Category: Grade 6 Math

Statistics and Probability

*if this unit is not covered, it will need to be covered in 7th grade

Unit description: In this unit students will learn to use statistical questions and methods for gathering data to answer questions. They will calculate measures of center and measures of variability and create different data displays. They will use theoretical and experimental probabilities to make predictions. They will implement the four-step investigative process by stating their statistical questions, explaining the plan they used to collect data, analyzing data numerically and with graphs, and interpreting their results as related to their questions.

Download the complete Grade 6 Math Unit 6 framework to customize for your own planning.

Essential Outcomes of the Unit

Statistics and Probability

Develop understanding of statistical variability

6.SP.1a Recognize that a statistical question is one that anticipates variability in the data related to the question and accounts for it in the answers.

6.SP.2 Understand that a set of quantitative data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.SP.3 Recognize that a measure of center for a quantitative 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.

Summarize and describe distributions

6.SP.4 Display quantitative data in plots on a number line, including dot plots, and histograms

6.SP.5c Calculate range and measures of center, as well as describe any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

6.SP.6  Understand that the probability of a chance event is a number between 0 and 1 inclusive, that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Other Standards Addressed in the Unit

Statistics and Probability

Develop understanding of statistical variability

6.SP.1b Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population.

6.SP.1c Understand that the method and sample size used to collect data for a particular question is intended to reduce the difference between a population and a sample taken from the population so valid inferences can be drawn about the population. Generate multiple samples (or simulated samples) of the same size to recognize the variation in estimates or predictions.

Summarize and describe distributions

6.SP.5 Summarize quantitative data sets in relation to their context.

6.SP.5a Report the number of observations.

6.SP.5b Describe the nature of the attribute under investigation, including how it was measured and its units of measurement.

6.SP.5d Relate the range and the choice of measures of center to the shape of the data distribution and the context in which the data were gathered. 

Investigate chance processes and develop, use, and evaluate probability models

6.SP.7 Approximate the probability of a simple event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.

6.SP.8 Develop a probability model and use it to find probabilities of simple events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy

6.SP.8a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of simple events.

6.SP.8b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

Essential Questions and Big Ideas

• What is a statistical question?  
  • Statistical questions are written to have variability. 
  • When asking statistical questions individuals examined should be representative of the larger population. 
  • The method used and sample size will impact the data collected from a statistical question.  

• How can the distribution of data be described? 
  • Data can be described by its center, spread, and overall shape.
  • A measure of center for a quantitative 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.
  • Mean, median, and mode represent center.  

  • Range represents the shape of a data set.  

• How can I represent quantitative data?  

  • Quantitative data can be represented with plots on a number line, including dot plots, and histograms.

• What is probability?    
  • The probability of a chance event is a number between 0 and 1 inclusive, that expresses the likelihood of the event occurring. 

  • Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 

Geometry

Unit description:  In this unit students will solve for the area and volume of a variety of polygons in real-world problems. The students will learn to create nets to represent three dimensional polygons.

Essential Outcomes of the Unit

Geometry: Solve real-world and mathematical problems involving area, surface area, and volume.

6.G.1 Find area of triangles, trapezoids, and other polygons by composing into rectangles or decomposing into triangles and quadrilaterals. Apply these techniques in the context of solving real-world and mathematical problems.

6.G.2 Find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Other Standards Addressed in the Unit

Geometry: Solve real-world and mathematical problems involving area, surface area, and volume.

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.

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.

6.G.5 Use area and volume models to explain perfect squares and perfect cubes.

Essential Questions and Big Ideas

How can I create and interpret geometric shapes on the coordinate plane?

• Lengths on a coordinate plane can be measured by counting spaces between points.  
• When two coordinates have an x or y coordinate in common, their distance is represented by the difference between the unique coordinates.    

How do I find the areas of geometric shapes using what I know about rectangles?

• Perfect squares are numbers that can represent the areas of squares, as they represent an amount times itself.  
• Right triangles can be thought of as halves of rectangles, so the area of a triangle can be found from ½ (l * w) or ½ (b * h).
• Trapezoids can be decomposed into rectangles and triangles to find area.

How do I find the volume of rectangular prisms with whole number and fractional edge lengths?  

• Volumes of rectangular prisms can be found by multiplying l * w * h.  
• Perfect cubes are numbers that can represent the volume of a cube, as they represent an amount times itself times itself.  
• Volumes of rectangular prisms with fractional edge lengths can be found in the same way as if they were whole number edge lengths.  

How do I represent 3-D figures using nets?  

• A net is a pattern obtained when a three-dimensional figure is laid out flat showing each face of the figure.  
• 3 – D figures can have more than one net depending on how it is laid out.  
• Nets can be used to determine surface area.  

Download the complete Grade 6 Math Unit 5 framework to customize for your own planning.

Expressions, Equations & Inequalities

In this unit the students will learn to recognize that variables are used to represent specific but unknown numbers and are used to make statements that are true for all numbers or a set of numbers. They will learn to read, write and evaluate expressions. The students will identify and generate equivalent expressions.

Essential Outcomes of the Unit

1. Recognize that variables are used to represent specific but unknown numbers and are used to make statements that are true for all numbers or a set of numbers.
2. Identify and generate equivalent expressions. 
3. Read, write and evaluate expressions in a variety of real world contexts.

Expressions, Equations, and Inequalities

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

6.EE.2a Write expressions that record operations with numbers and with letters standing for numbers.

6.EE.2b Identify parts of an expression using mathematical terms (term, coefficient, sum, difference, product, factor, and quotient); view one or more parts of an expression as a single entity.

6.EE.2c Evaluate expressions given specific values for their variables. Include expressions that arise from formulas in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order (Order of Operations).

6.EE.3 Apply the properties of operations to generate equivalent expressions.

6.EE.4 Identify when two expressions are equivalent.

Expressions, Equations, and Inequalities

Reason about and solve one-variable equations and inequalities.

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.

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.

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

6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another. Given a verbal context and an equation, identify 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.

Other Standards Addressed in the Unit

Geometry

Solve real-world and mathematical problems involving area, surface area, and volume.

6.G.5 Use area and volume models to explain perfect squares and perfect cubes.

Expressions, Equations, and Inequalities

Reason about and solve one-variable equations and inequalities.

6.EE.8 8. Write an inequality of the form x > c, x ≥ c, x ≤ c, or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of these forms have infinitely many solutions; represent solutions of such inequalities on a number line.

Essential Questions and Big Ideas

What are exponents?

• Exponents represent repeated multiplication of the same factor.  
• The term squared represents an exponent of 2.  The term cubed represents an exponent of 3.  

How can I write and evaluate expressions and equations?  

• Mathematical language such as sum, product, difference, etc. can be used to describe mathematical expressions.  
• When the value of a variable is known, it can be substituted into an expression or equation.  

How can I identify and create equivalent expressions?

• Like terms, terms with the same variable, can be combined or subtracted.  
• The distributive property can be used to simplify an expression with parentheses.  

How can I solve equations and inequalities?  

• Substitution can be used to solve equations and inequalities.  

Download the complete Grade 6 Math Unit 4 framework to customize for your own planning.

Rational Numbers

Unit description:  In this unit the students will learn to develop the concept of opposite numbers and absolute values, and that zero is its own opposite. They will use positive and negative numbers to represent real-world quantities and compare and order integers and rational numbers with and without number lines. The students will describe the relationship between rational numbers in real-world contexts through comparison and using understanding of absolute value. The students will learn to plot points in all four quadrants, find the distance between points, identify reflections across both axes, and create polygons. 

Download the complete Rational Numbers framework to customize for your own planning.

Essential Outcomes of the Unit

The Number System 

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

• 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. Use positive and negative numbers to represent quantities in real world contexts, explaining the meaning of 0 in each situation. 
• 6.NS.6 Understand a rational number as a point on the number line. Use number lines and coordinate axes to represent points on a number line and in the coordinate plane with negative number coordinates.
• 6.NS.6a 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, and that 0 is its own opposite.
• 6.NS.6c Find and position integers and other rational numbers on a horizontal or vertical number line. Find and position pairs of integers and other rational numbers on a coordinate plane.
• 6.NS.7 Understand ordering and absolute value of rational numbers. 

Other Standards Addressed in the Unit

The Number System 

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

• 6.NS.6b 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.NS.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line.
• 6.NS.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts
• 6.NS.7c 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.NS.7d Distinguish comparisons of absolute value from statements about order. 
• 6.NS.8 Solve real-world and mathematical problems by graphing points on a coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Geometry

Solve real-world and mathematical problems involving area, surface area, and volume

• 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.

Essential Questions and Big Ideas

How are positive and negative numbers used?

• Quantities having more or less than zero are described using positive and negative numbers.

How do rational numbers relate to integers?

• Number lines are visual models used to represent the density principle: between any two whole numbers are many rational numbers, including decimals and fractions.
• The rational numbers can extend to the left or to the right on the number line, with negative numbers going to the left of zero, and positive numbers going to the right of zero.

What is modeled on the coordinate plane?

• The coordinate plane is a tool for modeling real-world and mathematical situations and for solving problems.

Operations in Base 10 

Students will develop their skills at adding, subtracting, multiplying, and dividing multi-digit whole numbers and multi-digit decimals.

Essential Outcomes

• NY-6.NS.2: Fluently divide multi-digit numbers using a standard algorithm.
• NY-6.NS.3: Fluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation.

Essential Questions and Big Ideas

• How do I add and subtract multi-digit decimals?
  • Add or subtract like place values.  
  • Multi-digit decimals are added in the same way as multi-digit numbers. 
  • If a place value is empty, a placeholder of 0 can be used.  
• How do I multiply multi-digit decimals?
  • Multiply multi-digit decimals in the same way as multi-digit numbers. 
  • The number of decimal places in a product of multi-digit decimals is equivalent to the sum of the decimal places in both factors.  
• How do I divide multi-digit numbers and decimals?
  • When dividing by multi-digit numbers, you can use multiplication to find multiples of the divisor.  
  • When dividing multi-digit decimals, change the divisor into a whole number by multiplying by a power of 10.  Multiply the dividend by the same power of 10 to keep the relationship the same.  
  • When dividing a multi-digit decimal by a whole number, keep the decimal point in the same place in the quotient. 
• How do I solve problems with multi-digit whole numbers and multi-digit decimals?
  • To solve a problem, determine the unknown and the relationship between quantities in the problem.  

Download the complete Grade 6 Unit 2 framework to customize for your own planning.

Ratios, Unit Rate, and Percentages

Students will build on their knowledge of fractions as they use ratios and rates to describe relationships. Students will be able to describe ratios, unit rates, and percentages. Students will use tables and graphs to represent these relationships.

Essential Outcomes

Ratios and Proportional Relationships

Ratio

• 6.RP.1 – Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. e.g., “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received three votes.”
• 6.RP.3 – Use ratio and rate reasoning to solve real-world and mathematical problems. Note: Strategies may include but are not limited to the following: tables of equivalent ratios, tape diagrams, double number lines, and equations.
• 6.RP.3a – 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.RP.3d – Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Note: Conversion of units occur within a given measurement system, not across different measurement systems.

Unit Rate

• 6.RP.2 – Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 (b not equal to zero), and use rate language in the context of a ratio relationship. e.g., “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there are 3⁄4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” Note: Expectations for unit rates in this grade are limited to non-complex fractions.
• 6.RP.3 – Use ratio and rate reasoning to solve real-world and mathematical problems. Note: Strategies may include but are not limited to the following: tables of equivalent ratios, tape diagrams, double number lines, and equations.
• 6.RP.3b – Solve unit rate problems. e.g., If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? What is the unit rate? Note: Problems may include unit pricing and constant speed.

Proportional Reasoning with Percentages

• 6.RP.3 – Use ratio and rate reasoning to solve real-world and mathematical problems. Note: Strategies may include but are not limited to the following: tables of equivalent ratios, tape diagrams, double number lines, and equations.
• 6.RP.3c – Find a percent of a quantity as a rate per 100. Solve problems that involve finding the whole given a part and the percent, and finding a part of a whole given the percent. e.g., 30% of a quantity means 30/100 times the quantity.

Other Standards Addressed in this Unit

Expressions, Equations and Inequalities

• 6.EE.9- build base knowledge – Use variables to represent two quantities in a real-world problem that change in relationship to one another. Given a verbal context and an equation, identify 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.
  • e.g., In a problem involving motion at constant speed, list and graph ordered pairs of distances and times.
  • e.g., Given the equation d = 65t to represent the relationship between distance and time, identify t as the independent variable and d as the dependent variable.

Essential Questions and Big Ideas

• What is a ratio?
  • A ratio is a numerical relationship that represents how many times a number fits within another.
• What is a rate?
  • A rate is a special type of ratio that shows the relationship between two different units.
• What is a unit rate?
  • A unit rate represents the amount of a unit per one unit of another.
• How are percentages related to ratios?
  • A percentage represents a ratio where you’re considering how much of an amount within 100.
  • A percentage represents a part: whole ratio, not a part: part ratio.
• How do I solve problems with ratios, rates, or percentages?
  • Tables and graphs can be used to solve problems with ratios, rates, or percentages.
  • Fractions can be used to relate to ratios, rates, or percentages.

Download the complete Grade 6 Math Unit 1 framework to customize for your own planning.