## Grade 8 – CA Common Core Standards & Learning Objectives

### 8.8.NS The Number System

#### 8.8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Identify rational and irrational numbers (Eighth grade – D.1)

Convert between decimals and fractions or mixed numbers (Eighth grade – D.6)

#### 8.8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., pi²).

Estimate positive and negative square roots (Eighth grade – F.16)

Estimate cube roots (Eighth grade – F.21)

### 8.8.EE Expressions and Equations

#### 8.8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions.

Understanding exponents (Eighth grade – F.1)

Evaluate exponents (Eighth grade – F.2)

Solve equations with variable exponents (Eighth grade – F.3)

Exponents with negative bases (Eighth grade – F.4)

Exponents with decimal and fractional bases (Eighth grade – F.5)

Understanding negative exponents (Eighth grade – F.6)

Evaluate negative exponents (Eighth grade – F.7)

Multiplication with exponents (Eighth grade – F.8)

Division with exponents (Eighth grade – F.9)

Multiplication and division with exponents (Eighth grade – F.10)

Power rule (Eighth grade – F.11)

Evaluate expressions involving exponents (Eighth grade – F.12)

Identify equivalent expressions involving exponents (Eighth grade – F.13)

Multiply monomials (Eighth grade – Z.6)

Divide monomials (Eighth grade – Z.7)

Multiply and divide monomials (Eighth grade – Z.8)

Powers of monomials (Eighth grade – Z.9)

#### 8.8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that the square root of 2 is irrational.

Identify rational and irrational numbers (Eighth grade – D.1)

Square roots of perfect squares (Eighth grade – F.14)

Positive and negative square roots (Eighth grade – F.15)

Relationship between squares and square roots (Eighth grade – F.17)

Cube roots of perfect cubes (Eighth grade – F.19)

Solve equations involving cubes and cube roots (Eighth grade – F.20)

#### 8.8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

Convert between standard and scientific notation (Eighth grade – G.1)

Compare numbers written in scientific notation (Eighth grade – G.2)

#### 8.8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Convert between standard and scientific notation (Eighth grade – G.1)

Multiply numbers written in scientific notation (Eighth grade – G.3)

Divide numbers written in scientific notation (Eighth grade – G.4)

#### 8.8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

Unit rates (Eighth grade – H.5)

Do the ratios form a proportion? (Eighth grade – H.6)

Do the ratios form a proportion: word problems (Eighth grade – H.7)

Solve proportions (Eighth grade – H.8)

Solve proportions: word problems (Eighth grade – H.9)

Find the constant of proportionality from a graph (Eighth grade – I.3)

Graph proportional relationships (Eighth grade – I.5)

Solve problems involving proportional relationships (Eighth grade – I.8)

#### 8.8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Write equations for proportional relationships (Eighth grade – I.4)

Find the slope of a graph (Eighth grade – W.1)

Find the slope from two points (Eighth grade – W.2)

Find the slope of an equation (Eighth grade – W.4)

Graph a linear equation (Eighth grade – W.5)

Write a linear equation from a graph (Eighth grade – W.7)

Graph a line from an equation (Eighth grade – X.9)

#### 8.8.EE.7.a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Find the number of solutions (Eighth grade – U.12)

#### 8.8.EE.7.b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Solve equations involving squares and square roots (Eighth grade – F.18)

Model and solve equations using algebra tiles (Eighth grade – U.3)

Write and solve equations that represent diagrams (Eighth grade – U.4)

Solve one-step equations (Eighth grade – U.5)

Solve two-step equations (Eighth grade – U.6)

Solve multi-step equations (Eighth grade – U.7)

Solve equations involving like terms (Eighth grade – U.8)

Solve equations with variables on both sides (Eighth grade – U.9)

Solve equations: mixed review (Eighth grade – U.10)

Solve equations: word problems (Eighth grade – U.11)

#### 8.8.EE.8.a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Is (x, y) a solution to the system of equations? (Eighth grade – Y.1)

Solve a system of equations by graphing (Eighth grade – Y.2)

Find the number of solutions to a system of equations by graphing (Eighth grade – Y.4)

#### 8.8.EE.8.b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

Find the number of solutions to a system of equations (Eighth grade – Y.5)

Classify a system of equations by graphing (Eighth grade – Y.6)

Classify a system of equations (Eighth grade – Y.7)

Solve a system of equations using substitution (Eighth grade – Y.8)

Solve a system of equations using elimination (Eighth grade – Y.10)

#### 8.8.EE.8.c Solve real-world and mathematical problems leading to two linear equations in two variables.

Solve a system of equations by graphing: word problems (Eighth grade – Y.3)

Solve a system of equations using substitution: word problems (Eighth grade – Y.9)

Solve a system of equations using elimination: word problems (Eighth grade – Y.11)

### 8.8.F Functions

#### 8.8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Identify functions (Eighth grade – X.1)

Complete a table for a linear function (Eighth grade – X.7)

Graph a line from a function table (Eighth grade – X.8)

Evaluate a function graphically (Eighth grade – X.10)

#### 8.8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Graph a line from a function table (Eighth grade – X.8)

Graph a line from an equation (Eighth grade – X.9)

Write a linear function from a table (Eighth grade – X.11)

Identify linear and nonlinear functions (Eighth grade – X.14)

#### 8.8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

Graph a line from an equation (Eighth grade – X.9)

Identify linear and nonlinear functions (Eighth grade – X.14)

#### 8.8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Find the constant of proportionality from a graph (Eighth grade – I.3)

Write equations for proportional relationships (Eighth grade – I.4)

Find the constant of proportionality: word problems (Eighth grade – I.7)

Solve problems involving proportional relationships (Eighth grade – I.8)

Find the slope of a graph (Eighth grade – W.1)

Find the slope from two points (Eighth grade – W.2)

Find a missing coordinate using slope (Eighth grade – W.3)

Write a linear equation from a graph (Eighth grade – W.7)

Write a linear equation from two points (Eighth grade – W.9)

Rate of change (Eighth grade – X.4)

Constant rate of change (Eighth grade – X.5)

Write a linear function from a table (Eighth grade – X.11)

Write linear functions: word problems (Eighth grade – X.12)

#### 8.8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Write linear functions: word problems (Eighth grade – X.12)

### 8.8.G Geometry

#### 8.8.G.1.a Lines are taken to lines, and line segments to line segments of the same length.

Identify reflections, rotations, and translations (Eighth grade – Q.1)

Translations: graph the image (Eighth grade – Q.2)

Reflections: graph the image (Eighth grade – Q.4)

Rotations: graph the image (Eighth grade – Q.6)

#### 8.8.G.1.b Angles are taken to angles of the same measure.

Identify reflections, rotations, and translations (Eighth grade – Q.1)

Translations: graph the image (Eighth grade – Q.2)

Reflections: graph the image (Eighth grade – Q.4)

Rotations: graph the image (Eighth grade – Q.6)

#### 8.8.G.1.c Parallel lines are taken to parallel lines.

Identify reflections, rotations, and translations (Eighth grade – Q.1)

Translations: graph the image (Eighth grade – Q.2)

Reflections: graph the image (Eighth grade – Q.4)

Rotations: graph the image (Eighth grade – Q.6)

#### 8.8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Similar and congruent figures (Eighth grade – N.10)

Congruent figures: side lengths and angle measures (Eighth grade – N.12)

Congruence statements and corresponding parts (Eighth grade – N.13)

#### 8.8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Translations: find the coordinates (Eighth grade – Q.3)

Reflections: find the coordinates (Eighth grade – Q.5)

Rotations: find the coordinates (Eighth grade – Q.7)

Dilations: graph the image (Eighth grade – Q.8)

Dilations: find the coordinates (Eighth grade – Q.9)

#### 8.8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Similar and congruent figures (Eighth grade – N.10)

Similar figures: side lengths and angle measures (Eighth grade – N.11)

#### 8.8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Identify complementary, supplementary, vertical, adjacent, and congruent angles (Eighth grade – N.1)

Find measures of complementary, supplementary, vertical, and adjacent angles (Eighth grade – N.2)

Transversal of parallel lines (Eighth grade – N.3)

Exterior Angle Theorem (Eighth grade – N.7)

Interior angles of polygons (Eighth grade – N.9)

Congruent triangles: SSS, SAS, and ASA (Eighth grade – N.14)

#### 8.8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

Converse of the Pythagorean theorem: is it a right triangle? (Eighth grade – O.5)

#### 8.8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Pythagorean theorem: find the length of the hypotenuse (Eighth grade – O.1)

Pythagorean theorem: find the missing leg length (Eighth grade – O.2)

Pythagorean theorem: find the perimeter (Eighth grade – O.3)

Pythagorean theorem: word problems (Eighth grade – O.4)

#### 8.8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Distance between two points (Eighth grade – P.4)

#### 8.8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Volume of cylinders and cones (Eighth grade – N.31)

Volume of spheres (Eighth grade – N.32)

### 8.8.SP Statistics and Probability

#### 8.8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

Scatter plots (Eighth grade – AA.14)

Outliers in scatter plots (Eighth grade – BB.8)

#### 8.8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

Find the slope of a graph (Eighth grade – W.1)

Constant rate of change (Eighth grade – X.5)

Graph a line from an equation (Eighth grade – X.9)

Write linear functions: word problems (Eighth grade – X.12)

#### 8.8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

Interpret stem-and-leaf plots (Eighth grade – AA.9)

Interpret histograms (Eighth grade – AA.10)

Create histograms (Eighth grade – AA.11)

Create frequency charts (Eighth grade – AA.12)

## Arithmetic Operations Including Division of Fractions – Grade 6 Module 2

### OVERVIEW

In Module 1, students used their existing understanding of multiplication and division as they began their study of ratios and rates. In Module 2, students complete their understanding of the four operations as they study division of whole numbers, division by a fraction and operations on multi-digit decimals. This expanded understanding serves to complete their study of the four operations with positive rational numbers, thereby preparing students for understanding, locating, and ordering negative rational numbers (Module 3) and algebraic expressions (Module 4).

In Topic A, students extend their previous understanding of multiplication and division to divide fractions by fractions. They construct division stories and solve word problems involving division of fractions (6.NS.1). Through the context of word problems, students understand and use partitive division of fractions to determine how much is in each group. They explore real-life situations that require them to ask, “How much is one share?” and “What part of the unit is that share?”

Students use measurement to determine quotients of fractions. They are presented conceptual problems where they determine that the quotient represents how many of the divisor is in the dividend. For example, students understand that (6 cm)/(3 cm) derives a quotient of 3 because 2 cm divides into 6 centimeters three times. They apply this method to quotients of fractions, understanding 6/7 ÷ 2/7 = (6 sevenths)/(2 sevenths) = 3 because, again, two-sevenths divides into six-sevenths three times. Students look for and uncover patterns while modeling quotients of fractions to ultimately discover the relationship between multiplication and division. Using this relationship, students
create equations and formulas to represent and solve problems.

Later in the module, students learn to and apply the direct correlation of division of fractions to division of decimals. Prior to division of decimals, students will revisit all decimal operations in Topic B. Students have had extensive experience of decimal operations to the hundredths and thousandths (5.NBT.7), which prepares them to easily compute with more decimal places. Students begin by relating the first lesson in this topic to mixed numbers from the last lesson in Topic A. They find that sums and differences of large mixed numbers can sometimes be more efficiently determined by first converting the number to a decimal and then applying the standard algorithms (6.NS.3). They use estimation to justify their answers.

Within decimal multiplication, students begin to practice the distributive property. Students use arrays and partial products to understand and apply the distributive property as they solve multiplication problems involving decimals. By gaining fluency in the distributive property throughout this module and the next, students will be proficient in applying the distributive property in Module 4 (6.EE.3).

Estimation and place value enable students to determine the placement of the decimal point in products and recognize that the size of a product is relative to each factor. Students learn to use connections between fraction multiplication and decimal multiplication. In Grades 4 and 5, students used concrete models, pictorial representations, and properties to divide whole numbers (4.NBT.6, 5.NBT.6). They became efficient in applying the standard algorithm for long division. They broke dividends apart into like base-ten units, applying the distributive property to find quotients place by place.

In Topic C, students connect estimation to place value and determine that the standard algorithm is simply a tally system arranged in place value columns (6.NS.2). Students understand that when they “bring down” the next digit in the algorithm, they are essentially distributing, recording, and shifting to the next place value. They understand that the steps in the algorithm continually provide better approximations to the answer.

Students further their understanding of division as they develop fluency in the use of the standard algorithm to divide multi-digit decimals (6.NS.3). They make connections to division of fractions and rely on mental math strategies to implement the division algorithm when finding the quotients of decimals. In the final topic, students think logically about multiplicative arithmetic.

In Topic D, students apply odd and even number properties and divisibility rules to find factors and multiples. They extend this application to consider common factors and multiples and find greatest common factors and least common multiples. Students explore and discover that Euclid’s Algorithm is a more efficient way to find the greatest common factor of larger numbers and see that Euclid’s Algorithm is based on long division. The module comprises 21 lessons; four days are reserved for administering the Mid-and End-of-Module Assessments, returning the assessments, and re-mediating or providing further applications of the concepts.

The Mid-Module Assessment follows Topic B. The End-of-Module Assessment follows Topic C.

### Focus Standards

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc). How much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).

### Foundational Standards

Gain familiarity with factors and multiples.
4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1– 100 is prime or composite.

Understand the place value system.
5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Apply and extend previous understandings of multiplication and division to multiply and
divide fractions.
5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do
the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by fractions.

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷
(1/5) = 20 because 20 × (1/5) = 4.

### Focus Standards for Mathematical Practice

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

Students use concrete representations when understanding the meaning of division and apply it to the division of fractions. They ask themselves, “What is this problem asking me to find?” For instance, when determining the quotient of fractions, students ask themselves how many sets or groups of the divisor is in the dividend. That quantity is the quotient of the problem. They solve simpler problems to gain insight into the solution. They will confirm, for example, that 10 ÷ 2 can be found determining how many groups of two are in ten. They will apply that strategy to the division of fractions. Students may use pictorial representations such as area models, array models, number lines, and drawings to conceptualize and solve problems.

MP.2 Reason abstractly and quantitatively.

Students make sense of quantities and their relationships in problems. They understand “how many” as it pertains to the divisor in a quotient of fractions problem. They understand and use connections between divisibility and the greatest common factor to apply the distributive property. Students consider units and labels for numbers in contextual problems and consistently refer to what the labels represent to make sense in the problem. Students rely on estimation and properties of operations to justify the reason for their answers when manipulating decimal numbers and their operations. Students reason abstractly when applying place value and fraction sense when determining
the placement of a decimal point.

MP.6 Attend to Precision.

Students use precise language and place value when adding, subtracting, multiplying, and dividing by multi-digit decimal numbers. Students read decimal numbers using place value. For example, 326.31 is read as three hundred twenty-six and thirty-one hundredths. Students calculate sums, differences, products, and quotients of decimal numbers with a degree of precision appropriate to the problem context.

MP.7 Look for and make use of structure.

Students find patterns and connections when multiplying and dividing multi-digit decimals. For instance, they use place value to recognize that the quotient of: 22.5÷0.15, is the same as the quotient of: 2250 ÷ 15. Students recognize that when expressing the sum of two whole numbers using the distributive property, for example: 36 + 48 = 12(3+4), the number 12 represents the greatest common factor of 36 and 48 and that 36 and 48 are both multiples of 12. When dividing fractions, students recognize and make use of a related multiplication problem or create a number line and use skip counting to determine the number of times the divisor is added to obtain the dividend. Students use the familiar structure of long division to find the greatest common factor in another way, specifically the Euclidean Algorithm.

MP.8 Look for and express regularity in repeated reasoning.

Students determine reasonable answers to problems involving operations with decimals. Estimation skills and compatible numbers are used. For instance, when 24.385 is divided by 3.91, students determine that the answer will be close to the quotient of 24 ÷ 4, which equals 6. Students discover, relate, and apply strategies when problem-solving, such as the use of the distributive property to solve a multiplication problem involving fractions and/or decimals (e.g., 350 × 1.8= 350(1+0.8)= 350 + 280 = 630). When dividing fractions, students may use the following reasoning: Since 2/7+2/7+2/7=6/7, then 6/7÷2/7=3; and so I can solve fraction division problems by first getting common denominators and then solving the division problem created by the numerators. Students understand the long-division algorithm and the continual breakdown of the dividend into different place value units. Further, students use those repeated calculations and reasoning to determine the greatest common factor of two numbers using the Euclidean Algorithm.

### Terminology ` New or Recently Introduced Terms

 Greatest Common Factor

The largest quantity that factors evenly into two or more integers; the GCF of 24 and 36 is 12 because when all of the factors of 24 and 36 are listed, the largest factor they share is 12.

 Least Common Multiple

The smallest quantity that is divisible by two or more given quantities without a remainder; the LCM of 4 and 6 is 12 because when the multiples of 4 and 6 are listed, the
smallest or first multiple they share is 12.

 Multiplicative Inverses

Two numbers whose product is 1 are multiplicative inverses of one another. For example, 3/4  and 4/3 are multiplicative inverses of one another because  3/4 x 4/3 = 4/3 x 3/4 = 1.

### Topic A: Dividing Fractions by Fractions  6.NS.1

In Topic A, students extend their previous understanding of multiplication and division to divide fractions by fractions. Students determine quotients through visual models, such as bar diagrams, tape diagrams, arrays, and number line diagrams. They construct division stories and solve word problems involving division of fractions (6.NS.1). Students understand and apply partitive division of fractions to determine how much is in each group. They explore real-life situations that require them to ask themselves, “How much is one share?” and “What part of the unit is that share?” Students use measurement to determine quotients of fractions. They are presented conceptual problems where they determine that the quotient represents how many of the divisor is in the dividend. Students look for and uncover patterns while modeling quotients of fractions to ultimately discover the relationship between multiplication and division. Later in the module, students will understand and apply the direct correlation of division of fractions to division of decimals.

Lessons 1–2: Interpreting Division of a Whole Number by a Fraction—Visual Models

 Students use visual models such as fraction bars, number lines, and area models to show the quotient of whole numbers and fractions. Students use the models to show the connection between those models and the multiplication of fractions.
 Students divide a fraction by a whole number

 Students understand the difference between a whole number being divided by a fraction and a fraction being divided by a whole number

Lessons 3–4: Interpreting and Computing Division of a Fraction by a Fraction—More Models

 Students use visual models such as fraction bars and area models to show the division of fractions by fractions with common denominators.
 Students make connections to the multiplication of fractions. In addition, students understand that the division of fractions require students to ask, “How many groups of the divisor are in the dividend?” to get the quotient.

 Students use visual models such as fraction bars and area models to divide fractions by fractions with different denominators.
 Students make connections between visual models and multiplication of fractions

Lesson 5: Creating Division Stories

 Students demonstrate further understanding of division of fractions when they create their own word problems.
 Students choose a measurement division problem, draw a model, find the answer, choose a unit, and then set up a situation. Further, they discover that they must try several situations and units before finding which are realistic with given numbers.

Lesson 6: More Division Stories

 Students demonstrate further understanding of division of fractions when they create their own word problems.
 Students choose a partitive division problem, draw a model, find the answer, choose a unit, and then set up a situation. Further, they practice trying several situations and units before finding which are realistic with given numbers.

Lesson 7: The Relationship Between Visual Fraction Models and Equations

 Students formally connect models of fractions to multiplication through the use of multiplicative inverses as they are represented in models.

Lesson 8: Dividing Fractions and Mixed Numbers

 Students divide fractions by mixed numbers by first converting the mixed numbers into a fraction with a value larger than one.
 Students use equations to find quotients.

### Topic B: Multi‐Digit Decimal Operations—Adding, Subtracting, and Multiplying 6.NS.3

Prior to division of decimals, students will revisit all decimal operations in Topic B. Students have had extensive experience of decimal operations to the hundredths and thousandths (5.NBT.7), which prepares them to easily compute with more decimal places. Students begin by relating the first lesson in this topic to mixed numbers from the last lesson in Topic A. They find that sums and differences of large mixed numbers can be more efficiently determined by first converting to a decimal and then applying the standard algorithms (6.NS.3). Within decimal multiplication, students begin to practice the distributive property. Students use arrays and partial products to understand and apply the distributive property as they solve multiplication problems involving decimals. Place value enables students to determine the placement of the decimal point in products and recognize that the size of a product is relative to each factor. Students discover and use connections between fraction multiplication and decimal multiplication.

Lesson 9: Sums and Differences of Decimals

 Students relate decimals to mixed numbers and round addends, minuends, and subtrahends to whole numbers in order to predict reasonable answers.
 Students use their knowledge of adding and subtracting multi-digit numbers to find the sums and differences of decimals.
 Students understand the importance of place value and solve problems in real-world contexts.

Lesson 10: The Distributive Property and Products of Decimals

 Through the use of arrays and partial products, students strategize and apply the distributive property to find the product of decimals.

Lesson 11: Fraction Multiplication and the Products of Decimals

Students use estimation and place value to determine the placement of the decimal point in products and to determine that the size of the product is relative to each factor.
 Students discover and use connections between fraction multiplication and decimal multiplication.
 Students recognize that the sum of the number of decimal digits in the factors yields the decimal digits in the product.

### Topic C: Dividing Whole Numbers and Decimals 6.NS.2, 6.NS.3

In Topic C, students build from their previous learning to fluently divide numbers and decimals. They apply estimation to place value and determine that the standard algorithm is simply a tally system arranged in place value columns (6.NS.2). Students understand that when they “bring down” the next digit in the algorithm, they are distributing, recording, and shifting to the next place value. They understand that the steps in the algorithm continually provide better approximations to the answer. Students further their understanding of division as they develop fluency in the use of the standard algorithm to divide multi-digit decimals (6.NS.3). They make connections to division of fractions and rely on mental math strategies in order to implement the division algorithm when finding the quotients of decimals.

Lesson 12: Estimating Digits in a Quotient

 Students connect estimation with place value in order to determine the standard algorithm for division

Lesson 13: Dividing Multi-Digit Numbers Using the Algorithm

 Students understand that the standard algorithm of division is simply a tally system arranged in place value columns.

Lesson 14: The Division Algorithm—Converting Decimal Division into Whole Number Division Using Fractions

 Students use the algorithm to divide multi-digit numbers with and without remainders. Students compare their answer to estimates to justify reasonable quotients.
 Students understand that when they “bring down” the next digit in the algorithm, they are distributing, recording, and shifting to the next place value.

Lesson 15: The Division Algorithm—Converting Decimal Division into Whole Number Division Using Mental Math

 Students use their knowledge of dividing multi-digit numbers to solve for quotients of multi-digit decimals.
 Students understand the mathematical concept of decimal placement in the divisor and the dividend and its connection to multiplying by powers of 10.

### Topic D: Number Theory—Thinking Logically About Multiplicative Arithmetic 6.NS.4

Students have previously developed facility with multiplication and division. They now begin to reason logically about them in Topic D. Students apply odd and even number properties and divisibility rules to find factors and multiples. They extend this application to consider common factors and multiples and find greatest common factors and least common multiples. Students explore and discover that Euclid’s Algorithm is a more efficient means to finding the greatest common factor of larger numbers and determine that Euclid’s Algorithm is based on long division.

Lesson 16: Even and Odd Numbers

 Students apply odd and even numbers to understand factors and multiples

Lesson 17: Divisibility Tests for 3 and 9

 Students apply divisibility rules, specifically for 3 and 9, to understand factors and multiples

Lesson 18: Least Common Multiple and Greatest Common Factor

 Students find the least common multiple and greatest common factor and apply factors to the Distributive Property

Lesson 19: The Euclidean Algorithm as an Application of the Long Division Algorithm

 Students explore and discover that Euclid’s Algorithm is a more efficient means to finding the greatest common factor of larger numbers and determine that Euclid’s Algorithm is based on long division.

## Review 4

### 1.  Kelly used 9 centimeters of tape to wrap 3 presents. How many presents did Kelly wrap if she used 18 centimeters of tape? Solve using unit rates.

Unit Rates Problem — 6.RP.2 —First find the unit rate.  9 centimeters/ 3 presents – 3 centimeters per present,  Now use the unit rate in a formula: Total tape used = Tape used for 1 present x Number of presents or  18 = 3 x number of presents  — a missing factor problem — so Number of presents is 6 since 3 x 6 = 18

### 2.  Are these ratios equivalent? (yes/no why)    (a)  2/6  and 4/16     (b)  3/5 and  6/15    (c)  2/5 and 4/10

Equivalent Ratio Problem —6.RP.3  — Two ratios  a/b  and c/d are equivalent if a/b = c/d or that their cross-products  ad = bc are equal.  (a) No … 2 x 16 =32  is not equal to 6 x 4 = 24    (b) No … 5 x 6 = 30 is not equal to 3 x 15 = 45    (c)  Yes … 5 x 4 = 20 is equal to 2 x 10 =20

### 3.  Set up standard algorithms and show all work in finding:  (a)  123.08 + 2.1 = ?   (b)  302.45 – 123.57 = ?  ( c)  0.125 x 5.62 = ?  (d)  1.2 divided by 0.03

Set up standard algorithm problems — 6.NS.3 —

### 4.  Makiah buys a math text book online  for \$56.23.  If shipping costs are an additional 15% of the price   (a)  how much shipping will he pay? (b) what will be his total cost?  Show all work.

Percent problem — 6.RP.3.c — Convert 15% to 0.15.  Multiply 0.15 x  \$56.23

### 5.  Write equivalent expressions for: (a)   b+b+ x +x – f     (b)  2b +3c    (c)  g-d-d-d

(a)  2b-2x-f          (b)   b+b+c+c+c         (c)   g-3d

## Standards: California Common Core Content Standards

### 6.6.RP Ratios & Proportions  Ratios and Proportional Relationships

Understand ratio concepts and use ratio reasoning to solve problems.

6.6.RP.1  Ratio   Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

6.6.RP.2 Unit Rate Understand the concept of a unit rate a/b associated with a ratio a:b with b is not equal to 0, and use rate language in the context of a ratio relationship.

### 6.6.RP.3  Ratio Reasoning  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.[

6.6.RP.3.a Compare Ratios Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

6.6.RP.3.b Unit Rate Problems  Solve unit rate problems including those involving unit pricing and constant speed.

6.6.RP.3.c  Percent Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

6.6.RP.3.d  Measurement Units  Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities..ions involving fractions and mixed numbers .

### 6.6.NS The Number System Extension Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

6.6.NS.1  Quotients of Fractions  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

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

6.6.NS.2 Multi-digit Division  Fluently divide multi-digit numbers using the standard algorithm.

6.6.NS.3  Multi-digit Decimal Arithmetic  Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

6.6.NS.4  Greatest Common Factor  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

Apply and extend previous understandings of numbers to the system of rational numbers.

6.6.NS.5  Positive & Negative Numbers  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

### 6.6.NS.6  Signed Number Line Diagrams  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.

6.6.NS.6.a  Opposite Numbers  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.

6.6.NS.6.b  Ordered Pairs  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

6.6.NS.6.c  Read-Write Positions on Number Line  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.

### 6.6.NS.7  Order Rational Numbers  Understand ordering and absolute value of rational numbers.

6.6.NS.7.a   Inequalities  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

6.6.NS.7.b  Order Rational Numbers  Write, interpret, and explain statements of order for rational numbers in real-world contexts.

6.6.NS.7.c  Absolute Value  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

6.6.NS.7.d Absolute Value Order  Distinguish comparisons of absolute value from statements about order.

6.6.NS.8  Graph Points in Plane  Distance between Points  Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

### 6.6.EE Expressions and Equations Apply and extend previous understandings of arithmetic to algebraic expressions.

#### 6.6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

• Write variable expressions to represent word problems (Sixth grade – P.1)
• Does x satisfy the equation? (Sixth grade – P.5)
• Solve one-step equations with whole numbers (Sixth grade – P.6)
• Solutions to variable inequalities (Sixth grade – P.23)
• Solve one-step linear inequalities (Sixth grade – P.24)
• #### 6.6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

• Write variable expressions to represent word problems (Sixth grade – P.1)
• Solve word problems involving two-variable equations (Sixth grade – P.3)
• Convert between Celsius and Fahrenheit (Sixth grade – Y.10)
• #### 6.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

• Solve one-step equations with whole numbers (Sixth grade – P.6)
• Solve one-step equations involving decimals, fractions, and mixed numbers (Sixth grade – P.7)
• #### 6.6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

• Inequalities on number lines (Sixth grade – P.22)

• #### 6.6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

• Solve word problems involving two-variable equations (Sixth grade – P.3)
• Complete a function table (Sixth grade – P.10)
• Write linear functions (Sixth grade – P.11)
• Linear function word problems (Sixth grade – P.12)

### 6.6.G Geometry

• #### 6.6.G.1 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.

• Area (Sixth grade – Z.23)
• Area of compound figures (Sixth grade – Z.24)
• Compare area and perimeter of two figures (Sixth grade – Z.28)
• #### 6.6.G.2 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. 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.

• Volume of cubes and rectangular prisms (Sixth grade – Z.36)
• #### 6.6.G.3 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. Apply these techniques in the context of solving real-world and mathematical problems.

• Coordinate graphs review (Sixth grade – Q.1)
• Distance between two points (Sixth grade – Q.4)
• #### 6.6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

• Nets of 3-dimensional figures (Sixth grade – Z.35)
• Surface area of cubes and rectangular prisms (Sixth grade – Z.37)
• Volume and surface area of triangular prisms (Sixth grade – Z.38)

### 6.6.SP Statistics and Probability

• #### 6.6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

• Identify representative, random, and biased samples (Sixth grade – S.4)
• #### 6.6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

• Stem-and-leaf plots (Sixth grade – R.3)
• Create line plots (Sixth grade – R.5)
• Interpret box-and-whisker plots (Sixth grade – R.18)
• #### 6.6.SP.3 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.

• Calculate mean, median, mode, and range (Sixth grade – S.1)
• Interpret charts to find mean, median, mode, and range (Sixth grade – S.2)
• Mean, median, mode, and range: find the missing number (Sixth grade – S.3)

• #### 6.6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

• Interpret pictographs (Sixth grade – R.1)
• Create pictographs (Sixth grade – R.2)
• Stem-and-leaf plots (Sixth grade – R.3)
• Interpret line plots (Sixth grade – R.4)
• Create line plots (Sixth grade – R.5)
• Create frequency tables (Sixth grade – R.7)
• Interpret bar graphs (Sixth grade – R.8)
• Create bar graphs (Sixth grade – R.9)
• Interpret double bar graphs (Sixth grade – R.10)
• Create double bar graphs (Sixth grade – R.11)
• Create histograms (Sixth grade – R.12)
• Circle graphs with fractions (Sixth grade – R.13)
• Interpret line graphs (Sixth grade – R.14)
• Create line graphs (Sixth grade – R.15)
• Interpret double line graphs (Sixth grade – R.16)
• Create double line graphs (Sixth grade – R.17)
• Interpret box-and-whisker plots (Sixth grade – R.18)
• Choose the best type of graph (Sixth grade – R.19)

• #### 6.6.SP.5.a Reporting the number of observations.

• Create frequency tables (Sixth grade – R.7)
• Create histograms (Sixth grade – R.12)
• #### 6.6.SP.5.b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

• Identify representative, random, and biased samples (Sixth grade – S.4)
• #### 6.6.SP.5.c 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.

• Calculate mean, median, mode, and range (Sixth grade – S.1)
• Interpret charts to find mean, median, mode, and range (Sixth grade – S.2)
• Mean, median, mode, and range: find the missing number (Sixth grade – S.3)