5th Grade Mathematics - Investigations
.
Unit 2: Measurement and Geometry
Teacher Resource Guide
2012 - 2013
In Grade 5, instructional time should focus on three critical areas:
1. Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of
division of fractions (limited to unit fractions divided by whole numbers and whole numbers divided by unit fractions);
Students apply their understanding of fractions and fraction models (set model, area model, linear model) to represent addition
and subtraction of fractions with unlike denominators as equivalent calculations with like denominators (¼ + 2/8 = ¼ + ¼). They
develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the
meaning of fractions and the relationship between multiplication and division to explain why the procedures for multiplying and
dividing fractions make sense.
2. Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of
operations with decimals to hundredths, and developing fluency with whole number and decimal operations;
Students develop understanding of why division procedures work based on place value and properties of operations. They are
fluent with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals,
decimal notation, and properties of operations to add and subtract decimals to hundredths. Students use the relationship
between decimals and fractions, and the relationship between decimals and whole numbers (i.e., a decimal multiplied by
anpower of 10 is a whole number) to understand and explain why procedures for multiplying and dividing decimals make sense.
They compute products and quotients of decimals to hundredths efficiently and accurately.
3. Developing understanding of volume.
Students understand that volume is an attribute of three-dimensional space and can be measured by finding the total number of
same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit
cube is the standard unit for measuring volume. They decompose three-dimensional shapes and find volumes of right
rectangular prisms by viewing them as decomposed into layers of arrays of cubes.
Unit 2: Measurement and Geometry
October 15- November 9 (4 weeks)
TRIMESTER 2
TRIMESTER 1
5th Grade Mathematics 2012 - 2013
Unit
1: Multi-Digit Multiplication and Division
2: Measurement/Geometry
3: Addition and Subtraction of Fractions
TRIMESTER 3
4: Decimals
Time Frame
Testing Window
7 weeks
8/22 – 10/12
October 12
4 weeks
10/15 – 11/9
November 9
8 weeks
11/12 – 1/18
January 18
8 weeks
1/22 – 3/14
March 14
9 weeks
3/25 – 5/30
May 30
5: Multiplication and Division of Fractions
Math Wiki: http://dmps-mathematics.wikispaces.com
5th Grade
2012-2013
Page 2
Unit 2: Measurement and Geometry
October 15- November 9 (4 weeks)
Big Ideas
Essential Questions
The volume of a prism means to determine the number of cubes needed to fill the bottom layer. Then
multiply by the number of layers needed to fill the prism.
A shape can have more than one name, because each name is a more detailed description of the shape.
Identifier
5.MD.5
What does it mean to find the volume of a prism?
Why can a shape have more than one name?
Standards
Mathematical Practices
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.
STANDARDS
5.MD.3
5.MD.4
5.G.2 
5.G.1
5.OA.3
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a.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.
b.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.
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
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.
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. 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 (e.g., x-axis and x-coordinate, y-axis and ycoordinate).
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.
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. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number
0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding
terms in the other sequence. Explain informally why this is so.
5th Grade
2012-2013
Page 3
Unit 2: Measurement and Geometry
Identifier
5.G.3 
5.G.4
October 15- November 9 (4 weeks)
Standards
Mathematical Practices
Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that
category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right
angles.
Classify two-dimensional figures in a hierarchy based on properties.
5th Grade
2012-2013
Page 4
Unit 2: Measurement and Geometry
October 15- November 9 (4 weeks)
Instructional Strategies for ALL STUDENTS
Critical Reading for Teachers Prior to Instruction –
Wiki: Focus in Grade 5 Teaching with Curriculum Focal Points, NCTM, 2009, p. 80-83
Wiki: Teaching Student-Centered Mathematics Grades 3-5, Van de Walle&Lovin, Houghton Mifflin, 2006, p. 239
Wiki: Teaching Student-Centered Mathematics Grades 3-5, Van de Walle&Lovin, Houghton Mifflin, 2006, p. 224-226
Use models to develop concept of volume – Volume refers to the amount of space that an object takes up and is measured in cubic units such as cubic inches or cubic
centimeters. Students need to experience finding the volume of rectangular prisms by counting unit cubes, in metric and standard units of measure, before the formula is
presented. Provide multiple opportunities for students to develop the formula for the volume of a rectangular prism with activities similar to the one described below.
Give students one block (a I- or 2- cubic centimeter or cubic-inch cube), a ruler with the appropriate measure based on the type of cube, and a small rectangular box. Ask
students to determine the number of cubes needed to fill the box. Have students share their strategies with the class using words, drawings or numbers. Allow them to confirm
the volume of the box by filling the box with cubes of the same size.
Have students build a prism in layers. Then, have students determine the number of cubes in the bottom layer and share their strategies. Students should use multiplication
based on their knowledge of arrays and its use in multiplying two whole numbers. Ask what strategies can be used to determine the volume of the prism based on the number
of cubes in the bottom layer. Expect responses such as “adding the same number of cubes in each layer as were on the bottom layer” or multiply the number of cubes in one
layer times the number of layers.
The Priority Standard for volume of a right rectangular prism states that students apply V = l x w x h and V = b x h. By developing the concept of layering, the teacher can guide
students to see that l x w is the same thing as the base (b). Therefore, there are two commonly used formulas for volume of a rectangular prism. By understanding the concept
of layering, students will make connections to finding the volume of other 3-D figures in later grades.
Coordinate grid and plotting coordinates – Students need to understand the underlying structure of the coordinate system and see how axes make it possible to locate points
anywhere on a coordinate plane. This is the first time students are working with coordinate planes, and only in the first quadrant. It is important that students create the
coordinate grid themselves. This can be related to two number lines and reliance on previous experiences with moving along a number line. Multiple experiences with plotting
points are needed.
Provide points plotted on a grid and have students name and write the ordered pair. Have students describe how to get to the location. Encourage students to articulate
directions as they plot points.
Present real-world and mathematical problems and have students graph points in the first quadrant of the coordinate plane. Gathering and graphing data is a valuable
experience for students. It helps them to develop an understanding of coordinates and what the overall graph represents. Students also need to analyze the graph by
interpreting the coordinate values in the context of the situation.
When playing games with coordinates or looking at maps, students may think the order in plotting a coordinate point is not important. Have students plot
points so that the position of the coordinates is switched. For example, have students plot (3, 4) and (4, 3) and discuss the order used to plot the points. Have
students create directions for others to follow so that they become aware of the importance of direction and distance.
5th Grade
2012-2013
Page 5
Unit 2: Measurement and Geometry
October 15- November 9 (4 weeks)
Instructional Strategies for ALL STUDENTS (continued)
Classification This cluster builds from Grade 3 when students described, analyzed and compared properties of two-dimensional shapes. They compared and classified
shapes by their sides and angles, and connected these with definitions of shapes. In Grade 4 students built, drew and analyzed two-dimensional shapes to
deepen their understanding of the properties of two-dimensional shapes. They looked at the presence or absence of parallel and perpendicular lines or the
presence or absence of angles of a specified size to classify two-dimensional shapes. Now, students classify two-dimensional shapes in a hierarchy based on
properties. Details learned in earlier grades need to be used in the descriptions of the attributes of shapes. The more ways that students can classify and
discriminate shapes, the better they can understand them. The shapes are not limited to quadrilaterals.
Students can use graphic organizers such as flow charts or T-charts to compare and contrast the attributes of geometric figures. Have students create a Tchart with a shape on each side. Have them list attributes of the shapes, such as number of side, number of angles, types of lines, etc. they need to determine
what’s alike or different about the two shapes to get a larger classification for the shapes. Pose questions such as, “Why is a square always a rectangle?” and
“Why is a rectangle not always a square?”
5th Grade
2012-2013
Page 6
Unit 2: Measurement and Geometry
October 15- November 9 (4 weeks)
Routines/Meaningful Distributed Practice
Distributed Practice that is Meaningful and Purposeful
Practice is essential to learn mathematics. However, to be effective in improving student achievement, practice must be meaningful, purposeful, and
distributed.
 Meaningful: Builds on and extends understanding
 Purposeful: Links to curriculum goals and targets an identified need based on multiple data sources
 Distributed: Consists of short periods of systematic practice distributed over a long period of time
Routines are an excellent way to achieve the mandate of Meaningful Distributed Practice outlined in the Iowa Core Curriculum.. The skills presented
during routines do not necessarily reinforce the lesson concept for that day. Routines may be used to address a need for small increments of exposure
to a skill or review of skills already taught. Routine activities may be repeated several days in a row, allowing for a build-up of conceptual
understanding, or can be visited and re-visited over a period of time. Routines can be inserted as the schedule allows; in short intervals throughout
the day or as a lesson opener or closer. Selection of the routine should be made based on informal teacher observation and formative assessments.
Concepts taught through Meaningful Distributed Practice
Skill
Standard
Resource
Convert among different-sized standard measurement units
(May be taught in whole group and reinforced during MDP time)
Multi-digit multiplication
(
Whole-number
division
Interpreting remainders
5.MD.1
Math Expressions: Unit 6 p. 610
5.NBT.5
5.NBT.6
4.OA.3
Expressions: Unit 7; Resource Guide p. 13-14
Order of operations
5.OA.1
Expressions: Unit 8
Other skills students need to develop based on teacher observations and formative assessments.
5th Grade
2012-2013
Page 7
Unit 2: Measurement and Geometry
October 15- November 9 (4 weeks)
Investigations Resources for Unit 2 – 5th grade Geometry
Investigations
Standards
Addressed
5.MD.5a
5.MD.5b
5.MD.5c
Investigations
5.G.3
Supplement
5.G.1
5.G.1
5.G.1
5.G.2
Instructional Plan
Resource
Unit 2: Prisms and Pyramids
Note: Refer to CC10 for teaching note prior to teaching Unit 2.
 Investigation 1
Teach 1.5A before 1.5 p. CC14 – CC18, C9 – C12 (ICCSS)
 Investigation 2
2.3 additional resource C13
Teach 2.4A instead of 2.4, p. CC19 – CC22, C14 – C19 (ICCSS)
 Investigation 3
sessions 1, 2, 3, 4, omit 5
Unit 5: Measuring Polygons
 Investigation 1
sessions 2, 3
Additional Focus needed on:
 Coordinate Grids
 x-axis, coordinate
 y-axis, coordinate
 Representing real world and mathematical problems by graphing
points in the first quadrant of the coordinate plane, and
interpreting coordinate values within the context.
 Generate two numerical patterns when given a rule
 Form ordered pairs for the two patterns and graph the pairs on a
coordinate grid
5th Grade
2012-2013
5.OA.3
5.OA.3
Page 8
Unit 2: Measurement and Geometry
October 15- November 9 (4 weeks)
Additional Lesson Bank
These lessons are designed as supplemental lessons based on student’s needs.
Teacher’s
Edition
Pages
Lesson
Standards
Addressed
Resource Guide: Activity described in Instructional Strategies p. 4 (introduce volume)
Wiki: Activity – Fill the Box
Wiki: Activity – Volume of Prisms
Website: Activity – Explore the coordinate plane http://illuminations.nctm.org/LessonDetail.aspx?ID=L280
Website: Activity – Plotting points on coordinate plane http://illuminations.nctm.org/LessonDetail.aspx?id=L777
Website: Activity – Rectangles & Parallelograms http://illuminations.nctm.org/LessonDetail.aspx?ID=L350
Wiki: Activity – Mystery Definitions
http://illuminations.nctm.org/LessonDetail.aspx?ID=L277
Website: Activity – Classify Polygons http://illuminations.nctm.org/LessonDetail.aspx?ID=L270
Website: Activity – Classify Polygons
5th Grade
2012-2013
5.G.1
5.G.1
5.G.3
5.G.3
5.G.3
5.G.3
Page 9