Grade 2 Mathematics TEKS Connections: Grades K-2 805 Session 3: Addition and Subtraction Content Strand Grade 2: Measurement: What’s the Area? Addition and Subtraction in Other Strands Texas Essential Knowledge and Skills: (2.1) Number, operation, and quantitative reasoning. The student understands how place value is used to represent whole numbers. (A) Use concrete models of hundreds, tens, and ones to represent a given whole number (up to 999) in various ways; (B) Use place value to read, write, and describe the value of whole numbers to 999; and (C) Use place value to compare and order whole numbers to 999 and record the comparisons using numbers and symbols (<, =, >). (2.3) Number, operation, and quantitative reasoning. The student adds and subtracts whole numbers to solve problems. (A) Recall and apply basic addition and subtraction facts (to 18); (B) Model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers; and (C) Select addition or subtraction to solve problems using two-digit numbers, whether or not regrouping is necessary. TEKS Connections to other Strands: (2.7) Geometry and spatial reasoning. The student uses attributes to identify two- and three- dimensional geometric figures. The student compares and contrasts two- and three-dimensional geometric figures or both. (B) Use attributes to describe how 2 two-dimensional figures or 2 three-dimensional geometric figures are alike or different; and (C) Cut two-dimensional geometric figures apart and identify the new geometric figures formed. (2.9) Measurement. The student directly compares the attributes of length, area, weight/mass, and capacity, and uses comparative language to solve problems and answer questions. The student selects and uses nonstandard units to describe length, area, capacity, and weight/mass. The student recognizes and uses models that approximate standard units (from both SI, also known as metric, and customary systems) of length, weight/mass, capacity, and time. (A) Identify concrete models that approximate standard units of length and use them to measure length; and (B) Select a non-standard unit of measure such as square tiles to determine the area of a two- dimensional surface. (2.12) Underlying processes and mathematical tools. The student applies Grade 2 mathematics to solve problems connected to everyday experiences and activities in and outside of school. (A) Identify the mathematics in everyday situations; and (D) Use tools such as real objects, manipulatives, and technology to solve problems. (2.13) Underlying processes and mathematical tools. The student communicates about Grade 2 mathematics using informal language. (A) Explain and record observations using objects, words, pictures, numbers, and technology; and (B) Relate informal language to mathematical language and symbols.
Grade 2: Measurement: What’s the Area? Addition and Subtraction in Other Strands
Texas Essential Knowledge and Skills: (2.1) Number, operation, and quantitative reasoning. The student understands how place value is
used to represent whole numbers. (A) Use concrete models of hundreds, tens, and ones to represent a given whole number (up to
999) in various ways; (B) Use place value to read, write, and describe the value of whole numbers to 999; and (C) Use place value to compare and order whole numbers to 999 and record the comparisons
using numbers and symbols (<, =, >). (2.3) Number, operation, and quantitative reasoning. The student adds and subtracts whole
numbers to solve problems. (A) Recall and apply basic addition and subtraction facts (to 18); (B) Model addition and subtraction of two-digit numbers with objects, pictures, words, and
numbers; and (C) Select addition or subtraction to solve problems using two-digit numbers, whether or not
regrouping is necessary.
TEKS Connections to other Strands: (2.7) Geometry and spatial reasoning. The student uses attributes to identify two- and three-
dimensional geometric figures. The student compares and contrasts two- and three-dimensional geometric figures or both. (B) Use attributes to describe how 2 two-dimensional figures or 2 three-dimensional geometric
figures are alike or different; and (C) Cut two-dimensional geometric figures apart and identify the new geometric figures
formed. (2.9) Measurement. The student directly compares the attributes of length, area, weight/mass, and
capacity, and uses comparative language to solve problems and answer questions. The student selects and uses nonstandard units to describe length, area, capacity, and weight/mass. The student recognizes and uses models that approximate standard units (from both SI, also known as metric, and customary systems) of length, weight/mass, capacity, and time. (A) Identify concrete models that approximate standard units of length and use them to
measure length; and (B) Select a non-standard unit of measure such as square tiles to determine the area of a two-
dimensional surface. (2.12) Underlying processes and mathematical tools. The student applies Grade 2 mathematics to
solve problems connected to everyday experiences and activities in and outside of school. (A) Identify the mathematics in everyday situations; and (D) Use tools such as real objects, manipulatives, and technology to solve problems.
(2.13) Underlying processes and mathematical tools. The student communicates about Grade 2 mathematics using informal language. (A) Explain and record observations using objects, words, pictures, numbers, and technology;
and (B) Relate informal language to mathematical language and symbols.
(2.14) Underlying processes and mathematical tools. The student uses logical reasoning to justify his or her thinking using objects, words, pictures, numbers, and technology.
Purpose: The focus of this lesson is to use non-standard units to investigate and develop the concept of area. Measuring with concrete objects provides the opportunity for students to develop counting strategies as well as to reinforce the concepts of addition and subtraction. Grouping: Whole group, small group, pairs
Suggested Vocabulary:
actual area join separate add length square area measure subtract compare order triangle diagonal parallelogram units estimate rectangle estimated area rhombus
Materials:
• 1-inch square colored tiles such as Color tiles • Sticky notes (3-inch by 3-inch) • Index cards • Scissors • Tape or glue • Linking cubes • Tangrams pieces • Plastic Ten-Frames or ice cubes trays with 10 cells
Advanced Preparation:
• Pre-cut triangles for teacher For Each Student:
• Handout 2-1: Ten Frame Mat – several copies • Handout 2-2: Color Tile Recording Sheet – 1 transparency for teacher • Handout 2-3: Area Recording Sheet – 1 transparency for teacher • Handout 2-4: Part-Part-Whole Mat – several copies and 1 transparency for teacher • Handout 2-5: Part-Part-Whole Recording Sheet • Handout 2-6: Got You Covered • Paper cut into 9-inch by 6-inch two transparencies for teacher • Construction paper cut into 9-inch by 9-inch squares
Procedures Teacher Notes Engage: 1. Provide each pair of students with a 9-inch by 6-inch
piece of construction paper and a pair of scissors. • What shape is this piece of paper?
Rectangle • How do you know?
Responses may vary. Possible responses include:four sides, four corners/vertices, all the corners look square, 2 longer sides, 2 shorter sides, opposite sides have the same length
• What is a diagonal line? Responses may vary.
Teacher needs to explain that a diagonal line of a rectangle would be a line from one corner to the opposite corner of the rectangle. Have a transparency of the rectangle on the overhead and point to opposite corners.
2. Tell students you want to fold the paper and create a
different shape. Model and explain that you are folding the rectangle on the diagonal of the rectangle.
3. Instruct students to fold their rectangle the same way. Monitor the students and help those who are experiencing difficulty.
4. Demonstrate cutting the rectangle along the folded line. Ask the students to cut their paper along the fold line. Monitor as students cut the paper. • How many shapes do we have?
two • What shapes have we created?
triangles • How did we make two shapes out of one?
We cut it in half.
Prior to Engage Have several copies of the rectangle cut into equal triangles for step 13 of the Engage. This lesson provides the opportunity to reinforce the names of shapes and develops geometric vocabulary as the foundation for understanding the meaning of area. Note: Area is expressed with two dimensions being measured: length and width. Some students should be able to state that the line they use to make 5 with tally marks is a diagonal line. TEKS Connection: 2.7C As students create new shapes, this is the perfect time to introduce the formal vocabulary terms. Note: All of the 4- sided figures are called quadrilaterals. There are some quadrilaterals that have specific names. The quadrilateral listed below may also be called a kite if some students refer to it that way.
5. Ask the students how the two new shapes compare. • What do you notice about the two triangles?
Responses may vary. Possible responses include: They are the same size.
• How can you prove the triangles are the same size? Responses may vary. Encourage the students to move, flip, and rotate the shapes to compare. The teacher should be modeling this on the overhead.
Scaffolding Create a class-illustrated glossary that can be augmented and kept in the classroom library. Sheet protectors and a binder will make it easy to add illustrated math vocabulary throughout the year. Example
TEKS Connection: 2.7A, 2.7B Geometry and Spatial Reasoning. Integrating attributes of two-dimensional figures and area reasoning presents a bridge to the introduction of area for 2nd grade. Center Extension Set up center with Tangrams (a very old and popular set of puzzle shapes) or pattern blocks. Begin with just the two large triangles. Have students make all the shapes they can with the two large triangles. Are the shapes the same as the ones they made with the construction paper? If there are different shapes, how are they different? Add the remaining Tangram pieces or pattern blocks to the center. Create task cards for students to fill outlined areas with Tangrams or pattern blocks.
6. Instruct the students to use both triangles and to create as many shapes as possible. The only restriction is that the sides they put together must be the same length. Teacher models on the overhead.
7. Monitor as students create the new shapes. Have students trace the shapes into their math journal and share the different shapes created with the whole class.
8. As each shape is presented, ask the following questions: • What is the name of the shape?
Responses may vary. • How do you know?
Responses may vary. Possible responses include: All sides are equal in length. Students should name the attributes that make up the shape.
• Is that shape bigger, smaller, or the same as the original rectangle? Responses may vary.
• How can you prove the shape is bigger, smaller, or the same size? Responses may vary.
9. As the students share a new shape, the teacher will
take two pre-cut triangles and create the shape. Tape or glue the new shapes to paper and label with the appropriate name to add illustrated words to the math word wall. Engage students in discussion about the attributes of the different shapes. 3 sides and 3 corners, 4 sides, etc. Compare the angles of the parallelograms to the angles of squares and rectangles.
TEKS Connection: 2.1C As students create new shapes, this is the perfect time to introduce the formal vocabulary terms. Note: All of the 4- sided figures are called quadrilaterals. There are some quadrilaterals that have specified names. The quadrilateral listed below may also be called a kite if some students refer to it that way. Research Size in this context is a useful substitute for area with very young children, although size and area do not mean exactly the same thing (Van de Walle, 2004).
Explore: 1. Reflect with students about their thoughts
concerning the different shapes they made with the construction paper and whether the new shapes were the same size or different sizes.
Prior to Explore Students will need a 9-inch by 6-inch piece of paper and a pair of scissors. Units are introduced in this phase of the lesson to explore area.
2. Provide each pair of students with a 9-inch by 6-inch piece of paper and a pair of scissors.
3. Tell students to fold the paper in half and cut along
the fold. Teacher should not demonstrate. (See note.)
4. Have students create new shapes with the two pieces
of construction paper and tape the pieces together.
5. Select two students who have created rectangles differently to show their rectangles to the class. • What do you notice about the rectangles?
Responses may vary. • Which one is bigger? Why?
Responses may vary. Possible responses include: The sizes of the rectangles are different; one is larger/bigger than the other or wider/shorter than the other.
TEKS Connection: 2.1B, 2.1C, 2.3A, 2.3B, 2.3C Number Operations and Quantitative Reasoning. The task of estimating and counting units provides the opportunity to target number and operation skills. Note DO NOT TELL STUDENTS which way (vertically or horizontally) to fold. Both ways will be explored.
portrait view of paper
Only one new rectangle is possible for each of the vertical and horizontal folds. The other possibility is the original rectangle.
6. Provide students with colored tiles to use as a measurement tool to help them compare the size of the rectangles. • How can we use the color tiles to help us
prove the rectangles are the same size or different sizes? Responses may vary.
7. Lead students to conclude they can cover the
rectangles with tiles and count the tiles. Encourage students counting by 2’s, 3’s, etc. Tell students that when they completely cover the surface and count the tiles covering the surface, they are finding the area of the surface in tile units. When comparing the estimate to the actual area, some students may need to use Handout 2-1: Ten-Frame Mat. • Have you have seen a pan of brownies cut
into squares? Responses may vary.
The brownies are similar to the tiles. • Can you think of any other examples?
Responses may vary. Possible responses include: Tiles on the floor, etc.
8. Provide students with Handout 2-2: Color Tile
Recording Sheet. Have the students write their estimates on their recording sheet before actually covering the surface with color tiles. The teacher needs to model how to complete the recording sheet along with the students.
9. Ask students to compare their estimates and the actual number of tiles needed to cover the surface. Let the class know that the Ten-Frame mats are for them to use if needed. Observe the students as they find the difference. Listen to the conversations. • How close was your estimate and the actual
number? Responses may vary.
Handout 2-1: Ten-Frame Mat
Have multiple copies of Ten-Frames for each pair of students. If students are not familiar with the Ten-Frame mats, the teacher will need to model using them to complete an addition problem and a subtraction problem. The use of the Ten-Frames will allow the students to visualize as well as compare the two numbers.
10. Create a chart using chart paper or the board to record and discuss student results. Allow students to share strategies of how they handled the shorter wider rectangle and the tiles not fitting exactly. Allow students to share how they calculated the differences between the estimate and the actual measurement. Have the students share their different strategies for subtracting. Responses may vary. Possible responses include: My estimate was 60 and the number of tiles needed was 48. 48 is 2 from 50. 50 is 10 from 60. 2 + 10 = 12, so 60 – 48 = 12. My estimate was 60, which filled 6 Ten- Frames. My actual number was 48. I took 4 Ten-Frames away and I took 8 cubes from another Ten-Frame. That left 12 cubes.
Research The foundation of understanding of area in 2nd grade is the measure of covering. DO NOT introduce formulas. The exact answer is not the goal at this point (Van de Walle, 2004). Teachers can create a table to record student findings on the board, such as the one below.
Look for flexible strategies for comparing estimates and number of tile: • Counting down from the larger
number • Counting on from the smaller
number (Carpenter et al. 1999). Use of open number line to jump between numbers. • Example of open number line (start
at 72).
(Fosnot, 2001).
Explain: 1. Give each pair of students Handout 2-3: Area
Recording Sheet and explain each section.
Prior to Explain Set up the stations so that some tools are already chosen for the students. Tools to use can include: • Colored tiles • 3-inch by 3-inch sticky notes • Index cards • Construction paper 9-inch by 9-inch
Have at least two stations with objects that represent about the same area, but have different size unit tools such as sticky notes at one station and 4-inch x 6-inch index cards at the other station.
2. Point out the different area measurement stations around the room. Tell students that some stations have measurement tools ready for them to use and some stations let students choose from two or three different measurement tools. Tell students to record the name of the object measured, the measurement tool used, and the estimate and actual unit count on the recording sheet. The teacher needs to model the correct way to complete the recording sheet. Each student in the pair takes turns estimating, covering, and counting the units. The partner verifies the number sentence and solution with the estimate and count. Students take turns as they move from station to station. (Demonstrate how you want students to move from station to station.)
3. Monitor students as they cover the objects in the various stations. Support students with questions to gain insight into their strategies as they count the units when they measure. • How many units long is this side?
Responses may vary. • How many units across in this row?
Responses may vary. • Do you see a pattern that will help you count
all the units? Responses may vary.
• Can you show me how you covered this? Responses may vary.
• What is another way to count your units? Responses may vary.
Handout 2-3: Area Recording Sheet
Do not have students report their strategies at this time. The teacher needs to be observing the students and keeping track of different strategies the children use as they count the units for area. • Are students counting one unit at a
time? • Are they using grouping strategies
(by 2s, 5s, 10s)? • Do they use patterns of rows and
Elaborate: 1. Bring the class back together to compare estimates
and area measures. Students will need their recording sheets.
2. On chart paper or on the board, create a table to
write the students’ measurements. 3. Call on different pairs of students to complete the
chart. Ask students to demonstrate how they measured the area. Make sure they share different strategies. Have the class brainstorm possible reasons for different area measurements when they used the same object and unit tools. • Why do you think the area measurements are
different when the same object was measured using the same tool? Responses may vary. Possible responses include: Orientation of tool, error on the part of the person measuring, etc.
• Which stations were these? Identify the station by number. Stations ___ and ___ Responses may vary.
• How does the size of the measuring tool affect the number of units needed to find the area of the object? Responses may vary. Possible responses include: The smaller the measuring unit, the more units are needed to cover the object.
4. Have students share their number sentences and
strategies for determining the difference between their estimates and actual area measures.
Teachers can create a table to record student findings on the board such as the one below.
The completed chart will be used to set up addition and subtraction problems.
TEKS Connection: 1.7C: First grade lays the foundation for developing the understanding of the relationship between the size of the unit and the number of units needed for linear measurements. Second grade continues to develop the understanding of this relationship.
Exploring area provides a new dimension upon which to build the relational understanding. Students will need numerous linear and area measurement experiences throughout the year to develop and reinforce this relational understanding. Second grade continues to develop counting on from the first number. • Do students model both numbers,
then count on from the larger number?
• How do students compare numbers?
• Do they line the numbers side by side and count the non-paired counters?
• Do students model the larger number and take away the smaller number?
5. Refer to the specific objects for which students
found area measures. Tell students they will be solving addition and subtraction problems based on their recorded information. Give each pair of students multiple copies of Handout 2-4: Part-Part-Whole Mat. Tell the students that they are allowed to use any available work mats or manipulatives to solve the problems.
Research “Over time, these physical modeling strategies give way to more efficient counting strategies which are generally more abstract ways of modeling a problem. Eventually, children come to rely on number facts, but the learning of number facts is not necessarily rote skill. It can build upon an understanding of number relations, which are supported by a foundation of number sense developed through using modeling and counting strategies” (Carpenter et al., 1999, p. 4). Handout 2-4: Part-Part-Whole Mat When introducing the Part-Part-Whole mat, be sure to explain when comparing two numbers, (in this example 2 areas) the “whole” on the mat would represent the total number of units needed to cover both surface areas and each “part” would represent one surface area. Students can use each area for writing numbers and/or drawing pictures of the objects. Stress the need for labeling each section of the mat. Example: Have students complete the Part-Part-Whole mat using counters or manipulatives, such as linking cubes, for modeling addition and subtraction. After both “parts” on the mat have been completed, have the students slide their counters to the bottom rectangle representing the “whole” or total. Have students record each “part” in the trapezoid shape number sentence area.
Measurement Chart created in the Explain portion of this lesson. Provide students with counters, Part-Part-Whole Mats and Part-Part-Whole Recording Sheet. As the problems are presented by the teacher, the students will be completing the Part-Part-Whole recording sheet. Students may choose to use the large Part-Part-Whole mat and use manipulatives to work out a solution and then transfer their information to their recording sheet.
7. As the teacher presents the problems orally, the
teacher records the object and the total area of the object on the board. Present problems such as the problems below. Have students write a number sentence to fit the situation and solve the problem. Allow students to work in pairs while observing their use of the Part-Part-Whole Mat and strategies used.
After all tiles have been moved to the “whole” or total rectangle, students determine the total number of tiles used to measure both items. Then students record the total of both parts and complete the number sentence in the trapezoid number sentence area. Provide students with more problems. Observe the strategies students use to model addition and subtraction. • Do students count out each number
and then count each counter one by one?
• Do students regroup the counters to count by 2s, 5s, or 10s?
• Do students model the first number, then second number?
Handout 2-5: Part-Part-Whole Recording Sheet
Do not assume that all students are auditory learners. Therefore, record the given information on the board as present orally to students.
Object A (ex: The teacher’s desk) has an area of ______ units. Object B (ex: The shelf) has an area of ______ units. • If I wanted to entirely cover the top of the
desk and the shelf, what would be the total area covered? Responses may vary.
• Which object, A or B, has more units? Responses may vary.
• Which has less units? Responses may vary.
• How many more units does object __ have than object__? Responses may vary.
• Object B has _____ units in all. Object B and C have _____ units. How many units does Object C have? Responses may vary.
8. Continue creating problems using the information on
the Estimate and Area Recording Chart and observe each pair of students. Have students share various strategies they use to solve the problems.
Ex: Desk: 36 tiles Shelf: 24 tiles Find the total number of tiles needed for both the desk and the shelf. Shelf: 24 tiles Shelf and Chart Tablet: 49 tiles
• What is the area of the Chart Tablet?
Evaluate: 1. Provide students with the Handout 2-6: Got You
Covered. 2. Let’s read the directions on your handout together.
Fill in the chart below by finding the area of the objects listed. Be sure to write the name of the unit used to find the area of each object. Write the estimates before measuring the objects.
3. Use your answers in the chart to answer the
questions.
Provide students with Handout 2-6: Got You Covered and colored tiles or linking cubes for students to use to find the area of specific objects during the assessment.
Extension Set up a class area scavenger hunt. Create clue cards based on objects in your classroom. Have students read the cards (one at a time), then go find the object that will fit the description on the clue card. Example:
Handout 2-6: Got You Covered Directions: Fill in the chart below by finding the area of the objects listed. Be sure to write
the name of the unit used to find the area of each object. Write the estimates before measuring the objects.
Unit used to measure: ___________________
Object Estimate Actual Area A.
B.
Directions: Use your answers in the chart above to answer the questions below. 1. What is the difference between the estimate and the actual area for the objects
measured? Record your number sentence and circle the answer. Make sure you label your final answer.
A. _____________________ B. _____________________ 2. What is the total area for the two objects you measured? Record your number sentence
and circle the answer. Make sure you label your final answer. A. _____________________ B. _____________________ 3. Jack wants to find the area of his kitchen table. He has paper clips, index cards, and
construction paper that he can use as units. Which tool should he use? Write the name of the tool he should use and one or two sentences explaining why he should use the tool you chose.
