Middle School Mathematics UDL Instructional Unit- Lesson 1

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Objective

• Students will solve real-life and mathematical problems involving area, surface area, and volume.

Essential Questions

• How do we find the area of a square, rectangle, right triangle, and triangle?

Vocabulary

Acute triangle - a triangle where all three angles are acute (less than 90°)

Area - the measure of the size of the surface of a two-dimensional figure; measured in square units

Base - side of a plane figure or face of a solid, particularly one perpendicular to the direction height is measured

Complex shape - a figure that includes the elements of two or more simple shapes (triangles, rectangles, squares, circles)

Cubic centimeter - the volume equal to a cube one centimeter on each side

Cubic foot - the volume equal to a cube one foot on each side

Cubic inch - the volume equal to a cube one inch on each side

Cubic unit - a unit of measurement of volume or capacity

Edge - a line segment joining two adjacent vertices in a two-dimensional figure or a shared boundary between two faces of a three-dimensional figure

Equilateral triangle - triangle in which all three sides are equal in length

Face - a flat surface of a three dimensional figure

Height - the perpendicular distance of a two- or three- dimensional shape from the base to the opposite vertex

Length - a measurement of the distance from one point to another

Net - a flattened three-dimensional figure that can be turned into a solid by folding it

Obtuse triangle - a triangle that has one internal angle greater than 90°

Quadrilateral - a polygon that has four sides

Parallel lines - two lines that lie in the same plane and maintain the same distance apart over their entire length, and never intersect

Rectangle - a quadrilateral (four-sided figure) with four right angles

Rectangular prism - a three-dimensional figure of which all faces are rectangles

Right angle - an angle of exactly 90°

Right triangle - a triangle containing an angle of 90°

Square - a rectangle with four congruent sides

Square foot - the area equal to a square one foot on each side

Square inch - the area equal to a square one inch on each side

Square meter - the area equal to a square one meter on each side

Square unit - a unit of measurement of area

Surface area - the total area of all the faces or surfaces of a three-dimensional figure

Trapezoid - a quadrilateral that has exactly one pair of parallel sides

Vertices - plural for vertex, the common point where two sides or edges meet

Volume - a measure of the number of cubic units needed to fill the space inside an object

Width - the minimum distance between parallel lines of a figure

Materials

Blank grid paper so students can draw:

• A rectangle with the dimensions of 5 x 3
• A right triangle with sides measuring 3 x 5 x 6
• A square measuring 5 x 5

Calculator

Chart to be used for practice to reinforce skills.

Example of square inch

Example of square foot

Example of square meter

Formula sheet for the area of rectangles, squares, and triangles:

• Area of rectangles/squares A = l x w
• Area of triangles A = ½ b x h

Grid paper or a geoboard to provide students with the following drawn shapes:

• Square (2 x 2)
• Rectangle (3 x 7)
• Right triangle
• Acute triangle
• Obtuse triangle

Paper and pencils

Shape attribute blocks

Unit tiles

Worksheets

• Lesson 1 Body
• Lesson 1 Practice

Introduction

Activate Previous Knowledge

1. Lead a short discussion about the attributes of common shapes. During the discussion, have the students locate these shapes in their environment.
1. Review the attributes of common shapes:

• A square has 4 equal sides with opposite sides parallel and 4 right angles.
• A rectangle has 4 sides with opposite sides equal and parallel and 4 right angles.
• A triangle has 3 sides.
• A right triangle has 3 sides and one right angle.

2. Break the class into small groups to answer exercises:
1. Using figures (rectangle, square, triangle and right triangle) drawn on grid paper or formed on geoboards, have the students identify the shapes.
2. Using given dimensions for lengths of sides and the angles of particular shapes, have students draw the shapes on grid paper or form on geoboards.

Multiple means of representation: Use models and/or drawings during large group instruction. Allow students to have a copy of a drawing or a model at their desks.

Multiple means of expression: Provide a list of formulas to determine area or provide options for using manipulatives and/or computer models.

Multiple means of engagement: Allow students to use paper/pencil, manipulatives, computer, etc., to complete the exercises. Present information within the context of student interests (pets, gardening, new bedroom floor plan, etc.)

a. Activate Previous Knowledge

1. Lead a short discussion about the attributes of common shapes. During the discussion, have the students locate these shapes in their environment.
1. Provide graphic and/or digital representations of the four common shapes.

• Digital representations may include animations with sound to reinforce critical attributes.
• Graphic representations may include tactile and additional visual qualities, such as a purple right triangle with visually or tactilely represented numerals (1-3) marked on each side and the right angle indicated with an arrow labeled "90°".
• Graphic organizers may show shapes and critical attributes/details.
• Graphic representations may include tactile (outlines in glue, yarn, etc.) and additional visual qualities (color, pattern, etc.).
• Pre-teach concepts that the student may not have fully acquired. Use systematic instruction techniques as appropriate.

2. Provide photos of environmental shapes with or without visual or tactile outlines.

• Provide photos on digital display with or without speech generation.
• Preplan a shape for the student to contribute to the discussion. Allow the student to use his/her preferred mode of communication (verbal/vocal, sign, writing, drawing, typing/keyboarding, speech generating device, symbols, tactile symbols, AAC, etc.). Preplanning could include:

• raised line grid paper;
• enlarged cell/scale grid paper;
• outlined shapes with glue, yarn, etc.;
• student using representations, organizers, etc. from 1.a. to identify shapes;
• match shapes; and/or
• cut out grid paper figures and allowing the student to sort them.

2. Break the class into small groups to answer exercises:
1. Allow student to use representations, organizers, etc. from 1.a. to identify shapes.
2. Have the student measure shapes drawn on paper or cut from foam-board, and identify which one has the correct dimensions.

• Use adaptive measuring tools (such as a ruler with tactile qualities or a piece of paper laminated on card stock substituting for the paper shape if the student needs those; other rulers that accommodate the student more effectively may be used, such as:

• digital rulers, bendable/foldable rulers,
• tactile rulers,
• transparent/translucent rulers,
• simplified rulers with only inches or centimeters marked,
• rulers with hook-and-loop tape or other "handles") as necessary.

• After the student identifies a shape, have him/her trace the shape onto grid paper.
• Select digital shape from several options and drag to digital grid or geoboard. Add sound to digital activity.
• Have the student feel shapes created by partners on geoboard.
• Provide multiple means of representation: Use models and/or drawings during large group instruction. Allow students to have a copy of a drawing or a model at their desks.
• Provide multiple means of expression: Allow students to use paper and pencil, models, manipulatives, computers, etc., to complete exercises.
• Provide multiple means of engagement: Allow students to use paper/pencil, manipulatives, computer, etc.

b. Establish Goals/Objectives for the Lesson

Inform students that in this lesson they will make decisions as to the appropriate measurements and formulas to use in solving real world and mathematical problems involving the area of common shapes.

Explain that they will:

1. Describe and understand what area measures and the correct units for area measures.
2. Determine the area of rectangles and squares using graph paper/manipulatives and formulas.
3. Determine the area of right triangles using graph paper/manipulatives and formulas.
4. Determine the area of other triangles using graph paper/manipulatives and formulas.

Multiple means of representation: Along with posting lesson objectives in the classroom, provide individual copies for students.

Multiple means of expression: Allow students to record lesson objectives in different formats: mathematics journal, computer, graphic organizers (premade or original), etc.

Multiple means of engagement: Brainstorm ideas of how and when these skills might be relevant to "me".

Additional Considerations for Emerging Readers and Emerging Communicators

1. Provide the definition and unit measures for area with or without symbol-based text, with textures, other tactile qualities, colors, etc. (resources such as Standard Tactile Symbol List available from the Texas School for the Blind www.tsbvi.edu/tactile-symbols may be useful in determining representations; some representations may need to be pre-taught (Rowland, 2012), but use of the same representations/symbols \[or the system\] during this and other lessons will reduce the need for pre-teaching).
2. Provide definitions and images of squares and rectangles with or without symbol-based text, with textures, other tactile qualities, colors, etc. Provide:

• formulas and examples of calculating area of squares and rectangles with or without symbol-based text with textures, other tactile qualities, colors, etc.
• calculator (large button, handheld or computer-based).

3. Provide the definition and images of right triangles with or without symbol-based text, with textures, other tactile qualities, colors, etc. Provide:

• formula and examples of calculating area of right triangles with or without symbol-based text, with textures, other tactile qualities, colors, etc.
• calculator (large button, handheld or computer-based).

4. Provide the definition and images of other triangles with or without symbol-based text, with textures, other tactile qualities, colors, etc. Provide:

• formula and examples of calculating area of other triangles with or without symbol-based text, with textures, other tactile qualities, colors, etc.
• a calculator (large button, handheld or computer-based).

Body

1. Lead a discussion about what area is (the size of a surface; the amount of space inside the boundary of a flat, 2-dimensional shape such as a rectangle; the amount of carpet needed to cover the floor).

• Ask the students:

• What units (square units, square inches, square feet, square meters) are used to measure area? Use Lesson 1 Body worksheet, page 1.

2. Find the area of a rectangle and a square using grid paper/manipulatives and formulas. Use Lesson 1 Body worksheet, page 2.

• Working with rectangles, ask the students:

• What is the area of the rectangle by counting units?
• What is the length of the rectangle?
• What is the width of the rectangle?
• Show the students that they can find the area of the rectangle by multiplying length times width, e.g., 4 x 5 = 20 units2.

• Working with squares, ask the students:

• What is the area of the square by counting units?
• What is the length of the square?
• What is the width of the square?
• How do the length and width compare in a square?
• Show the students that they can find the area of the square by multiplying length times width, e.g., 5 x 5 = 25 units2.
• Ask the students: What would be the area of Sarah's rectangular garden if it has a length of 12 feet and a width of 3 feet?

3. Find the area of a right triangle using grid paper/manipulatives and formulas. Use Lesson 1 Body worksheet, page 3.

• Ask the students: "What is the area of the right triangle by counting units?"
• Demonstrate to the students how to use the formula for the area of a right triangle: (Area = ½ base times height). (Draw a square and find its area. Draw a diagonal through the square to form 2 right triangles. Each triangle is one-half the area of the square.)
• Ask the students:

• What is the height of the right triangle?
• What is the base of the right triangle?

4. Find the area of a triangle (not a right triangle) using grid paper/manipulatives and formulas. Use Lesson 1 Body worksheet, page 4.

• Show students the base and height of a triangle by using a right triangle and triangles that are not right triangles.
• Ask the students:

• Can any side of a triangle be the height and base?
• What is common about the height and base in the two triangles?
• What is the measure of the base of the triangle?
• What is the measure of the height of the triangle?
• What is the area of the triangle?

Multiple means of representation: Use models and/or drawings during large group instruction. Allow students to have a copy of a drawing or a model at their desks.

Multiple means of expression: Provide a list of formulas to determine area or provide options for using manipulatives and/or computer models.

Multiple means of engagement: Allow students to use paper/pencil, manipulatives, computer, etc., to complete the exercises. Present information within the context of student interests (pets, gardening, new bedroom floor plan, etc.)

Additional Considerations for Emerging Readers and Emerging Communicators

1. Demonstrate the definition and unit measures for area with or without symbol-based text, with textures, other tactile qualities, colors, etc. (resources such as Standard Tactile- Symbol List available from the Texas School for the Blind http://www.tsbvi.edu/). Tactile-symbols may be useful in determining representations; some representations may need to be pre-taught (Rowland, 2012), but use of the same representations/symbols \[or the system\] during this and other lessons will reduce the need for pre-teaching. Provide:

• choices from which the student can select. Depending upon the student's current understanding, distractors (incorrect answers) may be included. However, since this is instructional, the errorless learning technique in which all the choices would be correct is also appropriate;
• formulas and examples of calculating area of squares and rectangles with or without symbol-based text with textures, other tactile qualities, colors, etc.; and/or
• rectangle and square shapes on grid paper, geoboard, or digitally (use virtual manipulatives such as those from National Library of Virtual Manipulatives) -

• digital representations may include animations with sound to reinforce critical attributes. Graphic representations may include tactile and additional visual qualities.
• manipulatives can be used to mark each unit in the shapes.

2. Working with rectangles and/or squares, have the student either count each unit on the representation or place a manipulative on each unit and then count the manipulatives for the first three questions.

• Have the student count by using the "+1" function on the calculator (big button, hand-held or computer based) as he/she moves each manipulative.
• Place manipulatives on a number line to count.
• Provide a manipulative (string, pipe cleaner, strip of card stock paper) that is the length of the square and one that is the width of the square. Have the student compare the two manipulatives for the fourth question.
• Allow the student to use his/her preferred mode of communication (verbal, sign, writing, drawing, typing/keyboarding, speech generating device, symbols, etc.).
• Allow the student to use a calculator for the last question.

3. Provide the definition and unit measures for area with or without symbol-based text, with textures, other tactile qualities, colors, etc. (resources such as Standard Tactile Symbol List available from the Texas School for the Blind http://www.tsbvi.edu/tactile-symbols may be useful in determining representations; some representations may need to be pre-taught (Rowland, 2012), but use of the same representations/symbols \[or the system\] during this and other lessons will reduce the need for pre-teaching). Provide:

• choices from which the student can select. Depending upon the student's current understanding, distractors (incorrect answers) may be included. However, since this is instructional, the errorless learning technique in which all the choices would be correct is also appropriate;
• formulas and examples of calculating area of triangles with or without symbol-based text with textures, other tactile qualities, colors, etc. ;
• rectangles made from two triangles (which can be separated and put back together) so the student can better understand the concept of the area formula;
• the triangle shapes on grid paper, geoboard, or digitally (use virtual manipulatives such as those from National Library of Virtual Manipulatives). Digital representations may include animations with sound to reinforce critical attributes. Graphic representations may include tactile and additional visual qualities. Provide manipulatives that can be used to mark each unit in the shapes.

4. Have the student either count each unit on the representation or place a manipulative on each unit and then count the manipulatives for the first three questions (area height, base). For the last set of questions:

• Provide the same triangle representations with which the student has been working.
• Provide manipulatives corresponding to the height and base of both triangles.
• Use adaptive measuring tools (such as a ruler with tactile qualities or a piece of paper laminated on cardstock substituting for the paper shape if the student needs those; other rulers that accommodate the student more effectively can be used such as digital rulers, bendable/foldable rulers, tactile rulers, transparent/translucent rulers, simplified rulers with only inches or centimeters marked, rulers with hook-and-loop tape or other "handles") as necessary.
• Allow the student to use his/her preferred mode of communication (verbal, sign, writing, drawing, typing/keyboarding, speech generating device, symbols, etc.).
• Use calculator.

Practice

1. In small groups, have students work on a variety of problems involving area, some using different shapes (rectangles, squares, right triangles, and triangles) and some using dimensions only to determine area. Use Lesson 1 Practice worksheet.
2. Give each student the chart below (page 1 of worksheet). Students may use models to explore the various possibilities and complete the chart.

Problem	Figure	Formula	Formula with values	Area (unit2)
1	Rectangle	A = l x w	A = 2 x 7	14 units2
2
3
4
5
6
7
8
9

3. Bring the whole group back together. Ask one student from each group to discuss the results of a specific problem. Complete the chart and discuss any problems they may have had.

Multiple means of representation: Provide students with a copy of the problems, formulas, and the table. Have drawings and manipulatives available for students to use.

Multiple means of expression: Allow the students to solve the problem using formulas and/or models.

Multiple means of engagement: Ensure students are actively involved in their small groups. Present different problems related to student interests. As you observe group work, use questioning to encourage students to explain their strategies.

Additional Considerations for Emerging Readers and Emerging Communicators

1. Ensure that the student participates in all activities in the group – selecting problems, solving problems, contributing to discussions, etc.

• Allow the student to use his/her preferred mode of communication (verbal, sign, writing, drawing, typing/keyboarding, speech generating device, symbols, etc.). Other students will need to understand the student's communication mode and should be encouraged to use it themselves, serving as communication models/partners.
• Have the student solve the problems using any accommodations that have been successful or shown promise in previous activities (models, definitions, formulas, steps, tactile/visual cues, graphics (2- or 3- dimensional, digital, manipulatives, calculators, adapted measuring tools, etc.). It is important to maintain consistency in accommodations and supports, providing and using the ones with which the student has the most success.
• Since this practice is instructional, guidance in completing the chart is acceptable and expected. Errorless learning techniques and systematic instructional techniques may be helpful to use.
• Reduce the number of problems.
• Reduce the difficulty of some problems (e.g., focus on those with single digits only).

2. Preplan a problem for the student to present.

• Allow the student to use his/her preferred mode of communication (verbal, sign, writing, drawing, typing/keyboarding, speech generating device, symbols, etc.).

Closure

d. Revisit/Review Lesson and Objectives

Remind students of the lesson's objectives and expected student outcomes. Have students discuss which of the lesson's problems addressed the objectives below and what they learned from their experiences.

Students will solve real life and mathematical problems involving area, surface area, and volume.

1. Describe and understand what area represents and the correct units of measure for area.
2. Determine the area of rectangles and squares using graph paper/manipulatives and formulas.
3. Determine the area of right triangles using graph paper/manipulatives and formulas.
4. Determine the area of other triangles using graph paper/manipulatives and formulas.

Multiple means of representation: Along with posting lesson objectives in the classroom, students may refer to their individual copies.

Multiple means of expression: Students can share what they have learned in different formats: writing, drawing, creative expression, discussion, etc.

Multiple means of engagement: Brainstorm ideas of how and when these skills might be relevant to "me."

Additional Considerations for Emerging Readers and Emerging Communicators

Provide all materials and accommodations used throughout the lesson and use them to remind the student as the teacher reviews. Provide a graphic organizer (enhanced with additional tactile and visual qualities as appropriate) with categories of rectangles, squares, right triangles, and other triangles. Have the student sort (with guidance as necessary) characteristics, formulas, examples, problems, etc. into the categories (these may be symbolic, concrete, or a combination of both). Allow the student to use this to contribute to the discussion. Provide several choices the student can select from to answer the question(s). Preplan a contribution. Allow the student to use his/her preferred mode of communication (verbal, sign, writing, drawing, typing/keyboarding, speech generating device, symbols, etc.)

Exit Assessment

Give students a new area problem:

Mr. Hanks is choosing between laminate and tile for his floor and wants to use the material that has the largest area per piece so he can get the job done faster with fewer materials. The laminate comes in rectangular strips that are 36 inches by 8 inches. The tiles are 18 inches by 18 inches. Find the area of each piece to decide which one Mr. Hanks should buy.

Have students work independently to find the area. Students should use a similar table as was used during Practice.

Multiple means of representation: Provide students with a copy of the work problem and the table. Have drawings and manipulatives available for the students to use.

Multiple means of expression: Allow the students to solve the problem using formulas and/or models.

Multiple means of engagement: Present different problems related to student interests. As you observe students working, use questioning to encourage students to explain their strategies.

Additional Considerations for Emerging Readers and Emerging Communicators

1. Provide the problem with symbol-based text or text enhanced with tactile and visual qualities as necessary.

• Provide a photo related to the text.
• Have student highlight (color, underlined in glue, etc.) important parts of the text.

2. Provide graphic representations of materials (scaled, actual size, or actual samples of flooring materials).

• Provide any accommodations or supports used successfully in other activities in the lesson (formulas, calculator, manipulatives, definitions, graphics, etc.).
• Provide two graphic or digital floor plans divided into grids, one representing the laminate pieces and the other representing the tile pieces. Provide manipulatives for the student to place and count.
• Provide answer choices for the student to choose from.
• Simplify the language. For example, "…he can get the job done faster with fewer materials." could be changed to "…he can use a smaller number of pieces."

As this is now assessment as opposed to all previous activities in this lesson, the student's first independent attempts solving the two problems and answering the question should be observed and recorded for accuracy.

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