Middle School Mathematics UDL Instructional Unit- Lesson 2

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Objective

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

Essential Questions

• How can we find the area of quadrilaterals and complex shapes?

Vocabulary

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)

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

Quadrilateral - a polygon that has four sides

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

Right triangle - a triangle containing an angle of 90°

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

Width - the minimum distance between parallel lines of a figure

Materials

Chart to be used for practice reinforcing skills. Formula sheet for the area of rectangles, squares, and triangle:

• Area of rectangles $A = l * w$

• Area of triangles: $A = \frac{1}{2} b * h$

Geoboard Grid paper Square tiles Worksheets

• Lesson 2 Intro 2
• Lesson 2 Intro 3
• Lesson 2 Body
• Lesson 2 Practice

Introduction

Activate Previous Knowledge

1. Remind students that they previously determined the areas of shapes such as rectangles, squares, and triangles.
• Discuss with students that not all shapes or spaces are rectangles, squares, or triangles.
• Ask the students: What are some shapes you see around you that are not rectangles, squares, or triangles?
• What are some ways you might find the area of these quadrilaterals and complex shapes?
2. Lead a discussion about how to determine lengths from given dimensions and that lengths can be additive. Use Lesson 2 Intro 2 worksheet.
3. Provide practice worksheet on partitioning shapes and finding lengths. Use Lesson 2 Intro 3 worksheet.

Multiple means of representation: Provide drawings or models of partitioned shapes, shapes with some lengths and widths identified and lines with some measurements identified. Worksheets can be presented with fewer items per page.

• Worksheets may be printed on various colored sheets
• Worksheets may be presented on computer.

Multiple means of expression: Students may describe a situation where they have seen a shape partitioned or only knew some of the dimensions of a figure. Students may use drawings or models to complete worksheets. Students may complete work on a computer.

Multiple means of engagement: Provide examples of partitioned shapes and situations where they must determine measurements based on the dimensions they know based on interests of the students or real life situations in which they would be used.

Additional Considerations for Emerging Readers and Emerging Communicators

If the student has had little to no experience with the concept of area or the skill of calculating area, it might be helpful to provide instruction using the Lesson 2 Concept Reinforcement Activity (CRA) prior to teaching the Introduction to Lesson 2.

1. Provide the accommodations from Lesson 1 to remind students of shapes discussed and their details (length, width, base, height, angles, sides, area formulas, characteristics, etc.).
2. Provide representations (graphic or digital) of other shapes (trapezoids, quadrilaterals, complex shapes) found in the environment and examples of them (photos, icons, etc.).
• Enhance representations with tactile, visual, auditory qualities as necessary.
3. Preplan a shape of the student's choice to contribute to the discussion.
• Allow the student to use his/her preferred mode of communication (verbal, sign, writing, drawing, typing/keyboarding, speech generating device, symbols, etc.).
4. Provide the worksheet of a complex shape to each student and using the same shape/dimensions on the white board, smart board or computer projection, point out the different sides where the measurement is noted and sides without measurements.
1. Provide complex shapes composed of cut-outs of simple shapes.
• Provide a model of complex shapes that the student can use a "puzzle" template to construct a complex shape from end to end to see how the side lengths are additive. Or use cut-out equal sections of simple shapes, square sticky notes, unit cubes, etc. to demonstrate side lengths are additive. Use hook-and-loop tape if needed for manipulating pieces.
2. Provide grid (graphic or digital) to place shapes on for measurement calculations.
• Enhance grid with tactile, visual, auditory qualities.
• Provide manipulatives that can be used to measure/count on the grid.
• Provide measuring tools with accommodations used in Lesson 1.
5. Provide a worksheet of shapes divided into less complex shapes.
1. Provide complex shapes composed of cut-outs of simple shapes.
• Provide a model of complex shapes that the student can use a "puzzle" template to construct a complex shape from cut-out sections of simple shapes. Use hook-and-loop tape if necessary.
2. Provide grid (graphic or digital) to place shapes on for measurement calculations.
• Enhance grid with tactile, visual, auditory qualities.
• Provide manipulatives that can be used to measure/count on the grid.
• Provide measuring tools with accommodations used in Lesson 1.

Establish Goals/Objectives for the Lesson

Inform the students that in this lesson they may put quadrilaterals or complex shapes together or decompose them. They may also divide quadrilaterals and complex shapes into other shapes to find area using formulas and models. Explain that they will:

1. Use appropriate known formulas for area.
2. Determine the correct measurement to use.
3. Determine the area of quadrilaterals by composing and decomposing figures.
4. Determine the area of complex shapes by composing and decomposing figures.

Multiple means of representation: In addition to 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

Pre-teach concepts of composing and decomposing.

• Use other symbols and terms the student may be familiar with such as "+" and "-"
• Use quadrilaterals and complex shapes cut into simple shapes to illustrate the concepts.
• Make puzzle templates and use hook-and-loop tape.
• Add tactile, visual, and auditory qualities as necessary.

Provide any previous accommodations for measuring and determining area.

Body

Provide students with Lesson 2 Body worksheet.

1. Model how to find the area of the trapezoid.
1. Work with the students to decompose the figure into a rectangle and triangle and ask the students: What two figures have we made?
2. Work with the students to determine what dimensions they will need in order to find the area of the rectangle. Label the figure with any missing dimensions.
3. Find the area of the rectangle by filling in the Area Chart. (A completed chart is shown below for the trapezoid.)
4. Work with the students to determine what dimensions they will need to find the area of the triangle. Label the figure with any missing dimensions.
5. Find the area of the triangle by filling in the Area Chart.
6. Ask the students: We know the area of the rectangle and triangle, how can we now find the area of the trapezoid?

Figure the area of the trapezoid. See worksheet problem #1.

Figure	Area formula	Area formula with values	Area (unit2)
Rectangle	$A = l * w$	$A = 4 * 8$	32 inches2
Triangle	$A = \frac{1}{2} * b * h$	$A = \frac{1}{2} * 4 * 4$	8 inches2

Total area = 32 in² + 8 in² = 40 inches²

2. Model finding the area of the bedroom from the floor plan by first asking the students: What will be our first step in finding the area of the floor plan?
1. After selecting a decomposition, ask the students: Are there any dimensions we need to determine before we can find the area?
2. If there are missing dimensions, have the students determine and label those dimensions on the drawing.
3. Find the areas of the parts by filling in the Area Chart. (A completed chart is shown below for the bedroom floor plan.)
4. Find the area of the bedroom.
1. See worksheet problem #2.
5. To make sure that students realize there is more than one way to decompose the figure and solve the problem correctly, ask the students: Is there any other way to divide the figure to find the area?

Figure	Area formula	Area formula with values	Area (unit2)
Rectangle	$A = l * w$	$A = 14 * 6$	84 feet2
Rectangle	$A = l * w$	$A = 11 * 6$	66 feet2

Total Area = 84feet² + 66feet² = 150 feet²
3. Model finding the area of the shaded portion of the figure.
1. Have the students complete the Area Chart while answering the questions.
2. Ask the students: What is the area of the entire square, both shaded and non-shaded portions?
3. Ask the students: What is the area of the non-shaded portions?
4. Ask the students: Knowing the areas of both the entire figure and non-shaded portion, what is the best way to find the area of the shaded portion?
5. Ask the students: What is the area of the shaded portion of the figure?
1. See worksheet problem #3.

Figure	Area formula	Area formula with values	Area (unit2)
Square	$A = l * w$	$A = 20 * 20$	400 m2
Square	$A = l * w$	$A = 9 * 9$	81 m2

Total Area = 400m² – 81m² = 319 m²

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. Divide the trapezoid into a rectangle and triangle that the student can decompose.
1. Provide digitally or on paper/foam-board with additional tactile and visual qualities.
2. Provide the definitions and characteristics, graphic organizers, and shapes (paper or digital) from Lesson 1: Introduction for the student to use in identifying the shapes.
1. Provide formulas, measuring tools, etc. from Lesson 1: Introduction and Body. Use strategies from Lesson 1: Introduction and Body.
2. Have the student label missing dimensions by writing, keyboarding, cut and paste prewritten numerals, AT, etc.
3. Provide chart with accommodations consistent with those provided in Lesson 1: Practice, 2.
4. Provide formulas, measuring tools, etc. from Lesson 1: Introduction and Body. Use strategies from Lesson 1: Introduction and Body.
   1. Have the student label missing dimensions by writing, keyboarding, cut and paste prewritten numerals, AT, etc.
5. Provide a chart with accommodations consistent with those provided in Lesson 1: Practice, 2.
6. Provide graphic (paper or digital with additional tactile, visual, or auditory characteristics as necessary) showing that the rectangle + the triangle = the trapezoid.
1. Have the student connect the area measurements of the two shapes into the same formula so the student can calculate the area of the trapezoid.
2. Use calculator (talking, large button, handheld, or computer-based).
1. Provide paper or digital bedroom floor plan on a scaled grid with additional tactile, visual, or auditory qualities.
1. Show the student two ways it can be decomposed, and allow the student to choose, using preferred mode of communication including AAC/AT, the one with which he/she would like to work.
2. Look at each dimension individually and mark (sticky note, circle, highlight, etc.) any dimensions that are missing after decomposing.
2. Have the student determine missing dimensions using measuring tools and strategies provided in previous lesson activities.
1. Label the now-solved-for dimensions using previous strategies and tools.
3. Using strategies and tools for completing charts, have the student fill in the Area Chart.
4. Same as c.
5. Provide decomposition options from a. above and have the student demonstrate decomposition not used for Area Chart.
1. Follow steps 2.a.-d. above.
5. Using the chart, have the student answer the question using his/her preferred mode of communication.
1. Allow the student to select what he/she believes to be the best way to find the area of the shaded portion and use his/her preferred mode of communication to answer.
2. Preplan the answer to contribute.

Practice

1. Provide students with Lesson 2 Practice worksheet. Have students work in pairs on a variety of problems, using different complex figures and real-world situations.
2. Give each student an area chart to complete for each problem.
• Students may use models to explore the various possibilities to complete the charts.
• Worksheet answer key provides one possible way to find the area, students might determine a different way.
3. Bring the group back together.
• Ask one student from each pair to discuss the results.
• After reviewing the results, demonstrate to the students that even if the complex shape was decomposed in different ways, the area was always the same.

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

Multiple means of expression: Allow students to solve the problem using formulas and/or models. Allow students to solve the problem in more than one way.

Multiple means of engagement: Ensure each student is actively involved in his/her pair. Present different problems related to student interests. As you observe pair work, use questioning to encourage students to explain their strategies.

Additional Considerations for Emerging Readers and Emerging Communicators

1. Have the student work with a partner who understands the student's mode of communication.
• The partner might also use that same mode of communication so that he/she serves not only as a math partner but a communication model, too.
• Make sure that the student participates in all aspects of the activity.
2. Provide models, measuring tools, adapted worksheets using the same accommodations/supports/strategies used up to this point in the lesson.
3. Preplan a contribution to the discussion.
• Have the student demonstrate different ways that a complex shape might be decomposed, using tactile or digital means.

Closure

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. Use appropriate known formulas for area.
• Ask the students: What are the formulas we have been using to find area?
2. Determine the correct measurement to use.
• Ask the students: What is the measurement we use when measuring area?
3. Determine the area of quadrilaterals by composing and decomposing figures.
• Ask the students: How can we find the area of quadrilaterals that are not squares and rectangles?
4. Determine the area of complex shapes by composing and decomposing figures.
• Ask the students: How do we find the area of complex shapes?

Multiple means of representation: Along with posted 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

1. Provide all previous utilized accommodations/supports for using formulas for area.
2. Provide all previous utilized accommodations/supports for determining area.
3. Provide answer choices for the student to choose from.
4. Provide all previous utilized accommodations/supports for composing and decomposing complex shapes.
5. Provide all previously utilized accommodations/supports for determining area of complex shapes.

Exit Assessment

1. Have the students solve the following problem:

The local park is having a design contest for a garden it is going to build. You have decided to enter the design contest. Each design must meet the following conditions: The overall dimensions of the garden are 20 meters by 20 meters. The garden plan must include sidewalks or pathways and green areas, which will not be included in the square footage of the flowerbeds. A fountain may also be included but will not be included in the square footage of the flowerbeds. Indicate any flowerbeds on the garden plan by shading the area(s). The total area of the flowerbeds must be at least 150 square meters but cannot exceed 200 square meters. Have fun and be creative designing our park's new garden.

2. Have the students individually design/create a garden with beds that are complex shapes.
• The garden and beds need to have the dimensions indicated.
• The flowerbeds should be shaded.
• The total area of the garden, with flowerbeds, must be at least 150 square meters but cannot exceed 200 square meters.
• Provide the students with paper/pencil, graph paper, ruler, and area chart.

Multiple means of representation: Allow the students to have a written copy of the problem, area formulas, and area table. Have drawings and manipulatives available for students to use. Allow students to refer back to their work samples, models, drawings, notes, etc.

Multiple means of expression: Students may draw or use manipulatives or use the computer to model solutions.

Multiple means of engagement: Create situations that include areas of interests to the students.

Additional Considerations for Emerging Readers and Emerging Communicators

1. Provide the word problem with symbol-based text, tactile symbols, pictures, etc.; provide a text reader or read the problem to the student.
1. Simplify the wording; take out non-relevant information.
2. Highlight the important information.
2. Provide all accommodated materials and strategies including definitions, examples, formulas, measuring tools, charts, etc.
1. Provide several simple and complex shapes on grid paper (tactile/visual qualities added) or digitally (auditory qualities added) that the student can choose from for the flowerbeds. The student must calculate the area of each shape and determine which ones he/she can use to stay within the range given.
1. Provide cues to decomposing the complex shapes.
2. Simplify by using complex shapes that decompose into squares and rectangles only.
3. Indicate dimensions of shapes.
2. Secure selected flowerbeds with glue, tape, or hook-and-loop tape.

Return to Lesson 1 or Proceed to Lesson 3