Author: Alissa.Ouderkirk

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. 

Attributes of Polygons

Unit description: Students will extend their understanding of polygons as they classify them into categories such as triangles, quadrilaterals, pentagons, and hexagons.  Students will break shapes into fractional parts.  

Download the complete Grade 3  Unit 6 framework to customize for your own planning.

Essential Outcomes of the Unit

Geometry- Reason with shapes and their attributes.

• NY-3.G.1 Recognize and classify polygons based on the number of sides and vertices (triangles, quadrilaterals, pentagons, and hexagons). Identify shapes that do not belong to one of the given subcategories. Note: Include both regular and irregular polygons, however, students need not use formal terms “regular” and “irregular,” e.g., students should be able to classify an irregular pentagon as “a pentagon,” but do not need to classify it as an “irregular pentagon.
• NY-3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

Essential Questions and Big Ideas

How do I identify polygons?

• Polygons are figures with straight lines that are connected.  
• The type of polygon can be determined by counting the sides.

What are common types of polygons?

• A triangle is a shape with three sides and three angles
• A quadrilateral is a shape with four sides and four angles. 
• A pentagon is a shape with five sides and five angles. 
• A hexagon is a shape with six sides and six angles. 

How can shapes be partitioned?  

• A shape can be broken into parts of equal area which represent fractions.  

Download the complete Grade 3  Unit 6 framework to customize for your own planning.

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.

Algebra: Patterns and Graphing

Unit description: Students will begin to learn about the coordinate system.  Students will be able to graph an ordered pair in the first quadrant and consider what that ordered pair might represent.  Students will extend their understanding of patterns to interpret two connected numerical patterns with two rules. 

Essential Outcomes of the Unit  

Geometry- Graph points on the coordinate plane to solve real-world and mathematical problems

5.G.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Other Standards Addressed in the Unit

Operations and Algebraic Thinking- Analyze patterns and relationships.

5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

Geometry- Graph points on the coordinate plane to solve real-world and mathematical problems

5.G.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Coherence: NY-5.G.1 → NY-6.NS.6 Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond.

Essential Questions and Big Ideas

What is a coordinate system?

• A coordinate system is created with a pair of perpendicular lines called axes with the intersection of the lines arranged to coincide with 0 on each line (the origin).  
• A given point in the plane can be located by using an ordered pair of numbers called coordinates.  

How can I use the coordinate plane to represent and solve problems? 

• Points can be graphed on a coordinate plane to represent patterns and relationships.  
• The relationship between points on a coordinate plane can be used to solve problems.  

How can I interpret patterns?

• Patterns can be found by interpreting changes in values over time.  

Download the complete Grade 5 Math Unit 6 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.

Geometry and Volume

Unit description: Students will build on their understanding of area from fourth grade to consider 3-D shapes and their volumes.  Students will develop methods for finding the volume of rectangular prisms that include multiplying the area of the base times the height, multiplying length by the width by the height, and counting unit cubes. 

Essential Outcomes of the Unit

Measurement and Data- Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

• 5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
• 5.MD.5a Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base.
• 5.MD.5b Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
• 5.MD.5c Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Other Standards Addressed in the Unit

Measurement and Data- Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

• 5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
• 5.MD.3a Recognize that a cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume
• 5.MD.3b Recognize that a solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
• 5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in., cubic ft., and improvised units.

Geometry- Classify two-dimensional figures into categories based on their properties.

• 5.G.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
• 5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

Essential Questions and Big Ideas

• How do I classify quadrilaterals?
  • Quadrilaterals can be classified in multiple ways. 
  • Trapezoids and parallelograms are classified based on parallel sides.
  • Rectangles and squares are classified based on right angles. 
  • Squares and rhombi are classified based on equal side lengths. 
• What is volume?
  • Volume represents the amount of 3-D space a 3-D shape takes up. 
  • Volume is measured in cubic units. 
• What is a “unit cube” and how do I use it to measure volume?
  • A “unit cube” represents one cubic unit. 
  • “Unit cubes” can be combined together to represent a volume. 
• How is volume related to area?
  • Rectangular prisms are built of layers of areas of unit cubes. 
  • The area of a base can be multiplied by a height to find a volume.

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

Measurement and Data

Unit description: In this unit students will learn to recognize the need for standard units of measure (centimeter and inch) and apply this understanding to addition and subtraction problems involving length. The students will learn to recognize that the smaller the unit, the more iterations needed to cover a given length and they will learn to select appropriate tools to measure understanding that linear measure involves an iteration of units. The students will also learn to interpret and represent data in multiple ways, such as: line plot, bar graph, picture graph, and a tally chart.

Essential Outcomes of the Unit

Measurement and Data- Measure and estimate lengths in standard units.

2.MD.1 Measure the length of an object to the nearest whole by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

Measurement and Data- Represent and interpret data.

2.MD.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a picture graph or a bar graph.

Other Standards Addressed in the Unit

Measurement and Data- Measure and estimate lengths in standard units.

• 2.MD.2 Measure the length of an object twice, using different “length units” for the two measurements; describe how the two measurements relate to the size of the unit chosen.
• 2.MD.3 Estimate lengths using units of inches, feet, centimeters, and meters.
• 2.MD.4 Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard “length unit.”

Measurement and Data- Relate addition and subtraction to length.

• 2.MD.5 Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units.
• 2.MD.6 Represent whole numbers as lengths from 0 on a number line with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line.

Measurement and Data- Represent and interpret data.

2.MD.9 Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Present the measurement data in a line plot, where the horizontal scale is marked off in whole-number units.

Essential Questions and Big Ideas

• Why do standards units matter?
  • A standard unit of measure are the accepted, consistent increments we use to measure.
  • A standard unit of measure for length are centimeters and meters and inches and feet.
  • A nonstandard unit of measure is something that we use to measure, such as a paperclip or pencil,  if we want to compare the measure of two things without using a standard unit.
• How do we choose appropriate tools and use them to measure the length of an object?
  • Identify the appropriate tool to measure the length of various objects.
  • Measure various objects to the closest whole number length with correct units.
• How do we use picture graphs and bar graphs to display and analyze data?
  • Data collection is used to develop picture graphs and bar graphs with a single unit scale.
  • Picture and bar graphs can be used to analyze and interpret data and to answer questions about a data set.

Download the complete Grade 2 Math Unit 5 framework to customize for your own planning.

Conversions, Area, and Perimeter

Unit description: Students will learn how to complete metric conversions, as well as customary conversions for length and time.  Students will extend their knowledge of area and perimeter from third grade to complete multi-step real world and mathematical problems involving area, perimeter, and different types of rectangles. 

Essential Outcomes of the Unit

Measurement and Data- Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit

• NY-4.MD.1 Know relative sizes of measurement units: ft., in.; km, m, cm. Know the conversion factor and use it to convert measurements in a larger unit in terms of a smaller unit: ft., in.; km, m, cm; hr., min., sec. Given the conversion factor, convert all other measurements within a single system of measurement from a larger unit to a smaller unit. Record measurement equivalents in a two-column table.
• NY-4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. e.g., Find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

Other Standards Addressed in the Unit

Measurement and Data- Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit

• NY-4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money. 
• NY-4.MD.2a Solve problems involving fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. 
• NY-4.MD.2b Represent measurement quantities using diagrams that feature a measurement scale, such as number lines. Note: Grade 4 expectations are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Measurement and Data- Represent and interpret data

NY-4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit 1/2 , 1/4 , 1/8. Solve problems involving addition and subtraction of fractions by using information presented in line plots.

Essential Questions and Big Ideas

• How do I convert different sized units?
  • The metric system is based on powers of 10. 
  • There are 12 inches in 1 foot. 
  • There are 60 minutes in an hour.
  • There are 60 seconds in a minute. 
  • Conversions can be recorded in tables.  `
• How do I solve area and perimeter word problems involving rectangles? 
  • Area represents the total amount of space a flat shape takes up. 
  • Perimeter represents the distance around a shape. 
• How do I solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money?
  • Distances can be added or compared.
  • Times can be added or subtracted using understandings of minutes and hours. 
  • Volumes can be added or compared based on measurements in beakers or other tools. 
  • Masses can be added or compared based on measurements on scales or other tools. 
  • Money can be added or compared using understandings of cents and dollars. 
• How do I represent data using a line plot?
  • Line plots represent numerical data. 
  • Xs represent data points of a numerical value. 
  • The scale on a line plot must have equal sized units.

Download the complete Grade 4 Math Unit 5 framework to customize for your own planning.

Time, Liquid and Mass Measurement and Graphing

Unit description:  Students will develop an understanding of how to understand and relate time, liquid, and mass in word problems. Students will also build on their ability to create bar and picture graphs with scales other than one. They will also represent data using line plots.

Download the complete Grade 3  Unit 5 framework to customize for your own planning.

Essential Outcomes of the Unit

Measurement and Data- Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

• NY-3.MD.1 Tell and write time to the nearest minute and measure time intervals in minutes. Solve one-step word problems involving addition and subtraction of time intervals in minutes.

Measurement and Data- Represent and interpret data.

• NY-3.MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in a scaled picture graph or a scaled bar graph.
• NY-3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.

Other Standards Addressed in the Unit

Measurement and Data- Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

• NY-3.MD.2a Measure and estimate liquid volumes and masses of objects using grams (g), kilograms (kg), and liters (l).
  Note: Does not include compound units such as cm3 and finding the geometric volume of a container.
• NY-3.MD.2b Add, subtract, multiply, or divide to solve one-step word problems involving masses or liquid volumes that are given in the same units.Note: Does not include multiplicative comparison problems involving notions of “times as much.”

Essential Questions and Big Ideas

How do I solve word problems involving time?

• There are 60 minutes in an hour.
• Number lines can be used to represent time.

How do I solve word problems involving masses and volumes?

• Mass represents how much matter is in an item (similar to weight).
• Volume represents the amount of space a 3-D shape takes up.

How do I represent data using graphs?

• Graphs can model data using scales besides one.

How do I represent data using a line plot?

• Line plots represent numerical data using Xs.

Download the complete Grade 3  Unit 5 framework to customize for your own planning.