, multiplication, and division.

This unit includes problems with real-world contexts, formative assessment lessons, and Gallery problems.

Students use number lines to represent products of a negative integer and a positive integer, and they use patterns to understand products of two negative integers. Students write rules for products of integers.Key ConceptsThe product of a negative integer and a positive integer can be interpreted as repeated addition. For example, 4 • (–2) = (–2) + (–2) + (–2) + (–2). On a number line, this can be represented as four arrows of length 2 in a row, starting at 0 and pointing in the negative direction. The last arrow ends at –8, indicating that 4 • (–2) = –8. In general, the product of a negative integer and a positive integer is negative.The product of two negative integers is hard to interpret or visualize. In this lesson, we use patterns to help students see why a negative integer multiplied by a negative integer equals a positive integer. For example, students can compute the products in the pattern below.4 • (–3) = –123 • (–3) = –92 • (–3) = –61 • (–3) = –30 • (–3) = 0They can observe that, as the first factor decreases by 1, the product increases by 3. They can continue this pattern to find these products.–1 • (–3) = 3–2 • (–3) = 6–3 • (–3) = 9In the next lesson, we will prove that the rules for multiplying positive and negative integers extend to all rational numbers, including fractions and decimals.Goals and Learning ObjectivesRepresent multiplication of a negative integer and a positive integer on a number line.Use patterns to understand products of two negative integers.Write rules for multiplying integers.

Students solve division problems by changing them into multiplication problems. They then use the relationship between multiplication and division to determine the sign when dividing positive and negative numbers in general.Key ConceptsThe rules for determining the sign of a quotient are the same as those for a product: If the two numbers have the same sign, the quotient is positive; if they have different signs, the quotient is negative. This can be seen by rewriting a division problem as a multiplication of the inverse.For example, consider the division problem −27 ÷ 9. Here are two ways to use multiplication to determine the sign of the quotient:The quotient is the value of x in the multiplication problem 9 ⋅ x = −27. Because 9 is positive, the value of x must be negative in order to get the negative product.The division −27 ÷ 9 is equivalent to the multiplication −27 ⋅ 19. Because this is the product of a negative number and a positive number, the result must be negative.Goals and Learning ObjectivesUse the relationship between multiplication and division to solve division problems involving positive and negative numbers.Understand how to determine whether a quotient will be positive or negative.

Students use properties of multiplication to prove that the product of any two negative numbers is positive and the product of a positive number and a negative number is negative.Key ConceptsMultiplication properties can be used to develop the rules for multiplying positive and negative numbers.Students are familiar with the properties from earlier grades:Associative property of multiplication: Changing the grouping of factors does not change the product. For any numbers a, b, and c, (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).Commutative property of multiplication: Changing the order of factors does not change the product. For any numbers a and b, a ⋅ b = b ⋅ a.Multiplicative identity property of 1: The product of 1 and any number is that number. For any number a, a ⋅ 1 = 1 ⋅ a = a.Property of multiplication by 0: The product of 0 and any number is 0. For any number a, a ⋅ 0 = 0 ⋅ a = 0.Property of multiplication by −1: The product of −1 and a number is the opposite of that number. For any number a, (−1) ⋅ a = −a.Existence of multiplicative inverses: Dividing any number by the same number equals 1. Multiplying any number by its multiplicative inverse equals 1. For every number a ≠ 0, a ÷ a = a ⋅ 1a = 1a ⋅ a = 1.Distributive property: Multiplying a number by a sum is the same as multiplying the number by each term and then adding the products. For any numbers a, b, and c, a ⋅ (b + c) = a ⋅ b + a ⋅ c.In this lesson, students will encounter a proof showing that the product of a positive number and a negative number is negative and two different proofs that the product of two negative numbers is positive. Two alternate proofs are as follows.Proof that the product of two negative numbers is positive:Represent the negative numbers as −a and −b, where a and b are positive.(−a) ⋅ (−b)Original expression= ((−1) ⋅ a) ⋅ ((−1) ⋅ b)   Property of multiplication by −1= (−1) ⋅ (a ⋅ (−1)) ⋅ b   Associative property of multiplication= (−1) ⋅ ((−1) ⋅ a) ⋅ b   Commutative property of multiplication= ((−1) ⋅ (−1)) ⋅ (a ⋅ b)   Associative property of multiplication= 1 ⋅ (a ⋅ b)   Property of multiplication by −1= a ⋅ b   Multiplicative identity property of 1Because a and b are positive, a ⋅ b is positive.Proof that the product of a positive number and a negative number is negative:Let a be the positive number. Let −b be the negative number, where b is positive.a ⋅ (−b)Original expression= a ⋅ ((−1) ⋅ b)     Property of multiplication by −1= (a ⋅ (−1)) ⋅ b     Associative property of multiplication= ((−1) ⋅ a) ⋅ b     Commutative property of multiplication= (−1) ⋅ (a ⋅ b)     Associative property of multiplication= −(a ⋅ b)     Property of multiplication by −1Because a and b are positive, a ⋅ b is positive, so −(a ⋅ b) must be negative.Goals and Learning ObjectivesReview properties of multiplication.Explain why the product of two negative numbers is positive and the product of a negative number and a positive number is negative.

Students critique and improve their work on the Self Check. They then extend their knowledge with additional problems.Students solve problems that require them to apply their knowledge of multiplying and dividing positive and negative numbers. Students will then take a quiz.Key ConceptsTo solve the problems in the Self Check, students must apply their knowledge of multiplication and division of positive and negative numbers learned throughout the unit.Goals and Learning ObjectivesUse knowledge of multiplication and division of positive and negative numbers to solve problems.

Students solve fraction multiplication and division problems. To see the answers, learners need to roll their mouse over the colored area.

This website from The Math Page explains and defines how to multiply decimals. There is even a practice section at the very end of the lesson.

If you are looking for a website that explains and defines the basics of mental multiplication with helpful hints then you have arrived at an excellent resource. There are even practice problems at the very end of the lesson.

This lesson teaches students how to use colored blocks to model word problems. There are several sets of problems each for addition / subtraction, multiplication / division, and ratio word problems.

Download these free worksheets to sharpen your word problem skills. Sheets focus on addition, multiplication, subtraction, division, and other math topics.

Math variety of concepts, this colorful website is sure to give your students the chance show what they know. Concepts covered include rounding, division with decimals, order of operations, ratio, percent, multiplication, and division of fractions, simplifying fractions, and more. Not an interactive site, but great practice activities.

The purpose of the task is for students to solve a multi-step multiplication problem in a context that involves area. In addition, the numbers were chosen to determine if students have a common misconception related to multiplication.

Download these worksheets to sharpen your operations skills. Sheets focus on addition, subtraction, multiplication, division, mixed operations, input/output tables, the order of operations, and operations in other base number systems.

This video (2:53) illustrates a word problem involving the multiplication of fractions using an array to model the solution visually. The problem involves puffins, arctic birds that live in colonies and fish to feed their young. The accompanying classroom activity provides students with practice solving additional fraction multiplication problems using arrays. Student and teacher materials are included.

This video illustrates a word problem involving the multiplication of fractions using an array to model the solution visually. The problem involves puffins, arctic birds that live in colonies and fish to feed their young.

This video presents a visual model for solving a word problem involving the multiplication of fractions. The problem asks students to use the amount of bamboo that a panda eats in an hour to determine how much he eats in a quarter of an hour if he eats at a steady rate. The video ends with a challenge problem for students to model themselves. [2:32]