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Common Core Learning StandardsGRADE 6 Mathematics

EXPRESSIONS & EQUATIONS

Common Core Learning Standards Concepts Embedded Skills Vocabulary

Apply and extend previous understandings of arithmetic to algebraic expressions.

Numerical expressions

Evaluate numerical expressions with whole-number exponents.

numerical expressions

whole-number exponents

Write numerical expressions with whole-number exponents.

6.EE.1.Write and evaluate numerical expressions involving whole-number exponents.

SAMPLE TASKSI. Write a numerical expression from the words.

Nine less than thirteen squared.

A number cubed plus eleven.

Nine less than five squared.

Seven multiplied by the sum of five and four.

II. What is the value of all powers of 1? Explain.Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Alignment Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.

III. Use exponents to write the expression: 7×7×7

Copyright (c) 2011 by Erie 1 BOCES- Deep Curriculum Alignment Project for Mathematics-- Permission to use (not alter) and reproduce for educational purposes only.




algebraic expressions

Translate verbal expressions (word phrases) to algebraic expressions with letters standing for numbers.

verbal expressions


variables6.EE.2a.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.

SAMPLE TASKSI. Write an algebraic expression from the words.

Eight more than six divided by a number, n.

Erin divided 40 blueberries equally among f friends. How many blueberries did each friend get?

Jerry has five times as many albums as Mike, m. How many albums does Jerry have?

II. Kristen counted 30 people on her train when she got on it. At the next stop, m, more people got on the train, and six people got off. Write an expression that describes how many people were on Kristen’s train after the next stop. Explain how you wrote the expression.


III. Mary wrote “8 added to x” as 8 + x and “8 subtracted from x” as 8 – x. Why is the first expression correct but the second one incorrect? Explain when order is important when writing an expression.


Apply and extend previous parts of an expression

Identify parts of an expression using mathematical vocabulary.

sum term


understandings of arithmetic to algebraic expressions.

product factor quotient coefficient expression

6.EE.2b.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.

SAMPLE TASKSI. Identify the underlined part of the expression. Write sum, term, product, factor, quotient, variable, or coefficient.

4x + 3y + 2

6 (k + 3)

82 ∙7n

II. Write an algebraic expression that matches each description. A product of two variables.


An expression with four terms.

The difference of a product and a constant.


Apply and extend previous evaluating Evaluate expressions by substituting a numerical

variables


understandings of arithmetic to algebraic expressions.

expressions value for a variable. expression formula order of

operations

Simplify expressions using order of operations.

6.EE.2c.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 = s3and A = 6s2 to find the volume and surface area of a cube with sides of length s = 1/2.

Solve real-world problems when given a formula.

SAMPLE TASKSI. Evaluate 6(x + 3) for x = 8 and 6x + 3 for x = 8. Explain why the answers are not the same although the expressions are very

similar.

II. The volume of a cube is V=s3, where V is the volume and s is the length of one side. What is the volume of a cube if the length of a side, s, is 14cm.


III. A limo driver determines the cost using the formula C = 1.5 + 2.6m, where C is the total cost and m is the number of miles driven. What is the cost of a limo trip that is 25 miles?




equivalent expressions

Apply properties of operations to rewrite expressions.

expression equivalent distributive

property variable factor combine like

terms

Explain why an expression that is rewritten is equivalent to the original expression.

6.EE.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.

SAMPLE TASKSI. Are the following two expressions equivalent? 6(x +4) and 6x + 24. Explain how you determined your answer.

II. Are the following two expressions equivalent? t + t + t + t and t 4. Explain how you determined your answer.





Identify when two expressions are equivalent (one expression is the simplified version of the other one).


variableExplain why two expressions are equivalent regardless of the number that is substituted for the variable.6.EE.4.

Identify when two expressions are equivalent (i.e., 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.

SAMPLE TASKSI. Determine whether the given expressions are equivalent. Prove your reasoning.

2 y+3∙ (1+2) and 2y + 9

5(3 + 2) and 2(6 + 2) + 10

II. Simplify each expression.


8 + n + n + n + n + 7

9(4b + 2) + 3(2b + 6)



Reason about and solve one-variable equations and inequalities.

equations and

inequalities

Solve an equation or inequality by determining for which values of a set make the equation or inequality true.

equation inequality substitution solutionSubstitute a given number into an equation or

inequality to see if it makes the equation/inequality true.6.EE.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.

SAMPLE TASKSI. Determine whether the given value is a solution of the equation.

25 = x – 11; x = 14

n16

= 4; n = 64


II. Which of the following are possible solutions for the inequality below: 10.5, 25, or 30?4p < 100

III. Solve for the solution set of 5f > 75.





Write expressions with variables to represent numbers in a real-world problem.

expression variable set (of

numbers)Define a variable as a representation of an unknown number or numbers in a set. 6.EE.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.

SAMPLE TASKSI. Write an expression to represent the situation.

Mark had b books and received 7 new books for his birthday.

Aaron is one-third as old as his father, who is n years old.

Olivia bought n ride tickets at the carnival. Ellen bought four times as many tickets as Olivia. How many tickets were purchased?




equations Write and solve one-step equations with nonnegative rational numbers from real-world problems.

nonnegative rational numbers

one-step equations

6.EE.7.Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

SAMPLE TASKSI. Write and solve an equation to represent the situation.

Mark had b books and received 7 new books for his birthday. He now has 18 books. How many books did he originally have?

Aaron is one-third as old as his father, who is n years old. Aaron is 13 years old. How old is his father?

Olivia bought n ride tickets at the carnival. Ellen bought four times as many tickets as Olivia. Together they bought 35 tickets. How many tickets did each buy?


II. Solve and check to prove your answer: 2f = 32

29 + c = 72

Gary worked 16 hours last week and earned $192. The equation 16d = 192 can be used to find d, the number of dollars he earns per hour. What is Gavin’s hourly wage?




inequalities Write an inequality to represent a real-world condition or constraint.

constraint condition inequality solutions

(infinitely many)

Define inequalities as having infinitely many solutions.

6.EE.8.Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or 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.

Graph solutions to inequalities on number lines.

SAMPLE TASKSI. A pastry chef wants to spend no more than $100 on almonds. Almonds cost $4 per pound. Write an inequality to represent p, the

number of pounds of almonds the chef could buy.

II. Solve and graph: r + 9 < 12


http://www.bing.com/images/search?q=number+line&view=detail&id=A9260740F0BBAE76D73757D404F48E5AAF9124F3&first=31&FORM=IDFRIR

III. Mr. Smith hires a landscaper that charges $15 per hour. The landscaper says that total charge for the work Mr. Smith wants done will be at least $120.a. Write an inequality to show h, the number of hours it may take for the landscaper to do the work for Mr. Smith. Solve the

inequality and show your work.

b. Graph the inequality on a number line. Then, use the graph to explain if it could take 9.5 hours to complete the work for Mr. Smith.


http://www.mathematic.ws/wp-content/uploads/2009/04/number-line.png


Represent and analyze quantitative relationships between dependent and independent variables.

independent vs.

dependent

Write an equation to represent two variables, one dependent and one independent.

table independent dependent graph ordered pairs variables constant

Analyze the relationship between independent and dependent variables using graphs, tables, and equations.

6.EE.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.

List and graph ordered pairs and write the equation to represent the relationship.

SAMPLE TASKSI. Write an equation to show the relationship between x and y in the table below.


x 0 1 2 3y 0 3 6 9

II. Which ordered pair does not represent the equation y = 4x + 1?A. (0, 1) B. (1, 5) C. (2, 10) D. (3, 13)

Explain how you know your answer is correct.

III. Use the equation y = x + 2 to answer the questions below.A. Complete the table of values below for y = x + 2.

x y = x + 2 y (x, y)1357

B. Graph y = x + 2 on the coordinate grid below. Explain in words how you graphed the equation.


http://mathbits.com/MathBits/StudentResources/GraphPaper/quadone.gif

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scusd-math.wikispaces.com - … · Web viewCommon Core Learning Standards. Concepts. ... (word phrases) to algebraic expressions with ... Which of the following are possible solutions