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).

Content Standard(s):

Mathematics MA2019 (2019) Grade: 6

15. Write, read, and evaluate expressions in which letters represent numbers in real-world contexts.

a. Interpret a variable as an unknown value for any number in a specified set, depending on the context.

b. Write expressions to represent verbal statements and real-world scenarios.

c. Identify parts of an expression using mathematical terms such as sum, term, product, factor, quotient, and coefficient.

d. Evaluate expressions (which may include absolute value and whole number exponents) with respect to order of operations.

Unpacked Content

Evidence Of Student Attainment:

Students: Given contextual or mathematical problems both when known models exist (for example formulas) or algebraic models are unknown,

Interpret the parts of the model in the original context.

Create the algebraic model of the situation when appropriate.

Use appropriate mathematical terminology to communicate the meaning of the expression.

Evaluate the expressions for values of the variable including finding values following conventions of parentheses and order of operations.

Teacher Vocabulary:

Expressions

Term

Coefficient

Sum

Product

Factor

Quotient

Variable

Constant

Difference

Evaluate

Order of Operations

Exponent

Absolute Value

Knowledge:

Students know:

Correct usage of mathematical symbolism to model the terms sum, term, product, factor, quotient, variable, difference, constant, and coefficient when they appear in verbally stated contexts.

Conventions for order of operations.

Convention of using juxtaposition (5A or xy) to indicate multiplication.

Skills:

Students are able to:

Translate fluently between verbally stated situations and algebraic models of the situation.

Use operations (addition, subtraction, multiplication, division, and exponentiation) fluently with the conventions of parentheses and order of operations to evaluate expressions for specific values of variables in expressions.

Use terminology related to algebraic expressions such as sum, term, product, factor, quotient, or coefficient, to communicate the meanings of the expression and the parts of the expression.

Understanding:

Students understand that:

The structure of mathematics allows for terminology and techniques used with numerical expressions to be used in an analogous way with algebraic expressions, (the sum of 3 and 4 is written as 3 + 4, so the sum of 3 and y is written as 3 + y).

When language is ambiguous about the meaning of a mathematical expression grouping, symbols and order of operations conventions are used to communicate the meaning clearly.

Moving fluently among representations of mathematical situations (words, numbers, symbols, etc.), as needed for a given situation, allows a user of mathematics to make sense of the situation and choose appropriate and efficient paths to solutions.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.6.15.1: Define algebraic expression and variable.
M.6.15.2: Convert mathematical terms to mathematical symbols and numbers.
M.6.15.3: Translate verbal and numerical expression using all operations.
M.6.15.4: Define coefficient, constant and term.
M.6.15.5: Match mathematical terms with correct mathematical symbols.
M.6.15.6: Convert mathematical terms to mathematical symbols and numbers.
M.6.15.7: Calculate an expression in the correct order. with or without a calculator (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
M.6.15.8: Choose the correct value to replace each variable in the algebraic expression (Substitution).
M.6.15.9: Calculate a numerical expression, with or without a calculator (Ex. V=4x4x4).
M.6.15.10: Recognize the correct order to solve expressions with more than one operation.

Prior Knowledge Skills:

Recognize key terms to solve word problems.
Examples: times, every, at this rate, each, per, equal/equally, in all, total.

Recognize key terms to solve word problems.
Examples: times, every, at this rate, each, per, equal/equally, in all, total.

Define simple expression.

Recall simple equations.

Recognize properties of addition and multiplication.

Apply properties of operations as strategies to add and subtract.

Recall properties of operations as strategies to add and subtract.

Represent addition and subtraction with objects, mental images, drawings, expressions, or equations.

Use addition, subtraction, multiplication and division to solve one- and two-step word problems.

Recognize key terms to solve word problems.

Apply properties of operations as strategies to multiply and divide.

Apply the area and perimeter formulas for rectangles in real-world and mathematical problems.

Recall the formula for area (L × W).

Recognize that unit squares are equal.

Recall the formula for perimeter (P= L+L+W+W or P=2L + 2W).

Recall basic addition and multiplication facts.

Alabama Alternate Achievement Standards

AAS Standard: M.AAS.6.15 Evaluate algebraic expressions when given specific values for the variables (e.g. x + 2, where x = 4).

Mathematics MA2019 (2019) Grade: 6

17. Determine whether two expressions are equivalent and justify the reasoning.

Unpacked Content

Evidence Of Student Attainment:

Students: Given a contextual or mathematical situation that could be represented algebraically,

Explain by reasoning from the context why two expressions must be equivalent.

Use properties of operations and equality to verify if two algebraic expressions are equivalent or not.

Teacher Vocabulary:

Equivalent

Expressions

Knowledge:

Students know:

The properties of operations, including inverse, identity, commutative, associative, and distributive and their appropriate application to be able to determine whether two expressions are equivalent.

Conventions of order of operations.

Skills:

Students are able to:

Accurately use the properties of operations to produce equivalent forms of an algebraic expression when interpreting mathematical and contextual situations.

Use mathematical reasoning to communicate the relationships between equivalent algebraic expressions.

Understanding:

Students understand that:

Manipulation of expressions via properties of the operations verifies mathematically that two expressions are equivalent.

Reasoning about the context from which expressions arise allows for interpretation and meaning to be placed on each of the expressions and their equivalence.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: M.6.17.1: Define equivalent expressions.
M.6.17.2: Recognize equivalent expressions.
M.6.17.3: Substitute for the variable to find the value of a given expression.
M.6.17.4: Calculate a numerical expression.
M.6.17.5: Recognize that a variable without a written coefficient is understood to have a coefficient of one. (Ex. x = 1x).

Prior Knowledge Skills:

Use comparison symbols. Examples: >, =, or < .

Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits using >, =, and < symbols to record the results of comparisons.

Comparison symbols. Examples: >, =, and < .

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits using >, =, and < symbols to record the results of comparisons.

Compare two fractions with the same numerator or the same denominator by reasoning about their size.

Recognize that comparisons are valid only when the two fractions refer to the same whole.

Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Convert fractions to decimals.

Compare two decimals to tenths.

Compare whole numbers.

Identify comparison symbols. Examples: >, <, and = .