**Grade 6 Math Outline for the year 2017-2018**

Ratios

Ratios

In this unit, students develop a conceptual understanding of ratios. Throughout the unit students apply this understanding to solve problems involving ratios and rates. Students’ understanding of ratios is built upon their prior work with whole number multiplication and division. In this unit students begin to shift their focus from noticing more familiar additive relationships to noticing less familiar multiplicative relationships between quantities. Students explore relationships within and between equivalent ratios and use these relationships to find missing values in ratio tables. Students then apply this thinking to measurement systems and magnitude. The work that students engage with in this unit will be an integral component of their work later in Grade 6 as they begin to work with unit rates and percent, and later as they extend this understanding to proportionality, and eventually slope.

**Lessons**

Understand ratio relationships

Lesson objective: Understand the distinction between additive comparisons and multiplicative comparisons of quantities.

Identify and describe ratio relationships

Lesson objective: Represent a situation using a ratio, in multiple different ways.

Practice using ratios in different forms

Lesson objective: Create and simplify equivalent ratios using visual models, tables, and fractions.

Use ratio relationships to solve problems

Lesson objective: Apply creating and simplifying equivalent ratios to solve a problem.

Understand equivalent ratios

Lesson objective: Understand that multiplicative relationships exist both between and within equivalent ratios.

Find missing values in ratio relationships

Lesson objective: Use multiplicative "between" and "within" relationships to find missing values in a table.

Use equivalent ratios to solve problems

Lesson objective: Use "between" and "within" relationships in equivalent ratios in order to find missing values in a ratio table.

Connect measurement systems and ratios

Lesson objective: Understand that ratios can be used to convert between units of measurement.

Practice measurement conversions

Lesson objective: Convert measurements in one unit to another unit.

Solve problems using measurement conversions

Lesson objective: Apply conversions of measurement systems using ratio relationships to a real-life situation.

**Rates Including Percent**

In this unit, students extend their understanding of ratio and proportion. Ratio tables are the primary representation taught for scaling ratios up and down, although students are welcome to use other methods. Students learn to look for relationships “within” the components of a ratio and “between” components of ratios that are in a proportional relationship. They learn that every ratio can be associated with a unit rate and that there are two different ways to write a unit rate. Students are likely to struggle with the concept that a fraction bar, ratio bar, and division bar are the same. It is important to keep careful track of the units in ratios and rates in order to scale them up and down correctly. The final five lessons connect ratios to percents. Percent is a rate per 100. To find percent, the whole is divided equally into one hundred parts. Knowing the whole is divided into 100 parts, use of the given percentage can allow students to take a value (percent) of the 100. Three types of percent problem are: part unknown, percent (or fraction) unknown, and whole unknown.

**Lessons**

Compare ratios: Campers and pies

Lesson objective: Understand that two ratios can be scaled up or down in order to make a comparison.

Use ratio tables to create proportions

Lesson objective: Create and use ratio tables to identify quantities in a proportional relationship.

Resize photos using proportional reasoning

Lesson objective: Solve problems involving proportional relationships.

Compare ratios using unit rate: What's the best deal?

Lesson objective: Understand that every ratio can be written as a unit rate.

Use unit rate to compare and find other ratios

Lesson objective: Practice finding unit rates, and use unit rates to compare and write other ratios.

Use rates to compare growth: Who grew faster?

Lesson objective: Apply knowledge of unit rate to determine which sibling grew faster.

Write a unit rate in different ways: At the market

Lesson objective: Understand that there are two unit rates associated with quantities a and b in a proportional relationship: "a per one b" and "b per one a."

Express ratios as two different unit rates

Lesson objective: Practice writing two different unit rates for quantities in a proportional relationship.

Compare speeds using unit rate: Rainy Day Olympics

Lesson objective: Apply knowledge for writing unit rates in two different ways to decide who completed events the fastest.

Understand comparing ratios using percent

Lesson objective: Understand that percent means standardizing ratios as a comparison to 100.

Use double number lines to determine percent of a number

Lesson objective: Use a double number line to determine percent of a given number.

Use a double number line to determine a value related to 100%

Lesson objective: Practice using a double number line to determine a number when a percent of it is known.

Use ratio tables to determine what percent one number is of another

Lesson objective: Use ratio tables to organize values and practice scaling the values up or down to calculate percent.

Solve real-world percent problems

Lesson objective: Apply the use of a double number line to solving problems involving percent.

**Multi-digit computation and finding common factors and multiples**

In this unit, students will extend their knowledge of place value and standard algorithms for operations to include decimals and multi-digit division of whole numbers. This unit also builds on students’ previous knowledge of how to find factors and multiples of a given number to the new knowledge of the unit of finding the greatest common factor of two numbers and the least common multiple of two numbers. Students also use factors and multiples to apply the distributive property to write equivalent expressions. Students are likely to struggle with the abstractness of the distributive property; the unit builds an understanding of the property with the use of area models.

**Lessons**

Use the standard algorithm for operations with decimals

Lesson objective: Extend understanding of the rules for all operations to include decimals and to understand that, like whole-numbers, decimal operations are based on the place value system.

Add and subtract decimals using the standard algorithm

Lesson objective: Fluently add and subtract decimals using the standard algorithm.

Multiply and divide decimals using the standard algorithm

Lesson objective: Fluently multiply and divide decimals using the standard algorithm.

Divide multi-digit numbers using the standard algorithm

Lesson objective: Fluently divide multi-digit numbers using the standard algorithm.

Understand least common multiple and greatest common factor

Lesson objective: Understand that a common multiple of a set of numbers has each number in the set as a factor. Furthermore, the list of factors of the common multiple also includes every factor of the numbers in the original set.

Practice least common multiple and greatest common factor

Lesson objective: Find the least common multiple and greatest common factor of two given numbers.

Apply knowledge of least common multiples and greatest common factors

Lesson objective: Generate a rule for the relationship among the product, GCF and LCM of two given numbers.

Understand the distributive property

Lesson objective: Understand how we can use the distributive property and common factors to write equivalent expressions.

Use the distributive property

Lesson objective: Fluently use the distributive property and the greatest common factor to write equivalent expressions.

Use distributive property to solve problems

Lesson objective: Write equivalent expressions using the greatest common factor and the distributive property.

**Dividing fractions**

In this unit, students will be introduced to the conceptual idea of dividing a fraction by a fraction by connecting it to prior knowledge of multiplying fractions and dividing whole numbers and unit fractions, along with models, and situations in this unit. Students will use common denominators to divide fractions, and then use the multiplicative inverse, or reciprocal, to divide fractions. Students may struggle conceptually when the dividend is less than the divisor, as well as when the divisor does not divide evenly into the dividend.

**Lessons**

Divide a fraction by a fraction by connecting to multiplication

Lesson objective: Understand that the connection between multiplication and division extends to situations in which it is necessary to divide fractions by fractions.

Connect fraction division with multiplication

Lesson objective: Fluently connect fraction division with fraction multiplication and make sense of fraction division situations.

Divide a fraction by a fraction using common denominators

Lesson objective: Understand that measurement division extends to division of fractions, and may be done with common denominators. The result names the number of groups.

Use common denominators to divide fractions

Lesson objective: Fluently use the common denominator strategy to divide fractions. Understand how the value of the dividend in relation to the divisor affects the value of the quotient.

Make sense of fraction division

Lesson objective: Fluently compute and interpret the quotient and remainder in a fraction division situation.

Use common denominators to divide fractions

Lesson objective: Connect fraction division to make equal portions of snack mix with and without left over.

Use multiplication and division to divide whole numbers and fractions

Lesson objective: Understand that the relationship between multiplication and division can be extended to dividing whole numbers and non-unit fractions.

Use multiplication and division to divide fractions

Lesson objective: Extend understanding that the connection between division and multiplication can be extended to fraction division.

Divide a fraction by a fraction using the multiplicative inverse

Lesson objective: Fluently connect fraction division with multiplication by using multiplicative inverse.

Divide fractions using the multiplicative inverse

Lesson objective: Solve missing length area problems with fractional areas and side lengths using division.

**Representing relationships**

In this unit, students will extend their understanding of proportional relationships and unit rates to write expressions and equations that model the relationship between proportional quantities. Students will identify the independent and dependent quantities, model the relationship in multiple ways, and find missing values in the relationship.

**Lessons**

Understand that mathematical relationships can be represented as expressions

Lesson objective: Understand that numerical values can be represented as expressions using a ratio and multiplication.

Represent mathematical relationships as expressions

Lesson objective: Fluently multiply and divide within a table to find a ratio to help write an algebraic expression.

Use substitution to find the value of variables in an expression

Lesson objective: Fluently write expressions and use substitution to solve real-world mathematical problems.

Mathematical relationships can be represented as expressions

Lesson objective: Apply the understanding of writing expressions to represent recipe amounts.

Understand that the relationship between two quantities can be written as an equation

Lesson objective: Extend understanding that mathematical relationships can be written as expressions and we can show the relationship between the expressions with an equation using variables to represent quantities.

Show the relationship between two quantities as an equation

Lesson objective: Graph coordinate points from an equation using a table.

Write the relationship between two quantities as an equation

Lesson objective: Fluently write equations from tables, graphs, situations, and patterns.

Write equations that relate two quantities to solve problems

Lesson objective: Apply the knowledge of how equations, graphs, and tables relate.

Model the relationship between independent and dependent variables using equations

Lesson objective: Understand that we can model the relationship between independent and dependent variables using equations.

Model the relationship between independent and dependent variables using equations

Lesson objective: Fluently match equations with tables, graphs, situations, or patterns with the same relationship while playing a game.

Understand that equations can be used to find missing values in a data set

Lesson objective: Extend understanding that the relationship between variables can be used to write equations.

Where did Mr. Lee get his pencils? Find missing values in data set

Lesson objective: Solve problems using the relationship between variables.

Write equations to find missing values in tables

Lesson objective: Write equations that model the relationship between two proportional variables and use the equation to identify missing values within the relationship.

Use graphs to find missing values in a data set

Lesson objective: Find missing values in a data set using points on a graph.

Currency Conversion: Use knowledge of unit rates to find missing values

Lesson objective: Apply knowledge of unit rates to convert various currencies to US dollars.

**Extending the number system**

In this unit, students will extend their understanding of numbers to include all rational numbers. They will use rational numbers in real-world contexts and understand the meaning of 0 in any situation. Students will discover integers and rational numbers, how to determine their value, and where those numbers live on the number line in relation to other, more familiar numbers. Students are likely to struggle with comparing the values of negative integers and creating meaningful context for interpreting the values of integers.

**Lessons**

Use positive and negative numbers in opposite directions from zero

Lesson objective: Understand that a number's value is determined by its distance and direction from zero.

Use positive and negative numbers to describe real world quantities

Lesson objective: Fluently use integers to represent real life situations.

Use negative numbers on a number line

Lesson objective: Fluently use a number line to represent negative numbers.

Understand positive and negative numbers in real life situations

Lesson objective: Solve problems using positive and negative numbers in real life situations.

Use integers, opposites, and zero on the number line

Lesson objective: Understand that the set of integers includes whole numbers, their opposites, and zero.

Find opposites and the meaning of zero

Lesson objective: Fluently determine opposites and the meaning of zero in real life situations.

Find the opposite of an opposite

Lesson objective: Connect what we know about opposites to determine the opposite of the opposite of a number.

Find relationships between integers

Lesson objective: Compare integers by looking at their distance and direction from zero and each other.

Use inequalities to compare and order integers

Lesson objective: Fluently compare integers using inequalities in real life situations.

Use relationships between integers to compare and analyze situations

Lesson objective: Solve problems by ordering and comparing integers based on their distance and direction from zero.

Understand that rational numbers have opposites with the same absolute value

Lesson objective: Understand that rational numbers have opposites and that the absolute value of those rational numbers is equivalent.

Find opposite numbers on a number line

Lesson objective: Fluently locate opposite numbers on a number line.

Determine absolute value

Lesson objective: Fluently locate and compare rational numbers on a number line based on absolute value.

Solve problems using absolute value

Lesson objective: Apply the concepts of absolute value and opposite numbers.

Understand the relationships between two rational numbers

Lesson objective: Understand that relationships between two rational numbers result in three possibilities for a and b: a < b, a > b, or a = b.

Establish the relationships between two rational numbers

Lesson objective: Fluently compare the inequality of rational numbers.

Understand the relationships between two rational numbers

Lesson objective: Compare rational numbers by determining their values.

**Relationships in the coordinate plane**

In this unit, students will extend their knowledge of rational numbers and absolute value to include plotting points and exploring distance using coordinates having the same x or y coordinate on a four quadrant coordinate plane.

**Lessons**

Use two pieces of information to designate a position on the coordinate plane

Lesson objective: Understand that we need two pieces of information to designate a position on the coordinate plane.

Use two pieces of information to designate a position on the coordinate plane

Lesson objective: Fluently use two pieces of information to designate a position on the coordinate plane.

Use two pieces of information to designate a position on the coordinate plane

Lesson objective: Apply ability to use two pieces of information to designate a position on the coordinate plane and demonstrate an understanding of its structure.

Extend understanding of opposites as points reflected across the x or y axis

Lesson objective: Extend understanding of opposites as points reflected across the x or y axis.

Extend understanding of opposites as points reflected across the x or y axis

Lesson objective: Fluently extend understanding of opposites as points reflect across the x or y axis.

Extend understanding of opposites as points reflected across the x or y axis

Lesson objective: Apply an extended understanding of opposites as points reflected across the x or y axis.

Using absolute value to find the distance between points

Lesson objective: Use absolute value to find distance between points

Using absolute value to find the distance between points

Lesson objective: Fluently use absolute value to find the distance between points.

Using absolute value to find the distance between points

Lesson objective: Apply the ability to use absolute value to find the distance between points.

**Algebraic expressions**

In this unit, students will apply and extend previous understandings of arithmetic expressions to algebraic expressions. Students will understand that an expression can represent either a process or a product, that properties such as the associative, distributive, or commutative property can be used to generate equivalent algebraic expressions, and that equivalent algebraic expressions have the same value for all possible values of the variable. This will lay the foundation for more formal work writing and solving equations in later grades.

**Lessons**

Understand that algebraic expressions represent both instructions and a quantity

Lesson objective: Understand that algebraic expressions can represent both a “recipe” and a quantity (that we may or may not be able to name with a number).

Write algebraic expressions and identify their parts using mathematical terminology

Lesson objective: Fluently write algebraic expressions and identify their parts using mathematical terminology.

Write and evaluate algebraic expressions containing exponents

Lesson objective: Fluently evaluate expressions containing whole-number exponents by applying Order of Operations.

Write and evaluate algebraic expressions to solve problems

Lesson objective: Solve problems by writing and evaluating an algebraic expression.

Understand that the commutative and associative properties apply to algebraic expressions

Lesson objective: Understand that the commutative and associative properties apply to algebraic expressions just as they do to numeric expressions.

Apply the associative and commutative properties to algebraic expressions

Lesson objective: Fluently apply the associative and commutative properties to algebraic expressions.

Use the associative and commutative properties with algebraic expressions to solve real-world problems

Lesson objective: Solve problems using the commutative and associative properties with algebraic expressions.

Understand that the distributive property applies to algebraic expressions

Lesson objective: Understand that the distributive property applies to algebraic expressions.

Use the distributive property to rewrite algebraic expressions

Lesson objective: Fluently rewrite algebraic expressions using the distributive property.

Use the distributive property with algebraic expressions to solve real-world problems

Lesson objective: Solve real-world problems using the distributive property with algebraic expressions.

Understand that equivalent expressions have the same value

Lesson objective: Extend understanding of equivalence to include algebraic expressions.

Use models to identify equivalent expressions

Lesson objective: Use area models to determine the equivalency of expressions.

Use substitution to identify equivalent expressions

Lesson objective: Identify equivalent expressions by substituting values in place of the variable.

Write equivalent expressions based on patterns

Lesson objective: Solve mathematical problems that connect algebraic representations to a geometric pattern.

Use properties of operations to identify equivalent expressions

Lesson objective: Apply the properties of operations to identify equivalent expressions.

**Understanding, writing, and solving equations and inequalities**

In earlier units, students have written equivalent expressions and equations. In this unit, students will further develop their understanding by solving one-step equations and inequalities with nonnegative numbers. They will use a number line to represent their solutions to inequalities. Students may have difficulty understanding that problems and number lines with inequalities will have infinite solutions instead of just one. Students will continue to strengthen their understanding of variables as representations of unknown values.

**Lessons**

Understanding substitution and evaluating equations

Lesson objective: Understand how to substitute a given value for a variable into an equation and evaluate to determine if that value makes the equation true.

Practicing substituting and evaluating

Lesson objective: Fluently substitute and evaluate numbers in the specified set to determine which number makes the equation true.

Using substituting and evaluating in real world problems

Lesson objective: Apply understanding of substituting values in for variables and evaluting an inequality in order to solve real-world problems.

Identifying equivalent equations

Lesson objective: Understand that equivalent equations have the same solution.

Identifying equivalent equations

Lesson objective: Fluently identify equivalent equations.

Comparing equation solutions

Lesson objective: Apply understanding of equivalent equations to compare equation solutions.

Using equations to solve problems

Lesson objective: Use equations to solve problems.

Solving equations with inverse operations

Lesson objective: Fluently solve equations using inverse operations.

Solving equations with inverse operations

Lesson objective: Fluently solve equations with rational numbers by using inverse operations.

Using equations to solve problems

Lesson objective: Apply understanding of how to solve equations in order to solve real-world problems.

Understanding that inequalities show a relationship between expressions

Lesson objective: Understand that inequalities show a relationship between expressions.

Practice writing inequalities

Lesson objective: Write inequalities to model situations.

Solve one-step inequalities using properties of equality

Lesson objective: Solve one-step inequalities using the properties of equality for solving equations.

Understand how number lines represent inequalities

Lesson objective: Understand that number lines are a visual way to represent inequalities.

Graph inequalities on number lines

Lesson objective: Graph inequalities on a number line using word problems and inequalities.

**Write and graph inequalities using word problems**

Lesson objective: Write, solve and graph inequalites using word problems.

Problem solving with area in 2-D shapesThis unit builds on students’ prior knowledge of finding area of rectangles to develop an understanding that area is additive and to learn how to find the area of other figures. Students will learn to find area of triangles, special quadrilaterals and other polygons (by decomposing into rectangles or triangles). Students will learn to find area of irregular figures by decomposing the figure into regions of known or equal areas. Students sometimes struggle to find missing side lengths on these composite figures.

**Lessons**

Understand that area can be found by decomposing a figure into non-overlapping regions

Lesson objective: Understand that area is additive, meaning the area of a figure is the same as the sum of the area of non-overlapping regions of the figure

Decompose figures to find area

Lesson objective: Decompose figures to find the total area.

Pool Sidewalk: Find area by dividing into known parts

Lesson objective: Apply the method of decomposing a figure.

Find area by creating a new region of known area

Lesson objective: Understand that it is possible to find the area of a figure by creating a new region of known or equal area.

Develop and use the area formula for triangles

Lesson objective: Develop and use the area formula for triangles.

Develop and use the area formulas for parallelograms and trapezoids

Lesson objective: Develop and use the area formula for a parallelogram and trapezoid.

Park area: Find area by creating a new region or subtracting out a region

Lesson objective: Apply the formulas for finding the area of triangles, parallelograms, and trapezoids.

Find missing dimensions when area is known

Lesson objective: Understand that we can find unknown measurements if we know the area and sufficient information about the figure.

Find missing lengths of sides when area is known

Lesson objective: Find missing lengths of sides when the area is known.

Tents: Apply knowledge to find missing lengths

Lesson objective: Apply area formulas to determine areas and to find unknown measurements.

**Problem solving with volume and surface area**

Relationships exist between two-dimensional and three-dimensional shapes and their measures. In this unit, students will find volumes of rectangular prisms filled with fractional edge-length cubes and surface area of rectangular and triangular prisms and pyramids from their nets. Students will learn to identify and draw nets of prisms and pyramids. Students will apply the concept of additive area to find the surface area of geometric solids; this may introduce some confusion between 2- and 3-dimensional applications. Finding volume using unit cubes with fractional edge lengths may also challenge students since they are accustomed to unit cubes representing whole units.

**Lessons**

Compare length, area, and volume

Lesson objective: Differentiate among linear, area, and volume measurements and the units required for each.

Pack prisms with fractional edge length cubes

Lesson objective: Recognize that rectangular prisms can be built with unit cubes with fractional edge lengths which then can be used to determine the prisms' dimensions and volume.

Build prisms using fractional edge length cubes

Lesson objective: Find the dimensions and volumes of all unique prisms possible built with a given number of half-unit cubes.

Find volume and dimensions of rectangular prisms built with fractional edge length cubes

Lesson objective: Determine the volume and dimensions of rectangular prisms built with fractional edge-length cubes with fluency.

Find dimensions of rectangular prisms built with fractional edge length cubes

Lesson objective: Determine how many fractional-unit cubes it takes to fill a rectangular prism of a given volume.

Use nets of prisms and pyramids

Lesson objective: Identify three dimensional figures by their two dimensional nets and then use nets to distinguish between prisms and pyramids.

Draw nets of pyramids and prisms

Lesson objective: Draw nets of pyramids and prisms.

Use nets to find surface area

Lesson objective: Define surface area and determine a strategy for finding the surface area of three dimensional figures from their nets.

Use nets to find surface area of pyramids & prisms

Lesson objective: Find the surface area of three dimensional figures from their nets.

Find surface area of prisms and pyramids

Lesson objective: Solve a problem involving surface area.

**Data distributions**

In this unit, students will be introduced to the basic principles of descriptive statistics. They will expand their knowledge of representing data to include the use of boxplots and histograms. Students will identify and write statistical questions by understanding that a statistical question anticipates variability in its answers. They will describe numeric datasets using measures of center and variability so that they can later analyze and interpret datasets.

**Lessons**

Identify Statistical Questions

Lesson objective: Understand that a statistical question anticipates variability in the data related to the question and accounts for it in the answers.

Understand that a data display can give an informal representation of a dataset's shape, center, and spreadLesson objective: Understand that a graph can help us describe the center, shape, and spread of a dataset.

Describe the center, spread, and shape of data displays

Lesson objective: Find the center, spread, and shape of a graph.

Understand measures of center

Lesson objective: Understand that the arithmetic mean is a balance point of all the data.

Find measures of center

Lesson objective: Compute the mean, median, and mode of a dataset.

Find measures of center from data displays

Lesson objective: Calculate the mean, median, and mode using dot plots and bar charts.

Understand mean absolute deviation

Lesson objective: Understand that the mean absolute deviation is the mean distance from each data point in a given set to the mean of the data set.

Calculate mean absolute deviation

Lesson objective: Compute the mean absolute deviation of a dataset.

Calculate interquartile range

Lesson objective: Compute the five number summary, range, and interquartile range.

Solve problems with the MAD

Lesson objective: Use the Mean Absolute Deviation to compare two datasets.

**Analyzing data**

Students will extend their understanding of calculating the center, spread, and shape of a given data set and/or a given data display to represent the data set themselves using dot plots, histograms, and boxplots. Students will also continue describing the data set based on its center (median and/or mean), variability (interquartile range and/or mean absolute deviation), overall pattern, and any apparent deviations from the overall pattern. In this unit, students will be expected to use their analytical skills to decide which measure of center and variability is most appropriate for describing the data set. Finally, in this unit students will be expected to be able to determine the number of observations and describe how the data was measured, including the units of measure.

**Lessons**

Measure of center and variability can be used to describe a data set

Lesson objective: Understand that measures of center and variability can be used together to describe and analyze a data set.

Describe data using center, spread, and shape

Lesson objective: Describe data using measures of center, spread, and shape.

Interpret data based on the measures of center and spread

Lesson objective: Apply knowledge of center and spread to solve problems using a data set.

Choosing the best measure of center depends on the data collected

Lesson objective: Understand the best measure of center to represent a data set depends on the type and distribution of the data set.

Representing and analyzing data includes reporting context

Lesson objective: Understand that reporting data includes reporting the context in which the data was collected.

Report the context behind data collection

Lesson objective: Apply knowledge of reporting the context behind data collection to create a complete report to represent a data set.

Reporting results of data analysis includes a display of the data

Lesson objective: Extend the understanding that reporting data should include not only calculations, but an appropriate data display.

Create a histogram to represent data in a data set

Lesson objective: Create histograms to represent a data set.

Create a box plot to represent a data set

Lesson objective: Create a box plot to represent a data set.

Create the best visual representation for a data set

Lesson objective: Apply knowledge of data displays to create the best data display for the needs of the situation.