High School Mathematics UDL Instructional Unit-Lesson 6-Culminating Lesson

Overview

Grade Span: 9 - 10	Content Area: Investigating Measurement in the Real World – Culminating Activity
Lesson 6 of the Unit	Approximate Time Needed: 45 minutes

Objectives: Identify and quantify attributes of the problem that need to be measured. Determine a pattern. Generalize relationships. Determine percent of increase/decrease. Determine the precision of measurement.

Essential Questions: What are the relationships among the measurements of dimensions, area, and perimeter in problem solving situations? How can we use variable expressions to reflect relationships? How do we determine limitations of measurement

Materials Needed: Large and small grid graph paper Worksheets Lesson Vocabulary Area Centimeter Conversion Foot Inch Length Meter Perimeter Proportion Ratio Rectangles Similar Rectangles Unit of Measure Unit Rate Width Yards

Lesson 6: Culminating Activity - 45 minutes

CULMINATING ACTIVITY FOR THE UNIT. A. Revisit/Review Unit and Lesson Objectives Remind students that throughout these lessons 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. Set up and solve proportions. Convert units of measurement using standard/known conversions. Recognize when to multiply and when to divide in converting measurements. Use ratio and proportion to convert measurements. Use appropriate known formulas for area. Identify, quantify, and compare the attributes of the objects, situations, and/or events that need to be measured to solve the problem/situation. Use appropriate units of measure to identify, quantify, and compare objects, situations, and/or events to solve a real world problem. Convert units when necessary. Conduct a class discussion on which skills were used to solve different types of problems. Discuss the additional strategies they used to implement the skills and solve the problems. Multiple means of representation: Along with posted lesson objectives in the classroom, students may refer to their individual copies of the objectives and their mathematics journals. Multiple means of expression: Students share what they have learned or strategies they have used by showing different models, pictures, drawings, etc. used throughout the lessons. Multiple means of engagement: Share ideas of how these skills have been useful in solving the problems from previous lessons and what strategies were the most helpful.

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 to examples of their work to demonstrate how the skills and strategies were used. For example, a student could touch the tactile representations for area and orchard to state, "I used the concept of area to determine how much space a tree needs to grow."

B. Exit Assessment Scenario: The freshman class officers at Riverside High School are planning the annual Freshman Class Winter Dance. They have decided to hold the dance at the White Oak Country Club. The room they have reserved for the dance is carpeted, but for events such as this dance, the country club places parquet flooring over the carpet to make a dance floor. The parquet flooring is laid in interlocking sections, and so the dance floor can be arranged to be various sizes to accommodate the number of dancers attending the event. However, the class officers need to let the country club managers know a month in advance of the dance how big to make the dance floor so the club workers will have enough time to get the flooring laid out appropriately. The class officers decide to base that decision on their latest ticket sales. Things to Consider: The minimum size of the dance floor being considered by the class officers is 30 ft x 30 ft. The class officers assume 30 ft x 30 ft will be enough room for 100 couples to be on the dance floor at once. The country club workers can increase the sides of the dance floor in 5 ft increments in either direction, but the class officers want to maintain a square dance floor. The maximum the 30 ft sides of the dance floor can be increased is by 50%. Tasks: Determine how many square feet of dancing space each couple would have if the dance floor is 30 ft x 30 ft (i.e., 30 ft x 30 ft = 900 ft2 and 900 ft2 ÷ 100 couples = 9 ft2 for each couple). Determine the maximum length of the sides of the dance floor (i.e., 30 ft x 50% = 15 ft and 30 ft + 15 ft = 45 ft as the maximum length for the dance floor). Using n to represent the number of possible increases, solve for n having determined that the 50% increase to 30 ft would be an additional 15 ft and that the increments are made 5 ft at a time (i.e., n = 15 ft ÷ 5 ft increments = 3 increases [35, 40, and 45]). Determine the area of the dance floor for each of the 3 increases, and determine the number of couples that each would accommodate (i.e., 45 ft x 45 ft = 2,025 ft2 and 2,025 ft2 ÷ 9 ft2 needed per couple = enough space for 225 couples). Make a chart showing all of the dance floor size possibilities as well as the number of couples that could be accommodated by each. Side Lengths Area of the Dance Floor Number of Couples 30ft x 30ft 900ft2 100 couples 1st increase 35ft x 35ft 1,225ft2 136* couples 2nd increase 40ft x 40ft 1,600ft2 177* couples 3rd increase 45ft x 45ft 2,025t2 225 couples * 1,225 ft2 ÷ 9 ft2 actually = 136.1 and 1,600 ft2 ÷ 9 ft2 actually = 177.777778 so, there would be a little space left over but not enough for another couple. Create a graph that shows the relationship between the increase in the size of the dance floor and the number of couples who can attend the dance. Make sure to scale and label your axes (i.e., (x,y) where x = dance floor area and y = number of couples). Given that 165 couples' tickets have been sold, suggest how large the class officers should tell the country club managers the floor will need to be and explain why it should be 40' x 40' because that area can accommodate up to 177 couples whereas the 35' x 35' could only accommodate 136 couples. Multiple means of representation: Allow students to refer to their mathematics journals and other notes as they solve the problem. 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 and record information into the tables using various formats: computer, paper pencil, drawings, etc. Multiple means of engagement: Ensure all students are actively involved in solving the problem. Encourage students to consider options for solving the problem that will engage them. Use questioning to encourage students to explain their strategies.

Additional Considerations for Emerging Readers

Provide the written problem to students paired with picture symbols and/or tactile cues as well as the things to consider list. Allow students to use a text reader to initially access the problem and to go back and review the problem as needed. Provide the written questions (1-7) to students paired with picture symbols and/or tactile cues. Provide students with a variety of formulas, including the ratio for the unit rate $\frac{area\ of\ floor}{\#\ of\ couples}$, area, and perimeter as well as a copy of the table below. Students should have grid paper to create a model of the area of the dance floor as well as manipulatives representing both variables (area of dance floor and number of couples). Students first need to determine the area of the dance floor by choosing and using the formula for area. Students need to determine the unit rate. Students use the formula chosen from options ($\frac{area\ of\ floor}{\#\ of\ couples} = \frac{area\ of\ floor}{1\ couple} ;\ \frac{900ft^2}{100ft}=\frac{area\ of\ floor}{1}$), or manipulatives, etc. Side Lengths Area of the Dance Floor Number of Couples Unit Rate area/couple 30ft x 30ft Question 1 100 couples 1st increase 2nd increase 3rd increase Determine the maximum length of the sides of the dance floor (i.e., 30 ft x 50% = 15 ft and 30 ft + 15 ft = 45 ft as the maximum length for the dance floor). Be sure students have the following "things to consider" in picture format and or electronic text reader to refer to when answering question 2: The country club workers can increase the sides of the dance floor in 5 ft increments in either direction, but the class officers want to maintain a square dance floor. Students create a unit length representing 5 ft or are given choices of various unit lengths, one of which is 5 ft that can be used to extend the dance floor. Students could also have a template that represents + 5 to use to complete the column in the chart on floor lengths. The maximum the 30 ft sides of the dance floor can be increased is by 50%. Provide students with a variety of strategies for determining half of 30 (divide by two, fold representation of 30 ft x 30 ft dance floor in half and count half side length, etc.) Using n to represent the number of possible increases, solve for n having determined that the 50% increase to 30 ft would be an additional 15 ft and that the increments are made 5 ft at a time (i.e., n = 15 ft ÷ 5 ft increments = 3 increases [35, 40, and 45]). Students may use a drawing of the 30 ft x 30 ft dance floor on grid paper and divide it evenly in half to determine what 50% more would be. Students use the 5 ft unit length template to determine how many increments of 5 the floor can be increased and what the measurement would be for each increase (i.e. 30 ft + 5 ft = 35 ft; 35 ft + 5 ft = 40 ft, etc.). Students may also determine the increase by counting by 5s from 30. Determine the area of the dance floor for each of the 3 increases and determine the number of couples that each would accommodate (i.e., 45 ft x 45 ft = 2,025 ft2 and 2,025 ft2 ÷ 9 ft2 needed per couple = enough space for 225 couples). Students must remember to increase both the length and width by 5 ft and use the formula and/or manipulatives to determine the area of each floor increase. Using the unit rate determined in step 1, students choose the correct given formula to use to determine how many couples can be accommodated for each increase, and/or students use manipulatives of the different floor measurements and the unit rate to determine how many couples can dance per floor size. Make a chart showing all of the dance floor size possibilities as well as the number of couples that could be accommodated by each. Students can be given the above chart to complete from results found in steps 1, 2, and 3. Create a graph that shows the relationship between the increase in the side length of the dance floor and the number of couples that can attend the dance. Students should scale and label the coordinate grid (first quadrant), using the side length of each dance floor, to keep the numbers manageable. (i.e., (x,y) where x = side length of the dance floor and y = number of couples; where the scale of x increases by 5's to 50 and the scale of y increases by 25 to 250 ). Students can be given choices of scaled and labeled coordinate grids and choose the best representation for the problem and then plot the points on the chosen grid. Students should use the table to determine the ordered pairs needed to plot the points. Students identify and highlight the two variables used for graphing (side length and number of couples). Finally, given that 165 couples' tickets have been sold, suggest how large the class officers should tell the country club managers the floor will need to be and explain why the dance floor should be 40' x 40' because that area can accommodate up to 177 couples whereas the 35' x 35' could only accommodate 136 couples. Students should use their table and graph to determine the appropriate floor size. Students explain using closed sentences (e.g., The dance floor should be ______________ because that area can accommodate up to _______ couples whereas the _____________ could only accommodate ________ couples.)

Additional Considerations for Emerging Communicators

Provide the written problem to students paired with picture symbols and/or tactile cues. Allow students to use a text reader to initially access the problem and to go back and review the problem as needed. Provide models of the initial problem as well as for the things to consider list. Modify the things to consider section to use smaller numbers for the side lengths and number of couples to start the problem. Things to Consider: The minimum size of the dance floor being considered by the class officers is 12 ft x 12 ft. The class officers assume 12 ft x 12 ft will be enough room for 16 couples to be on the dance floor at once. The country club workers can increase the sides of the dance floor in 2 ft increments in either direction, but the class officers want to maintain a square dance floor. The maximum the 12 ft sides of the dance floor can be increased by is 50%. Tasks: Provide students with a variety of formulas, including the ratio for the unit rate ($\frac{area\ of\ floor}{\#\ of\ couples}$), area, and perimeter as well as a copy of the table. Students should have grid paper to create a model of the area of the dance floor in question one as well as manipulatives representing both variables (area of dance floor and number of couples). Students first need to determine the area of the dance floor by choosing and using the formula for area. Side Lengths(x) Area of the Dance Floor Number of Couples(y) Unit Rate area/couple 12ft x 12ft Question 1 16 couples 1st increase 2nd increase 3rd increase Determine the maximum length of the sides of the dance floor (i.e., 12 ft x 50% = 6 ft and 12 ft + 6 ft = 18 ft as the maximum length for the dance floor). Be sure students have the following things to consider in picture format and/or electronic text reader to refer to when answering question 2: The country club workers can increase the sides of the dance floor in 2 ft increments in either direction, but the class officers want to maintain a square dance floor. Students can be given choices of various unit lengths, one of which is 2 ft that can be used to extend the dance floor. The maximum the 12 ft sides of the dance floor can be increased by is 50%. Provide students with a variety of strategies for determining half of 12 (divide by two, fold representation of 12 ft x 12 ft dance floor in half and count half side length, etc.). Using n to represent the number of possible increases, solve for n having determined that the 50% increase to 12 ft would be an additional 6 ft and that the increments are made 2 ft at a time (i.e., n = 6 ft ÷ 2 ft increments = 3 increases [14, 16, and 18]). Determine the area of the dance floor for each of the 3 increases, and determine the number of couples that each would accommodate (i.e., 18 ft x 18 ft = 324 ft2 and 324 ft2 ÷ 9 ft2 needed per couple = enough space for 36 couples). Students must remember to increase both the length and width by 2 ft and use manipulatives and/or computer with virtual manipulatives to determine the area of each floor increase. Using the unit rate determined in step 1, students should us the manipulative/template of the different floor measurements and the unit rate to determine how many couples can dance per floor size. Make a chart showing all of the dance floor size possibilities as well as the number of couples that could be accommodated by each. Students can be given the above chart to complete from results found in steps 1, 2, and 3 Students should refer to the representations used/created in each step above and multiple choice options to complete the table. Create a graph that shows the relationship between the increase in the size of the dance floor and the number of couples that can attend the dance (make sure to scale and label axes) \[i.e., (x,y) where x = dance floor area and y = number of couples). The graph should be scaled and labeled using the side length of each dance floor, to keep the numbers manageable. \[i.e., (x,y) where x = side length of the dance floor and y = number of couples; where the scale of x increases by 2's to 20 and the scale of y increases by 2 or 4 to 40 ). Students can be given choices of scaled and labeled coordinate grids and choose the best representation for the problem and plot the points on the chosen grid. Students should use their completed table to determine the ordered pairs needed to plot the points. The x and y variables can be highlighted for graphing (side length and number of couples). Students identify where the points should go by first identifying the number representing the independent variable (x) and the correct corresponding number representing the dependent variable (y), or students use a computer program to graph by indicating the numbers in an order pair. See Example: PowerPoint Lesson 5, slide 6 Finally, given that 32 couples' tickets have been sold, suggest how large the class officers should tell the country club managers the floor will need to be and explain why it should be 18' x 18' because that area can accommodate up to 36 couples whereas the 16' x 16' could only accommodate 28 couples. Students should use their table and graph to determine the appropriate floor size. Students indicate which floor size by comparing the number of couples attending to the number of couples that can fit on the dance floor and choosing the floor with immediately higher/larger number. Students explain using closed sentences (e.g., The dance floor should be ______________ because that area can accommodate up to _______ couples whereas the _____________ could only accommodate ________ couples.)

Lesson 6: Resources

UDL_HS_Math_Lesson_6_Resources.pdf