All the resources on this page come from the GADOE Framework 5th Grade Unit 6 Overview. Reviewing this information before or during planning can help ensure your instruction is aligned with the standards and is at the depth expected.

Unit 6 Overview

CONVERT LIKE MEASUREMENT UNITS WITHIN A GIVEN MEASUREMENT SYSTEM

Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: conversion/convert, metric and customary measurements. From previous grades: relative size, liquid volume, mass, weight, length, kilometer (km), meter (m), centimeter (cm), kilogram (kg), gram (g), liter (L), milliliter (mL), inch (in), foot (ft), yard (yd), mile (mi), ounce (oz), pound (lb), cup (c), pint (pt), quart (qt), gallon (gal), hour, minute, second

REPRESENT AND INTERPRET DATA

Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: line plot, length, mass, liquid volume.

GEOMETRIC MEASUREMENT: UNDERSTAND CONCEPTS OF VOLUME AND RELATE VOLUME TO MULTIPLICATION AND TO ADDITION

Students recognize volume as an attribute of three-dimensional space. They understand that volume 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 select appropriate units, strategies, and tools for solving problems that involve estimating and 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. They measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems. Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: measurement, attribute, volume, solid figure, right rectangular prism, unit, unit cube, gap, overlap, cubic units (cubic cm, cubic in. cubic ft. nonstandard cubic units), multiplication, addition, edge lengths, height, area of base.

In this unit students will:

• change units to related units within the same measurement system by multiplying or dividing using conversion factors.

• use line plots to display a data set of measurements that includes fractions.

• use operations to solve problems based on data displayed in a line plot.

• recognize volume as an attribute of three-dimensional space.

• understand that volume can be measured by finding the total number of same size units of volume required to fill the space without gaps or overlaps.

• understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume.

• select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume.

• decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes.

• measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems.

• communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language.

Students convert measurements within the same system of measurement in the context of multistep, real world problems. Both metric and customary measurement systems are included, but the emphasis in the standards is on metric measure. Although students should be familiar with the relationships between units within either system, the conversion may be provided to them when they are solving problems. For example, when determining the number of feet there are in 28 inches, students may be provided with 12 inches = 1 foot. Students will explore how the base ten system supports conversions within the metric system. For example, 100 cm = 1 meter; 1.5 m = 150 cm. This builds on previous knowledge of placement of the decimal point when multiplying and dividing by powers of 10.

Students use measurements with fractions to collect data and graph it on a line plot. Data may include measures of length, weight, mass, liquid volume and time. Students will use data on the line plots to solve problems that may require application of operations used with fractions in this grade level. Operations with fractions may include addition and subtraction with unlike denominators, fraction multiplication, and fraction division which involve a whole number and a unit fraction.

Students recognize volume as an attribute of three-dimensional space. They understand that volume 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 select appropriate units, strategies, and tools for solving problems that involve estimating and 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. They measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems.

For more detailed information about unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the Grade Level Overview.

STANDARDS FOR MATHEMATICAL PRACTICE

This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

1. Make sense of problems and persevere in solving them. Students make sense that square units are used to measure 2-dimensional objects which have both length and width, and cubic units are used to measure 3-dimensional objects which have length, width, and height.

2. Reason abstractly and quantitatively. Students use reasoning skills to determine an average time by analyzing data and equally redistributing each data point. Students demonstrate abstract reasoning to create a display of square and cubic units in order to compare/contrast the measures of area and volume.

3. Construct viable arguments and critique the reasoning of others. Students construct and critique arguments regarding their knowledge of what they know about measurement, area and volume.

4. Model with mathematics. Students use line plots to show time measurements. Students use snap cubes to build cubes and rectangular prisms in order to generalize a formula for the volume of rectangular prisms.

5. Use appropriate tools strategically. Students select measurement tools to use for measuring length, weight, mass and liquid volume. Students also select and use tools such as tables, cubes, and other manipulatives to represent situations involving the relationship between volume and area.

6. Attend to precision. Students select appropriate scales and units to use for measuring length, weight, mass and liquid volume. Students attend to the precision when comparing and contrasting the prisms made using the same amount of cubes.

7. Look for and make use of structure. Students use their understanding of number lines to apply the construction of line plots. Students recognize volume as an attribute of solid figures and understand concepts of volume measurement. Students use their understanding of the mathematical structure of area and apply that knowledge to volume.

8. Look for and express regularity in repeated reasoning. Through experiences measuring different types of attributes, students realize that measurements in larger units always produce smaller measures and vice versa. Students relate new experiences to experiences with similar contexts when studying a solid figure that can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

***Mathematical Practices 1 and 6 should be evident in EVERY lesson***

STANDARDS FOR MATHEMATICAL CONTENT

MGSE5.MD.1 Convert among different-sized standard measurement units (mass, weight, length, time, etc.) within a given measurement system (customary and metric) (e.g., convert 5cm to 0.05m), and use these conversions in solving multi-step, real word problems.

MGSE5. MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

MGSE5.MD.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.

MGSE5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

MGSE5.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.

BIG IDEAS

From Teaching Student Centered Mathematics, Van de Walle & Lovin, 2006.

• When changing from smaller units to larger related units within the same measurement system, there will be fewer larger units.

• A line plot can provide a sense of the shape of the data, including how spread out or how clustered the data points are. Each data point is displayed on the line plot along a continuous numeric scale, similar to a number line. • Three-dimensional (3-D) figures are described by their faces (surfaces), edges, and vertices (singular is “vertex”).

• Volume can be expressed in both customary and metric units.

• Volume is represented in cubic units – cubic inches, cubic centimeters, cubic feet, etc.

• Volume refers to the space taken up by an object itself.

• Measurement involves a comparison of an attribute of an item with a unit that has the same attribute. Lengths are compared to units of length, areas to units of area, time to units of time, and so on.

• Data sets can be analyzed in various ways to provide a sense of the shape of the data, including how spread out they are (range, variance).

• Volume is a term for measures of the “size” of three-dimensional regions.

• Volume typically refers to the amount of space that an object takes up.

• Volume is measured with units such as cubic inches or cubic centimeters-units that are based on linear measures.

• Two types of units can be used to measure volume: solid units and containers.

ESSENTIAL QUESTIONS

• What strategies can you use to estimate measurements?

• What happens to a measurement when you change its unit of measure to a related unit? • How is data collected and displayed on a line plot?

• What strategies help when solving problems with line plots?

• How do we measure volume?

• How are area and volume alike and different?

• How can you find the volume of cubes and rectangular prisms?

• What is the relationship between the volumes of geometric solids?

• Why are some tools better to use than others when measuring volume?

• Why is volume represented with cubic units and area represented with square units?

CONCEPTS/SKILLS TO MAINTAIN

MGSE5.MD.1: Students progress through the underlying concepts of the measurement trajectory with the use of non-standard and standard units of measurement interchangeably. By the end of third grade, they will have learned how to use tools and appropriate units to measure metric and customary length, time, liquid volume, mass and weight. In fourth grade, students make conversions from larger units to smaller related units within the same measurement system by multiplying. All of these skills will be needed when they begin to make conversions from smaller units to larger units by dividing in fifth grade.

MGSE5.MD.2: In kindergarten, students begin working with categorical data. By the end of second grade, they will have learned how to draw line plots, picture graphs and bar graphs. In third and fourth grades, student draw scaled picture and bar graphs, graph measurement data on line plots, and solve problems using information from all three types of graphs. All of these skills will be applied in fifth grade when students use measurement data from line plots to solve problems.

MGSE5.MD.3, MGSE5.MD.4, and MGSE5.MD.5: These standards represent the first time that students begin exploring the concept of volume. In third grade, students begin working with area and covering spaces. The concept of volume should be extended from area with the idea that students are covering an area (the bottom of the cube) with a layer of unit cubes and then adding layers of unit cubes on top of the bottom layer. Students should have ample experiences with concrete manipulatives before moving to pictorial representations. Students’ prior experiences with volume were restricted to liquid volume. As students develop their understanding volume they understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. This cube has a length of 1 unit, a width of 1 unit and a height of 1 unit and is called a cubic unit. This cubic unit is written with an exponent of 3 (e.g., in3 , m3 ). Students connect this notation to their understanding of powers of 10 in our place value system. Models of cubic inches, centimeters, cubic feet, etc. are helpful in developing an image of a cubic unit. Students estimate how many cubic yards would be needed to fill the classroom or how many cubic centimeters would be needed to fill a pencil box.

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

• number sense

• computation with whole numbers and decimals, including application of order of operations

• addition and subtraction of common fractions with like denominators

• angle measurement

• measuring length and finding perimeter and area of rectangles and squares

• characteristics of 2-D and 3-D shapes

• data usage and representations, including line plots, bar graphs and picture graphs

• convert from larger units to smaller metric or customary units using previous knowledge of relationships between related units

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.

Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore, students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.

Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency. Fluency implies a much richer kind of mathematical knowledge and experience.

Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context.

Fluent students:

• flexibly use a combination of deep understanding, number sense, and memorization.

• are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.

• are able to articulate their reasoning.

• find solutions through a number of different paths.

For more about fluency, see: http://www.youcubed.org/wpcontent/uploads/2015/03/FluencyWithoutFear-2015.pdf and: https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/nctm-timedtests.pdf

STRATEGIES FOR TEACHING AND LEARNING

Convert like measurement units within a given measurement system.

MGSE5.MD.1 This standard calls for students to convert measurements within the same system of measurement in the context of multi-step, real-world problems. Both customary and standard measurement systems are included; students worked with both metric and customary units of length in second grade. In third grade, students work with metric units of mass and liquid volume. In fourth grade, students work with both systems and begin conversions within systems in length, mass and liquid volume.

To convert from one unit to another unit in the standard and metric system, the relationship between the units must be known. In order for students to have a better understanding of the relationships between units, they need to use measuring tools in class. The number of units must relate to the size of the unit.

Example 1: 100 cm = 1 meter

Example 2: 12 inches = 1 foot and 3 feet = 1 yard

When converting in the metric system, have students extend their prior knowledge of the baseten system as they multiply or divide by powers of ten (as referenced in Units 1 and 2). Teaching conversions should focus on the relationship of the measurements, not merely rote memorization. The questions ask the student to find out the size of each of the subsets. Students are not expected to know e.g. that there are 5280 feet in a mile. If this is to be used as an assessment task, the conversion factors should be given to the students. However, in a teaching situation it is worth having them realize that they need that information rather than giving it to them upfront; having students identify what information they need to have to solve the problem and knowing where to go to find it allows them to engage in Standard for Mathematical Practice 5, Use appropriate tools strategically.

Retrieved from Illustrative Mathematics
http://www.illustrativemathematics.org/standards/k8

Represent and interpret data.

MGSE5.MD.2 This standard provides a context for students to work with fractions by measuring objects to one eighth of a unit. This includes length, mass, and liquid volume. Students are making a line plot of this data and then adding and subtracting fractions based on data in the line plot.

Example 1: Students measured objects in their desk to the nearest 1/2, 1/4, or 1/8 of an inch then displayed data collected on a line plot. How many objects measured 1/4? 1/2? If you put all the objects together end to end what would be the total length of all the objects?

Geometric measurement: Understand Concepts of volume and relate volume to multiplication and to addition.

MGSE.MD.3 – MGSE.MD.4 – MGSE.MD.5

These standards involve finding the volume of right rectangular prisms and extend their understanding of finding the area of composite figures into the context of volume. Students should have experiences to describe and reason about why the formula is true. Specifically, that they are covering the bottom of a right rectangular prism (length x width) with multiple layers (height). Therefore, the formula (length width height) is an extension of the formula for the area of a rectangle.

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 1- 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.

By stacking geometric solids with cubic units in layers, students can begin understanding the concept of how addition plays a part in finding volume. This will lead to an understanding of the formula for the volume of a right rectangular prism, b x h, where b is the area of the base. A right rectangular prism has three pairs of parallel faces that are all rectangles.

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.

Instructional Resources/Tools

• Cubes

• Rulers (marked in standard or metric units)

• Grid paper

http://illuminations.nctm.org/ActivityDetail.aspx?ID=6: Determining the Volume of a Box by Filling It with Cubes, Rows of Cubes, or Layers of Cubes This cluster is connected to the third Critical Area of Focus for Grade 5, Developing understanding of volume.

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

The terms below are for teacher reference only and are not to be memorized by the students.

• measurement

• attribute

• conversion/convert

• metric and customary systems

• metric and customary units of measure

• line plot

• length

• mass

• weight

• liquid volume

• volume

• solid figure

• right rectangular prism

• unit

• unit cube

• gap

• overlap

• cubic units (cubic cm, cubic in, cubic ft, nonstandard cubic units)

• edge lengths

• height

• area of base