All the resources on this page come from the GADOE Framework 2nd Grade Unit 5 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 5 Overview

In this unit students will cultivate spatial awareness by:

• further developing understandings of basic geometric figures

• identifying plane figures and solid figures based on geometric properties

• describing plane figures and solid figures according to geometric properties

• expanding the ability to see geometry in the real world

• partitioning shapes into equal shares by cutting, slicing, or dividing

• represent halves, thirds, and fourths using rectangles and circles to create fraction models

• compare fractions created through partitioning same-sized rectangular or circular wholes in different ways

• understand what an array is and how it can be used as a model for repeated addition

• organize and record data using tallies, simple tables and charts, picture graphs, and bar graphs

Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as counting, time, money, positional words, patterns, and tallying should be addressed on an ongoing basis through the use of calendar, centers, and games.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important to study the tasks in this unit early in the planning process. The tasks in this unit illustrates the types of learning activities that should be utilized from a variety of sources in order for students to gain a solid foundation in geometry to meet or exceed grade level standards.

NUMBER TALKS

Between 5 and 15 minutes each day should be dedicated to “Number Talks” in order to build students’ mental math capabilities and reasoning skills. Sherry Parrish’s book Number Talks provides examples of K-5 number talks. The following video clip from Math Solutions is an excellent example of a number talk in action. http://www.mathsolutions.com/videopage/videos/Final/Classroom_NumberTalk_Gr3.swf

During the Number Talk, the teacher is not the definitive authority. The teacher is the facilitator and is listening for and building on the students’ natural mathematical thinking. The teacher writes a problem horizontally on the board in whole group or a small setting. The students mentally solve the problem and share with the whole group how they derived the answer. They must justify and defend their reasoning. The teacher simply records the students’ thinking and poses extended questions to draw out deeper understanding for all.

The effectiveness of Numbers Talks depends on the routines and environment that is established by the teacher. Students must be given time to think quietly without pressure from their peers. To develop this, the teacher should establish a signal, other than a raised hand, of some sort to identify that one has a strategy to share. One way to do this is to place a finger on their chest indicating that they have one strategy to share. If they have two strategies to share, they place out two fingers on their chest and so on. Number Talk problem possible student responses:

Number talks often have a focus strategy such as “making tens” or “compensation.” Providing students with a string of related problems, allows students to apply a strategy from a previous problem to subsequent problems. Some units lend themselves well to certain Number Talk topics. For example, the place value unit may coordinate well with the Number Talk strategy of “making ten.” For additional information 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. 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 will use nets to create cubes and discover different attributes about them.

2. Reason abstractly and quantitatively. Students will use partitioning and equal groups to break shapes into different pieces.

3. Construct viable arguments and critique the reasoning of others. Students will use known information/attributes of different shapes to construct viable arguments about them.

4. Model with mathematics. Students will create cubes to learn about sides, edges, vertices and angles.

5. Use appropriate tools strategically. Students use tangrams to help make/create different shapes.

6. Attend to precision. Students will create and draw shapes and will have to make sure to keep their lines straight, form the correct angles, and keep lines congruent if needed.

7. Look for and make use of structure. Students will use different shapes to create another object.

8. Look for and express regularity in repeated reasoning. Students will use knowledge of equal parts and portioning to develop strategies for sharing and grouping items.

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

STANDARDS FOR MATHEMATICAL CONTENT

Reason with shapes and their attributes.
MGSE2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MGSE2.G.2 Partition a rectangle into rows and columns of same-size squares to find the total number of them.

MGSE2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Represent and interpret data.
MGSE2.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 bar graph.

BIG IDEAS

By the conclusion of this unit, students should be able to demonstrate the following competencies:

• Describe plane figures according to their characteristics (sides, corners, angles).

• Describe solid figures according to their characteristics (faces, edges, vertices).

• Describe and understand the relationships (similarities and differences) between solid figures and plane figures.

• Recognize the relationship between geometry and the environment.

• Compare geometric figures to similar objects in everyday life.

• Identify and represent the fractional parts of a whole or of a set (halves, thirds, fourths).

• Recognize and represent that differently partitioned fractional parts of same-sized rectangles or circles are equal.

• Identify the number of rows and columns in an array and count the same-size squares to find the total.

• Pose questions that will result in data that can be shown on a bar graph or picture graphs.

• Use charts, simple tables, and surveys to collect data that can be shown on a bar graph or picture graph.

• Graph data on a bar graph or picture graph and in a simple table. Interpret data shown on a bar graph or picture graph.

• identify plane figures and solid or hollow figures according to geometric properties

• describe plane figures and solid or hollow figures according to geometric properties

• develop an understanding of the relationship between solid or hollow figures and plane figures

• understand that the faces of solid or hollow figures are plane figures

• further develop spatial awareness of geometric solids and figures

• investigate what happens when geometric figures are combined

• investigate what happens when geometric figures are cut apart

• recognize plane and solid figures in the real world

• Repeatedly adding the same quantity or forming a rectangular array are strategies for repeated addition.

• Fractional parts are equal shares of a whole number, whole object, or a whole set.

• The more equal sized pieces that form a whole, the smaller the pieces (fraction) will be.

• When the numerator and denominator are the same number, the fraction equals the number one or one whole (entire object or set).

• The fraction name (half, third, fourth) indicates the number of equal parts in the whole.

• Equal shares of identical wholes may not have the same shape. For example, fourths can be represented in multiple ways (i.e. with diagonal, horizontal, vertical cuts) and although they look different they represent the same amount/size piece.

ESSENTIAL QUESTIONS

• How do we describe geometric figures?

• Where can we find geometric figures in the world around us?

• How do we use the following terms: angle, vertex, face, side, and edge to describe geometric figures?

• How do we apply the use of fractions in everyday life?

• How do we know how many fractional parts make a whole?

• When is it appropriate to use fractions?

• How can we use a picture graph, bar graph, chart, or table to organize data and answer questions?

CONCEPTS/SKILLS TO MAINTAIN

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 and automaticity. 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, and 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 varieties 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/wp-content/uploads/2015/03/FluencyWithoutFear-2015.pdf and https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/nctm-timedtests.pdf

Skills from Grade 1:

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.

• Developing understanding of linear measurement and measuring lengths as iterating length units.

Second Grade Year Long Concepts:

Organizing and graphing data as stated in MGSE2.MD.10 should be incorporated in activities throughout the year. Students should be able to draw a picture graph and a bar graph to represent a data set with up to four categories as well as solve simple puttogether, take-apart, and compare problems using information presented in a bar graph.

Specifically, it is expected that students will have prior knowledge/experience related to 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 understanding of these ideas.

• Composition and decomposition of two- dimensional shapes

• Recognition of shapes from different perspectives and orientations

• Basic geometric figures and spatial relationships

• Sides, vertices, and other geometric attributes

• Fractions: halves, fourths

• Tally marks

• Picture graphs

STRATEGIES FOR TEACHING AND LEARNING
(Information adapted from the North Carolina DPI Instructional Support Tools)

General Strategies:

• Students should be actively engaged by developing their own understanding.

• Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols, and words.

• Appropriate manipulatives and technology should be used to enhance student learning.

• Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition which includes self-assessment and reflection.

• Math journals are an excellent way for students to show what they are learning about a concept. These could be spiral bound notebooks that students could draw or write in to describe the day’s math lesson. Second graders love to go back and look at things they have done, so journals could also serve as a tool for a nine-week review, parent conferencing, etc.

Reason with shapes and their attributes
MGSE2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MGSE2.G.2 Partition a rectangle into rows and columns of same-size squares to find the total number of them.

MGSE2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Instructional Strategies (Information quoted from Van de Walle and Lovin, Teaching Student-Centered Mathematics: Grades K-3, page 188)

“Not all people think about geometric ideas in the same manner. Certainly, we are all not alike, but we are all capable of growing and developing in our ability to think and reason in geometric contexts. The research of two Dutch educators, Pierre van Hiele and Dina van Hiele-Geldof, has provided insight into the differences in geometric thinking and how the differences come to be.

The most prominent feature of the model is a five-level hierarchy of ways of understanding spatial ideas. Each of the five levels describes the thinking processes used in geometric contexts. The levels describe how we think and what types of geometric ideas we think about, rather than how much knowledge we have. A significant difference from one level to the next is the objects of thought-what we are able to think about geometrically.”

• Level 0: Visualization

• Level 1: Analysis

• Level 2: Informal Deduction

• Level 3: Deduction

• Level 4: Rigor

For more information on the van Hiele Levels, refer to Van de Walle and Lovin, Teaching Student-Centered Mathematics: Grades K-3, Chapter 7.

MGSE2.G.1 Calls for students to identify (recognize) and draw shapes based on a given set of attributes. These include triangles, quadrilaterals (squares, rectangles, and trapezoids), pentagons, hexagons and cubes. Example: Draw a closed shape that has five sides. What is the name of the shape? Student 1 I drew a shape with 5 sides. It is called a pentagon.

Prior to teaching the unit, you can use the plane shapes graphic organizer as a whole class assessment or give each child a copy and have them list everything they know about the given shapes.

MGSE2.G.2 calls for students to partition a rectangle into squares (or square-like regions) and then determine the total number of squares. This relates to the standard 2.OA.4 where students are arranging objects in an array of rows and columns. Modeling repeated addition with partitioned rectangles provides the foundation for student understanding of multiplication. While discussions of multiplication may arise as an offshoot of work in this standard, the emphasis should be on understanding repeated addition through the array model. Tell students that they will be drawing a square on grid paper. The length of each side is equal to 2 units. Ask them to guess how many 1 unit by 1 unit squares will be inside this 2-unit by 2-unit square. Students now draw this square and count the 1 by 1 unit squares inside it. They compare this number to their guess. Next, students draw a 2-unit by 3-unit rectangle and count how many 1 unit by 1 unit squares are inside. Now they choose the two dimensions for a rectangle, predict the number of 1 unit by 1 unit squares inside, draw the rectangle, count the number of 1 unit by 1 unit squares inside and compare this number to their guess. Students repeat this process for different-size rectangles. Finally, ask them to share what they observed as they worked on the task.

Next example in the series: Split the rectangle into 2 rows and 4 columns. How many small squares did you make?

Note: This standard is laying the foundation for student understanding of area which will be studied in 3rd grade. They are creating an area model when they partition a rectangle into squares. It is important to help them see that for this there should be no gaps or overlaps between squares.

MGSE2.G.3 calls for students to partition (split) circles and rectangles into 2, 3 or 4 equal shares (regions). Students should be given ample experiences to explore this concept with paper strips and pictorial representations. Students should also work with the vocabulary terms halves, thirds, half of, third of, and fourth (or quarter) of. While students are working on this standard, teachers should help them to make the connection that a ―whole‖ is composed of two halves, three thirds, or four fourths.

This standard also addresses the idea that equal shares of identical wholes may not have the same shape.

Example: Divide each rectangle into fourths a different way.

It is vital that students understand different representations of fair shares. Provide a collection of different-size circles and rectangles cut from paper. Ask students to fold some shapes into halves, some into thirds, and some into fourths. They compare the locations of the folds in their shapes as a class and discuss the different representations for the fractional parts. To fold rectangles into thirds, ask students if they have ever seen how letters are folded to be placed in envelopes. Have them fold the paper very carefully to make sure the three parts are the same size. Ask them to discuss why the same process does not work to fold a circle into thirds. Use an analog clock as a model and allow children to draw a line from the center of the clock to the place where the 12, 4 and 8 are on the clock face. This will divide the circle into three equal sections. This clock connection can also be made for discussing halves and fourths and discovering to which numbers (or hours) you would draw the lines to in order to create two or four equal parts/pieces.

Represent and Interpret Data

MGSE2.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 bar graph.

INSTRUCTIONAL STRATEGIES

At first students should create real object and picture graphs so each row or bar consists of countable parts. These graphs show items in a category and do not have a numerical scale. For example, a real object graph could show the students’ shoes (one shoe per student) lined end to end in horizontal or vertical rows by their color. Students would simply count to find how many shoes are in each row or bar. The graphs should be limited to 2 to 4 rows or bars. Students would then move to making horizontal or vertical bar graphs with two to four categories and a single unit scale.

Students display their data using a picture graph or bar graph using a single unit scale.

As students continue to develop their use of reading and interpreting data it is highly suggested to incorporate these standards into daily routines. It is not merely the making or filling out of the graph but the connections made from the data represented that builds and strengthens mathematical reasoning.

SELECTED TERMS AND SYMBOLS

The following terms and symbols are not an inclusive list and should not be taught in isolation. Instructors should pay particular attention to them and how their students are able to explain and apply them (i.e. students should not be told to memorize these terms).

Teachers should present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

For specific definitions, please reference the Georgia Standards of Excellence Glossary. Note: GA uses the inclusive definition of a trapezoid. This card set contains the exclusive definition. • angle • attribute • bar graph • circle • column • cone • cube • cylinder • data set • edge • face • fourths • fraction • halves • hexagon • irregular polygon • partition • pentagon • picture graph • plane figure • polygon • quadrilateral • rectangle • regular polygon • row • scale • shapes• solid figure • square • thirds • trapezoid • triangle • unit fraction • vertex/vertices • whole