Unit 5

Area, Surface Area, and Volume Problems

 

Grade 6

Math

Unit Length and Description:

 

25 days

 

In Unit 5, students apply their knowledge of expressions and equations to solve for unknowns in area, surface area, and volume problems. They find the area of triangles and other two-dimensional figures and use the formulas to find the volumes of right rectangular prisms with fractional edge lengths. Students use negative numbers in coordinates as they draw lines and polygons in the coordinate plane. They also find the lengths of sides of figures, joining points with the same first coordinate or the same second coordinate and apply these techniques to solve realworld and mathematical problems.

 

Standards:

 

Major Cluster Standards

Expressions and Equations

Apply and extend previous understandings of arithmetic to algebraic expressions.

6.EE.A.2

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

a.   Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract 𝑦 from 5” as 5 − 𝑦.

b.   Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

c.    Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas 𝑉=𝑠3 and 𝐴=6𝑠2 to find the volume and surface area of a cube with sides of length .

Reason about and solve one-variable equations and inequalities.

6.EE.B.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.B.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.B.7

Solve real-world and mathematical problems by writing and solving equations in the form 𝑥+𝑝=𝑞 and px=𝑞 for cases in which 𝑝, 𝑞 and 𝑥 are all nonnegative rational numbers.

Supporting Clusters

Geometry

6.G.1

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

6.G.2

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

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.

Foundational Standards

4.MD.3

 

Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, 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.

5.NF.4

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.  Interpret the product of (a/b) × q as a part of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.  For example, use a visual fraction model to show (2/3 × 4 = 8/3, and create a story context for this equation.  Do the same with (2/3) × (4/5) = 8/15.  (In general, (a/b) × (c/d) = ac/bd.)

5.MD.5

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

a.   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. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

b.   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.

c.    Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world 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.

Standards for Mathematical Practices

1.   Make sense of problems and persevere in solving them.

2.   Reason abstractly and quantitatively.

3.   Construct viable arguments and critique the reasoning of others.

4.   Model with mathematics.

5.   Use appropriate tools strategically.

6.   Attend to precision.

7.   Look for and make use of structure.

8.   Look for and express regularity in repeated reasoning.

Instructional Outcomes

6.G.A.1

·        I can recognize and know how to compose and decompose polygons into triangles and rectangles.

·        I can compare the area of a triangle to the area of the composed rectangle.

·        I can apply the techniques of composing and/or decomposing to find the area of triangles, special quadrilaterals and polygons to solve mathematical and real world problems.

·        I can discuss, develop and justify formulas for triangles and parallelograms.

6.G.2

·        I can calculate the volume of a right rectangular prism.

·        I can apply volume formulas for right rectangular prisms to solve real-world and mathematical problems involving rectangular prisms with fractional edge lengths.

·        I can model the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths.

6.G.3

·        I can draw polygons in the coordinate plane.

·        I can use coordinates (with the same x-coordinate or the same y-coordinate) to find the length of a side of a polygon.

·        I can apply the techniques of finding polygon side lengths in real world and mathematical problems.

6.G.4

·        I can recognize that 3-D figures can be represented by nets.

·        I can represent three-dimensional figures using nets made up of rectangles and triangles.

·        I can apply knowledge of calculating the area of rectangles and triangles to a net.

·        I can combine the areas for rectangles and triangles in the net to find the surface area of a 3-dimensional figure.

·        I can solve real-world and mathematical problems involving surface area using nets.

 

6.EE.A.2:

 

6.EE.A.2a

·        I can evaluate expressions with variables.

·        I can evaluate expressions using specific values for variables.

·        I can translate written phrases into algebraic expressions.

·        I can translate algebraic expressions into written phrases.

6.EE.A.2b

·        I can identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient).

·        I can understand that parts of an expression can have more than one name.

6.EE.A.2c

·        I can substitute specific values for variables.

·        I can write real world and mathematical problems.

6.EE.B.5

·        I can determine which values make an equation or inequality true.

·        I can use the solution to an equation or inequality to prove that the answer is correct. 

·        I can use substitution to decide if a number makes an equation or inequality true.

6.EE.B.6

·        I can recognize that a variable can represent one number, or, a set of numbers.

·        I can use variables to represent numbers.

·        I can write expressions when solving a real-world or mathematical problem.

6.EE.B.7

·        I can define an inverse operation.

·        I can use inverse operations to solve equations.

·        I can apply rules of the form x + p = q and px = q, for cases in which p, q and x are all positive rational numbers, to solve real world and mathematical problems. (There is only one unknown quantity).

·        I can solve real world and mathematical problems by writing and solving equations.

 

 

 

Enduring Understandings:

 

·        Decomposing and rearranging provide a geometric way of both seeing that a measurement formula is the right one and seeing why it is the right one.

·        Tools provide new sources of imagery as well as specific ways of thinking about geometric objects and processes.

·        Geometry and special sense offer ways to visualize, to interpret, and to reflect on our physical environment.

·        A net is a plane figure that can be folded to make a solid figure.

·        Solid figures can be identified and classified by the number of faces, edges, and vertices.

·        A visual representation of a shape hierarchy is an efficient way to describe the relationship among shapes with similar attributes.

Essential Questions:

 

·        What is a real world application of surface area?

·        How is a net utilized to represent a 3D figure?

·        How is volume affected by a change in one dimension?

·        What are the similarities and differences between area and surface area?

·        What are the properties of two- and three- dimensional figures?

·        What is the relationship between the areas of rectangles and triangles?

·        How is the formula for the area of a rectangle used to find the volume of a rectangular prism?

·        How is finding the volume of a rectangular prism similar to finding the volume of a pyramid?

·        How does the change in height affect the volume or surface area of a prism?

·       How can you estimate the volume or surface area of a prism or a pyramid?