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3rd Grade MathematicsUnit 3: Geometry/Measurement
Teacher Resource Guide
2012 - 2013
In Grade 3, instructional time should focus on four critical areas:
- Developing understanding of multiplication and division and strategies for multiplication and division within 100;
Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations.
- Developing understanding of fractions, especially unit fractions (fractions with a numerator of 1);
Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.
- Developing understanding of the structure of rectangular arrays and of area;
Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.
- Describing and analyzing two-dimensional shapes;
Students compare and classify shapes by their sides and angles, and connect these with definitions of shapes. They also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.
3rd Grade 2012-2013Page 1
Unit / Time Frame / Test ByTRIMESTER 1 / 1: Addition and Subtraction
(Within 1,000) / 7 weeks / 8/27 – 10/12 / October 12
2: Multiplication and Division:
Models within 100 / 5 weeks / 10/15 – 11/16 / November 16
TRIMESTER 2 / 3: Geometry/Measurement / 4 weeks / 11/19-12/21 / December 21
4: Multiplication and Division:
Properties within 100 / 5 weeks / 1/2 – 2/8 / February 8
5: Fractions / 8 weeks / 2/11 – 4/12 / April 12
TRIMESTER 3
6: Multiplication and Division:
Application & Fluency within 100 / 7 weeks / 4/15 – 5/30 / May 30
3rd Grade Mathematics 2012-2013
Math Wiki:
3rd Grade 2012-2013Page 1
Unit 3: Geometry/Measurement November 19-December 21 (4 weeks)
Big Ideas / Essential QuestionsA two-dimensional shape can have more than one name when it belongs to more than one category based on its attributes. / When can a two-dimensional shape have more than one name?
Area measures the number of same sized squares used to cover a 2-D figure. / What does area measure?
Perimeter measures the distance around a 2-D figure. / What does perimeter measure?
Identifier / Standards / Mathematical Practices
STANDARDS / 3.G.1 / Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. / 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.
3.MD.7a
3.MD.5
3.MD.6
5.MD.7 / Relate area to the operations of multiplication and addition.
- Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
- A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area.
- A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
Relate area to the operations of multiplication and addition.
b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths aandb + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non- overlapping parts, applying this technique to solve real world problems.
3.MD.8
3.MD.2 / Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).[1] Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.[2]
Identifier / Standards / Bloom’s / Skills / Concepts
STANDARDS / 3.G.1 / Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. / Understand (2)
Remember (1) / Understand (shapes share attributes)
Recognize (quadrilaterals) / shape
attribute
category
rhombus
rectangle
square
3.MD.7a
3.MD.5
3.MD.6
5.MD.7 / Relate area to the operations of multiplication and addition.
- Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
a. A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
Relate area to the operations of multiplication and addition.
b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths aandb + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non- overlapping parts, applying this technique to solve real world problems. / Remember (1) / Find (area of rectangle) / area
rectangle
tiling
side length
3.MD.8
3.MD.2 / Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).[3] Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.[4] / Apply (3) / Solve (problems involving perimeter) / polygons
side length
rectangle
Instructional Strategies forALL STUDENTS
Sorting and Classifying – In earlier grades, students have experiences with informal reasoning about particular shapes through sorting and classifying using their geometric attributes. Student have built and drawn shapes given the number of faces, number of angles, and number of sides. The focus now is on identifying and describing properties of two-dimensional shapes in more precise ways using properties that are shared rather than the appearances of the individual shapes. These properties allow for generalizations of all shapes that fit a particular classification. Development in focusing on the identification and description of shapes’ properties should include examples and nonexamples, as well as examples and nonexamples drawn by students of shapes in a particular category. For example, students could start with indentifying shapes with right angles. An explanation as to why the remaining shapes do not fit this category should be discussed. Students should determine common characteristics of the remaining shapes.
Activities for sorting and classifying are listed in the resource section of this guide. The actual activities are on the wiki.
Important note: Students may identify a square as a “nonrectangle” or a “nonrhombus” based on limited images they see. They do not recognize that a square is a rectangle because it has all of the properties of a rectangle. They may list properties of each shape separately, but not see the interrelationships between the shapes. For example, students do not look at the properties of a square that are characteristic of other figures as well. Using straws to make four congruent figures have students change the angles to see the relationships between a rhombus and a square. As students develop definitions for these shapes, relationships between the properties will be understood.
Develop conceptual understanding for area of rectangles–Area is a measure of space inside a region or how much it takes to cover a region. As with other attributes, students must first understand the attribute of area before measuring. The Priority Standard states that students will find the area of a rectangle by tiling and show that the area can be found by multiplying the side lengths. This should be considered the end result of a study of area rather than the only objective. This is students’ first exposure to area, so in order to develop conceptual understanding, several experiences related to area are recommended prior to working with rectangles and multiplying side lengths. The activities recommended give students experiences in comparison of area and developing an understanding of units of area. These activities are found in Teaching Student-Centered Mathematics K-3, Van de Walle & Lovin, Pearson, 2006, p. 234-238. The activities are listed in the Lesson Bank and on the wiki.
Instructional Strategies for ALL STUDENTS (continued)
Develop conceptual understanding for perimeter of rectangles – Students have created rectangles before when finding the area of rectangles and connecting them to using arrays in the multiplication of whole numbers. To explore finding the perimeter of a rectangle, have students use nonstretchy string. They should measure the string and create a rectangle before cutting it into four pieces. Then, have student use four pieces of nonstretchy string to make a rectangle. Two pieces of the string should be of the same length and the other two pieces should have a different length that is the same. Students should be able to make the connection that perimeter is the total distance around the rectangle.
Geoboards can be used to find the perimeter of rectangles also. Provide students with different perimeters and have them create the rectangles on the geoboards. (Blackline masters of paper geoboards are available on the wiki if actual geoboards are not available.) Have students share their rectangles with the class. Have discussions about how different rectangles can have the same perimeter with different side lengths.
Once students know how to find the perimeter of a rectangle, they can find the perimeter of rectangular-shaped objects in their environment. The can use appropriate measuring tools to find lengths of rectangular-shaped objects in the classroom. Present problem situations involving perimeter, such as finding the amount of fencing needed to enclose a rectangular-shaped park, or how much ribbon is needed to decorate the edges of a picture frame. Also present problem situations in which perimeter and two or three of the side lengths are known, requiring students to find the unknown side length.
Students need to explore how measurements are affected when one attribute to be measured is held constant and the other is changed. Using square tiles, students can discover that the area of rectangles may be the same, but the perimeter of the rectangles varies. Geoboards can also be used to explore this same concept.
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 during Unit 3:
Skill / StandardA square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area. / 3.MD.5a
Addition and Subtraction / 3.NBT.2
Multiplication and Division facts (word problems) / 3.OA.3
Tell and write time / 3.MD.1
Graphing / 3.MD.3
Other skills students need to develop based on teacher observation and formative assessments.
Resource Bank
Activities and Lessons / Teacher’s Edition
Pages / Standards
Addressed
Activities for perimeter: See Instructional Strategies on p. 5 of this guide for explanation of 2-3 different activities to develop conceptual understanding for perimeter. Do these activities first. / 3.MD.8
Math Expressions, Unit 2, Lesson 1, Activity 1-2 / 138 / 3.MD.8
Math Expressions, Unit 2, Lesson 2, Activity 1-3 / 146 / 3.MD.8
Website: Link for lesson also listed on Wiki Unit 3(This can be done as a whole class or in a computer lab setting individually or with partners.) / 3.G.1
Math Expressions, Unit 2, Lesson 3, Activities 1-4 (This should come after the online lesson.) / 152 / 3.MD.8, 3.G.1
Math Expressions, Unit 2, Lesson 4, Activities 1-2 / 160 / 3.G.1
Math Expressions, Unit 2, Lesson 5, Activity 1 / 166 / 3.G.1
Wiki: Two-Piece Shapes / 3.MD.7
Wiki: Rectangle Comparison – No Units / 3.MD.7
Wiki: Rectangle Compare – Square Units / 3.MD.7
Math Expressions, Unit 8, Lesson 1, Activities 1-3, Homework – can be done in class (This has more practice with decomposing shapes to find area and area word problems.) / 610 / 3.MD.7, 3.MD.8, 3.MD.5, 3.MD.6
Math Expressions, Unit 8, Lesson 2, Activities 1-2, Homework – can be done in class (This has more practice with area.) / 620 / 3.MD.8, 3.MD.7b
Math Expressions, Unit 8, Lesson 3, Activities 1-3, Homework – can be done in class (This has more practice with area word problems.) / 626 / 3.MD.7, 3.MD.8
Math Expressions, Unit 13, Lesson 6, Activities 1-3 (only focus on metric measures) / 1032 / 3.MD.2
Math Expressions, Unit 13, Lesson 9, Activities 2-3 (only focus on metric measures) / 1055 / 3.MD.2
Note: The Activities and Lessons listed below in the Resource Bank should be done in the order they are listed.
Math Expressions Activities(use as centers, re-teaching/extension support, etc.)
Activity / Standards
Activity Card 2-1 On-Level / 3.MD.8
Activity Card 2-2 Intervention, On-Level (use after the online lesson) / 3.G.1
Activity Card 2-3 Intervention, On-Level / 3.G.1
Activity Card 2-4 On-Level / 3.G.1
Activity Card 2-5 Intervention / 3.G.1
Activity Card 8-1 Intervention, On-Level / 3.MD.7a, 3.MD.8
Activity Card 8-2 Intervention, On-Level, Challenge / 3.MD.7c
Activity Card 8-3 Challenge / 3.MD.8
3rd Grade 2012-2013Page 1
1Excludes compound units such as cm3 and finding the geometric volume of a container.
2Excludes multiplicative comparison problems (problems involving notions of “times as much.”
1Excludes compound units such as cm3 and finding the geometric volume of a container.
2Excludes multiplicative comparison problems (problems involving notions of “times as much.”