Unit

Unit Storyline

Full Objectives

Unit 1

Comparing & Ordering Rational Numbers

Timing

1 - 2 weeks

Objectives

6.N.1.5
6.N.1.6
6.N.1.4
6.N.1.1
6.N.1.2

Before comparing and ordering rational numbers it is important to have a deep understanding of equivalent fractions. This unit begins by using area arrays to teach factors, primes and composites, GCF and LCM. Students build upon these concepts with prime factorization with exponents to prepare to manipulate fractions using multiple representations for the purpose of ordering and comparing on a number line and finally with comparison symbols. Students build efficient and appropriate strategies for comparing and ordering positive rational numbers. Students build upon their fifth-grade experiences with decimal place value and converting fractions to decimals to begin exploring placing these rational numbers on a number line.

6.N.1.5 Factor whole numbers and express prime and composite numbers as a product of prime factors with exponents.

6.N.1.6 Determine the greatest common factors and least common multiples. Use common factors and multiples to calculate with fractions, find equivalent fractions and express the sum of two-digit numbers with a common factor using the distributive property.

*6.N.1.4 Determine equivalencies among fractions, decimals, and percents. Select among these representations to solve problems.

*6.N.1.1 Represent integers with counters and on a number line and positive rational numbers on a number line, recognizing the concepts of opposites, direction, and magnitude; use integers and rational numbers in real-world and mathematical situations, explaining the meaning of 0 in each situation.

*6.N.1.2 Compare and order positive rational numbers, represented in various forms, or integers using the symbols <, >, and =.

Unit 2

Operation with Fractions

Timing

3 - 4 weeks

6.N.4.2

Now that students are comfortable with manipulating fractions students are able to explore multiplication of fractions with area arrays. By using fractional areas students are able to develop a conceptual understanding of ½ of ¼ etc. Dividing fractions is best represented with fraction strips to show how many groups of ¼ are in ½ . Benchmark fractions are used to estimate solutions to check the reasonableness of real world situations.

*6.N.4.2 Illustrate multiplication and division of fractions and decimals to show connections to fractions, whole number multiplication, and inverse relationships. 

*6.N.4.1 Estimate solutions to problems with whole numbers, decimals, fractions, and mixed numbers and use the estimates to assess the reasonableness of results in the context of the problem.

*6.N.4.3 Multiply and divide fractions and decimals using efficient and generalizable procedures.

*6.N.4.4 Solve and interpret real-world and mathematical problems including those involving money, measurement, geometry, and data requiring arithmetic with decimals, fractions, and mixed numbers. 

Unit 3

Introduction to Algebra

Timing

1 - 2 weeks

Objectives

6.A.1.1

6.A.1.2

Before introducing the concept of “integers”, students will explore negative numbers in the simple, visual context of a coordinate system. This will equip the students with a concept of negative numbers on a number line. Having this directly after fractions on a number line will help with plotting halves and fourths. This follows well with varying quantities represented as multiple coordinate points on a graph. Linking rules, graphs and tables will build the foundation of graphing equations in Algebra I.

*6.A.1.1 Plot integer- and rational-valued (limited to halves and fourths) ordered-pairs as coordinates in all four quadrants and recognize the reflective relationships among coordinates that differ only by their signs.

6.A.1.2 Represent relationships between two varying quantities involving no more than two operations with rules, graphs, and tables, translate between any two of these representations.

Unit 4

Operations with Decimals

Timing

2 - 3 weeks

Objectives

6.N.4.2

6.N.4.3

6.N.4.1

6.N.4.4

Area array models are the best way to illustrate multiplication of decimals. This method works well for tenths to allow students to develop efficient algorithms. Division of decimals is a complex and sometimes abstract process that requires the use of visual and graphical representations. It is also useful to integrate estimation to ensure the place value of the product and quotient are correct.

*6.N.4.2 Illustrate multiplication and division of fractions and decimals to show connections to fractions, whole number multiplication, and inverse relationships. 

*6.N.4.3 Multiply and divide fractions and decimals using efficient and generalizable procedures.

*6.N.4.1 Estimate solutions to problems with whole numbers, decimals, fractions, and mixed numbers and use the estimates to assess the reasonableness of results in the context of the problem.

*6.N.4.4 Solve and interpret real-world and mathematical problems including those involving money, measurement, geometry, and data requiring arithmetic with decimals, fractions, and mixed numbers.

Unit 5

Measures of Central Tendency

Timing

1 - 2 weeks

Objectives

6.D.1.1

6.D.1.2

6.D.1.3

6.N.4.4

Students need lots of practice multiplying and dividing decimals. Problems involving measures of central tendency provide an excellent opportunity to practice these skills. Newspaper graphs and data tables are excellent sources of real world data to practice algorithms and analyze data. Box-and-whisker plots can also be created from this data.

6.D.1.1 Calculate the mean, median, and mode for a set of real-world data

6.D.1.2 Explain and justify which measure of central tendency (mean, median, or mode) would provide the most descriptive information for a given set of data.

6.D.1.3 Create and analyze box and whisker plots exploring how each segment contains one-quarter of the data.

*6.N.4.4 Solve and interpret real-world and mathematical problems including those involving money, measurement, geometry, and data requiring arithmetic with decimals, fractions, and mixed numbers.

Unit 6

Operations with Integers

Timing

1 - 3 weeks

Objectives

At this point, students have placed positive whole numbers, fractions, and decimals on a number line. They have explored the negative side of a number line with the coordinate plane. Now they are ready to extend this to the concept of integers. Concepts such as opposites, direction,and magnitude can be easily integrated into the conversation. Real world situations are perfect for estimating solutions to integer problems. Ex. If the temperature outside is 5 degrees and drops by 10 degrees will the temperature be above or below zero. Counters, number lines, algorithms and visual representations are excellent ways to form an understanding of the algorithms used to add and subtract integers.

*6.N.1.1 Represent integers with counters and on a number line and rational numbers on a number line, recognizing the concepts of opposites, direction, and magnitude; use integers and rational numbers in real-world and mathematical situations, explaining the meaning of 0 in each situation

6.N.2.1 Estimate solutions to addition and subtraction of integers problems in order to assess the reasonableness of results.

6.N.2.2 Illustrate addition and subtraction of integers using a variety of representations.

6.N.2.3 Add and subtract integers; use efficient and generalizable procedures including but not limited to standard algorithms.

Unit 7

Order of Operations & Substitution

Timing

1 - 2 weeks

Objectives

6.A.2.1

6.A.1.3

6.A.3.1

Beginning with writing expressions for real-world situations, students will learn to use mathematical properties (commutative, associative and distributive) to evaluate expressions for a given variable. By linking these three objectives together, you add relevance to each step by creating a process.

6.A.2.1 Generate equivalent expressions and evaluate expressions involving positive rational numbers by applying the commutative, associative, and distributive properties and order of operations to solve real-world and mathematical problems.

*6.A.1.3 Use and evaluate variables in expressions, equations, and inequalities that arise from various contexts, including determining when or if, for a given value of the variable, an equation or inequality involving a variable is true or false. 

*6.A.3.1 Represent real-world or mathematical situations using expressions, equations, and inequalities involving variables and rational numbers.

Unit 8

Equations & Inequalities

Timing

2 - 3 weeks

Objectives

6.A.1.3

6.A.3.2

6.A.3.1

Building off of the experience of using mathematical properties and operations to evaluate expressions, we will now extend this knowledge to include equations and inequalities. Continuing to use real-world situations to create equations and inequalities students will solve one-step equations. Variables may represent positive rational numbers including decimals and fractions.

*6.A.1.3 Use and evaluate variables in expressions, equations, and inequalities that arise from various contexts, including determining when or if, for a given value of the variable, an equation or inequality involving a variable is true or false.

6.A.3.2 Use number sense and properties of operations and equality to solve real-world and mathematical problems involving equations in the form x + p = q and px = q, where x, p, and q are non-negative rational numbers. Graph the solution on a number line, interpret the solution in the original context, and assess the reasonableness of the solution.

*6.A.3.1 Represent real-world or mathematical situations using expressions, equations, and inequalities involving variables and rational numbers.

Unit 9

Ratios

Timing

1 - 2 weeks

Objectives

6.N.3.1

6.N.3.2

6.N.3.4

6.N.3.3

Students will use their fluency with fractions to process ratio problems. Using concepts of equivalency and basic comparison procedures, ratios become an extension of fractions.

6.N.3.1 Identify and use ratios to compare quantities. Recognize that multiplicative comparison and additive comparison are different.

6.N.3.2 Determine the unit rate for ratios.

6.N.3.4 Use multiplicative reasoning and representations to solve ratio and unit rate problems.

*6.N.3.3 Apply the relationship between ratios, and equivalent fractions, and percents to solve problems in various contexts, including those involving mixture and concentrations.

Unit 10

Converting Measurements

Timing

1 - 2 weeks

Objectives

6.GM.3.1

6.GM.3.2

Students must have a solid foundation of unit awareness in order to accurately process unit conversions. Students may process conversions from the perspective of equivalent ratios or they may process as a multiplicative comparison. Ex.  For every 1 ft there will be 12 in.

6.GM.3.1 Estimate weights, capacities and geometric measurements using benchmarks in customary and metric measurement systems with appropriate units.

6.GM.3.2 Solve problems in various real-world and mathematical contexts that require the conversion of weights, capacities, geometric measurements, and time within the same measurements systems using appropriate units.

Unit 11

Percents

Timing

1 - 2 weeks

Objectives

Students will begin to explore percents using visual and graphical representations as well as real-world situations that students are familiar with. Students will add percents to their cache of comparing, ordering and equivalent conversions among representations. Understanding which representation is the most appropriate for a given situation will come with repeated exposure to real-world problems.

6.N.1.3 Explain that a percent represents parts “out of 100” and ratios “to 100”.

*6.N.1.4 Determine equivalencies among fractions, decimals, and percents. Select among these representations to solve problems.

*6.N.1.2 Compare and order positive rational numbers, represented in various forms, or integers using the symbols <, >, and =.

*6.N.3.3 Apply the relationship between ratios, equivalent fractions, and percents to solve problems in various contexts, including those involving mixture and concentrations.

*6.N.4.4 Solve and interpret real-world and mathematical problems including those involving money, measurement, geometry, and data requiring arithmetic with decimals, fractions, and mixed numbers.

Unit 12

Angle Relationships

Timing

1 - 2 weeks

Objectives

6.GM.2.1

6.GM.2.2

6.A.3.1

Begin by letting students explore angles with protractors and creating polygons from triangles. Students should have mastered one step equations at this point so reinforce this process by setting up missing angle problems as an algebraic equation. Ex.  x + 20 + 60 = 180. Solve for x.

6.GM.2.1 Solve problems using the relationships between the angles (vertical, complementary, and supplementary) formed by intersection lines.

6.GM.2.2 Develop and use the fact that the sum of interior angles of a triangle is 180 degrees to determine missing angle measures in a triangle.

*6.A.3.1 Represent real-world or mathematical situations using expressions, equations, and inequalities involving variables and rational numbers.

Unit 13

Transformations

Timing

2 - 3 weeks

Objectives

6.A.1.1

6.GM.4.1

6.GM.4.2

6.GM.4.3

This unit focuses on using the coordinate system to discuss transformations and distance measurements. It should be introduced after integers and fractions on a number line to assist with plotting ordered-pairs with integers and rational numbers.

*6.A.1.1 Plot integer- and rational-valued (limited to halves and fourths) ordered-pairs as coordinates in all four quadrants and recognize the reflective relationships among coordinates that differ only by their signs.

6.GM.4.1 Predict, describe, and apply translations (slides), reflections (flips), and rotations (turns) to a two-dimensional figure.

6.GM.4.2 Recognize that translations, reflections, and rotations preserve congruency and use them to show that two figures are congruent.

6.GM.4.3 Use the distances between two points that are either vertical or horizontal to each other (not requiring the distance formula) to solve real-world and mathematical problems about congruent two-dimensional figures.

Unit 14

Area

Timing

2 - 4 weeks

Objectives

6.GM.1.1

6.GM.4.4

6.GM.1.2

6.GM.1.3

6.N.4.4

This unit will use a common method of exploration of area using tangrams and simple shapes to develop formulas for parallelograms and triangles. Once students have a grasp on area and its’ additive properties, solving for polygons should become intuitive.

6.GM.1.1 Develop and use formulas for the area of squares and parallelograms using a variety of methods including but not limited to the standard algorithm.

6.GM.4.4 Identify and describe the lines of symmetry in two-dimensional shapes.

6.GM.1.2 Develop and use formulas to determine the area of triangles.

6.GM.1.3 Find the area of right triangles, other triangles, special quadrilaterals and polygons that can be decomposed into triangles and other shapes to solve real-world and mathematical problems.

*6.N.4.4 Solve and interpret real-world and mathematical problems including those involving money, measurement, geometry, and data requiring arithmetic with decimals, fractions, and mixed numbers.

Unit 15

Probability

Timing

1 - 2 weeks

Objectives

6.D.2.1

6.D.2.2

6.D.2.3

This unit starts by introducing students to probability on a probability continuum in which students decide where the probability for a given situation lies on the continuum from impossible to certain. Probabilities associated with the continuum can be expressed as fractions, decimals, or percents. Students also learn how to determine a sample space for a given experiment and which members of the sample space are included in a certain event for the experiment. Simple experiments are also used to compare relative frequencies to the known probability for the experiment. For example, the relative frequency for getting tails while flipping a coin will approach the known probability of 50% the more times you flip the coin.

6.D.2.1 Represent possible outcomes using a probability continuum from impossible to certain. 

6.D.2.2 Determine the sample space for a given experiment and determine which members of the sample space are related to certain events. Sample space may be determined by the use of tree diagrams, tables or pictorial representations.

6.D.2.3 Demonstrate simple experiments in which the probabilities are known and compare the resulting relative frequencies with the known probabilities, recognizing that there may be differences between the two results.