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.

This lesson unit is intended to help you assess how well students are able to: Calculate the mean, median, mode, and range from a frequency chart; and to use a frequency chart to describe a possible data set, given information on the mean, median, mode, and range.

This video lesson examines the meaning of the equal sign as we work to solve unknowns in multiplication and division equations.

This short video and interactive assessment activity is designed to teach fourth graders about multiples - word problems.

This short video and interactive assessment activity is designed to teach second graders about multiplication sentences from illustrations (numbers 2, 3, 4, 5, and 10).

This short video and interactive assessment activity is designed to teach second graders about multiplication sentences from illustrations II (numbers 6, 7, 8, 9, 11 and 12).

This short video and interactive assessment activity is designed to teach second graders about multiplication stories and illustrations i - word problems.

This short video and interactive assessment activity is designed to teach third graders about multiplication stories and illustrations II - word problems.

The area model can be used when multiplying any two numbers, including decimals. This video demonstrates how to multiply using the area model when a decimal is involved.

This short video and interactive assessment activity is designed to teach fourth graders about shortening multiplication expressions.

This short video and interactive assessment activity is designed to teach third graders about multiplying 2 digits by 1 digit.

This short video and interactive assessment activity is designed to teach fifth graders about multiplying 2-digit by 2-digit numbers in columns.

This short video and interactive assessment activity is designed to teach fourth graders about multiplying 2-digit by 2-digit numbers in columns.

This short video and interactive assessment activity is designed to teach third graders about calculating given a statement.

This short video and interactive assessment activity is designed to teach fourth graders about comparing using greater than, less than, and equal to.

This lesson's target is multiplying a fraction by a whole number by drawing a visual area model. Students will engage in one problem as a group, then try four problems on their own or in a partnership. There are suggestions for summarizing questions included, as well as a formative assessment. The student activity page includes both the exploration questions (front) and the formative assessment (back).

This short video and interactive assessment activity is designed to teach second graders about multiplying by 10.