Unit 4

Expressions and Equations

Grade 6

Math

Unit Length and Description:

 

40 days

 

In Unit 4, students develop their understanding of variables in mathematical expressions.  Students will generate expressions and equations that correspond to given situations.  They will also evaluate expressions and use expressions and formulas to solve real-world problems.  Students will use properties of operations to understand that expressions in different forms can be equivalent, to generate equivalent expressions, and to solve simple one-step equations.  Students will use values of variables to solve equations.  Finally, students will construct and analyze tables and use equations to describe relationships between quantities.

 

Standards:

 

Major Cluster Standards

Apply and extend previous understandings of arithmetic to algebraic expressions.2

6.EE.A.1

Write and evaluate numeric expressions involving whole-number exponents.

6.EE.A.2

Write, read, and evaluate expressions in which letters stand for numbers.

a.    Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract 𝑦 from 5” as 5 − 𝑦.

b.    Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

c.    Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas 𝑉=𝑠3 and 𝐴𝐴=6𝑠2 to find the volume and surface area of a cube with sides of length .

6.EE.A.3

Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2+𝑥) to produce the equivalent expression 6+3𝑥; apply the distributive property to the expression 24𝑥+18𝑦 to produce the equivalent expression 6(4𝑥+3𝑦); apply properties of operations to 𝑦+𝑦+𝑦 to produce the equivalent expression 3𝑦.

6.EE.A.4

Identify when two expressions are equivalent (when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions 𝑦+𝑦+𝑦 and 3𝑦 are equivalent because they name the same number regardless of which number 𝑦 stands for.

Reason about and solve one-variable equations and inequalities.3

6.EE.B.5

Understand solving an equation or inequality as a process of answering a question; which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

                                 

6.EE.B.7

Solve real-world and mathematical problems by writing and solving equations in the form 𝑥+𝑝=𝑞 and px=𝑞 for cases in which 𝑝, 𝑞 and 𝑥 are all nonnegative rational numbers.

6.EE.B.8

Write an inequality of the form 𝑥>𝑐 or 𝑥<𝑐 to represent a constraint or condition in a real-world mathematical problem. Recognize that inequalities of the form 𝑥>𝑐 or 𝑥<𝑐 have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Represent and analyze quantitative relationships between dependent and independent variables.

6.EE.C.9

 

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation 𝑑=65𝑡 to represent the relationship between distance and time.

Foundational Standards

Understand and apply properties of operations and the relationship between addition and subtraction.

1.OA.B.3

 

Apply properties of operations as strategies to add and subtract.4 Examples: If 8+3=11 is known, then 3+8=11 is also known. (Commutative property of addition.) To add 2+6+4, the second two numbers can be added to make a ten, so 2+6+4=2+10=12. (Associative property of addition.)

Understand properties of multiplication and the relationship between multiplication and division.

3.OA.B.5

Apply properties of operations as strategies to multiply and divide. Examples: If 6×4=24 is known, then 4×6=24 is also known. (Commutative property of multiplication.) 3×5×2 can be found by 3×5=15, then 15×2=30, or by 5×2=10, then 3×10=30.(Associative property of multiplication.) Knowing that 8×5=40 and 8×2=16, one can find 8×7 as 8×(5+2)=(8×5)+(8×2)=40+16=56 (Distributive property.)

Gain familiarity with factors and multiples.

4.OA.B.4

Find all factors for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

Geometric measurement: understand concepts of angle and measure angles.

4.MD.C.5

Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

a.    An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through  of a circle is called a “one-degree angle,” and can be used to measure angles.

b.    An angle that turns through 𝑛 one-degree angles is said to have an angle measure of 𝑛 degrees.

4.MD.C.6

Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

4.MD.C.7

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

Write and interpret numerical expressions.

5.OA.A.2

Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Analyze patterns and relationships.

5.OA.B.3

Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Understand the place value system.

5.NBT.A.2

Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Graph points on the coordinate plane to solve real-world and mathematical problems.

5.G.A.1

Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., 𝑥-axis and 𝑥-coordinate, 𝑦-axis and 𝑦-coordinate).

5.G.A.2

Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Understand ratio concepts and use ratio reasoning to solve problems.

6.RP.A.3

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a.    Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b.    Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

Compute fluently with multi-digit numbers and find common factors and multiples.

6.NS.B.4

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9+2).

Standards for Mathematical Practices

1.   Make sense of problems and persevere in solving them.

2.   Reason abstractly and quantitatively.

3.   Construct viable arguments and critique the reasoning of others.

4.   Model with mathematics.

5.   Use appropriate tools strategically.

6.   Attend to precision.

7.   Look for and make use of structure.

8.   Look for and express regularity in repeated reasoning.

Instructional Outcomes

·         6.EE.A.1:

o   I can write numerical expressions involving whole number exponents.

o   I can evaluate numerical expressions involving whole number exponents.

o   I can evaluate expressions using the order of operations.

·         6.EE.A.2:

·         6.EE.A.2a

o   I can evaluate expressions with variables.

o   I can evaluate expressions using specific values for variables.

o   I can translate written phrases into algebraic expressions.

o   I can translate algebraic expressions into written phrases.

·         6.EE.A.2b

o   I can identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient).

o   I can understand that parts of an expression can have more than one name.

·         6.EE.A.2c

o   I can substitute specific values for variables.

o   I can write real world and mathematical problems.

·         6.EE.A.3

o   I can create equivalent expressions using the properties of operations (e.g. distributive property, associative property, adding like terms with the addition property or equality, etc.).

o   I can apply the properties of operations to create equivalent expressions.

·         6.EE.A.4

o   I can recognize when two expressions are equivalent.

o   I can prove (using various strategies) that two expressions are equivalent no matter what number is substituted.

·         6.EE.B.5

o   I can determine which values make an equation or inequality true.

o   I can use the solution to an equation or inequality to prove that the answer is correct. 

o   I can use substitution to decide if a number makes an equation or inequality true.

·         6.EE.B.6

o   I can recognize that a variable can represent one number, or, a set of numbers.

o   I can use variables to represent numbers.

o   I can write expressions when solving a real-world or mathematical problem.

·         6.EE.B.7

o   I can define an inverse operation.

o   I can use inverse operations to solve equations.

o   I can apply rules of the form x + p = q and px = q, for cases in which p, q and x are all positive rational numbers, to solve real world and mathematical problems. (There is only one unknown quantity).

o   I can solve real world and mathematical problems by writing and solving equations.

·         6.EE.B.8

o   I can identify the constraint or condition in a real-world or mathematical problem in order to set up an inequality.

o   I can recognize that inequalities of the form x>c or x<c have an infinite number of solutions.

o   I can write an inequality of the form x>c or x<c to represent a set of solutions for a real-world or mathematical problem.

o   I can graph solutions to inequalities on a number line.

·         6.EE.C.9

o   I can define independent and dependent variables .

o   I can use variables to represent two quantities in a real-world problem that change in relationship to one another.

o   I can write an equation to express one quantity (dependent) in terms of the other quantity (independent).

o   I can analyze how dependent and independent variables change in tables and graphs.

o   I can understand that a graph, table and an equation can all represent the same real world problem.

 

 

 

Enduring Understandings:

 

·         Variables can be used as unique unknown values or as quantities that vary.

·         Exponential notation is a way to express repeated products of the same number.

·         Expressions can be written with variables to represent real-world problems.

·         Numerical expressions can be written and evaluated using whole number exponents.

·         Properties of operations can be used to generate, simplify and evaluate equivalent expressions.

·         Two equivalent expressions form an equation.

Essential Questions:

 

·         What is the purpose of an exponent?

·         How do I generate expressions involving whole number exponents?

·         What order of operations do I follow when evaluating expressions?

·         How do I evaluate expressions with variables?

·         How do I identify the parts of an expression?

·         How do I know if two or more expressions are equivalent?

·         How can knowing the properties of operations help me generate expressions?

·         How do I determine which values make an equation or inequality true?

·         How do I solve real-world problems by writing and solving equations?

·         How do I represent solutions of inequalities?

·         What strategies can I use to help me understand and represent real situations using algebraic expressions?