High School Mathematics UDL Instructional Unit-Lesson 1

Lesson 1: Objective

Grade Span: 9-10	Content Area: Mathematics – Measurement Investigating Measurement in the Real World
Lesson 1 of the Unit	Approximate Time Needed: 50 minutes

Objective: Student will make decisions about units and scales that are appropriate for problem solving situations involving mathematics within mathematics or across disciplines or contexts, and: Convert units using standard/known conversion units. Use appropriate known formulas for the area. Solve multistep problems involving one unit of measure.

Essential Question: What are the relationships among the measurements of dimensions, area, and perimeter in problem solving situations?

Lesson 1: Materials

Materials Needed: Using grid paper or a Geoboard, provide students with rectangles having different dimensions, some of which are similar. Some of the rectangles should have the same perimeter but different areas. For example: 9 x 12, P = 42; A = 108 8 x 13 P = 42; A = 104 6 x 15 P = 42; A = 90 Some should have the same area but different perimeters. For example: 4 x 5; A = 20; P = 18 10 x 2; A = 20; P = 24 Make chart as used in practice note below to reinforce skills. Modify chart to include picture and/or representations of headings. Provide chart in electronic format for access with switch or alternate keyboard. Materials needed: Grid paper Grid paper with raised lines Grid paper created on overhead transparencies for use with light boards Virtual grid paper Geoboards Virtual Geoboards Calculator Paper and pencils Ruler Yardstick Foldable ruler Conversion charts(for inches to feet, feet to yards, centimeters to meters) See Resources: See Lesson 1 Resources for example exercises/images. Lesson Vocabulary Area Centimeter Foot Inch Length Meter Perimeter Rectangles Similar Rectangles Width Yards

Lesson 1: Introduction – 10 minutes

A. Activate Previous Knowledge Lead a short discussion about how to find perimeter and area of rectangles. Review the concepts of perimeter and area. Discuss how these concepts are used in real life examples. Example 1: A runner is practicing by running along the fence line of a parking lot. -Is he running the perimeter of the parking lot or is he running the area? Example 2: The school is getting new carpet in the classroom. -Will the workers need to figure out the area of the classroom or the perimeter? Break class into small groups to answer exercises. Using figures (rectangles and squares) drawn on grid paper or formed on Geoboards, find the perimeters and areas. Remind students that answers should/must include the appropriate units of measure. 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 and perimeter 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 exercises.

Additional Considerations for Emerging Readers and Emerging Communicators

Provide picture and/or tactile and/or object representations of relevant vocabulary paired with the written word as it is mentioned during presentation or discussion for rectangle, area, perimeter as well as the meanings of each word. Create math journals to record vocabulary, formulas, and notes. Provide the formulas for area and perimeter as the concepts of each are discussed. During discussion, provide picture representation of real world uses for area and perimeter. As students work in small groups or pairs, ensure they have a means for gaining their group members' or partner's attention and a means for contributing to the discussion. Students may use their math journals or a graphic organizer to collect/store information gathered during group. To find area and perimeter, use grid paper, count/mark/tally each unit along the length of the figure to determine length and count/mark/tally each unit along the width of the figure to determine the width. Use the formulas to determine area and perimeter. A list of formulas may be used by the student as a reference. Student may be presented with manipulatives of a unit and the rectangle drawn on grid paper. Students determine area and perimeter by placing the manipulative units on each unit around the rectangle on the grid paper to demonstrate perimeter as well as within the rectangle to demonstrate area. Using manipulatives may be demonstrated electronically by using a computer program or PowerPoint to count units virtually to determine area and perimeter. Each time the student hits the switch, the computer program counts each unit around the rectangle to determine perimeter. To determine area, each time the student hits the switch, the program counts the units within the rectangle or for larger numbers, highlights a row or column of units and skip counts by 5s to determine the total number of units. See Resources: See PowerPoint, Slides 1 and 2. As answers are reviewed, be sure to reference the appropriate units of measure. For example, if students determine the perimeter of a 3inch by 4inch figure is 14, reply, "That is correct. It is 14 inches." If they determine the area is 12, reply, "That is correct. It is 12 inches square." Remind students to record the appropriate unit. Model how to write the appropriate units. Present students with an alternative representation of unit to record in their math journals or graphic organizers. Important Note for Communicators Considered Pre-Symbolic: Be sure students have a way to attain peer attention as well as to share and receive information. Limit measurements to one type: standard or metric unit.

B. Establish Goals/Objectives for the Lesson Inform students that in this lesson they will make decisions about units and scales that are appropriate for problem solving situations within mathematics or across disciplines or contexts, and: Convert units using standard/known conversion units. Use appropriate known formulas for the area. Solve multistep problems involving one unit of measure. 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 journals, computer, premade or original graphic organizers, 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

Inform students of expected outcomes. Provide keys words paired with symbols/images/tactile representations. Provide the key words in the lesson objectives paired with images or tactile representations, record into mathematics journals, or students may use an electronic picture writer paired with text-to-speech to record the lesson's objectives. Provide photographs, models or tactile representations of examples of situations in which these concepts are used.

Lesson 1: Body – 15 minutes

Direct Instruction and/or Facilitation of the Lesson Ask students: For two rectangles of the same (similar) shape, how do their sizes compare? How do their perimeters compare? How do their areas compare? See Resources: Lesson 1, Pages 15 and 16. Identify rectangles that have the same areas. Ask students: "Do they have the same perimeter?" See Resources: Lesson 1, Page 17. Lead a discussion on how to find perimeter and area of rectangles when units are different. (e.g., a rectangle measures 8 inches wide, 2 feet long or 80 cm wide, 1 meter long). Review converting units: From inches to feet From feet to inches From feet to yards From yards to feet From centimeters to meters From meters to centimeters See Resources: Lesson 1, Page 18. 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 and perimeter 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 exercises. Present information within the context of students' interests such as pets, gardening, new bedroom floor plan, etc.

Additional Considerations for Emerging Readers

Provide picture representations of relevant vocabulary: similar rectangle, area, perimeter as well as the meanings of each word. Provide students with grid paper with two similar rectangular figures printed on it or grid paper with manipulatives of the figures. Ensure students have a means for sharing how the rectangles are the same and/or different. Students use their math journals or a graphic organizer to record information about the attributes and measurements of each rectangle. Students use the same strategies as were used to determine area and perimeter in the introduction. Students share how the perimeters are the same or different using the information recorded in the journals or the graphic organizer. Students share how the areas are the same or different using the information recorded in the journals or the graphic organizer. Demonstrate how ratios are used to compare the area of each figure and the perimeter of each figure. See Example: Lesson 1, Body 1, Similar Figures. Provide students with two rectangles that have different measurements but the same area drawn on grid paper or as manipulatives of the figures. Students verify that the area is the same and determine the perimeter of each figure. See Example: Lesson 1, Body 2, Same Area. Provide students with picture representations of relevant vocabulary for discussion: inches, feet, yard, centimeter, and meter as well as the meanings of each word. When discussing different measurements within the same system (inches, feet, and yards, or centimeters and meters), present students with picture/object/tactile representations of the different measures. To build understanding of the relationship between the different measurements, have students measure: a book using inches marked on a ruler; a table using feet while attending to how many inches that would be; the length of the chalkboard/ whiteboard using a yardstick while attending to how many feet that would be. Repeat measurements using centimeters and meters. Emphasize the difference in size of object in relation to the size of unit used. Students measure a given piece of paper that is 8 inches wide and 2 feet long. Allow students to explore other units that can be used. Discuss how the paper can be measured using the same unit of measure (i.e., Inches, 8 inches wide and 24 inches long). Provide students with a calculator, the formulas, and task-analyzed steps for converting from one unit to another. Or provide students with a conversion chart with which to match the measured unit to a converted unit. As a whole class or in small groups, work together to convert the units of measure for a rectangle that measures 8 inches wide and 2 feet long. Students use the conversion formulas or create the figures on grid paper. Students count the units on the grid paper to determine the conversion. See Example: Lesson 1, Body 4, Converting Units.

Additional Considerations for Emerging Communicators

Provide picture and tactile representations of relevant vocabulary: similar rectangle, area, perimeter as well as the meanings of each word. Using manipulatives or a computer program or PowerPoint that contains 2 similar figures (Figure A and Figure B) in a grid, students determine the area of Figures A and B by counting or using one-to-one correspondence to determine the number of units within the figure (area) and number of units around the figure (perimeter). Students compare the areas by moving the smaller figure (Figure B) into the larger figure (Figure A) to see how many are needed to completely cover the larger figure. This activity demonstrates how many times bigger Figure A is than Figure B. See Example: PowerPoint Lesson 1, Slides 3 and 4. Using a computer program or PowerPoint that contains 2 figures (Figure A and Figure B) with the same area in a grid, students use a switch to determine the area for each figure and verify that they are the same. Students determine if the perimeters are also the same. See Example: PowerPoint Lesson 1, Slides 5 and 6. Provide picture and tactile/object representations of relevant vocabulary: inches, feet, yard, centimeter, meter as well as the meanings of each word. When discussing different measurements within the same system (inches, feet, and yards, or centimeters and meters), present students with tactile representations or objects of the different measures. Students work with peer partners to measure a book using inches marked on a foldable ruler so students feel the inches. Students measure a table using feet while attending to how many inches that would be. Again use the foldable ruler. To build understanding of the relationship between the different measurements, students measure the length of the chalkboard/whiteboard using a yardstick while attending to how many feet that would be. Emphasize the difference in the size of an object in relation to the size of the unit used. Students measure a given piece of paper that is 8 inches wide and 2 feet long. To emphasize the inches within the foot, use a foldable ruler to show students that the piece of paper can also be measured as 8 inches wide and 24 inches long. As a whole class or in small groups, work together to convert the units of measure for a rectangle that measures 8 inches wide and 2 feet long. Provide students with the figure represented on grid paper. Tactilely represent the figure with the original units of inches across and feet down as well as with a rectangle with all measurements converted to inches. Represent the different units of measure (1foot and 12 inches) tactilely with different thicknesses, so when students are tactilely scanning the inches, they feel the difference when a foot has been reached. Students use a computer program with an alternate keyboard or multi-switch access with each tactile representation on the keyboard or one per switch. Students locate the original figure and press it so it is represented on the computer screen. The computer program states the original measurements (8 inches by 2 feet), and students locate and press the tactile representation that has been converted to inches. The figure is displayed on the screen, and the computer states the converted measurements (8 inches by 24 inches). See Example: PowerPoint Lesson 1, Slide 7. Important Note for Communicators Considered Pre-Symbolic: Work with only one system: standard or metric units.

Lesson 1: Practice – 20 minutes

In small groups, students work on a variety of problems using different given dimensions such as: Alex has 140 feet of fencing to place around a rectangular garden he is making. He wants the area of the garden to be as large as possible. What should the length and width of the garden be? Give each student the chart below. Students may use models to explore the various possibilities and complete the chart. Rectangle Length Width Perimeter Area A 40ft 30ft 140ft 1200ft2 See Resources: Pages 20 and 21. Bring the whole group back together. Ask one student from each group to discuss the results. After reviewing students' results, help students realize that when a perimeter is 140 units, length plus width always equals 70 units, but length times width varies. Make a chart with students of the possible lengths and widths of a rectangle in increments of 5 when the perimeter is 140 units. Use information from their charts and organize it in order from smallest length to largest length. L + W L x W 70 units = 5 units + 65 units 5 units x 65 units = 325 units2 70 units = 10 units + 60 units 10 units x 60 units = 600 units22 70 units = 20 units + 50 units 20 units x 50 units = 1000 units22 70 units = 35 units + 35 units 35 units x 35 units = 1225 units22 70 units = 50 units + 20 units 50 units x 20 units = 1000 units22 Demonstrate to students that as the dimensions change, the area gets larger, reaching the highest value, and then gets smaller. Students should conclude that when asked to find the largest area when given a perimeter, a square would always have the greatest area. Multiple means of representation: Provide students with a copy of the word problem and the table. Have drawings and manipulatives available for students to use. Multiple means of expression: Allow students to solve the problem using formulas and/or models. Multiple means of engagement: Ensure each student is actively involved in the small groups. Present different problems related to students' interests. Use questioning to encourage students to explain their strategies in their groups.

Additional Considerations for Emerging Readers

Provide picture representations of the word problem to students as it is being read. Present word problem electronically using text-to-speech. Label each column of the table using picture symbols and include measurements for length and width needed for a perimeter of 140 ft. Remind students how to determine perimeter and area. Provide formulas and task-analyzed steps for using the formulas: -Perimeter = L+L+W+W or 2L + 2W -Area = L x W -Perimeter = 140ft: 2L+2W = 140ft or L+W = 70ft Provide students with models of the various fenced-in areas (rectangles). -Students use the given formula and calculators to determine area. -Students use the grid paper to identify the rectangle that provides the most area. For steps 3-6, students discuss results by presenting the length and width of the rectangle with the largest area that they discovered in their group, or students point to the actual picture representation of the rectangle with the largest area.

Additional Considerations for Emerging Communicators

Modify the word problem to include smaller whole numbers. For example, Alex has 16 yards of fencing to place around a rectangular garden he is making. He wants the area of the garden to be as large as possible. What should the length and width of the garden be? Provide students with picture and/or tactile representations of the important aspects of the problem as it is being read. As information is provided, students organize the information in a graphic organizer, baskets, etc. Provide students with a computer program or pre-made PowerPoint as used in previous portions of the lesson to explore rectangles with the same perimeter but different areas. Using the computer program or PowerPoint with switch access, students verify that the perimeter is 16 yd. in the modified word problem and count the area units to determine the area. After students have verified the perimeter or determined the area for a particular rectangle, the program should insert the number of units into the correct portion of the chart. Remember, the alternate keyboard or switches used should have picture or tactile representations of area and perimeter so students determine the correct measurement. See Example: PowerPoint Lesson 1, Slide 8. Students indicate the rectangle with the largest area by choosing rectangle A or B.

Lesson 1: Closure – 5 minutes

A. Review Lesson and Objectives Remind students that they were to make decisions about units and scales that are appropriate for problem solving situations within mathematics or across disciplines or contexts, and: Convert units using standard/known conversion units. Use appropriate known formulas for the area. Solve multistep problems involving one unit of measure. Multiple means of representation: Along with posting lesson objectives in the classroom, students may refer to their individual copies. Multiple means of expression: Students share what they have learned in different formats: writing, drawing, creative expression, 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

When reviewing the expected outcomes, have students refer to the lesson objectives they recorded in their mathematics journals or their electronic picture versions. Students use the information recorded in their journals to refer back to the lesson objectives key words paired with images. From that information, they share what they have learned based on each of the expectations. For example, the student may grab the tactile cue for area to state, "I have learned that the area is all the space measured within a figure." Students refer back to the photographs or models/tactile representations of examples of real-life situations in which these concepts are used to share when they could use these new skills. For example, the student could touch the tactile representations for area and table top to state, "I can use area to determine if an object will fit on my desk."

B. Exit Assessment Students are given a new word problem to solve that includes mixed measurements within the same system. Students work independently to find perimeter and area of rectangles and solve for the situation. Students should use a similar table as that used during practice. New Problem Example: Josh is designing a display for his Science Fair project. His display must have a perimeter of 120 inches or 10 feet. He found two display boards: board one measures 24 in x 3 ft and board two measures 30 in x 30 in. Which display meets the criteria? Is it display 1? Is it display 2? Both? None? Which display gives Josh the largest display area? Multiple means of representation: Provide students with a copy of the word problem and the table. Have drawings and manipulatives available for students to use. Multiple means of expression: Allow 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 students' interests. Use questioning to encourage students to explain their strategies.

Additional Considerations for Emerging Readers and Emerging Communicators

Use the same supports as used in the practice section to solve for the given problem. Use the supports described in Lesson Body, Step 4: Converting Units of Measure.

Lesson 1: Resources

UDL_HS_Math_Lesson_1_Resources.pdf

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