· Rangely School District Re4 6th Grade Mathematics Curriculum . Content Area . Mathematics ....

Rangely School District Re4 6th Grade Mathematics Curriculum

Content Area Mathematics Grade Level 6th Grade Course Name/Course Code 6th Grade Math

Standard Grade Level Expectations (GLE) GLE Code

1. Number Sense, Properties, and Operations

1. Quantities can be expressed and compared using ratios and rates MA10-GR.6-S.1-GLE.1

2. Formulate, represent, and use algorithms with positive rational numbers with flexibility, accuracy, and efficiency

MA10-GR.6-S.1-GLE.2

3. In the real number system, rational numbers have a unique location on the number line and in space MA10-GR.6-S.1-GLE.3

2. Patterns, Functions, and Algebraic Structures

1. Algebraic expressions can be used to generalize properties of arithmetic MA10-GR.6-S.2-GLE.1

2. Variables are used to represent unknown quantities within equations and inequalities MA10-GR.6-S.2-GLE.2

3. Data Analysis, Statistics, and Probability

1. Visual displays and summary statistics of one-variable data condense the information in data sets into usable knowledge

MA10-GR.6-S.3-GLE.1

4. Shape, Dimension, and Geometric Relationships

1. Objects in space and their parts and attributes can be measured and analyzed MA10-GR.6-S.4-GLE.1

Colorado 21st Century Skills

Critical Thinking and Reasoning: Thinking Deeply, Thinking Differently

Information Literacy: Untangling the Web

Collaboration: Working Together, Learning Together

Self-Direction: Own Your Learning

Invention: Creating Solutions

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.

Unit Titles Length of Unit/Contact Hours Unit Number/Sequence Very Variable 6 weeks 2 No Drama Be Rational 5 weeks 3 (continue to practice fluency skills

throughout the year) Rockin’ Ratios 8 weeks 4 Know the 411 5 weeks 5 Go Figure! 4 weeks 6

Invention

December 10, 2013 of 64 Rangely School District Re4


Unit Title Very Variable Length of Unit 6 weeks

Focusing Lens(es) Structure Modeling

Standards and Grade Level Expectations Addressed in this Unit

MA10-GR.6-S.2-GLE.1 MA10-GR.6-S.2-GLE.2

Inquiry Questions (Engaging- Debatable):

• Why do we call letters meant to symbolize unknown quantities variables, do they always vary? (MA10-GR.6-S.2-GLE.2-EO.b)

Unit Strands Expressions and Equations

Concepts exponents, multiplication, symbolizing, base, substitution, variable, inequalities, equations, solutions, values, order of operations, commutative, associate, distributive, equivalent, numeric expressions, algebraic expressions, characteristics, term, coefficient, sum, factor, quotient, product, constraint, condition, infinite, number line diagram, independent, dependent

Generalizations

My students will Understand that…

Guiding Questions Factual Conceptual

Exponents symbolize repeated multiplication; the base represents the quantity being repeatedly multiplied and the exponent defines the number of repeated multiplications. (MA10-GR.6-S.2-GLE.1-EO.a)

What does a base and exponent represent? In order of operations, which comes should be

performed first exponents or multiplication and division?

How is multiplication symbolizing repeated addition similar and different from exponents symbolizing repeated multiplication?

Why isn’t (2+3)3 equal to 23+ 33?

Using order of operations and the commutative, associate, and distributive properties generates equivalent expressions. (MA10-GR.6-S.2-GLE.1-EO.b.iv, c)

Which property justifies that 2k+6 is equivalent to 2(k+3)?

How do you simplify a given expression?

Why do the order of operations exist? (MA10-GR.6-S.2-GLE.1-IQ.4)

Algebraic expressions contain variables that represent either a single unknown or a quantity that varies (changes). (MA10-GR.6-S.2-GLE.1-EO.b.i, c)

What is the difference between an algebraic and numeric expression?

If we didn’t have variables, what would we use? (MA10-GR.6-S.2-GLE.1-IQ.1)

What purpose do algebraic expressions serve? (MA10-GR.6-S.2-GLE.1-IQ.2)

Why is it helpful to record operations with numbers and letters standing for numbers?

Why are there advantages to being able to describe a pattern using variables? (MA10-GR.6-S.2-GLE.1-IQ.3)

The structure and characteristics of mathematical expressions (term, coefficient, sum, factor, quotient and product and equations) help predict mathematical behaviors and model mathematical situations. (MA10-GR.6-S.2-GLE.1-EO.b.ii)

What does the word term mean in relation to expressions?

How can you determine the number of terms in an expression?

What is a coefficient? What is a constant?

How does understanding the structure of expressions support the simplification of expressions?

Why is it helpful to sometimes view terms of an expression as a single entity and other times as separate entities?


Mathematicians can ascertain the equivalence of two algebraic expressions if, for any value substituted for the variable, the values of two expressions are equal. (MA10-GR.6-S.2-GLE.1-EO.d) and (MA10-GR.6-S.2-GLE.2-EO.a)

How can you determine if two expressions are equivalent?

If an equation has multiple instances of the same variable, does the same value need to be substituted for all of them?

Do all equations have one unique solution? (2.2.IQ1)

Why is only one counter-example (a value for the variable that yields unequal results for the two expressions) needed to disprove the equivalence of two expressions?

How might simplifying two expressions help determine if they are equivalent?

Why is a solution(s) to an equation or inequality the value(s) that make them true?

An inequality of the form x > c or x < c represents a constraint or condition in a real-world or mathematical problem and inequalities of this form have infinitely many solutions. (MA10-GR.6-S.2-GLE.2-EO.e, f)

How can you show an infinite amount of solutions on a number line?

Why does an inequality have an infinite number of solutions?

Dependent and independent variables represent two quantities that change in relationship to one another. (MA10-GR.6-S.2-GLE.2-EO.g)

How can you represent the relationship between dependent and independent variables in a table? graph?

Why are coordinate pairs used to represent solutions to two variable equations?

How can you determine if a variable is independent or dependent? (MA10-GR.6-S.2-GLE.2-IQ.2)


Key Knowledge and Skills: My students will…

What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the mathematics samples what students should know and do are combined.

• Write and evaluate numerical expressions involving whole-number exponents. (MA10-GR.6-S.2-GLE.1-EO.a) o Write numerical expressions involving whole-number exponents PARCC

Tasks involve expressing b-fold products 𝑎 ∙ 𝑎 ∙ 𝑎⋯𝑎 in the form 𝑎𝑏, where a and b are non-zero whole numbers. Tasks do not require use of the laws of exponents. PARCC

• Students use exponents to code repeated multiplication, distinguishing the base from the exponent, and evaluate expressions involving whole-number exponents. For example, 34= 3 · 3 · 3 · 3= 81. (CC 6.EE.1)

• Students evaluate expressions such as 23 + (5 + 3)2 - 43 by first adding the 5 and 3 in parentheses (8), followed by evaluating the terms with exponents (8 + 64 – 64), and then finally adding and subtracting from left to right to get 8. (CC 6.EE.1)

• Example: Students apply order of operations to evaluate the expressions on each side of the equations below and make a judgment about the equality: (CC 6.EE.1)

o Is (2 + 3)3 = 23 + 33 False o Is (2 ∙ 5)3 = 23 ∙ 53 True o Is 23 + 24 = 27 True

• Write expressions that record operations with numbers and with letters standing for numbers. (MA10-GR.6-S.2-GLE.1-EO.b.i) o Write, read, and evaluate expressions in which letters stand for numbers. PARCC

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.

Tasks do not have a context. Numerical values in these expressions may include whole numbers, fractions, and decimals. The testing interface can provide students with a calculation aid of the specified kind for these tasks.

• For example, in the equation 5 + b = 9, the symbol b is a variable that is equal to 4. (CC 6.EE.2a) • In the expression 7 + x, the symbol x is a variable that can change. When x equals 1, the expression has a value of 8. When x equals 12, the

expression has a value of 19. (CC 6.EE.2a) • To summarize, students can make distinctions among:

(CC 6.EE.2a)

Expression Equation Numeric 8 + 7 8 + 7 = 15 Algebraic 8 + x 8 + x = 15



• Students write algebraic expressions that represent word phrases such as: Phrase Expression

Five increased by/more than a number x 5 + x Fifteen decreased by a number y 15 − y

Seven less than a number p p − 7 The product of 8 and a number q 8 · q or 8q

Twenty divided by a number n 20 ÷ n or 20/n

Ten more than twice a number 2x + 10 Five times a number decreased by 4 5y − 4

Ten divided by twice a number 10 ÷ 2r or 10/2r

(CC 6.EE.2a) • 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

(MA10-GR.6-S.2-GLE.1-EO.b.ii) o Write, read, and evaluate expressions in which letters stand for numbers. PARCC o Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expressions 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. PARCC A term is a part of an expression separated from others by an addition or subtraction sign. The number of terms in an expression is often identified. For

example, the expression 5 + 7x consists of two terms: 5 and 7x. (CC 6.EE.2b) A coefficient is the numeric factor in a product that includes a variable. For example, in the expression above, 7 is the coefficient in the product 7x. In the

equation 5y + 10 = 30, 5 is the coefficient in 5y. (CC 6.EE.2b) A constant is a fixed, numeric term, in contrast to a variable term. For example, in the expression x + 3, the number 3 is a constant, while x is a variable. (CC

6.EE.2b) • Evaluate expressions at specific values of their variables; including expressions that arise from formulas used in real-world problems and perform arithmetic operations,

involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order. (MA10-GR.6-S.2-GLE.1-EO.b.iii, b.iv) o Evaluate expressions at specific values of their variables. 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). PARCC o Evaluate expressions that arise from formulas used in real-world problems at specific values of their variables. For example, use the formulas𝑉 = 𝑠3 and𝐴 = 6𝑠2 to

find the volume and surface area of a cube with sides of length 𝑠 = 12 PARCC

Students evaluate expressions at specific values by using the order of operations. For instance, given the expression 5x + 3y, find the value of the expression when x is equal to 2 and y is equal to 3.8. Students are required to understand that 5x and 3yrepresent multiplication and to use the order of operations to evaluate the expression:

5 • 2 + 3 • 3.8 = 10 + 11.4 = 21.4 (CC 6.EE.2c)

Given a context and the formula arising from the context, students write an expression and then evaluate for any value required. Here are two examples: • “The distance traveled by a car can be found by the formula: D = s • t, in which s is the speed of the car, and t is the total time traveled.

Suppose, a car traveled 40km/h for 3 hours. How far did the car travel?” (CC 6.EE.2c) • “To convert Fahrenheit (F) to Celsius (C), one can use the formula C = 5/9 (F − 32). If the temperature is 72° F, what is the temperature in



degrees Celsius?” (CC 6.EE.2c) • Apply the properties of operations to generate equivalent expressions. (MA10-GR.6-S.2-GLE.1-EO.c)

o Apply the properties of operations to generate equivalent expressions. PARCC o Students apply the commutative, associative, and distributive properties to generate equivalent expressions. PARCC

Students generate x + 8 when given 13 + x − 5, by using the commutative property of addition on 13 + x − 5 to change the order of the operations in that expression to 13 − 5 + x. (CC 6.EE.3)

Students generate 21x when given 7(3x), using the associative property of multiplication to multiply (7 • 3) x. (CC 6.EE.3) The distributive property states that a product can be found by decomposing a number and multiplying each addend separately, and then adding their

products. For example, 5 • 32 can be broken into 5 • (30 + 2) = (5 • 30) + (5 • 2). In either case the product equals 160. (CC 6.EE.3) Students have experience factoring expressions, such as the following examples:

2k + 6 = 2 (k + 3) -n − 5 = - (n + 5) 4s − 4 = 4 (s − 1)

(a − 3) = -1 (- a + 3)

• This last example is particularly challenging for students and merits careful discussion and analysis, because it is later used frequently as a strategy to equate expressions. (CC 6.EE.3)

• Identify when two expressions are equivalent. (MA10-GR.6-S.2-GLE.1-EO.d) o 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. PARCC Students know that two algebraic expressions are equivalent if, for any value substituted for the variable, the values of the two expressions are equal, i.e.

evaluating the two expressions using the value for the variable gives the same result for both expressions. This implies that a single counterexample (a value for the variable that yields unequal results for the two expressions) disproves the equivalence of expressions. (CC 6.EE.4)

• Students identify like terms. Like terms are terms that contain the same variable with the same whole number. For example, 3x and 5x are like terms (both expressions contain x), while 3x and 5y are not (they contain different variables). (CC 6.EE.4)

• It may help students to recognize that 5x − x contains two like terms, i.e. 5x and 1x, and then more easily recognize the applicability of the distributive property, whereby 5x − 1x = x(5 − 1), or 4x. Students might also code 5x − x as 5 x’s minus 1 x gives 4 x’s. (CC 6.EE.4)

• 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. (MA10-GR.6-S.2-GLE.2-EO.c, c.i)

o Tasks may require students to write an expression to represent a real-world or mathematical problem. Tasks do not require students to find a solution. PARCC o Tasks may require students to interpret a variable as a specific unknown number, or, as a number that could represent any number in a specified set. PARCC

Students write expressions to represent word problems. For example, “A plumber charges $50 per hour but has to spend $20 on gasoline in a day for traveling. Write an algebraic expression to represent his earnings for one day.” Students may represent the number of hours the plumber works in one day with the letter x and represent his earnings per day with the expression 50x – 20. (CC 6.EE.6)

Students may also give possible values to x and evaluate the expression. For instance, if the plumber works 6 hours, then, by evaluating [(50 · 6) − 20], one can tell that he will earn $280. If he works 8 hours, then by evaluating [(50 · 8) − 20], one can find that he will earn $380. While acknowledging that the number of hours varies, students also recognize the reasonableness of the values that x can take on. For instance, a plumber may work between 1 and 10 hours per day. (CC 6.EE.6)

An example of a problem that students can solve with a variable as an unknown is: “Suppose the plumber wants to earn $330 in a day. How many hours does he need to work?” They would set up the problem as 50x − 20 = 330, with x representing the number of hours. (CC 6.EE.6)



• 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? (MA10-GR.6-S.2-GLE.2-EO.a)

o The testing interface can provide students with a calculation aid of the specified kind for these tasks. PARCC In this standard, students are given a specified set of numbers and determine which values make an equation or an inequality in one variable a true (or

false) statement when substituted for the variable. For equations, students correctly apply rules for order of operations to obtain the answer. (CC 6.EE.5-1) The following table provides the cases that are included in this standard. Asterisks are placed by the cases students typically find more difficult. Students

verify whether specific values of x are solutions to the following types of equations by substitution:

Types of Equations Features of the equation 1) x + 2 = 5; x − 2 = 6

Addition/Subtraction: x is an unknown addend/minuend in the equation

2) 3x = 9; x/4 = 7

Multiplication/Division: x is an unknown factor/dividend in the equation

*3) -x = 5 Negative coefficient: x has a negative coefficient *4) 9 − x = 7 Subtraction: the unknown quantity x is subtracted from a

number *5) 12/x = 4 Division: x is an unknown divisor 6) 3(2 + x) = 11 Distributive property: x is an unknown term in a factor *7) 5 − (x + 1) = 3 36/(2x) = 6

Order of operations and parentheses: is an unknown term in the subtrahend or divisor

*8) 3x = x + 1 Variables on both sides: the unknown quantity x appears on both sides of the equation

*cases students typically find more difficult (CC 6.EE.5-1) o The testing interface can provide students with a calculation aid of the specified kind for these tasks. PARCC

Students show their solutions using set notation or a number line diagram. They list the values of a set using the braces “{“ and “},” and use the symbol “∈” (is an element of) to express the idea of “x is a member of the set,” e. g., “x∈{1, 2, 3, …, 7, 8, 9}.” This notation indicates that the elements of this set are the solutions to the equation. For example, when students find that x = 1 is a solution to the equation x − 1= 0, they write “x ∈ {1}”. They represent the solution on the number line by placing a black circle on “1”.

• Use substitution to determine whether a given number in a specified set makes an equation or inequality true. (MA10-GR.6-S.2-GLE.2-EO.b) o 80% of tasks involve values from an infinite set of nonnegative numbers (e.g., even numbers; whole numbers; fractions). PARCC o 20% of tasks involve values from a finite set of nonnegative numbers e.g., {2, 5, 7, 9}). PARCC

Students are also asked to solve an equation with no solution. For example, they interpret the equation x + 2 = x as asking “what number is two more than itself?” They provide the empty set symbol ∅ or { } as an answer to indicate that there is no such value of x [39]. (CC 6.EE.5-2)

A common misconception expressed by students is that the empty set∅ is the same as the solution x = 0 [37]. This misconception may be caused by equating 0 as “nothing” in earlier grades and the conventional use of ∅ as zero to distinguish from the alphabet “O”. To help students correct this misconception, they are asked to determine which of the following two equations are true, given than x = 0:


http://turnonccmath.net/ajax.php?key=linear-simultaneous-eqns&fmt=html%23_ENREF_39

http://turnonccmath.net/ajax.php?key=linear-simultaneous-eqns&fmt=html%23_ENREF_37

(1) x + 2 = x (2) x + 2 = 2

• In equation (1), there is no solution because no value can be substituted for the variable to make the equation true. The solution is therefore the empty set ∅ or { }. In equation (2), on the other hand, students can substitute the value of zero to make the equation true, and thus the solution set is 0 or x ∈ {0}. (CC 6.EE.5-2)

• 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. (MA10-GR.6-S.2-GLE.2-EO.d)

o Problem situations are of “algebraic” type, not “arithmetic” type. PARCC o 50% of tasks involve whole number values of p, q, and/or x; 50% of tasks involve fraction or decimal value of p, q, and/or x. Fractions and decimals should not appear

in the same problem. PARCC o A valid equation and the correct answer are both required for full credit. PARCC o The testing interface can provide students with a calculation aid of the specified kind for these tasks. PARCC

Note to Teachers: It is important to note that because variables represent a specific numerical value or a range of numerical values, they can be treated like numbers, i.e., applying operations and inverse operations. (CC 6.EE.7)

The following table provides the cases that are included in this standard. Students solve the following types of equations systematically by applying the axioms of equivalence (for equations) to produce the solution of the equation in the form of x = c or c = x:

Types of Equations Operations on both sides of the equation

to preserve equivalence 1) x + 2 = 5 Subtraction: (x + 2) − 2 = (5) – 2 2) x − 3 = 7 Addition: (x − 3) + 3 = (7) + 3 3) 3x = 12 Division: (3x)/3 = (12)/3 4)

1/5 x = 6 Multiplication: 5(1/5 x) = 5 (6) (CC 6.EE.5-2)

Students develop familiarity and fluency with notation. For an example, see Standard 5.OA.1 in the Early Equations and Expressions LT, which introduces the notion that (2)(3) means 2 • 3. (CC 6.EE.5-2)

• Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem and recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. (MA10-GR.6-S.2-GLE.2-EO.e, f)

o Constraint values (denoted c in standard 6.EE.8) are not limited to integers. PARCC Students also solve inequalities of the form x ≥ c or x ≤ c which are associated with phrases of comparison such as “greater than or equal to” or “at least,”

and “less than or equal to” or “at most.” These inequalities include the particular solution x = c as part of the solution set for x. (CC 6.EE.8) Number lines are useful representations to aid students in solving simple inequalities. For example, students use the number line to represent x > 5, as

shown below, to solve the problem “Solve the inequality x > 5.”

Students also interpret the inequality statement “7 is greater than x” and write 7 > x. They recognize it is equivalent to x < 7 by comparing their number line diagrams. (CC 6.EE.8)

Students work with inequalities in contexts such as “The lengths of two sides of a triangle are 3 inches and 4 inches. What is the length of the third side?”


Students are provided with manipulatives (e.g., strings) or dynamic geometry software to explore what values are possible for the third side length.

Students notice that the third length must be smaller than 7 inches, which is the sum of the other two lengths (3 inches + 4 inches). (CC 6.EE.8)

If the third length is 7 inches, that side and the other two sides collapse into a straight line, and the figure is no longer a triangle. (CC 6.EE.8)

They also notice that the third length, x, must be longer than 1 inch, which is the difference between the other two sides (4 inches – 3 inches). (CC 6.EE.8)

If the third side is 1 inch, the 3-inch side and the 1-inch side overlap with the 4-inch side, and the figure is no longer a triangle. (CC 6.EE.8)

Therefore, students conclude that the third side would be greater than 1 and less than 7. This is an example of the “Triangle Inequality”, which illustrats how geometry and

algebra can be related using inequalities. (CC 6.EE.8) The solution set can be represented by the following number line. (CC 6.EE.8)

• 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. (MA10-GR.6-S.2-GLE.2-EO.g, g.i, g.ii) o Students extend their understanding of a single-variable linear equation to an equation with two unknowns, often x and y. Students explore the relationship

between the two variables, where one variable (dependent variable, y) is dependent on the other variable (independent variable, x). (CC 6.EE.9) o Students use letters to represent two variables, time traveled and distance, in the following problem: “When running at its maximum power, an Amtrak Express



train travels at 160 miles per hour. Describe the distance traveled by the train in relation to the amount of time it travels.” (CC 6.EE.9) Students assign x to represent the amount of time, and y to represent the distance traveled. (CC 6.EE.9)

o Because there are two variables, students learn to select a value for one variable and figure out the corresponding value of the second variable. An equation in two variables thus describes a relation between the variables. (CC 6.EE.9)

o Below is a list of different representations that students use to create the solution set for two-variable equation for the example above. (CC 6.EE.9) (1) Set notation using ordered pairs: (x, y) ∈ {(1, 160), (2, 320),(3, 480), (4, 640), (5, 800), …} (2) A table of two columns, one for the time spent traveling (x hours) and the other for the distance traveled (y miles)

y = 160x x

(hours) y

(miles) 1 160 2 320 3 480 4 640 5 800

• In working with tables of values, students explore patterns in the data both as x corresponds

to y (correspondence description: multiplication by 160) and as covariation (the value of x increases by 1 and the value of y increases by 160 simultaneously).

(3) A graph of the distance traveled (y in miles) versus the time traveled (x in hours).

When expressing the relationship between two variables x and y, students reason and identify which variable, for example, time or distance, is the independent or dependent variable. Conventionally, the variable placed on the x-axis, as the input values, is considered the independent variable, and the variable placed on the y-axis, as the output variable, is considered the dependent variable. (CC 6.EE.9)

• Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. (MA10-GR.6-S.2-GLE.2-EO.g.iii) o The choice of independent and dependent variables is not inherent to the problem context but depends on the question asked. In this problem, one can say that the

calculated distance (output) is dependent on time (input). Similarly, the distance to be traveled (input) could be the independent variable and the calculation of time (output) would be the dependent variable. (CC 6.EE.9)

Students determine from the problem context whether the possible values for x and y are continuous quantities. In the problem above, students also locate



the coordinates of the points that are non-whole numbers that satisfy the equation. For example, when the amount of time traveled is 1.5 hours, the distance traveled is 240 miles. As they plot more points on the graph, students recognize that these plotted points collectively form a straight line that passes through the origin with a unit ratio of 160 miles per hour. (CC 6.EE.9)

Through working with tables and graphs, students identify the line as the set of all the points (x, y) that satisfy the equation y = 160x, consistent with what they learned about ratios and direct variation. (CC 6.EE.9)

o Students solve a variety of equations in two variables, such as representing the solution set for the length and width of rectangles whose perimeter is 12 cm, using sketches, tables and graphs. (CC 6.EE.9)

Critical Language: includes the Academic and Technical vocabulary, semantics, and discourse which are particular to and necessary for accessing a given discipline.

EXAMPLE: A student in Language Arts can demonstrate the ability to apply and comprehend critical language through the following statement: “Mark Twain exposes the hypocrisy of slavery through the use of satire.”

A student in ______________ can demonstrate the ability to apply and comprehend critical language through the following statement(s):

Algebraic inequalities have infinitely many solutions because there are infinitely many values on a number line between any two points.

Academic Vocabulary: Multiplication, characteristics, identify, apply, solve, represent

Technical Vocabulary: Exponents, base, substitution, variable, inequalities, equations, solutions, order of operations, commutative, associate, distributive, numeric expressions, algebraic expressions, term, coefficient, sum, factor, quotient, product, constraint, condition, infinite, number line diagram, values, equivalent, evaluate, independent, dependent


Curriculum Development Overview Unit Planning for 6th Grade Mathematics

Unit Title No Drama Be Rational Length of Unit 5 weeks

Focusing Lens(es) Comparison Location




• Why are there negative numbers? (MA10-GR.6-S.1-GLE.3-IQ.1) • Are there more rational numbers or integers? (MA10-GR.6-S.1-GLE.3-IQ.3)

Unit Strands The Number System

Concepts Fluency, operations, multi-digit numbers, decimal numbers, standard algorithm, estimation, reasonableness, positive, negative, quantities, opposite, direction, values, magnitude, vertical, horizontal, number lines, rational numbers, inequalities, number line diagrams, absolute value, ordered pairs, location, reflection, axes

Generalizations



Fluency with the four operations on multi-digit number and decimal numbers requires strategic application of the standard algorithm and other tools including estimation strategies to assess reasonableness of answers. (MA10-GR.6-S.1-GLE.2-EO.a, b)

How do the algorithms for decimals compare to the algorithms for whole numbers? (MA10-GR.6-S.1-GLE.2-IQ.2)

Why is estimating to assess the reasonableness of answers an important when using standard algorithms?

Positive and negative numbers can represent quantities in real world contexts having opposite directions or values. (MA10-GR.6-S.1-GLE.3-EO.a.i)

When a + or – sign is not used in front a number is the number positive or negative?

What are examples of real world situations involving positive and negative numbers?

Why is it important to explain the meaning of 0 in situations involving positive and negative numbers?

Locating rational numbers on horizontal and vertical number lines requires attention to both components of a number, the magnitude (distance from zero) and direction. (MA10-GR.6-S.1-GLE.3-EO.b.i, b.ii, b.iii, b.iv, b.v)

How is directionality represented in rational numbers? What does it mean for two numbers to be opposites? What number is its own opposite? How do you locate rational numbers on a horizontal and

vertical number line? What is a rational number?

Why is it helpful for numbers to represent both magnitudes and directionality?

Why are -1.5 and 1.5 an equal distance from zero on a number line?

Why is the opposite of the opposite of a number the number itself?


Statements of inequalities represent the relative position of two numbers on a number line diagram. (MA10-GR.6-S.1-GLE.3-EO.c.i, c.ii)

When comparing two numbers how can you tell by the direction they are from each other, which is smaller?

How can you represent the comparison of two different quantities with inequalities?

Why is attention to the magnitude and directionality of a number important when making comparisons?

Interpreting absolute value of a rational number (e.g., its distance from zero) requires an examination of the magnitude of a number without its directionality. (MA10-GR.6-S.1-GLE.3-EO.c.iii)

What is the definition of absolute value?

Why is the absolute value of a positive number the number itself?

Why is absolute value connected to real world contexts involving distance?

Taking absolute values of numbers in an inequality can reverse the inequality. (MA10-GR.6-S.1-GLE.3-EO.c.iv)

If a |b|? If a > b when is the |a| > |b| and when is the |a| < |b|?

Why does taking the absolute value of numbers in an equality sometimes affect the inequality?

Signs of numbers in ordered pairs indicate locations in one of four quadrants of the coordinate plane (MA10-GR.6-S.1-GLE.3-EO.b.iv, b.vi, d)

What is true of all coordinates in the 1st quadrant? 2nd quadrant? 3rd quadrant? 4th quadrant?

How do the signs of coordinates differ when reflected across the horizontal axis? Vertical axis? Both axes?

Why are four quadrants necessary to represent ordered pairs of rational numbers?

Why can changing the signs of numbers in an ordered pair be interpreted as a reflection across an axis on a coordinate plane?



• Fluently divide multi-digit numbers using the standard algorithm (MA10-GR.6-S.1-GLE.2-EO.a) o The given dividend and divisor are such as to require an efficient/standard algorithm (e.g., 4058476).÷ Numbers in the task do not suggest any obvious ad hoc or

mental strategy (as would be present for example in a case such as 4006416÷). PARCC Tasks do not have a context. PARCC Only the answer is required. PARCC Tasks have five-digit dividends and two-digit divisors, with or without remainders. PARCC

• Students use all of their previously learned division strategies to help them divide multi-digit numbers using the standard algorithm and select and use appropriate tools strategically. They fluently connect the computation used with the context of the problem and apply estimation strategies and assess for reasonableness. (CC.6.NS.2)

• It is also crucial that students are NOT taught only how to procedurally multiply and to divide fractions. Cute rhymes to teach division of fractions such as “Ours is not to reason why, just invert and multiply” may appear amusing, but are detrimental to the development of students’ mathematical reasoning. Procedures and concepts complement each other and should be taught hand-in-hand. Previous multiplication and division provide considerable resources including: 1) simple primitive meanings of a × b as a b times (b is the multiplier of the multiplicand a)


and a ÷ b as a divided into b equipartitions (b is the divisor of the dividend a), 2) established inverses of division and multiplication, 3) three models, and 4) properties to apply and extend. (CC.6.NS.2)

• To extend to multiplication and division by fractions, students recall that multiplication and division, as operators, start with one number and produce another one. This can be pictured as:

Figure 25

Likewise division can be viewed as:

Figure 26

(CC.6.NS.2)

• Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation (MA10-GR.6-S.1-GLE.2-EO.b) o Tasks do not have a context. PARCC o Only the difference is required. PARCC o Prompts do not include visual models. The answer sought is a number, not a picture. PARCC o The given subtrahend and minuend are such as to require an efficient/standard algorithm (e.g., 177.3-72.635). The subtrahend and minuend do not suggest any

obvious ad hoc or mental strategy (as would be present for example in a case such as 20.5-3.50). PARCC o The subtrahend and minuend are each greater than or equal to 0.001 and less than or equal to 99.999. Positive differences only. PARCC o The given dividend and divisor are such as to require an efficient/standard algorithm (e.g., 177.3 ÷ 0.36). The dividend and divisor do not suggest any obvious ad hoc

or mental strategy. PARCC o Problems are effectively 4-digit divided by 2-digit or 3-digit÷3-digit. (For example, 14.28 ÷ 0.68 or 2.394 ÷ 0.684.) PARCC o Every quotient is a whole number or a decimal terminating at the tenths, hundredths, or thousandths place. PARCC

Students discover that solving division problems involving either a decimal dividend or decimal divisor (or both) can always be translated to an equivalent division problem where the divisor is a whole number. For example, the quotient of 354.6 ÷ 2.34 is equivalent to the quotient 35,460 ÷ 234. Students learn to work problems out to a specified or desired number of decimal places, rounding when appropriate. (CC.6.NS.3)

Students notice that for multiplication problems involving one or more decimal numbers, the total number of decimal places in the product is equivalent to



the sum of the decimal places in each factor. For example, the product of 6.79 • 12.105 would have 5 (2+3) decimal places. Therefore, students solve this problem by finding the product of 679 • 12,105 (= 8,219,295), and then move the decimal point five places to the left to get 82.19295. (CC.6.NS.3)

One misconception that may arise is in finding the product of 4.2 • 7.5 students will multiply 42 by 75 to get 3,150. When they move the decimal place, some students may decide that the 0 does not count once it is on the right side of the decimal point because it does not indicate any additional value. Therefore, those students may claim the answer is 3.15 because it must have two decimal places; however, this behavior is easily curbed when they consider the values of the factors in the problem, which should yield a product around 30. (CC.6.NS.3)

Another misconception can arise if students move the decimal from the front of the number to the right. It is important to remind these students that the answer must be consistent with the result of multiplying the fractions. (CC.6.NS.3)

• Understand positive and negative numbers are used together to describe quantities having opposite directions or values (MA10-GR.6-S.1-GLE.3-EO.a) o Tasks do not require students to perform any computations. PARCC o Students may be asked to recognize the meaning of 0 in the situation, but will not be asked to explain. PARCC

Students recognize that numbers have two components, a magnitude and a direction. For example, the number 3, on the horizontal number line, is located at a distance of three units (magnitude) to the right of zero (direction). The plus sign (+) can be used to signify positive numbers; when there is no sign indicated explicitly with a number, the number (with the exception of zero) is assumed to be positive. (CC.6.NS.5)

Students extend the horizontal number line to the left of zero and use negative numbers to describe quantities that have a certain magnitude and a direction to the left of zero. The negative sign (- or –) is used to signify the negative number. They learn to write a negative sign in front of numbers to differentiate negative numbers from positive numbers and read, for example, –3 as “negative three.” Students know conventionally that negative numbers are “below 0,” especially in a vertical number line (such as on a thermometer). (CC.6.NS.5)

Students should avoid reading -3 as “minus three” because this may be confused with subtraction.

• Use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation (MA10-GR.6-S.1-GLE.3-EO.a.i) o Students relate negative numbers to everyday experiences and explore what negative, positive, and zero mean in their examples. Students might mention

temperature, elevation (relative to sea level), money, above and below par in golf, gains and losses, and time (before and after a given starting point). In the case of elevation, they recognize that 0 represents sea level, positive elevations refer to heights above sea level, and negative elevations refer to heights or depths below sea level. (CC.6.NS.5)

o Students know that if positive and negative whole numbers and zero are called integers, and, conversely, that the integers are the set of all positive and negative numbers and zero. They learn that the term rational numbers is used to describe the set of all positive and negative fractions (including whole numbers), and that therefore, the integers are a subset of the set of rational numbers. (CC.6.NS.5)

o Students may exhibit a misconception as they extend inequalities to non-integer rational numbers. If asked to plot x = –1.5, many will correctly locate –1 as one unit to the left of zero, but then move .5 units to the right to reach –1.5, thus mistaking the point –.5 for –1.5. It is important for them to understand to separate the



magnitude of a negative number from its direction relative to 0, hence to go 1.5 units to the left of zero to locate –1.5. (CC.6.NS.5) • Understand a rational number as a point on the number line (MA10-GR.6-S.1-GLE.3-EO.b.i)

o Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. PARCC

o Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g.,-(-3)=3, and that 0 is its own opposite. PARCC

• Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself (MA10-GR.6-S.1-GLE.3-EO.b.ii, b.iii)

o Students learn that two rational numbers that have the same magnitude but have opposite signs are called opposite numbers (e.g., 5 and –5). They make observations about opposite numbers and their locations on the number line. They may notice that:

• The number line is like a mirror and opposite numbers “reflect about 0.” That is, opposite points are equidistant from 0, and they have the same magnitude, but opposite signs.

• The sum of two opposite numbers is 0. • 0 is its own opposite and is neither a positive nor a negative number. • The opposite of a negative number is positive (e.g., the opposite of –10.5 is 10.5). Students learn to code this as −(−10.5) = 10.5 and read it as

either, “The negative of negative 10.5 equals 10.5,” or “The opposite of the opposite of 10.5 equals 10.5. (CC.6.NS.6a) • Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane (MA10-GR.6-S.1-GLE.3-EO.b.iv)

o Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. PARCC

o Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane. PARCC Students extend the coordinate plane to all four quadrants by extending the x- and y-axes of the quadrant I coordinate plane (see Standard 5.G.1 earlier in

this LT) and by constructing two perpendicular number lines (that include all rational numbers) that intersect at the 0 point of each number line. The quadrants are labeled quadrants I, II, III, and IV as shown in the figure below, starting with the quadrant containing pairs of positive numbers. Points located on the x-axis or the y-axis do not belong to any of the four quadrants. (CC.6.NS.6b)



Students learn that when plotting coordinate pairs, a negative x-value shows how far left from the origin to move and a negative y-value shows how far down from the origin to move. (CC.6.NS.6b)

As they may do when working in the 1st quadrant, students may reverse the x-values with the y-values when plotting or reading coordinates. Another common student misconception is to believe that a single number is sufficient to designate a point located on one of the axes. For example, for point (0,−8), on the y-axis, students may just write (−8), ignoring the x-value. (CC.6.NS.6b)

• Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes (MA10-GR.6-S.1-GLE.3-EO.b.iv) o Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. PARCC

Research showed that students may also encounter difficulty in recognizing they-axis as a vertical number line. For example, when working in the 2nd quadrant, students may not recognize that only the x-coordinate is negative, not both coordinates. To avoid this misconception, students can be asked, “What is true and in common for all coordinate points in the 1st quadrant? The 2nd quadrant? The 3rdquadrant? The 4th quadrant?” They observe that:

• Each coordinate points in the 1st quadrant has a positive x- and y-coordinates. • Each coordinate point in the 2nd quadrant has a negative x-coordinate and a positive y-coordinate. • Each coordinate point in the 3rd quadrant has negative x- and y-coordinates. • Each coordinate point in the 4th quadrant has a positive x-coordinate and a negative y-coordinate. (CC.6.NS.6b)

• Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane (MA10-GR.6-S.1-GLE.3-EO.b.v, b.vi)

o Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. Find and position integers and other rational numbers on a horizontal or vertical number line diagram. PARCC

Students extend their understanding from Standard 6.NS.6.a in the Integers, Number Lines, and Coordinate Planes LT to be able to plot any rational number on a number line (both a horizontal and vertical number line). They can also work in reverse by naming the location of plotted points. (CC.6.NS.6c)

Students can be asked to order mixed sets of numbers on the number line, such as -3, 5, ¾, 1.2, -.8, -6.1, .5, -2/5, -7/4. Coordinating fractions, mixed numbers, decimals and integers, all of which are members of the set of rational numbers, is an important skill for students to learn. (CC.6.NS.6c)

A second important skill in ordering rational numbers is to be able to name a number between any two arbitrarily close rational numbers. This develops their sense that between any two numbers there is an infinite number of rational numbers. Students may be asked to write a number between, for example, -1.99 and -2. They should be able to visualize how to add a set of equipartitions between these two numbers and name the numbers that lie between the two given numbers, as in the figure below. (CC.6.NS.6c)

• Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram (MA10-GR.6-S.1-GLE.3-EO.c.i)

o Understand ordering and absolute value of rational numbers. PARCC


o Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret −3 > −7 as a statement that −3 is located to the right of −7 on a number line oriented from left to right. PARCC

Students compare integers, including positive to positive, negative to positive, and negative to negative. They learn that a smaller number is positioned to the left of a larger number and that a larger number is positioned to the right of a smaller number on the horizontal number line. They apply this idea to interpret and write statements of inequality involving two integers. (CC.6.NS.7a)

Students often express a misconception, when comparing two negative integers, that the negative number with the larger magnitude is the greater number. They avoid the misconception that, for example –5 is greater than –4 by plotting negative numbers on a number line. They also can relate –5 and –4 to a context [6]. For example, someone who owes 5 dollars is “more” in debt than someone who owes 4 dollars and, hence, has “less” money, so –5 is less than –4. They can state that although the magnitude of 5 is greater than the magnitude of 4, (hence –5 is further from zero than –4), the fact that –5 is to the left of –4 makes –5 < –4 the correctly stated inequality. (CC.6.NS.7a)

Students develop a mental model of the number line expressed by locating numbers on a number line, such as locating –4.5 and –2 on a number as

shown in the figure below.

They observe that since –4.5 is positioned to the left of –2, –4.5 < –2 and –2 > –4.5. (CC.6.NS.7a)

Students are also asked whether a given inequality is a true or false statement, and use the number line to answer. If the inequality is false, students make changes to make the inequality true. For example, students determine that –10 > 5 is false, and correct the inequality to read –10 < 5. (CC.6.NS.7a)

Finally, students can be challenged to make generalizations about inequalities involving positive and negative numbers. For example, they may observe (including reasoning with a number line diagram) that: 1. A positive number is always larger than a negative number, regardless of their magnitudes. Likewise, a negative number is always less than a

positive number, regardless of their magnitudes. 2. If a and b are positive numbers and a < b, then –a > –b (NOT –a < –b). 3. If a and b are negative numbers and a < b, then –a > –b. (CC.6.NS.7a)

• Write, interpret, and explain statements of order for rational numbers in real-world contexts (MA10-GR.6-S.1-GLE.3-EO.c.ii) o Understand ordering and absolute value of rational numbers. PARCC o Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write −3C° >−7C° to express the fact that −3C° is warmer

than −7C°. PARCC o Tasks are not limited to integers. PARCC o Tasks do not require students to explain. PARCC

Students extend Standard 6.NS.7.a of the Integers, Number Lines and Coordinate Planes LT to real world contexts. (CC.6.NS.7b) For example, given the scenario: “Suppose that at the end of 2011, the town of Bleakville’s bank balance is at -$607.89, and the town of Aphasia’s bank


http://turnonccmath.net/ajax.php?key=integers-numberlines-coordinateplanes&fmt=html%23_ENREF_6

balance is at -$457.23,” students are asked to write two inequalities describing the scenario and to interpret the inequalities in context. Students write that -$607.89 < -$457.23 or that $-457.23 >-$607.89. They explain that the inequality “-$607.89 is less than -$457.23” can be interpreted as, “The town of Bleakville owes more money than the town of Aphasia” or, “The town of Aphasia owes less money than the town of Bleakville.” (CC.6.NS.7b)

• Understand the absolute value of a rational number as its distance from 0 on the number line (MA10-GR.6-S.1-GLE.3-EO.c.iii) o Students learn that the absolute value of a number a is defined as the distance between a and 0 on the number line and is written |a|. They observe that the

absolute value of a non-negative number is the number itself (sometimes written |a| = a for a 0) and the absolute value of a negative number is its opposite (sometimes written |a| = –a for a < 0). For example, |5| = 5 and |–7| = 7. (CC.6.NS.7c)

o Students express four common misconceptions: 1. “The absolute value of a number is the number without its sign.” This states a procedure but fails to demonstrate an understanding of

the meaning of absolute valu. (CC.6.NS.7c) 2. “The absolute value is the positive value of an expression.” This leads to problems later when students solve equations and

overgeneralize that the solution to an equation involving absolute value is always positive. (CC.6.NS.7c) 3. “Absolute value means the answer is always positive.” This is an incomplete statement that can lead to students believing that |a| = –

a is never true (it IS true only when a < 0). (CC.6.NS.7c) 4. “If a 

|8|.(CC.6.NS.7c) • Interpret absolute value as magnitude for a positive or negative quantity in a real-world situation (MA10-GR.6-S.1-GLE.3-EO.c.iii)

o Students use the idea that the absolute value of a number is its distance from 0 to interpret absolute value in real world contexts. For example, students are asked to determine the difference in altitude between Long Beach, California, with an elevation of 7 feet below sea level, and Denver, Colorado, which lies 5100 feet above sea level. They reason that because Long Beach is a distance of 7 feet from sea level (the 0 point) and Denver is 5100 feet from 0 that the difference in altitude is 5107 feet or |–7| + |5100|. Some students may write that the difference in altitude is |5100 – (–7)| or |(–7) – 5100|. Students are challenged to explain why these three number sentences produce the same result, and are all correct ways to find the difference in altitude. (CC.6.NS.7c)

• Distinguish comparisons of absolute value from statements about order (MA10-GR.6-S.1-GLE.3-EO.c.iv) o Students distinguish comparisons of absolute value from statements about order by constructing a table and seeing how taking the absolute value of numbers in an

inequality affects the inequality. In the table below, the signs of a and b are given in the first two columns, and a true inequality about a and b is given in the third column. Students are asked to determine if the absolute value inequality in the fourth column is always true, always false, or sometimes true and sometimes false (and to give the relevant conditions) by filling in the fifth column. The first three rows of the table are completed below. (CC.6.NS.7d)

a b Inequality Absolute Value Statement

T or F? Conditions?

Positive Positive a b |a| > |b| ? Always True Negative Positive a |a| Positive Negative a > b |a| > |b| ? Negative Negative a b |a| > |b| ?

(CC.6.NS.7d)

• Students provide justification for absolute value statements that they think are always true or false, and counterexamples for absolute value statements that they think are sometimes false. For example, for the third row students give the counterexample a = –20, b = 5 since –20 < 5 but |–20| > |5|. They reason that the third row is only true when b is greater than the magnitude of a, that is, if b > |a|.(CC.6.NS.7d)

• Students are also challenged to use the number line and the concept of absolute value to distinguish between accumulated changes in values and net change, for instance to distinguish between the distance traveled in a sequence of steps, and the net change those steps represent. For instance, if a courier makes several deliveries along a street by traveling east and west, and she travels, in order, 2 blocks west, then 5 blocks east, then another 3 blocks east, and finally 4 blocks west to make her final delivery, students represent these deliveries as directed distances on a number line, determine how far she traveled (the sum of the absolute values of the distances), and determine the net change in position from her starting point. (CC.6.NS.7d)




Every rational number is a point on a number line and has an opposite number that is equal distance from zero on the number line and can compared with other rational numbers.

Academic Vocabulary: Fluency, multi-digit numbers, reasonableness, decimal numbers, coordinate axes, opposite, opposite of the opposite, positive and negative numbers, direction, vertical, horizontal, location, interpret, compare,

Technical Vocabulary: operations, standard algorithm, estimation, quantities, values, magnitude, number lines, rational numbers, inequalities, number line diagrams, absolute value, ordered pairs, reflection, axes, order


Unit Title Rockin’ Ratios Length of Unit 8 weeks

Focusing Lens(es) Interpretation Relationships


MA10-GR.6-S.1-GLE.1 MA10-GR.6-S.1-GLE.2 MA10-GR.6-S.2-GLE.2


• At 10% simple interest, which will earn you more money investing $1000 all at once or $100 a month for one year? Two years? (MA10-GR.6-S.1-GLE.1-EO.c.vi)

Unit Strands Ratios and Proportional Relationships, The Number System, Expressions and Equations, Personal Financial Literacy

Concepts Ratios, rates, relationships, quantity, multiplicative, ratio tables, graphs, percents, (greatest) common factor, (least) common multiple, comparison, additive, factor, multiple, constant, unit ratio, proportion, comparison, speed, equivalence, coordinate grid, measurement, conversion, division, fractions, quotient, product, sum, distributive property, operators, fair sharing, rate, measurement, scaling, array, area, equal groups, number line

Generalizations



Ratios represent a constant multiplicative relationship between two quantities. (MA10-GR.6-S.1-GLE.1-EO.a)

What differentiates fractions from ratios? What type of relationship does a ratio describe? What are two ways to compare quantities? What types of actions on a ratio preserve the

multiplicative relationship?

Why can the relationship between two quantities be the same (constant) even though the amount of each of the two quantities gets larger or smaller?

Why would you compare quantities as ratios rather than as a difference?

Ratios typically express a relationship between two quantities with different units. (MA10-GR.6-S.1-GLE.1-EO.a, c.i, c.ii, c.viii)

How can ratio reasoning be used to convert measurements?

What is a ratio table and how can you use it to make a graph?

How ratio relationships be represented in multiple ways?

Unit rates operate as single quantities. (MA10-GR.6-S.1-GLE.1-EO.b, c.ii, c.vi, c.vii)

What is a unit rate? What are methods for finding a unit rate? How does your choice of unit rate to calculate depend

on the situation?

Why is it important to focus on the units for each part of a unit rate?

How does a unit rate apply to saving and investing? Why are unit rates useful when solving word problems?



Real world problems for division of fractions often involve contexts such as fair sharing, rate, measurement, scaling, array, area, and equal groups. (MA10-GR.6-S.1-GLE.2-EO.f, g, h)

What are contexts involving division of fractions? How does interpreting division of fraction problems within a context lead to multiple solution strategies?

The distributive property uses the greatest common factor (GCF) to express the sums of whole numbers as multiple of a sum with no common factor (MA10-GR.6-S.1-GLE.2-EO.c, e)

How can a Venn diagram of the prime factorization of two numbers provides a means for easily finding the GCF of two numbers?

Why might it be helpful to express the sum of two numbers as the multiple of a sum with no common factors?

The least common multiple (LCM) provides an efficient means to add and subtract fractions with unlike denominators (MA10-GR.6-S.1-GLE.2-EO.d)

How can you use LCM to create equivalent fractions and ratios?

How can a Venn diagram of the prime factorization of two numbers provides a means for easily finding the LCM of two numbers?

How does the LCM support the addition and subtraction of fractions?

Why does the product of the LCM and GCF equal the product of the original two numbers?

As with unit rates, percents require a standardized value for one of the compared quantities. (MA10-GR.6-S.1-GLE.1-EO.c.iv, c.v)

How can you use a ratio table to solve percent problems How is a percent a ratio?

Why is it possible to have a percent greater than 100? Why is the second quantity in a percent standardized to

100?

Number lines express quantities with a referent unit of one and cannot express the quantitative relationships of ratios and percents. (MA10-GR.6-S.1-GLE.1-EO.a)

How are ratios different from fractions? (MA10-GR.6-S.1-GLE.1-IQ.1)

What does it mean that fractions and decimals refer to a single amount of a quantity?

What is the difference between quantity and number? (MA10-GR.6-S.1-GLE.1-IQ.2)

Why do ratios and percents not refer to single amounts of a quantity?

Why is it misleading to say 75% = 0.75 or that 3:4 is always the same as the fraction ¾?

Why do fractions, and not ratios, imply a referent of one?





• Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities (MA10-GR.6-S.1-GLE.1-EO.a) o Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS p 42.) The initial numerator and denominator should be

whole numbers. PARCC In sixth grade, students begin to formalize ratio as a particular kind of comparison between two quantities. A ratio is a comparison between two values of

those quantities (numbers) that does not change if the values are varied multiplicatively by the same factor. Thus, 6 : 8 is a ratio because 6/8 = (6 * 2)/(8 * 2) = 12/16: the relationship between 6 and 8 is the same as the relationship between 12 and 16. Varying the values multiplicatively by the same factor is said to “preserve the ratio.” It follows that the ratio is also preserved if both numbers are divided or “split” by the same number. Thus, 6/8 = (6 ÷ 2)/(8 ÷ 2) = 3/4. The relationship between the pair of numbers does not change as its values are varied multiplicatively. (CC.RP.1)

What is fundamentally different and new for students is the idea that the amount of two quantities can be in the same relationship (and hence the relationship be referred to as “equal”) even though the amount of each of the two quantities gets larger or smaller. The understanding that equal means “preserving the particular kind of relationship” between the values is an important cognitive achievement for students. (CC.RP.1)

Too often, students experience difficulty with ratios due to a failure to recognize that when they write a ratio as a : b, they must clearly delineate the order of the quantities and represent their units (i.e., distance (miles) : time (hours)). This eliminates ambiguity later when discussing unit ratios and rates. (CC.RP.1)

A common misconception can be demonstrated using the number pairs (2, 3) and (5, 6). The comparison between the numbers based on subtraction is the same (they both differ by one: 2 + 1 = 3 and 5 + 1 = 6) but 2/3 ≠ 5/6. In context, this misconception could occur if a student reasons that if 2 dogs eat 3 bones, and if there were 3 more dogs (for 5 dogs total), then there should be 3 more bones (for 6 total bones). This is a misconception because it assumes that the ratio is preserved across additive, rather than multiplicative, changes to each of the two quantities. That is, 2/3 ≠ (2 + 3)/(3 + 3). (CC.6.RP.1)

This new idea of equality, namely, that a relationship can stay the same, even when the values change by a common factor, differs from the related statement about fractions. For fractions, the claim that 3/4 = 6/8 is a claim that a region can be partitioned further to show an equivalent way of naming it. In this case, the region of the whole remains the same, with finer equipartitions. Similarly, the points 6/8 and 3/4 lie on the same point on a number line. What differentiates fractions from ratios is that when given two fractions, one can assume that they share a common unit of one, a referent unit. With ratios on the other hand, the two quantities represented by the values typically refer to different referent units, objects, numbers, qualities, or measurable things. (CC.RP.1)

• Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship (MA10-GR.6-S.1-GLE.1-EO.b) o Understand the concept of a unit rate a/b associated with a ratio a:b with b≠0, and use rate language in the context of a ratio relationship. For example, “This

recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” PARCC

A unit rate is a unit ratio that can change as a single quantity, in order to reflect changes in a situation or condition. For instance, the unit price for a food item in a grocery store--a unit rate—can increase or decrease. The per-pound cost of pecans can increase, for example, from $7.25 : 1 pound to $8.14 : 1 pound. (CC.6.RP.2)

Note to teachers (1): Ratio can be described as a comparison that is invariant across a set of proportions (as discussed above, a/b = na/nb, for instance). A unit ratio is one of the equivalent proportions for a ratio, in which one of values is 1. For a given situation or condition, especially when considering



equivalence relationships, ratios (and therefore unit ratios) are constant—they do not change. Some unit ratios are unique because they characterize particular objects or phenomena. For example, the ratio of the perimeter of a square to the length of a single side is 4 : 1 and cannot be increased or decreased; π is the ratio of the circumference of a circle to its diameter (for every one unit increase in a circle’s diameter, the circle’s circumference increases by π units). (CC.6.RP.2)

Note to teachers (2): Unit rates, on the other hand, can change, and are used extensively in mathematically describing certain kinds of change. They are unit ratios (one of the values is 1), and they can be used in describing equivalence among proportions. For example, in the unit pricing example above, at any single unit rate, the same rate applies to any amount of pecans (the relationship between pounds of pecans and price is the same for a given unit price). But they also can be incremented or decremented (increased or decreased). In the unit price example above, the change in the unit price represents the establishment of a new multiplicative relationship between the quantities of price and pounds of pecans. In the CCSS-M, no distinction is made between unit rates and unit ratios. In these descriptors, unit rate is used intentionally as a unit ratio that can change as a single quantity. These descriptors explore the understanding of—and the relationships among—the concepts of ratio, ratio unit, unit ratio, and unit rate, because together they provide a more robust foundation for proportional reasoning, slope, and functions. (CC.6.RP.2)

Students find unit ratios (sometimes known as “per one” ratios) useful because they make solutions to many word problems easier. For example, if students are given the fact that: “3 pounds of tuna contain 210 micrograms of mercury,” students might construct the following ratio table:

Pounds of Tuna Micrograms of Mercury

3 210 1 70

1/70 1 (CC.6.RP.2)

Students find it helpful to include both unit ratios in their tables. In this example, it allows them to assert that there are 70 micrograms of mercury per 1 pound of tuna or that there is 1 microgram of mercury per 1/70 pound of tuna. (CC.6.RP.2)

Having flexibility in describing unit ratios helps students to solve problems such as, “Over a year’s time, people in Iceland consume 1.6 pounds of tuna per week. How much mercury do they consume in a year?” The total amount of mercury per year could be calculated by a simple product of 52 * 1.6 pounds of tuna * 70 micrograms of mercury. Another question to ask students could be, “If eating over 2160 micrograms of mercury per year is dangerous, what is the maximum number of pounds of tuna a person should eat in a year?” (CC.6.RP.2)

Therefore, unit rates help students interpret ratios as operators. If one knows the total micrograms of mercury, then the unit rate of 1/70 : 1 operates on the total number of micrograms to produce the number of pounds of tuna. If one knows the total pounds of tuna, then the unit rate of 70 : 1 operates on the total pounds of tuna to produce the micrograms of mercury. (CC.6.RP.2)

• 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. (MA10-GR.6-S.1-GLE.1-EO.c) o The testing interface can provide students with a calculation aid of the specified kind for these tasks. PARCC o Expectations for ratios in this grade are limited to ratios of non-complex fractions. PARCC



o The initial numerator and denominator should be whole numbers. PARCC • Make tables of 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, equivalent ratios relating quantities with whole number measurements and plot the pairs of values on the coordinate plane. (MA10-GR.6-S.1-GLE.1-EO.c.i)

o Students use a number of strategies to represent equivalent ratios relating quantities with whole-number measurements. Consider the following example: Suppose that to maintain a healthy lifestyle, a person needs to use 18 gallons of water every 4 days. How many gallons of water are needed every week? (CC.6.RP.3a)

o Students build a ratio table, order the values in the table from high to low, and graph it. (CC.6.RP.3a)

Gallons of Water Days

36 8

18 4

9 2

9/2 1

In order to predict how many gallons of water are needed in 7 days, students can use a variety of strategies: 1. Build up from the ratio table (for instance 8 – 1 = 7 and 36 – 9/2 = 63/2, giving the ordered pair (7, 63/2)). 2. Use the unit ratio as an operator, and multiply (9/2 gallons per day* 7 days =63/2 gallons). 3. Use a graph to predict the values. 4. Use a ratio box created from the ratio table.

• For any two pairs of rows (x1, y1) and (x2, y2) in the table, given three of the values, students can predict the fourth value. They know how to use multiplication and division to describe both of the correspondence relationships of x1 to y1, and x2 to y2. They also know that the relationships are identical. That is, the correspondence between x1 and y1 is the same as the correspondence between x2 and y2 . Likewise, they know how to use multiplication and division to describe the covariation relationships of x1 to x2, and y1 to y2. They also know that these relationships are identical as well, but not necessarily the same factors as in the correspondence set. (CC.6.RP.3a)

o For example, students use the ratio box below to predict how much water is needed every 7 days:



In the ratio box above, students recognize that 4 x 9 = 36 and 36/2 = 18. Applying this same sequence to the number 7 produces 7 x 9 = 63 and 63/2 = 31.5. That is, each week, a person needs 31.5 gallons of water to be healthy. The problem could have also been solved by multiplying 18 by 7 and then dividing by 4. (CC.6.RP.3a)

Conventionally, this approach is often written as

An advantage of the ratio box is that it does not preference numerators or denominators in equating or comparing

ratios. Using a ratio box, it is easy for students to observe the horizontal and vertical multiplicative relationships. Furthermore, for this Standard, there is no expectation for students to solve this as an equation. (CC.6.RP.3a)

• Find missing values in tables of equivalent ratios and use the tables to compare ratios. (MA10-GR.6-S.1-GLE.1-EO.c.i, c.ii) • This standard asks students to compare ratios using tables, but they will also use graphs for this purpose. For example, suppose Luis mixed 6 ounces of cherry syrup

with 48 ounces of water to make a cherry-flavored drink. Martin mixed 8 ounces of the same cherry syrup with 60 ounces of water. Students are asked: “Who made the drink with the stronger cherry flavor?” (CC.6.RP.3a)

Luis’s Drink

Cherry Syrup (Ounces) Water (ounces)

24 192

6 48

3 24

1 8



Martin’s Drink

Cherry Syrup (Ounces) Water (ounces)

24 180

8 60

4 30

2 15

By this time, students have many strategies and resources for solving this problem. Some of these are listed below. 1. Students double the (1, 8) in Luis’s drink to obtain (2, 16) and compare this with (2, 15) in Martin’s drink to draw the conclusion that Luis’s

drink is more “watery.” This strategy allows students to compare two ratios by finding rows that contain a common value for one of the quantities, or extending the values in the table to create a common value.

2. Students halve the (2, 15) in Martin’s drink to obtain the unit ratio (1, 15/2) and compare it to (1, 8) drawing the conclusion that Luis’s drink is more “watery.” This strategy represents the use of a unit ratio or rate.

3. Students identify 24 as the least common multiple of 6 and 8, and build up the ratio tables appropriately to compare and conclude that Martin’s drink is slightly weaker than Luis’s drink.

4. Students graph the two tables and visually observe that the ray representing Luis’s mixture is “rotated” so that the mixture is more watery than the ray that represents Martin’s mixture.

5. Also using the graph, students can find the common value of one quantity and compare the associated values of the other either horizontally or vertically as in the figure below.



6. Students convert the given ratio to a unit ratio or rate by dividing the ounces of water by the ounces of cherry syrup to produce a decimal and

interpreting this as 7.5 oz. of water to 1 oz. of cherry syrup in Martin’s mixture, compared to 8 oz. of water to 1 oz. of cherry syrup in Luis’s mixture. The division can also be done the other way. (That is, dividing the ounces of cherry syrup by the ounces of water produces a decimal representing the number of ounces of cherry syrup to 1 ounce of water, i.e., 0.133… compared to 0.125). (CC.6.RP.3a)

• Solve unit rate problems including those involving unit pricing and constant speed. (MA10-GR.6-S.1-GLE.1-EO.c.iii) o The testing interface can provide students with a calculation aid of the specified kind for these tasks. PARCC o Expectations for unit rates in this grade are limited to non-complex fractions. PARCC o The initial numerator and denominator should be whole numbers. PARCC o Note to teachers: Students’ understanding and use of ratio units prior to focusing on unit ratios (unit rates) strengthen their ratio reasoning by preventing bias

toward only the per-one versions of ratios, and sets up a strong foundation for the use of slope triangles and slope in later grades. PARCC Students use unit rates to help them solve rate problems. The most common problems involve shopping or travel. For example, suppose students are

informed that it takes Tasha 20 minutes to ride 8 miles on her scooter and are then asked to determine how long it would take her to ride 100 miles at this rate. To solve this problem, students could construct a ratio table starting with (8, 20) and perform two 2-splits to obtain the values (4, 10) and (2, 5) as in the ratio table below. (CC.6.RP.3b)

Distance (in miles) Time (in minutes) 8 20 4 10 2 5

They then build up by using the ratio unit (2, 5) and multiply each coordinate by 50 to obtain (100, 250) and conclude that it will Tasha 250 minutes

to ride 100 miles. (CC.6.RP.3b)



Alternatively, students have been observed adding new rows to a ratio table to create a new value, which is a new strategy. This new strategy should be justified by considering how it can be represented on the graph. This strategy could produce (10, 25) by adding the first (8, 20) and last (2, 5) pairs of values in the above ratio table. Then, they would multiply each coordinate by 10 to obtain (100, 250). (CC.6.RP.3b)

Another approach would be to extend the table to create the associated unit ratios of 1 mile : 5/2 minutes or 2/5 miles : 1 minute. Students could multiply the coordinates for the first unit ratio by 100 to obtain the result of (100, 250). They could also use the operator construct to obtain that 100 miles * 5/2 minutes per mile yields 250 minutes. (CC.6.RP.3b)

• Find a percent of a quantity as a rate per 100 and solve problems involving finding the whole, given a part and the percent. (MA10-GR.6-S.1-GLE.1-EO.c.iv, c.v) o The testing interface can provide students with a calculation aid of the specified kind for these tasks. PARCC o Pool should contain tasks with and without contexts PARCC o Expectations for ratios in this grade are limited to ratios of non-complex fractions. (See footnote, CCSS p 42.) The initial numerator and denominator should be

whole numbers. PARCC Percent involves two equivalent proportions for the same ratio, with one of the proportions indexed to the value 100. Another, somewhat more formal

way to describe a percent is as a way to describe a ratio of one value of a quantity to another value of the same quantity, with the second value standardized to 100. (CC.6.RP.3c)

• Thus, percent is a rate per one hundred (recall that “cent” is a Latin term for 100 as in century, cents, centipede). Just as a unit rate requires a standardized value of 1 for one of the quantities, percents have a standardized value of 100 in the second value of the quantity, such that 50 : 100 describes half of a whole, and 100 : 100 describes the whole. This facilitates an easy way to compare ratios: thus, 30% is a ratio of 30 : 100, 45% is a ratio of 45 : 100, and 45% > 30%. (CC.6.RP.3c)

o Students often carry the misconception that percents cannot exceed 100% or 100 : 100. It is important that they become comfortable with percents such as 200% and can simplify the ratio from 200 : 100 to 2 : 1. Likewise 150% implies a ratio of 1.5 : 1 and 325% implies a ratio of 3.25 : 1. (CC.6.RP.3c)

Most simple percent problems can be easily solved using a ratio box. For example, if 30% of students get less than 4 hours of sleep per night and there is a total of 250 students, how many students get less than 4 hours of sleep? (CC.6.RP.3c)

Note to teachers: In the ratio tables for percents, it can help students keep track of the values that represent percents if percent (%) and number (#) symbols are added to the appropriate cells of the ratio tables, as shown in the examples in this standard. (CC.6.RP.3c)

Students with less

than 4 hours of sleep per night

Total number of students

(%) 250 (%) 100 (#) x (#) 3.8



(CC.6.RP.3c) For this type of percentage problem, the headers will always include two different descriptions of the same underlying quantity, in this case, “students.”

Students describe the percentage (30%), the “whole” (250 students) and the “part of the whole” (x = 75). They can also be asked to generate problems in which the x is in the other positions (except for the cell with 100, because percent is always a rate per one hundred). For instance, the problem “If the number of students getting less than 4 hours of sleep is 60 out of 300, then what percent of students are getting less than four hours of sleep?” can be solved using the following ratio box:

(CC.6.RP.3c) Students can be challenged to find a situation in which the percentage exceeds 100. Usually, this is found in a growth relationship, such as

population projections. For example, if a country has a population of 3.8 million and the population grew 250%, what is the new population? (CC.6.RP.3c)



New Population (millions)

Old population (millions)

(%) 250 (%) 100 (#) x (#) 3.8

x = 9.5 million people Many students initially will have trouble putting the correct values in each column. They should be asked to reason qualitatively by looking at the

ratio table and its multiplicative relations and justify whether the unknown value will increase or decrease to assess the reasonableness of their responses. In the example above, they would initially reason that the new population should be more than 3.8 million. Therefore, the answer of 9.5 million people makes sense. (CC.6.RP.3c)

Another situation that students have trouble with is distinguishing between growth by a factor (multiplicative) and growing (incrementing) or decreasing (decrementing or discounting) by a percentage of an original amount. For example, they may be asked: “A prom dress costs $88. What would the dress cost if the price increased by 33%?” In terms of the population example above, this growth is equivalent to growing the original amount 133%. But in common usage (pricing, for instance), what is being set up in this example is a price increasing by 33% in addition to the original price. Ask students to consider whether an “increase” in price by 33% would result in a higher or lower price. This is an example where mathematics and language becomes slightly problematic because an “increase” by a percentage less than 100 would appear to “lower” the price, but conventionally we use it in English as increasing the price by a percentage of the original. There are two ways to solve this problem. The ratio box solution is given below. (CC.6.RP.3c)

Original price

(dollars) New price (dollars)

(%) 100 (%) 133 ($) 88 ($) x

x = $117.04 • Another way for students to reason about solving this problem is to add 33% of $88 to the original price ($88) giving $88 + .33($88) =

$117.04. For this, students could construct a ratio box to calculate the price increase, as shown below on the left, and then construct the summary table at the right to display how to obtain new price of the dress).

Original price

(dollars) Price increase

(dollars) New price

(dollars) (%) 100 (%) 33 (%) 100 + 33 ($) 88 ($) x ($) 88 + x x = $29.04 $88 + x = $117.04 (CC.6.RP.3c)



Another difficult problem for students to consider is what the original cost of a prom dress would have been if the price of $88 represents a 33%

discount (decrease) from the original price. Again, students must be very careful in constructing the ratio box. Many students will want to construct the ratio box with $100 in the “old price” column and $33 in the “new price” column. However, students must recognize that if a price is lowered by 33%, then the final price is one pays 67% (that is, 100% - 33%) of the original price. Therefore, if the old price was $100, the new price would be $67, not $33. Corresponding to this context, $88 is the new price while x in this example is the old price. The ratio box solution is shown below. (CC.6.RP.3c)

Original price

(dollars)

New price

(dollars) (%) 100 (%) 67 ($) x ($) 88

x = $131.34 (CC.6.RP.3c)

• For completeness, the ratio box and summary table approach, as in the previous example, is shown. In this instance, the summary table shows the price decrease. (CC.6.RP.3c)

Original price (dollars)

New price (dollars)

Price decrease (dollars)

(%) 100 (%) 67 (%) 33

($) x ($) 88 ($) x - 88

x = $131.34 $x - 88 = $33.34

Students should compare the two examples, and recognize that in the two problems, the 33% were taken relative to different values: in the first, there was a 33% price increase relative to the original price of $88 (33% of $88 is $29.04). This is different from the second example, in which a price discount of 33%, resulting in a final price of $88, represented 33% relative to an originally higher value of $131.34. (CC.6.RP.3c)

Another common way to represent percentage problems uses the concept of percentage (or ratio) as an operator. Similar to ratios, percents often work as operators. Standard 6.RP.2 and Bridging Standard 6.RPP.E in this LT dealing with ratios as operators are important to set up this idea. Also see Standard 6.RP.3.a in this LT where this idea is addressed as a way to solve a unit rate problem. (CC.6.RP.3c)

Consider again the scenario in which a country has a population of 3.8 million and the population grows by 250%. In solving this percentage problem using percent as an operator, students will typically learn to convert the percentage to a unit ratio or rate of p : 1 where p is the percent. Because this can be accomplished by dividing the original percent by 100, it appears that it is as simple as converting the percentage to a decimal. However, students must keep in mind that the percentage is actually being converted to a unit ratio or rate, so it can act as an operator. (CC.6.RP.3c)



• Note to Teachers: See also the end of this standard for a discussion of fractions versus decimals versus percents versus ratio). o Thus, the problem is solved by converting the percent to the unit ratio or rate 2.5 : 1 and multiplying it to the old population of 3.8.

Old population Percentages as operator New population

3.8 250% (250:100 or 2.5:1) x

x = 3.8 * 2.5 = 9.5 million people (CC.6.RP.3c)

Students solve these types of problems where the x is in any position, even the unit rate or ratio. For example, if a country’s population grows from 4.5 million to 5.7 million, what is the growth rate, expressed as a percent? (CC.6.RP.3c) • Confusion often surrounds the relationships among ratios, decimals, percents and fractions. For example, the following questions might be

posed to students: “Is a ratio of 3 : 4 equal to a fraction 3/4? Is a decimal .75 equal to 75%?”(CC.6.RP.3c) The answers to these questions depend on the way in which these terms (ratio, decimals, fraction, percent) are being used as numbers. Numbers

can be viewed as single amounts of a quantity, as locations on a number line, or as operators that can act on other quantities. (CC.6.RP.3c) Ratios and percents are not numbers that can be placed on a number line, nor are they single amounts of a quantity, because they do not imply a

referent unit (of one). Ratios and percents always describe some amount of something in relation to another amount (sometimes of the same thing, and sometimes of something different) and therefore cannot specify a particular amount by themselves. They are used as operators (in both multiplication and division) to produce numbers or amounts of a quantity as a result. (CC.6.RP.3c)

• In contrast, fractions and decimals as numbers can be located on a number line because they are defined in relation to a referent unit of one. Confusion arises because fractions and decimals can also be operators that act on other quantities. For example, in the problem, “A rectangle has one side equal to .75 inches and the other equal to 6 inches. What is its area?” uses the decimal length, .75 inch, as an operator. When fractions and decimals are used as operators, computationally, they behave the same way as ratios and percents. However, this does not imply that one could say that area was calculated by multiplying 75% * 6. (CC.6.RP.3c)

o Therefore, it is misleading to claim that 75% = .75 or to say that 3 : 4 is unequivocally the same as the fraction 3/4. It is permissible to say that in calculating 75% of 240, one can do the calculation .75 * 240 to get the correct result. (CC.6.RP.3c)

• Use ratio reasoning to convert measurement units and manipulate and transform units appropriately when multiplying or dividing quantities. (MA10-GR.6-S.1-GLE.1-EO.c.viii)

o Tasks require students to multiply and/or divide dimensioned quantities. PARCC o 50% of tasks require students to correctly express the units of the result. PARCC o The testing interface can provide students with a calculation aid of the specified kind for these tasks. PARCC o Expectations for ratios in this grade are limited to ratios of non-complex fractions. PARCC o The initial numerator and denominator should be whole numbers. PARCC

Students apply the strategies from the previous standard to solve conversion problems. For example, in converting from yards to feet, the following representations can be used: (CC.6.RP.3d)



x (yards)

y (feet)

1 3 2 6 3 9 4 12 5 15

(CC.6.RP.3d)

• It is important to note that these representations can also be used to convert feet to yards. A challenge can be presented to students by asking them to predict, for example, how many yards are in 5 feet. Graphs are good for estimation and can be used to create precise descriptions; however ratio boxes and tables are typically more efficient. Students observe from the graph that the number of yards in 5 feet is between 1 and 2. Then, students use a ratio table or ratio box to find that there are 5/3yards in 5 feet, as shown below:

(CC.6.RP.3d)

x (yards)

y (feet)

1 3 1/3 1 2/3 2 1 3

4/3 4 5/3 5



When students learn to fill in values of a data table or graph between two given values, it is called interpolation. When students learn to extend values of a data table or graph, it is called extrapolation. (CC.6.RP.3d)

• Interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions by using visual fraction models and equations to represent the problem. (MA10-GR.6-S.1-GLE.2-EO.f, g, h)

o Only the answer is required; explanations and representations are not assessed here. PARCC o Note that the italicized examples correspond to three meanings/uses of division: (1) equal sharing; (2) measurement; (3) unknown factor. These meanings/uses of

division should be sampled equally. PARCC o Tasks may involve fractions and mixed numbers but not decimals. PARCC

When faced with the problem of dividing two generalized, non-unit fractions, students can use the variety of strategies they have developed to simplify the problem. They write it as a complex fraction and then multiply the numerator and the denominator by the inverse of the denominator and simplify. (CC.NS.1)

(CC.NS.1) • While this procedure produces an effective way to solve the problems, they also can solve the problems in different contexts using a variety of

models. (CC.NS.1) The D/M box below can be used to explore division of fractions by fractions.

In the following D/M box below, students work on the division 1/4 ÷ 2/3. (CC.NS.1)

Figure 54 (CC.NS.1)

• When dividing 1/4 by 2/3, students apply the inverse relationship between division and multiplication to reason that dividing by 2/3 is the same as multiplying by 3/2. They recognize that finding the outcome of dividing 1/4 by 2/3 is equivalent to determine the outcome such that 2/3 of it is 1/4. This implies that 1/3 of the outcome equals to 1/2 of1/4 and thus the outcome is 3 times as large as 1/2 of 1/4, i.e., 1/4 × 1/2 × 3. Therefore,1/4 ÷ 2/3 = 1/4 × 3/2. (CC.NS.1)



There are three types of dividing fractions by fractions (1) Dividing two "normal" fractions, (2) Dividing a mixed number by a fraction, and (3) Dividing two mixed numbers. (CC.NS.1)

Division problems are classified in three models depending on their nature:

Model 1: Referent Transforming

Model 2: Referent Preserving

Model 3: Referent

Composing Fair Sharing (Partitive)

Measurement (Quotative) Arrays

Rate Scaling Area Equal Groups Cartesian Product

(CC.NS.1) Note that Arrays and Cartesian products are not viable models for interpreting division of fractions or mixed numbers in this standard. Referent Transforming (CC.NS.1)

1. A fair-sharing example might be: “Ally made 4/5 of a pound of trail mix to be shared among her friends. She places them into bags of each having 2/25 pounds for each person. How much trail mix will there be in each bag?” (Solution: 4/5pounds ÷ 2/25 pounds per person = 4/5 × 25/2 people = 100/10 people = 10 people)

2. An example of rate problems involving partitive division might be: “A long distance runner covers 2 3/4 miles in 2/5 hour. How many miles can the runner cover in 1 hour?” (Solution: 2 3/4 miles ÷ 2/5 = 9/4 ÷ 2/5 = 9/4 × 5/2 = 45/8 miles) (For rate problems involving unit rates, see Standard 6.RP.2 of the Ratio, Proportion, and Percents LT.

3. An example of Equal Groups problem involving division with transformation of units might be: “If a box of cereal holds 28 1/2 ounces, then how many servings will there be if each serving is 31/2 ounces?” (Solution: 28 1/2 ÷ 3 1/2 = (28 1/2 × 2) ÷ (3 1/2 × 2) = 8 1/7)

Referent Preserving (CC.NS.1) 1. A measurement example might be, “A whole ribbon measuring 3/4 foot in length is cut into 3/8 -foot long strips. How many strips can be made from

the whole ribbon?” (Solution: 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2) The whole ribbon is twice as long as the strip (An analysis of the units in this problem anticipates scale factors at 8th grade: inches ÷ inches = a scale factor)

2. Scaling problems involve the division of a quantity with a scale factor. An example of a scaling problem involving division might be: “After shrinking a square by a scale factor of 1/4, the length of the square is 12/15 feet long, what will the length of a side of a square before enlarging?” (Solution: 12/15 ÷ 1/4 =12/60 = 1/5 feet long)

3. An example of Equal Groups problem involving division with preservation of unit units might be: “A bottle of medicine contains 8 2/3 oz. You can have 12 1/2doses from this bottle. How many ounces are in each dose?” (Solution: 8 2/3 ÷121/2 = 26/3 ÷ 25/2= 26/3 × 2/25 = 52/75 ounces in each dose)

Referent Composing (CC.NS.1)



1. An example of an area problem involving division is: “Mr. Jones plans to build a rectangular garden outside of his house that will cover 2/3 of a square mile. One dimension of the garden area will be determined by a fence that is 3/4 of a mile long. What is the other dimension of the garden area?” (Solution: 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9)

A common way to approach division by fractions is by using the “invert and multiply rule”. However, this procedure is often taught without context. (CC.NS.1)

Multiplication and division of decimals are unpacked in the Place Value and Decimals LT (see Standards 4.NF.6, 5.NBT.1, 5.NBT.2, 5.NBT.3.a, and 5.NBT.7) (CC.NS.1)

• 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. (MA10-GR.6-S.1-GLE.2-EO.d)

o Tasks do not have a context. PARCC o Tasks require students to find the greatest common factor or the least common multiple only. PARCC

Students are introduced to the greatest common factor (GCF) and the least common multiple (LCM), using contextual examples. (CC.NS.4) • The GCF is the largest factor that divides into any set of numbers. • The LCM is the smallest multiple of a set of numbers.

o For example, a student is asked to clap every fourth beat. Another student is asked to clap every sixth beat. The class is asked to observe and determine when the two students clap at the same time. The answer (12) is the least common multiple (LCM) of 4 and 6. Then, the class is asked how often a third student would have to clap every time each of the other two students claps (without clapping every single beat). The answer (2) is a common factor of 4 and 6. If this is the largest common factor, the number is the GCF, or Greatest Common Factor. (CC.NS.4)

(CC.NS.4)

• Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. (MA10-GR.6-S.1-GLE.2-EO.e)

Venn diagrams are effective ways to display GCF and LCM. Each of the circles of the Venn diagram represents a number, and contains the prime factorization of that number. In the intersection of any two circles, the common factors are listed once, and the rest of the factors are placed in the non-intersecting portion of the circle. The product of the factors in the intersection gives the GCF. The product of all the factors on the diagram, known as the “union” of all the factors, gives the LCM. (CC.NS.4)



Figure 20 (CC.NS.4)

• Students can be asked to prove that the product of the LCM and GCF equals the product of the original two numbers. The Venn diagram is a powerful tool in proving this from elementary number theory. (CC.NS.4)

LCM provides an efficient means to add and subtract fractions with unlike denominators (see Standard 5.NF.1 in the Fractions LT for an example of using Venn diagrams to find the common denominator). (CC.NS.4)

• Use common fractions and percents to calculate parts of whole numbers in problem situations, including comparisons of savings rates a different financial institutions. (MA10-GR.6-S.1-GLE.1-EO.c.iv)*

o Percent involves two equivalent proportions for the same ratio, with one of the proportions indexed to the value 100. Another, somewhat more formal way to describe a percent is as a way to describe a ratio of one value of a quantity to another value of the same quantity, with the second value standardized to 100. (CC.6.RP.3c)

o Thus, percent is a rate per one hundred (recall that “cent” is a Latin term for 100 as in century, cents, centipede). Just as a unit rate requires a standardized value of 1 for one of the quantities, percents have a standardized value of 100 in the second value of the quantity, such that 50 : 100 describes half of a whole, and 100 : 100 describes the whole. This facilitates an easy way to compare ratios: thus, 30% is a ratio of 30 : 100, 45% is a ratio of 45 : 100, and 45% > 30%. (CC.6.RP.3c)

o Students often carry the misconception that percents cannot exceed 100% or 100 : 100. It is important that they become comfortable with percents such as 200% and can simplify the ratio from 200 : 100 to 2 : 1. Likewise 150% implies a ratio of 1.5 : 1 and 325% implies a ratio of 3.25 : 1. (CC.6.RP.3c)

o A difficult problem for students to consider is what the original cost of a prom dress would have been if the price of $88 represents a 33% discount (decrease) from the original price. Again, students must be very careful in constructing the ratio box. Many students will want to construct the ratio box with $100 in the “old price” column and $33 in the “new price” column. However, students must recognize that if a price is lowered by 33%, then the final price is one pays 67% (that is, 100% - 33%) of the original price. Therefore, if the old price was $100, the new price would be $67, not $33. Corresponding to this context, $88 is the new price while x in this example is the old price. The ratio box solution is shown below. (CC.6.RP.3c)

Original price

(dollars)

New price

(dollars) (%) 100 (%) 67 ($) x ($) 88

x = $131.34 (CC.6.RP.3c)



• For completeness, the ratio box and summary table approach, as in the previous example, is shown. In this instance, the summary table shows the price decrease. (CC.6.RP.3c)

Original price (dollars)

New price (dollars)

Price decrease (dollars)

(%) 100 (%) 67 (%) 33

($) x ($) 88 ($) x - 88

x = $131.34 $x - 88 = $33.34

o Students should compare the two examples, and recognize that in the two problems, the 33% were taken relative to different values: in the first, there was a 33% price increase relative to the original price of $88 (33% of $88 is $29.04). This is different from the second example, in which a price discount of 33%, resulting in a final price of $88, represented 33% relative to an originally higher value of $131.34. (CC.6.RP.3c)

• Express the comparison of two whole number quantities using differences, part-to-part ratios, and part to whole ratios in real contexts, including investing and saving. (MA10-GR.6-S.1-GLE.1-EO.c.vii)*




When comparing the prices of two items it is helpful to look at the unit ratio or unit price for each item and ratio tables can help me find the unit prices for each item.

Academic Vocabulary: savings rate, simple interest, percent, compare, multiply, divide, graphs, speed, measurement, fractions

Technical Vocabulary: ratios, rates, part-to-part ratios, part-to-whole ratio, ratio tables, (greatest) common factor, (least) common multiple, factor multiple, unit ratio, unit rate, equivalence, fair sharing, rate, measurement, scaling, array, area, equal groups, coordinate grid, conversion, quotient, product, sum, distributive property, number line

*Denotes connection to Personal Financial Literacy (PFL)



Unit Title Know the 411 Length of Unit 5 weeks

Focusing Lens(es) Interpretation Validity


MA10-GR.6-S.3-GLE.1


• Why is the more than one type of average? (MA10-GR.6-S.3-GLE.1.EO.a)

Unit Strands Statistics and Probability

Concepts Statistical question, data, data collection, measures of center (mean, median), spread, measures of variability (interquartile range, mean absolute deviation), distribution, overall shape, visual displays, graphs (number lines, dot plots, histograms, box plots), equipartition, fair share, typicality, balance point, absolute value, range, outliers, interpretation, attributes, units, measurement

Generalizations



A statistical question anticipates variability in data and the data collected. (MA10-GR.6-S.3-GLE.1-EO.a, b)

What are ways of describing variability in data? How is the shape of a data distribution connected to the

location of the mean and median? When is one data display better than another? (MA10-

GR.6-S.3-GLE.1-IQ.2) What makes a good statistical question? (MA10-GR.6-

S.3-GLE.1-IQ.4)

How do different types of visual displays show the overall shape, center and spread of a set of data?

Why are there so many ways to describe data? (MA10-GR.6-S.3-GLE.1-IQ.1)

Single numbers can describe measures of center and measures of variation (how the values of the data set vary) for a numerical data set. (MA10-GR.6-S.3-GLE.1-EO.c)

What are ways of describing measures of center with a single number? Variation?

When is one statistical measure better than another? (MA10-GR.6-S.3-GLE.1-IQ.3)

Why is it helpful to describe numerous data values with single numbers?



The median represents a measure of center that equipartitions the total number of data points in a data set into two equal groups. (MA10-GR.6-S.3-GLE.1-EO.d.ii.3)

What happens to the median when you either remove or add data points smaller, lager or the same as the median?

What percent of data is below the median?

When is the median a better measure of center for a data set than the mean?

The mean brings together ideas of fair share, typicality, and balance point among the values of a data set. (MA10-GR.6-S.3-GLE.1-EO.d.ii.3)

What happens to the mean when you either remove or add a data point smaller, larger or the same as the mean?

Why is the mean described as a balance point of a data set?

Mean absolute deviation, as a measure of variability, provides the foundation for standard deviation. (MA10-GR.6-S.3-GLE.1-EO.d.ii.3)

What is the mean absolute deviation? Why is it necessary to sum the absolute values differences to the mean rather than positive and negative differences to create a measure of variability?

Why is the mean absolute deviation a measure of variability?

Box plots provide a visual representation of the spread (range) of a set data which facilitates assessing the data’s variability (MA10-GR.6-S.3-GLE.1-EO.d.i)

How is the “lower quartile” like a median of the lower half a data set?

How is the “upper quartile” like a median of the upper half a data set?

How does the interquartile range provide a measure of variability that is based on the median?

How is a box plot representation of data similar/different from a histogram?

Meaningfully interpreting data and avoiding errors in interpretation requires an analysis of the attributes measured, methods of measurement, measurement units, and the number of measurements/observations. (MA10-GR.6-S.3-GLE.1-EO.d.ii.1, d.ii.2, d.ii.3, d.ii.4)

Which types of data displays allow you to determine the number of observations in a data set?

Why is it important to ensure attributes are measured with the same units and under the same conditions?

Why is it important to know the number of observations when considering a statistical question?


What students will know and be able to do are so closely linked in the concept-based discipline of mathematics. Therefore, in the mathematics samples of what students should know and do are combined.

• Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. (MA10-GR.6-S.3-GLE.1-EO.a) o Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am

I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages. PARCC o In the elementary grades, students recognized the distinction betweennatural/experimental variation (e.g., how individuals are different from each other)

and measurement-related variation (e.g., measurement error). Bridging Standard 2.MD.A and Standard 3.MD.4 in the Elementary Data and Modeling LT discusses



how these ideas are addressed in the elementary grades. In the middle grades, students learn the formal names for the two kinds of variation. They also identify which kind(s) of variation is relevant to a particular statistical investigation or statistical question. (CC.6.SP.1)

Natural/experimental variation describes variation that occurs in nature or experimental due to varied treatments. For example, if the statistical question is, “Do tulips grow better in sunlight or in shade?”, students understand that the kind of variation to anticipate is experimental variation that arises from the conditions of the experiment. Natural variation describes the differences that occur in nature concerning when, for example, maple trees lose their leaves. (CC.6.SP.1)

Measurement-related variation that occurs as a result to variations in the practices of measuring. For example, if the statistical investigation involves finding the “exact” measurement of the circumference of their teacher’s head, students understand that the kind of variation to anticipate is measurement-related variation that can arise from human error or imprecise units. For example, if students measure using a rigid ruler versus a tape measure or measure in inches versus centimeters, the variation of the measurements among these different approaches would be attributable to measurement-related variation. (CC.6.SP.1)

Research shows that a deep understanding of variation is very difficult, even for college students. Students need to avoid the misconception that variability is associated with a lack of structure. For example, many students believe that variation in data is just mere differences among data. They may not connect the variation when summing the roll of two dice with probability but rather interpret it as haphazard chance. A well-structured trajectory through variation and distribution helps them to understand that variability is structured as distribution and a distribution reflects the operation of a repeated random process, variability is connected with chance, and patterns can be found in variability. An example of a statistical investigation that addresses these three ideas is given here and will be extended in Standard 6.SP.2 which addresses distribution and 6.SP.3 which addresses center. (CC.6.SP.1)

o The statistical investigation begins with asking all the students in a class to measure an object, for example, the circumference of their teacher’s head. Before beginning the investigation, students discuss whether or not everyone’s measurements will be the same. After all students have completed their measurements, they display their results and see that not all of their measures are the same. Students discuss the question, “Why didn’t everyone get the same measurement?” as a starting point to investigate variation. (CC.6.SP.1)

o Students are asked to conjecture what might have accounted for the variation in their measurements. They might conjecture that the variation arose because of human error, imprecise units of measure, a lack of a standard on where on their teacher’s head to start measuring, etc. Students ask themselves to imagine how to measure or “mathematize” variability. They come up with ideas such as counting the number of different values on the scale and associating that count with variation, or looking at the difference between the extreme measurements. These ideas anticipate spread and range. (CC.6.SP.1)

• Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. (MA10-GR.6-S.3-GLE.1-EO.c)

• Tasks might ask student to rate statements True/False/Not Enough Information, such as, “The average height of trees in Watson Park is 65 feet. Are there any trees in Watson Park taller than 65 feet?” PARCC

In this standard, students invent or create a statistic where a single number is used to represent a measure of central tendency or of variation. The challenge of representing characteristics of a distribution made up of numerous data values with a single number is a fundamental element of statistical reasoning and students need to share in their inventions. (CC.6.SP.3)

One way for students to gain entry into concepts of central tendency and variation is to pose an investigation in which students recognize two different ways of talking about a set of measurements which can lead them to establishing the need to characterize a set of data with two different statistics, a measure of central tendency and a measure of variation. (CC.6.SP.3)

One way statisticians judge the quality of a statistic is to explore the degree to which it endures with changes in unit size, use of intervals, and



characteristics of tools. This can be referred to as the “robustness of the measure with different data samples.” For example, if students re-measure the circumference of the teacher’s head using a more precise unit, such as to the nearest a centimeter rather than to the nearest inch or they use a flexible measuring tape instead of a rigid ruler, they produce related data representations. They explore how these changes affect their proposed measures of center and variability. (CC.6.SP.3)

• Understand a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape (MA10-GR.6-S.3-GLE.1-EO.b)

o Students know how to construct and interpret dot plots and histograms (see Standard 6.SP.4 Part 1). While this section is focused on characteristics of a single distribution, informal comparisons of two or more distributions can be effective in generating discussions of characteristics of a single distribution. For example, the two histograms below show how long (in seconds) a group of boys and a group of girls can hold their breath underwater. (CC.6.SP.2)

(CC.6.SP.2)

• In comparing the two distributions, students ask questions and make a number of observations. (CC.6.SP.2) i. Do the graphs have the same number of observations? Are their intervals the same?

ii. Do they have the same range? iii. Do they have the same overall shape?



iv. Who holds their breath the longest, a boy or girl? v. Who holds their breath the shortest, a boy or girl?

o These questions lead students to distinguish among shape, center, and spread for the two graphs. (CC.6.SP.2) o Shape. Students can think about shape by drawing a “skyline silhouette” around dot plots and histograms. The skyline silhouettes for the underwater breath

data is shown in the figure below. (CC.6.SP.2)

(CC.6.SP.2)

• Students identify important shapes that histograms can have and learn the names. Examples of common shapes are shown below. (CC.6.SP.2)

Mounded Constant or Plateau Skewed Right (The tail stretches out to the right)



Skewed Left (The tail stretches out to the left) Bimodal

• Students notice that for the underwater breath data set, the histogram for girls is skewed right while the histogram for boys is mounded. o Center: Students can be asked about what is the most typical value of a variable as a way to discuss the idea of center. This leads students to circle the clumps

of data in the “middle” or to visualize various meanings of balance which will be explored further in the next standard. Students recognize that highly distributed data may not appear to have a “typical” value, or that highly skewed data may not appear to have a “center.” (CC.6.SP.2)

Spread: Students describe the way the data are distributed. For example, the histograms shown below are both mounded. (CC.6.SP.2)



o When asked to compare the differences between the two histograms, students may say that the one on the bottom is more “spread out” or “dispersed” than the one on the top. Some students may also use an informal description of the distance from the lowest value to the highest value to describe the spread or dispersion. (CC.6.SP.2)

For the underwater breath data set, students notice that the spread of the histogram is greater for the boys than for the girls. (CC.6.SP.2) • Display numerical data in plots on a number line, including dot plots, histograms, and box plots. (MA10-GR.6-S.3-GLE.1-EO.d.i)

o Tasks are technology-enhanced to make creation of the plots as quick and effortless as possible; or tasks ask the student to identify which display corresponds to a given set of data. PARCC

o Histograms should be continuous (bars must be touching); the data points are not part of one of the interval endpoints, for example, if intervals are 0-3, 3-6, 6-9 then none of the data points should be 0, 3, 6, 9. PARCC

Students in the middle school grades work with dot plots (called line plots in the elementary grade CCSS-M standards) and histograms as representations of data. This builds on their previous understanding of organizing and displaying data on pictographs, line plots, and bar graphs. Using these displays to explore the distribution of a data set, including center, spread, and shape, is discussed in the next standard (6.SP.2). (CC.6.SP.4)

• Based on Standards 2.EDM.A, 3.MD.4, and 4.MD.4, and 5.MD.2 in the Elementary Data and Modeling LT, students have developed an understanding that line plots consist of an attribute display on one axis which is used to display a number line (possibly scaled) and a frequency display on the other axis. In the middle grades, the term line plot is replaced with dot plot. Dots are now commonly used to display counts on the frequency display. (CC.6.SP.4)

Students propose and invent ways of representing data and by critiquing each other’s representations. Common trends in students’ data displays include: (CC.6.SP.4)

1. Displays that only highlight order, such as lists or case value plots. For example, students may simply order all of their measurements and list them. They may also represent each case with a bar (using the bar’s height to denote the case’s value) and put the bars side-by-side. See Standard 2.EDM.A in the Early Data and Modeling LT for examples of these student generated displays.

2. Displays that show order and frequency, but do not display missing values. For example, students might create a horizontal axis and list all of the observed measurements in order and place icons (e.g., boxes) on top of the measurements to denote the frequency of the observed measurement.

3. Displays that highlight order, frequency, and display places for missing values. These can include displays that look similar to dot plots or line plots, but can also be lists where students try to describe where there are missing values in the data.

4. Displays that use bins or intervals to avoid listing every value on the attribute axis. Helping students to compare their representations is crucial to highlight choices and decisions. Otherwise, students will tend to focus on the types of icons

used or the readability of a display. Students: (CC.6.SP.4) 1. Discuss what each display “shows” or “hides” about the data. This helps them to compare and contrast across displays rather than focusing on

features of each individual display. 2. Discuss how to find relevant information in a display and compare it with a different display. For example, the frequency of a particular value

is seen as the length of a plateau in a case value plot, while it is seen as the height in a dot plot or histogram. 3. Transfer cluster of cases from one display to another and compare how the shape changes. 4. Explore how the size of bins, if they exist, affects the shape of the display.

The discussion of the displays provides students a broad set of experiences to refer to as they to discuss center, spread, and shape. (CC.6.SP.4)



• For example, suppose that there are 28 students in a class and their test scores (in points out of a possible 100 points) are ordered in the following list: 45, 57, 70, 73, 73, 73, 76, 76, 80, 80, 80, 80, 80, 83, 85, 85, 87, 89, 91, 91, 91, 94, 94, 94, 98, 98, 100, 100. Below, the data are displayed on a dot plot.

(CC.6.SP.4) The attribute display is on the horizontal axis and is a scaled number line. Each tick mark on the number line represents 1 point. Each dot in the dot plot

represents a student, so the five dots above 80 indicate that 5 students scored an 80 on the test. (CC.6.SP.4) Students should have learned by now that bar graphs have a second scale [axis] on the frequency display to display the data values and when they

contain numerical data, the data can be organized into “bins.” There are two differences between bar graphs and histograms. First, unlike in bar graphs, the attribute display in a histogram must be a scale [axis] and cannot be used to display nominal/categorical data. Second, the bars in a histogram are adjacent, thus the values represented by a first bar lead in ordered fashion to those represented by the next. The test data example is displayed in a histogram below. (CC.6.SP.4)

The table below summarizes the characteristics of the four data representations from the elementary and middle grades. Note that the term line

plot has been replaced with dot plot and the histogram has been added to the table. (CC.6.SP.4)



Axis 1 Axis 2 Attribute Display Frequency

Display Categories Scaled

Axis Count Scaled

Axis Unordered Ordered Pictograph X X X

Dot Plot X X Bar Graph X X X X X Histogram X X X

(CC.6.SP.4) • Reporting the number of observations. (MA10-GR.6-S.3-GLE.1-EO.d.ii.1)

o Tasks provide students with a text-based and graphics-based overview of a numerical data set. This overview includes the necessary information for (a) and (b). Students must extract this information from the overview and enter or identify/select it as part of the task. PARCC

o Tasks require students to choose a measure of center and a measure of variability; tasks are technology-enhanced to allow for rapid computation of the chosen measures. PARCC

o With reference to the second clause in 6.SP.5c, tasks are technology-enhanced, e.g., to allow students to “tag” outliers, circle the bulk of the observations, etc. PARCC

Students learn how to report the number of observations in a data set. They can do this when presented with numeric data sets, and data displays including dot plots and histograms by counting the number of dots on the plot or adding the height of all of the bars in a histogram. Students recognize that unless they are given the number of data points in one of the quartiles (in which case they multiply the number of data points in a quartile by 4 to determine the number of data points in the data set), they cannot determine the total number of data points from looking at a box plot. (CC.6.SP.5a)

They discuss why knowing the number of observations is important when considering a statistical question. For example, suppose that students wanted to know if Stephanie or Tyrone was a better free throw shooter. In her last game, Stephanie made 15 out of 20 free throws and Tyrone made 4 out of 5 free throws. Although a preliminary analysis of this data might suggest that Tyrone is the better free throw shooter because he made 80% of his free throws while Stephanie made 75% of her free throws, Stephanie was observed attempting 20 free throws while Tyrone was only observed attempting 5 free throws. Students discuss whether it is possible or not to make valid conclusions when the number of observations are low. (CC.6.SP.5a)

• Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. (MA10-GR.6-S.3-GLE.1-EO.d.ii.2) o Students have a number of tools for conducting statistical investigations including dot plots, histograms, and box plots. They also have a number of statistics for central

tendency (mean, median, and mode) and variation (lower quartile, upper quartile, interquartile range, range, and mean absolute deviation). In the next three standards, they apply these tools and statistics to ensure that they use meaningful statistics to answer the questions in a statistical investigation. (CC.6.SP.5b)

o Students learn that before summarizing or analyzing a data set, it is important to understand the attributes that were measured, how they were measured, and what units were used to measure them. Without these understandings, it is difficult to meaningfully interpret data and errors in interpretation can be made. (CC.6.SP.5b)



For example, suppose that a statistical investigation involves finding the effects of a certain brand of fertilizer on tulip growth but the heights are measured using a yard stick. Because the height of tulips is relatively small in comparison to a yard, and because a yard stick is generally marked off only by quarter inches, it may be an imprecise measuring instrument for the investigation. Therefore, differences in the height of the tulips may be due to measurement error and not on the type of fertilizer. It is important to recognize these possible problems and modify the investigation appropriately such as using a more appropriate tool to measure. (CC.6.SP.5b)

Students also learn to avoid common errors that arise from not understanding the nature of the attribute under investigation such as: (CC.6.SP.5b) 1. Comparing two attributes that were measured using a different unit. 2. Measuring an attribute under different conditions.

For example, it would not be feasible to test the effectiveness of fertilizer on plant height by performing half the experiment on a rainy day and half the experiment on a sunny day since differences in plant height may be the result of weather, and not the fertilizer. (CC.6.SP.5b)

• Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. (MA10-GR.6-S.3-GLE.1-EO.d.ii.3)

o Eventually, students shift from invented data displays and invented measures of center and variability to the statistical measures that are used by statisticians. These measures are more meaningful to students who have had a chance to invent and discuss their own. (CC.6.SP.5c)

o Measures of center summarize the aggregate with a single number. Sixth graders explore different ways of finding the center, which lead to mode, median, and

mean. (CC.6.SP.5c) o Students often start with the mode, which is the most frequent value, as a measure of the centeredness of the distribution of data, including categorical data. They

compare data displays where the mode is in the center of the distribution, when it is not in the center of the distribution, and when there is more than one mode. They consider advantages and disadvantages of the mode as a measure of central tendency. (CC.6.SP.5c)

o In addition to the mode, students use the median, which is the value of the center of an ordered numerical data set. Students recognize that the median of an odd number of data values is derived differently from that of an even number of data values. Of the former, the median is the data value in the middle of a list of ordered numbers. For example, the median of the data set {2, 3, 5, 5, 7, 12, 14} is the value 5. Of the latter, the median is the value midway between the two data values in the middle of the list of ordered data set. For example, the median of the data set {4, 5, 8, 10, 12, 17, 18, 19} is 11, which is midway between 10 and 12. (CC.6.SP.5c)

o Sixth graders also describe the center of a distribution using the mean or average. Bringing together the idea of fair share, typicality, and the balance point among the values of the data produces a more robust understanding of the mean. For example, students interpret the mean as a “fair share” by exploring data sets using concrete objects, such as snap cubes. In the drawing below, 10 stacks of cubes represent the number of mini-pies for 10 students. Each snap cube represents one pie. (CC.6.SP.5c)



Students are asked, “What if we combined all our pies and redistributed them one at time to each of the nine students until all the pies are gone? How

many pies would each student get?” Students redistribute the cubes one at a time until each student has the same number. Each student would get 4 pies. This activity demonstrates the idea of “fair share” and brings meaning to the way that the mean is calculated by adding all the cases in a data set and dividing by the number of cases. (CC.6.SP.5c)

The display with the unifix cubes should not be confused with the histogram for these data. The histogram, as shown below with the mean (4) depicted with a blue line, represents the distribution of the values of the data. If the area between the “skyline silhouette” and the axis of the distribution were cut out of cardboard, the cutout would balance on the mean. (CC.6.SP.5c)

o Measures of variability summarize with a single number how the values of a data set vary. Students often start with the range, which is the difference between the

largest value in data set and the smallest value in the data set. (CC.6.SP.5c) o Students understand that the median equipartitions the total number of data points in a data set into two equal groups, resulting in 50% of the data points below

the median and 50% above the median. Students apply a similar procedure to find the median of the lower half of the data set. This median is referred to as the “lower quartile” because 25% of the data values are lower than this point. It follows that 25% of the data values also reside between the lower quartile and the median. Students apply the same procedure to find the median of the upper half of the data set. This median is referred to as the “upper quartile” because 25% of the data values are higher than this point. It follows that 25% of the data values also reside between the median and the upper quartile. (CC.6.SP.5c)

o The figure below shows a case list. Each quartile contains three data points. The median of the data set is 14 (there are 6 data points above and below 14), the lower quartile is 9, and the upper quartile is 18.5. The range of each quartile is displayed above the data values. (CC.6.SP.5c)



(CC.6.SP.5c)

Students avoid the misconception that they are dividing the values on the scale into equal sized intervals, which is typically done in building histograms. Rather, they are equipartitioning the number of data points into equal sized groups and identifying the ranges to create and name the intervals. In a histogram, the ranges are chosen ahead of time and are equal. These are called “bins” and the bins do not have to correspond to the data. (CC.6.SP.5c)

o For illustration, a histogram of the above data is shown below with 5 bins (8-10, 11-13, 14-16, 17-19, and 20-22). The data’s “box plot” (formally introduced in

Standard 6.SP.4 Part 2) is drawn above the histogram. What is important to note is the “box plot” shows the ranges of the four quartiles and they are not the same (e.g., the first quartile’s range is 1 and the second quartile’s range is 5). However, each quartile contains the same number of data points. (CC.6.SP.5c)

(CC.6.SP.5c) For large data sets (such as hundreds of student scores on an exam), the process of equipartitioning the number of data points into any desired number of

bins can be undertaken. That is, instead of creating quartiles (4 bins), they could create quintiles (5 bins), octiles (8 bins), or even percentiles (100 bins). These bins of equal numbers of data points can be called “quantiles” to indicate that they represent an equipartitioned quantity of data. These will be discussed further in Standard 6.SP.5.d. (CC.6.SP.5c)

The act of equipartitioning the number of data points helps to display important markers in the data set but does not provide a single statistic to describe the spread of data. This is accomplished by a measure called theinterquartile range. The interquartile range is the difference between the upper and lower quartiles of a data set. The interquartile range for the above data set is 18.5 – 9 = 9.5. By definition, half of the data points are contained in the interquartile range. (CC.6.SP.5c)

o When students are challenged to create a measure of variation, they may propose measuring the distances of each data point to the mean, giving positive values if the data point is greater than the mean and negative values if the data point is less than the mean. Students are surprised to find that these values always sum to 0. It is important for them to explain why this result is inevitable. (CC.6.SP.5c)



o Students adjust their approach to take the absolute value of the distances, and if they then divide by the number of cases, this results in a statistic calledmean absolute deviation which is the average distance to the mean. Thus, students can calculate this by adding up the distance between every point in the data set and the mean and dividing by the number of cases. For example, data is displayed in a bar graph below with the mean (14) marked. (CC.6.SP.5c)

To calculate the mean average deviation, students find the absolute value of each data point to the mean (14) as shown in the table below.

Data Point Absolute Value of

Distance to the Mean (14) 8 6 8 6 9 5 9 5

11 3 13 1 15 1 17 3 18 4 19 5 19 5 22 8

(CC.6.SP.5c) They then sum up the absolute values to obtain 52 and divide by the number of data points (12) to obtain 13/3 or 4.333. Therefore, the mean absolute

deviation or average distance of each data point to the mean is 4.333. (CC.6.SP.5c) o Students also identify “unusual data points” in data sets. This anticipates the introduction of outliers. For example, the dot plot below shows the distribution of

predicted life spans, in years, of various species of mammals. Students identify the case with a lifespan of 69 years as an outlier. (CC.6.SP.5c)



(CC.6.SP.5c) • Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. (MA10-GR.6-S.3-GLE.1-

EO.d.ii.4) o Students can locate the mean, median, and mode on a data set’s histogram. They can further describe how moving data points, adding data points, or taking away

data points will affect the mean, median, and mode (increase, decrease, or stay the same) without needing to do any calculations. (CC.6.SP.5d) o The examples below describe the location of the mean, median, and mode for common histogram shapes. (CC.6.SP.5d)

Example 1: Mounded (Symmetric): Students recognize that for mounded (and symmetric) distributions, the median is located symmetrically to the other data values and that the mean is very near (or equal to, in the case of perfectly symmetric data) the median. The mode is also near (or equal to) the mean and median. The figure below shows a mounded distribution with perfect symmetry where the mean, median, and mode are all equal to 6. (CC.6.SP.5d)

Example 2: Skewed Left (no image available): Students recognize that for skewed left distributions, the median is located to the right of the mean because there are more data points in the right of the distribution than the tail and the lower values in the tail of the distribution pull the mean toward the left of the number line. The mode is located on the right side of the distribution. An example of a skewed left distribution is shown below with the mean plotted with a blue line and the median plotted with a yellow line. (CC.6.SP.5d)

Example 3: Skewed Right: Students recognize that for skewed right distributions, the median is located to the left of the mean because there are more data points in the left of the distribution than the tail and the higher values in the tail of the distribution pull the mean toward the right of the number line. The mode is located to the left of the data set. An example of a skewed right distribution is shown below with the mean plotted with a blue line and the median plotted with a yellow line.



(CC.6.SP.5d)

o For all distributions, students recognize how the mean and median changes (increases, deceases, or stays the same) when data points are moved, added, or taken away from a data set. The table below summarizes the possible effects of certain actions on the median. Students are asked to generate the table for themselves and construct examples. (CC.6.SP.5d)

Effects on the Median Action Increases Decreases Stays the Same

Move a data point “within” the left or the right of the median X Move a data point from the left of the median to the right of the median X X Move a data point from the right of the median to the left of the median X X Add a data point to the left of the median X X Add a data point to the right of the median X X Add a data point at the median X Remove a data point to the left of the median X X Remove a data point to the right of the median X X Remove a data point at the median X X X

(CC.6.SP.5d) Students avoid the misconception that moving a data point from the left to the right of the median or adding a data point to the right of the

median always increases the median and generate examples to show that this is not always so. (CC.6.SP.5d) The table below summarizes the possible effects of certain actions on the mean. Students are asked to generate the table for themselves and construct

examples.



Effects on the Mean Action Increases Decreases Stays the Same

Move a data point to the left X Move a data point to the right X Add a data point to the left of the mean X Add a data point to the right of the mean X Add a data point at the mean X Remove a data point to the left of the mean X Remove a data point to the right of the mean X Remove a data point at the mean X

(CC.6.SP.5d) Unlike with the median, students recognize that moving a data point to the right, adding a data point to the right of the mean, and removing a data

point to the left of the mean always increases the mean. They can informally explain why by using the definition of the mean. (CC.6.SP.5d) Students apply this knowledge along with their previous work on variation to contextual problems to explain when it might be better to use the mean,

median, or mode to describe the center or typical value of a data set. (CC.6.SP.5d) o For example, suppose that students are given a data set that consists of a randomly drawn sample of 500 yearly wages and salaries in Texas collected during the

2000 census. They begin by using digital tools to display the data in a histogram. (CC.6.SP.5d) Note to teachers: Students learn more about the importance of drawing random samples in Standards 7.SP.2 and 7.SP.4 later.

(CC.6.SP.5d)

o Students discuss which measure of center to use to summarize the data given the shape of the graph and the context. They know from their work in this standard that the histogram of wage and salary is skewed right and, therefore, the median is located to the left of the mean. Students debate whether the



mean (which they calculate to be $17,331) or the median (which they calculate to be $9,550) would be a better description of the typical wage and salary. They may reason that the few very large values in the right tail of the distribution are shifting the mean to the right and, thus the mean may be an

overestimate of the typical value. Choosing the median overcomes this misrepresentation in this situation and they understand that 250 people (half) in the sample had a yearly wage of less than $9,550. (CC.6.SP.5d)

They also discuss and interpret variability. They note the right skewness of the data indicating that the “bins” for higher salaries have fewer and fewer people. They compare different measures for variability noting that while the range of the data is $120,000, the absolute mean deviation is $16,731. They interpret this to mean that while the data has a large overall spread as indicated by the range, the average distance from the mean ($17,331) is $16,731. This captures the fact that most of the data points in the data set are clustered around the mean. In context, this means that most people in Texas made approximately between $0 and $33,000 in 2000. (CC.6.SP.5d)




If I add a data point below the mean it will lower the mean but the median may also lower but it could also stay the same if the median is a data point and there are several data points of that value.

Academic Vocabulary: data, data collection, average, distribution, spread, overall shape, visual displays, graphs, fair share, typicality, balance point, interpret, attributes, measurement, display, summarize,

Technical Vocabulary: statistical question, measures of center (mean, median), measures of variability (interquartile range, mean absolute deviation), number lines, dot plots, histograms, box plots, equipartition, absolute value, range, outliers, units,

Unit Title Go Figure! Length of Unit 4 weeks

Focusing Lens(es) Application Tools




• Can a triangle and a parallelogram have the same area? (MA10-GR.6-S.4-GLE.1-EO.a)

Unit Strands The Number System, Expressions and Equations, Geometry

Concepts Volume, three-dimensional figures, surface area, composing, decomposing, shapes, area, formula, x- and y- coordinates, coordinate plane, horizontal, vertical, line, graphing, distance, difference, fractional part, unit cube, equipartitioning, dimensions (length, width, height)



Generalizations



Decomposing three-dimensional figures into nets facilitates calculation of surface area. (MA10-GR.6-S.4-GLE.1-EO.d.i, d.ii, d.iii)

What is a net? What is an edge? Face? Vertices?

Decomposing three-dimensional figures into nets facilitates calculation of surface area. (MA10-GR.6-S.4-GLE.1-EO.d.i, d.ii, d.iii)

Composing and decomposing shapes maintains the attribute of area. (MA10-GR.6-S.4-GLE.1-EO.a.i, a.ii)

How can every parallelogram be decomposed and rearranged into a rectangle with the same height and base?

Composing and decomposing shapes maintains the attribute of area. (MA10-GR.6-S.4-GLE.1-EO.a.i, a.ii)

On a coordinate plane, points with the same x-coordinate form a vertical line when connected, and points with the same y-coordinate form a horizontal line when connected. (MA10-GR.6-S.4-GLE.1-EO.c.ii)

How can you find the distance between two points that form a vertical or horizontal line on a coordinate plane?

Without graphing, how can you tell if two points will form a vertical or horizontal line?

On a coordinate plane, points with the same x-coordinate form a vertical line when connected, and points with the same y-coordinate form a horizontal line when connected. (MA10-GR.6-S.4-GLE.1-EO.c.ii)

The equipartitions of a unit cube into smaller components using a composition of splits along each dimension provides a visual representation of the value of fractional amounts (length, width, and height). (MA10-GR.6-S.4-GLE.1-EO.b.i, b.ii)

What is the size of a right rectangular prism with side lengths of 1/5, 1/2, and 1/4 units?

What is the formula of right rectangular prism volume?

The equipartitions of a unit cube into smaller components using a composition of splits along each dimension provides a visual representation of the value of fractional amounts (length, width, and height). (MA10-GR.6-S.4-GLE.1-EO.b.i, b.ii)



• Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane and include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinates. (MA10-GR.6-S.1-GLE.3-EO.d)

o Students use their knowledge of graphing in all four quadrants to solve contextual problems including finding the distances between points with the same x-coordinate (form a vertical line when connected) and points with the same y-coordinate (form a horizontal line when connected). (CC.6.NS.8)

o For example, students are asked to find the length of the sides of a polygon with vertices A (–1.5, 2.5), B (7, 2.5), C (7, –5), and D (–1.5, –5), as shown in the figure below, by visual observation of the plotted points and by using only the coordinate points. (CC.6.NS.8)



(CC.6.NS.8)

Note for teachers: A common student misconception is the belief that the coordinate points can only have integer values. Therefore, it is important to include examples, like the one above, in which the values are non-integers.

o Another common misconception is exhibited when students try to find the length of a side of a polygon: they may subtract within a coordinate point, for instance by subtracting the x-value of a coordinate pair from the y-value of the same coordinate pair. For example, to find the length of the side AB above, students may subtract –1.5 – 2.5 within the coordinate point A (–1.5, 2.5) ignoring the coordinate point B. (CC.6.NS.8)

o In Standard 7.NS.1.c in the Rational and Irrational Numbers LT, students will learn that subtracting a negative is equivalent to adding that number’s opposite, and that the distance between any two points on the number line can be found by subtracting their coordinates and taking the absolute value even if the numbers lie across 0 on the number line. Before formalizing these two ideas, however, students use visual inspection of the coordinate plane and absolute value to figure out, for example, the length of side AB which “crosses the y-axis.” They observe that the length of AB is equal to the distance between the y-axis and the point B, plus the distance between the point A and the y-axis. They reason that the distance between the y-axis and B is 7 units because the x-coordinate of B is 7. They reason that the distance between Aand the y-axis is the absolute value of the x-coordinate of A which is |–1.5| = 1.5. Therefore, the length of AB is 1.5 + 7 = 8.5. (CC.6.NS.8)

• Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. (MA10-GR.6-S.4-GLE.1-EO.a.i, a.ii)

o Students are introduced to the area of non-rectangular 2-D shapes including triangles (including right triangles), special quadrilaterals, and other polygons. They do this by using ideas of dissection theory where they decompose irregular shapes into more familiar shapes. (CC.6.G.1)

o An understanding of the additivity of area is critical at this stage in order to understand area. For example, in order to understand the area of a triangle, students understand that two triangles can also be formed from a rectangle by splitting the rectangle into 2 with a diagonal cut. From their understanding of equipartitioning and the area formula for a rectangle, they can come to understand that the area of a triangle with base b and height h must be half that of a



rectangle or ½ * b * h. (CC.6.G.1)

(CC.6.G.1) • Note to teachers: The above figures shows three different cases of finding the area of a triangle. By recognizing the parts outlined by the

same color are having equal area, students can understand why the area formula of a triangle applies to all cases. o Students use similar reasoning to decompose an area into one rectangle and two triangles (e.g., the area of a trapezoid). (CC.6.G.1)

(CC.6.G.1)

o Students apply the composition and decomposition of areas into rectangles and triangles to determine the area problems in the real world. For example, “Martha has a garden measuring measuring 15 yards by 8 yards. If she builds a playground and a sand pit in the garden with the side lengths shown in the land plan below, what area of the garden will be covered with grass?” (CC.6.G.1)



• Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism and apply the formulas V = l w h and V = B h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. (MA10-GR.6-S.4-GLE.1-EO.b.i, b.ii, b.ii)

o The testing interface can provide students with a calculation aid of the specified kind for these tasks. PARCC o A trapezoid is defined as “A quadrilateral with at least one pair of parallel sides. PARCC

Students understand and use procedures and formulas (e.g., V = l w h or V = B h) to calculate the volume of rectangular prisms with full understanding of the 3-D array structure underlying these procedures and formulas. They can explain howmultiplication creates a measure of volume (i.e., connecting the arithmetic operations to mental operation of spatial structuring). (CC.6.G.2)

Similarly to the fluency standards in number, students not only apply and use volume formulas, but they explain why the formulas work and understand the context of their answer in relation to the unit cube. (CC.6.G.2)

For rectangular prisms with fractional edge lengths, students learn that the fractional parts of the volumes are composed of smaller components whose lengths are unit fractions. They created these components by equipartitioning of a unit cube into smaller components using a composition of splits along each dimension (length, width, and height). For example, students are asked to find the volume of a rectangular mailing box of length 8 1/2 inches, width 51/4 inches and height 4 1/5 inches. They equipartition the unit cube along the length by 1/2‘s, along the width by 1/4 ’s, and along the height by 1/5’s, and find that the volume has fractional parts made up by smaller components. Each component is 1/40 of an inch-cube, a unit fractional volume, which is the product of multiplying the unit fractional edge lengths of 1/2 inch, 1/4 inch, and 1/5 inch. (CC.6.G.2)

(CC.6.G.2) Note to Teacher: “Unit fractional edge lengths” refers to the unit fraction of the fractional parts of the edge lengths. For example, if the height of a prism is 4 3/8 feet, the unit

fractional edge length for the height is 1/8 foot. (CC.6.G.2)



• Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate and apply these techniques in the context of solving real-world and mathematical problems. (MA10-GR.6-S.4-GLE.1-EO.c.i, c.ii)

o Students draw polygons on the coordinate plane, assigning the vertices of the polygon to coordinate points. By drawing on the coordinate plane, they identify properties of the polygons such as equal sides, equal angles, perpendicular sides, and parallel sides (see Standard 4.G.2 in the Shapes and Angles LT). They make polygons on grids, using geoboards, graph paper, and dynamic geometry software. (CC.6.G.3)

o For example, students are asked to draw and name the polygon with vertices (4, 3), (4, 8), (9, 3), and (9, 5.75) as shown in the figure below. (CC.6.G.3)

(CC.6.G.3)

o By studying their drawing on the coordinate plane, they observe that the polygon with vertices (4, 3), (4, 8), (9, 3), and (9, 5.75) has: (CC.6.G.3) 1. Two sides, each with length 5: the side that connects the vertices (4, 3) and (9, 3), and the side that connects the vertices (4, 3) and (4, 8). 2. Two right (90 degree) angles (and therefore two sets of perpendicular sides) at the vertices (4, 3) and (9, 3). 3. One acute angle at the vertex (4, 8). 4. One obtuse angle at the vertex (9, 5.75). 5. One pair of parallel sides: The side connecting the vertices 6. (4, 3) and (4, 8), and the side connecting the vertices (9, 3) and (9, 5.75).

Students conclude that the polygon is a trapezoid, because it has four sides and exactly one pair of parallel sides (or at least one pair of parallel lines, see Standard 3.G.1 in the Shape and Angles LT for the two definitions of trapezoid). Furthermore, it is a right trapezoid because it has two right angles. (CC.6.G.3)

o Students avoid the misconception that they can describe the length of the diagonal side by counting the number of squares through which it passes. They can explain why this argument is incorrect by showing a variety of unequal line segments whose measurements would all be the same if the length was determined by counting the number of grid squares the segment passed through. (CC.6.G.3)

o After graphing a variety of polygons and observing the coordinate points of two vertices that, when connected, form a vertical or a horizontal line, students are asked if they can tell whether two coordinate points form a vertical or horizontal line without plotting the points. They observe that horizontal lines are formed by connecting coordinate points with the same y-coordinate and that vertical lines are formed by connecting coordinate points with the same x-coordinate. They apply their knowledge of finding the distance between two points on a number line (see Standard 2.MD.6 in the Length, Area, and Volume LT) to find the length of the non-



diagonal sides of a polygon using the coordinate points of its vertices. For example, in the right trapezoid example above, students know that the vertices (9, 3) and (9, 5.75) form a vertical line when connected because the x-coordinates are the same. They find the length of the side by subtracting the y-coordinates: 5.75 – 3 = 2.75. For now, students subtract the smaller y-coordinate from the larger y-coordinate. Later, they learn that they can subtract in any order and take the absolute value of the difference to find the side length. (CC.6.G.3)

o Students can be asked to predict what happens to the trapezoid above if 6 is added to the x-coordinate of each of its vertices. They reason that leaving the y-coordinates constant and adding 6 to each of the x-coordinates of the vertices will cause the right trapezoid to “slide” or translate 6 units in the x direction (to the right). They verify their prediction by drawing the original trapezoid, adding 6 to the x-coordinate of each of the vertices to obtain (10, 3), (10, 8), (15, 3), and (15, 5.75), and drawing the original and translated trapezoids as shown in the figure below:

(CC.6.G.3)

o They can do this for the cases in which one or more of the following are done: (CC.6.G.3) 1. a number is (a) added to or (b) subtracted from all the x-coordinates (horizontal translation) 2. a number is (a) added to or (b) subtracted from all the y-coordinates (vertical translation), and 3. a number is added to or subtracted from all the x-coordinates, and a number is added to or subtracted from all the y-coordinates (diagonal

translation) o They can also determine how addition or subtraction determines the direction of the translation. They are equally comfortable with using negative rational numbers

to describe movement as positive ones, in both the x and y directions. (CC.6.G.3) • Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures and apply these techniques in

the context of solving real-world and mathematical problems. (MA10-GR.6-S.4-GLE.1-EO.d.i, d.ii, d.iii) o Building on previous exploration of three-dimensional shapes, students work with 3-D models (excluding cones, cylinders, and spheres) to identify faces, edges, and

vertices. As their spatial reasoning develops, students recognize that certain two-dimensional representations (nets) could be cut and folded to form a particular three-dimensional shape. (CC.6.G.4)

o In this standard, students compose 2-D shapes to form a 3-D shape and decompose 3-D shapes to its 2-D components. For example, students may recognize that the cube can be created by folding 6 squares, and thus being introduced to the idea of “nets” for producing 3D shapes: (CC.6.G.4)



(CC.6.G.4)

o Students begin to identify nets that can be assembled to form 3-D shapes. For example, which of the following nets makes a cube?

(CC.6.G.4)

o Dynamic geometry software may be used to help students to visualize the composition and decomposition of 3-D shapes:

(CC.6.G.4)

o Students also use the nets to determine the surface area of the 3-D figures which are made up of triangular and rectangular faces. For example, “The city of Raleigh is erecting a triangular prism sculpture. The base frame of the sculpture is a right-angle triangle of base 6 meters, height 8 meters and the longest side 10 meters. The sculpture stands tall at 12 meters. How much area of thick aluminum sheet is needed to cover the sculpture?” (CC.6.G.4)






When comparing the prices of two items it is helpful to look at the unit ratio or unit price for each item and ratio tables can help me find the unit prices for each item.

Academic Vocabulary: savings rate, simple interest, percent, compare, multiply, divide, graphs, speed, measurement, fractions

Technical Vocabulary: ratios, rates, part-to-part ratios, part-to-whole ratio, ratio tables, (greatest) common factor, (least) common multiple, factor multiple, unit ratio, unit rate, equivalence, fair sharing, rate, measurement, scaling, array, area, equal groups, coordinate grid, conversion, quotient, product, sum, distributive property, number line

*Denotes connection to Personal Financial Literacy (PFL

(CC.6.G.4) o Students construct the net of the triangular prism to determine that the shapes of the net include two right-angle triangles of base 6 meter by height 8 meters

and three rectangles of dimensions, 12 meters by 10 meters. 12 meters by 8 meters, and 12 meters by 6 meters. They recognize that the triangular face at the bottom of the prism is excluded and apply the area formulae of triangle and rectangle (see the Standards 4.MD.3 and 6.G.1 earlier) to calculate the total area of the other faces. (CC.6.G.4)


· Rangely School District Re4 6th Grade Mathematics Curriculum . Content Area . Mathematics ....

Documents

Transcript of · Rangely School District Re4 6th Grade Mathematics Curriculum . Content Area . Mathematics ....