In this video lesson, students consolidate their equation writing and solving skills. They solve a variety of equations with different structures. Then they match equations to situations and solve them. Students may choose any strategy to solve equations, including drawing diagrams to reason about unknown quantities, looking at the structure of the equation, or doing the same thing to each side of the equation. Students choose efficient tools and strategies for specific problems, helping them develop flexibility and fluency in writing and solving equations.
The focus of this video lesson is writing expressions to represent situations. Students write expressions to represent operations with numbers and with letters standing in for numbers. Students can also choose to represent expressions with tape diagrams (MP5).
Solve a linear equation that has negative numbers and a variable. This video focuses on using inverse operations to solve for a variable.
This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers and is part of the Math at the Core: Middle School collection.
Watch and listen to a recitation of the order of operations set to hip-hop music. This video focuses on PEMDAS as the acronym for order of operations and demonstrates the concept by walking through a problem in the correct order.
Apply your critical thinking skills to learn about multiplication and division of exponents. This interactive exercise focuses on positive and negative exponents and combining exponents in an effort to get you to recognize patterns and determine a rule.
This resource is part of the Math at the Core: Middle School collection.
In Module 4, Topic B, students experience special notations of operations. They determine that 3x = x + x + x is not the same as x3, which is x times x times x. Applying their prior knowledge from Grade 5, where whole-number exponents were used to express powers of ten (5.NBT.A.2), students examine exponents and carry out the order of operations, including exponents. Students demonstrate the meaning of exponents to write and evaluate numerical expressions with whole-number exponents (6.EE.A.1).
Students represent letters with numbers and numbers with letters in Module 4, Topic C. In past grades, students discovered properties of operations through example (1.OA.B.3, 3.OA.B.5). Now, they use letters to represent numbers in order to write the properties precisely. Students realize that nothing has changed because the properties still remain statements about numbers. They are not properties of letters, nor are they new rules introduced for the first time. Now, students can extend arithmetic properties from manipulating numbers to manipulating expressions. In particular, they develop the following identities: a times b = b times a, a + b = b + a, g times 1 = g, g + 0 = g, g divided by 1 = g, g divided by g = 1, and 1 divided by g = 1/g. Students understand that a letter in an expression represents a number. When that number replaces that letter, the expression can be evaluated to one number. Similarly, they understand that a letter in an expression can represent a number. When that number is replaced by a letter, an expression is stated (6.EE.A.2).
In Module 4, Topic D, students become comfortable with new notations of multiplication and division and recognize their equivalence to the familiar notations of the prior grades. The expression 2 × b is exactly the same as 2 · b and both are exactly the same as 2b. Similarly, 6 ÷ 2 is exactly the same as 6/2. These new conventions are practiced to automaticity, both with and without variables. Students extend their knowledge of the greatest common factor and the distributive property from Module 2 to expand, factor and distribute expressions using new notation (6.NS.B.4). In particular, students are introduced to factoring and distributing as algebraic identities. These include: a + a = 2 · a = 2a, (a + b) + (a + b) = 2 · (a + b) = 2(a + b) = 2a + 2b, and a ÷ b = a/b.
In Module 4, Topic E, students express operations in algebraic form. They read and write expressions in which letters stand for and represent numbers (6.EE.A.2). They also learn to use the correct terminology for operation symbols when reading expressions. For example, the expression 3/(2x - 4) is read as “the quotient of three and the difference of twice a number and four.” Similarly, students write algebraic expressions that record operations with numbers and letters that stand for numbers. Students determine that 3a + b can represent the story “Martina tripled her money and added it to her sister’s money” (6.EE.A.2b). A Mid-Module Assessment follows Topic E.
Students write and evaluate expressions and formulas in Module 4, Topic F. They use variables to write expressions and evaluate those expressions when given the value of the variable (6.EE.A.2). From there, students create formulas by setting expressions equal to another variable. For example, if there are 4 bags containing c colored cubes in each bag with 3 additional cubes, students use this information to express the total number of cubes as 4c + 3. From this expression, students develop the formula t = 4c + 3, where t is the total number of cubes. Once provided with a value for the amount of cubes in each bag (c = 12 cubes), students can evaluate the formula for t: t = 4(12) = 3, t = 48 + 3, t = 51. Students continue to evaluate given formulas such as the volume of a cube, V = s3 given the side length, or the volume of a rectangular prism, V = lwh given those dimensions (6.EE.A.2c).
