Author: Capital Region BOCES

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.

Linear Equations

Unit description: Students will extend their understanding of equations to represent real world contexts to consider linear equations.  Students will leave this unit able to solve linear equations and recognize the number of solutions to a linear equation.  

Download the complete Linear Equations framework to customize for your own planning.

Essential Outcomes of the Unit

Expressions and Equations

Understand the connections between proportional relationships, lines, and linear equations.

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

Analyze and solve linear equations and pairs of simultaneous linear equations.

8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms. Note: This includes equations that contain variables on both sides of the equation.

Other Standards Addressed in the Unit

Expressions and Equations

Understand the connections between proportional relationships, lines, and linear equations.

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

Analyze and solve linear equations and pairs of simultaneous linear equations.

8.EE.7 Solve linear equations in one variable.

8.EE.7a Recognize when linear equations in one variable have one solution, infinitely many solutions, or no solutions. Give examples and show which of these possibilities is the case by successively transforming the given equation into simpler forms

8.EE.8 . Analyze and solve pairs of simultaneous linear equations.

8.EE.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Recognize when the system has one solution, no solution, or infinitely many solutions

8.EE.8b Solve systems of two linear equations in two variables with integer coefficients: graphically, numerically using a table, and algebraically. Solve simple cases by inspection.

8.EE.8c Solve real-world and mathematical problems involving systems of two linear equations in two variables with integer coefficients

Essential Questions and Big Ideas

What is a linear equation in one variable? 

• A linear equation in one variable, is one where there is only one variable and that variable does not have an exponent.  

How do I simplify terms in an equation?   

• Like terms can be combined or subtracted. 
• Like terms are terms with the same variable.  
• The distributive property can be used to expand an expression or equation.  

How can I determine the number of solutions to a linear equation?  

• If an equation ends with an untrue statement, there are no solutions. 
• If an equation ends with a true statement, there are infinite solutions.  
• If an equation ends with an x = statement, there is one solution.  

Download the complete Linear Equations 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.

Similarity

Unit description: In this unit the students will  learn to identify properties of dilations and compositions of dilations, describe the effect of dilations on two-dimensional figures in general and using coordinates. They will apply the Pythagorean Theorem to two and three dimensions using real world examples.

Download the complete Number Pairs Similarity framework to customize for your own planning.

Essential Outcomes of the Unit

Geometry

Understand congruence and similarity using physical models, transparencies, or geometry software

• 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
• 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Understand and apply the Pythagorean Theorem

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

Other Standards Addressed in the Unit

Geometry

Understand congruence and similarity using physical models, transparencies, or geometry software

• 8.G.4 Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane.

Understand and apply the Pythagorean Theorem

• 8.G.6 Understand a proof of the Pythagorean Theorem and its converse.
• 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Essential Questions and Big Ideas

How can you change a figure’s position or size on a 2-D coordinate plane?

• A figure can be moved using translation, rotation, reflection without changing its size or shape.
• A figure can change its size through dilation without changing its shape.

How do you recognize congruence and similarity in figures?

• Figures that are congruent if the corresponding angles and sides are the same. 
• Figures are similar if the corresponding angles are the same but the sides may be different.

How can angle relationships be used to find missing angle measures?

• Complementary angles have a sum of 90 degrees.
• Supplementary angles have a sum of 180 degrees.
• Vertical angles are equivalent.
• The sum of the interior angles of a triangle is 180 degrees.
• The sum of the exterior angles of a triangle is 360 degrees.
• Alternate interior angles are equal.
• Alternate exterior angles are equal.

What is the Pythagorean Theorem?  

• The Pythagorean Theorem is the way to solve for a missing side length in a right triangle. 
• The Pythagorean Theorem can be proved geometrically or algebraically. 

Download the complete Number Pairs Similarity 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.