High School Mathematics UDL Instructional Unit-Lesson 3

Lesson 3:Objective

Grade Span: 9 - 10	Content Area: Mathematics – Measurement Investigating Measurement in the Real World
Lesson 3 of the Unit	Approximate Time Needed: 90 minutes

Objective: Students will make decisions about units and scales that are appropriate for problem solving situations within mathematics or across disciplines or contexts.

Essential Question: How can we use proportion to convert measurements from one unit to another in the same system?

Materials Set Up: Provide practice and review worksheets or class discussion about ratio and proportion. Activities will be varied and should include individual and group worksheets, text exercises, and hands-on problem solving activities. Lesson 3:Materials Needed: Worksheets Grid paper Grid paper with raised lines Grid paper copied on transparencies for light boxes Square tiles Geoboard Pencils Overhead projector Transparencies 12-inch ruler Yardstick Poster board Chalk board, white board, or Smart Board Lesson Vocabulary: Area Centimeter Conversion Foot Inch Length Meter Proportion Ratio Width Yards

Lesson 3: Introduction – 15 minutes

A. Activate Previous Knowledge Lead a discussion on the meaning of ratio and proportion. Provide a practice worksheet(s) on finding ratios and proportions. Multiple means of representation: Present illustrations or models of ratio and proportions during discussions. Multiple means of expression: Allow students to use paper and pencil, models, computers, etc. to practice the concepts of ratios and proportions. Multiple means of engagement: Present real life uses for ratios and proportions related to students' interests.

Additional Considerations for Emerging Readers and Emerging Communicators

Provide picture and/or tactile representations of ratio and proportion as well as concrete examples throughout the discussion. Examples can be provided using a computer program and adaptive software with a talking word processor. Refer to ratios and examples used in Lesson 2. Include picture representations of key words in directions and word problems. Provide graphic or manipulative representations of proportional relationships or use a computer program with a switch or alternate keyboard access and talking word processor.

B. Establish Goals/Objectives for the Lesson Inform students that they will make decisions about units and scales that are appropriate for problem solving situations within mathematics or across disciplines or contexts and: 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. Solve problems requiring calculations that involve different units of measure within a measurement system. 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

Provide students with keys words paired with symbols/images/tactile representations. Provide the key words in the lesson objectives paired with images/symbols/tactile representations, record into mathematics journals or use an electronic picture writer to record the lesson objectives. Provide students with photographs/objects/tactile representations of examples of situations when these concepts are used.

Lesson 3: Body – 30 minutes

Direct Instruction and/or Facilitation of the Lesson Lead the discussion on the types of customary units used to measure length: inches, feet, and yards. List the common conversions: 12 in = 1 ft; 3 ft = 1 yd; 36 in = 1 yd. Show that when converting a larger unit to a smaller unit, we multiply (e.g., 4 ft = 4 x 12 = 48 in) or set up and solve a proportion: $\frac{1 ft}{12 in} = \frac{4 ft}{x in}$ $\frac{1 yd}{36 in} = \frac{2 yd}{x in}$ $\frac{1 yd}{3 ft} = \frac{2 yd}{x ft}$ Show that when converting a smaller unit to a larger unit, we divide: (e.g., $6 in = 6/12 =1/2 ft$): Convert feet to yards: Number of feet $\div 3 =$ Number of yds e.g., $2 ft \div 3 = 2/3 yds$ Convert inches to yards: Number of inches $\div 36 =$Number of yds $20 in \div 36 = 20/36$ or $5/9 yd$ Use ratio and proportion: $\frac{1 ft}{12 in} = \frac{x ft}{6 in}$ $\frac{3 ft}{1 yd} = \frac{2ft }{x yd}$ Convert units of measure for area: Show using grid paper, tiles, or Geoboard that a square measuring 12 in on a side is the same size as a square measuring 1 ft on a side. The area of a 12-in by 12-in square = 144 sq. in. or in2 The area of a 1 ft square = 1 sq ft or ft2 Therefore, 144 sq in = 1 sq ft or ft2 In like manner, show 9 sq ft = 1 sq yd or yd2 Discuss how measurements and area can be used in real world situations. For example, area of floor for tile or carpet, length and width of pictures for frames, area of table tops for tile or tablecloth, length and width of windows for curtains, etc. Provide students an opportunity to communicate ideas with the class. Multiple means of representation: Use models and/or drawings on grid paper during large group instruction. Allow students to have a copy of a drawing or a model at their desks. Provide examples of measuring tools. Provide a list of conversion formulas to convert between measurements within the same system. Multiple means of expression: Allow students to use paper/pencil, manipulatives, computer, etc., to complete exercises. Multiple means of engagement: Allow students to brainstorm ideas by writing descriptions of examples, drawing examples, acting out examples, etc.

Additional Considerations for Emerging Readers

Provide picture representations of inch, foot, and yard as well as concrete examples throughout discussion. Remind students that when the measurements increase, use multiplication. Remind students how to make sure the ratio remains balanced: Whatever is multiplied on the top must be multiplied on the bottom. Refer back to supports used in the introduction. $\frac{1ft}{12in} * 4 = \frac{4ft}{xin}$ Remind students that when the measurements decrease, use division. Remind students how to make sure the ratio remains balanced: Whatever is divided on the top must be divided on the bottom. Refer back to supports used in the introduction. $\frac{1ft}{12in} \div 2 = \frac{xft}{6in}$ Students measure and draw a square that is 12 inches by 12 inches. Using the formula for area, students determine the area in square inches. Students measure the square again using the measurement of foot. Students should see that it is 1ft x 1ft and equals 1 ft2. Students should conclude that 144in2 = 1ft2. Students measure and create a square that is 3 ft x 3ft by taping it out on the floor and determine the area. Students measure the same square using a yardstick. Students should determine the square is also 1yd x 1 yd and equals 1 yd2. Therefore, 9ft2 and 1yd2 are equal. See Example: Lesson 3 Conversions.

Additional Considerations for Emerging Communicators

Provide picture and tactile/object representations of inch, foot, and yard as well as concrete examples of measurement tools for the different units throughout discussion. Provide students with concrete examples of the ratios when increasing measurements. Provide students with concrete examples of the ratios when converting to smaller units. See Resources: Lesson 3 tools to use, pages 47- 49.

Lesson 3: Practice – 30 minutes

Model a problem with the class that involves making decisions about units and scales, and determine various ways to solve it. Problem 1: A floor is 9 ft wide and 12 ft long. How many tiles (12" on a side) are needed to completely cover the floor? Draw a rectangle to represent dimensions 9 ft by 12 ft or make a scale drawing of it. Model a second problem for students. Problem 2: A floor is 9 ft wide and 12 ft long. How many tiles (18" on a side) are needed to completely cover the floor? Students recommend whether to convert the floor plan to inches or the tiles to feet. If students recommend converting 9 ft and 12 ft to inches, then find the area of the rectangle in square inches (108in x 144in) and divide by the area of a tile (12in x 12in). For example: 9 x12in = 108 in 12 x 12in = 144 inches Area of the floor in inches: 108 in x 144 in = 15,522 in2 Area of the tile in inches: 18 in x 18 in = 324 in2 1552 in2 ÷ 324 in2 = 48 in2, so 48 tiles are needed Multiple means of representation: Allow students to have a written copy of the problem, drawn models of the situation, and/or conversion formulas as needed/requested. Multiple means of expression: Students may draw or use manipulatives to model solutions or use the computer. Multiple means of engagement: Create situations that include areas of interest to students.

Additional Considerations for Emerging Readers

Provide the written problem to include picture representations of relevant words so students can follow along as the problem is introduced. Provide students a scale drawing of the rectangle (floor) on grid paper measuring 9 units by 12 units with each unit representing a foot. Label the rectangle "floor" using word and picture representation. Each square in the grid represents 1 inch in length and 1 inch in width. Highlight around 12 in x 12 in to represent 1 ft x 1 ft. Provide students with paper or object squares that equal the size of a square foot on the graph paper. Use these to represent the 12" tiles. Allow students to use the conversion chart from previous lessons to determine how to best convert the units. Convert the floor to inches or convert the tiles to feet. Once students have made the conversion, determine the area of the tile using the equation length x width. Students should discover that the area of the tile is 1ft2. Use the ratio $\frac{1 tile}{1ft^2}$ to determine how many tiles are needed to cover the area of the floor: $\frac{1 tile}{1ft^2} = \frac{? tiles}{108 ft^2}$ Students use a calculator to determine and solve the proportions. Or students determine how many tiles are needed to complete the length of one side of the floor by placing a manipulative tile on the floor plan and count 9 tiles needed and repeat for the width, counting 12 tiles. Students multiply 9x12 to determine the number of tiles needed to complete the floor. Provide the written problem to students to include picture representations of relevant words so students can follow along as the problem is introduced. Be sure students have the picture representations of inches, feet, tile, and floor so they can give an opinion as to which rectangle should be converted to which unit of measure. Students should still have the scale drawing of the rectangle (floor) on grid paper measuring 9 units by 12 units with each unit representing a foot and the rectangle labeled "floor" using word and picture representations. Provide students with paper or object squares that represent 18" scaled to one and a half the size of the grid paper unit. Given a model of the multiplication problem for converting feet to inches (___ft x 12 inches) and a calculator, students convert the length and the width of the floor from feet to inches. 9 x 12in = 108 in 12 x 12in = 144 inches Using the formula for area, students use the calculator to determine the area of the floor (108 in x 144 in) and the area of the tile (18 in x 18 in). Students create a ratio of the area of the floor to the area of the tile then divide: $\frac{15522in^2}{324in^2}=48$

Additional Considerations for Emerging Communicators

Provide students with relevant picture or tactile/object representations of relevant words/concepts as the problem is introduced. Use real or replicas of 12-inch tiles and create a 9 ft by 12 ft rectangle on the floor using colored tape or a computer program to model the floor plan. Using the foldable ruler, review that 12 inches is the same as one foot. Convert scale representation of tiles from 12 inches to one foot. Place manipulative tiles down the length of one side of the taped floor plan and count how many tiles are needed to cover the length of that side. Repeat for one side the width of the floor. Multiply the two numbers to determine the number of tiles needed to cover the floor or use a computer program to input the tiles and input the numbers into a multiplication problem to solve the problem. Provide students with picture or tactile/object representations of relevant words/concepts as the problem is introduced. Use real or replicas of 12-inch tiles and create a 9 ft by 12 ft rectangle on the floor using colored tape or a computer program to model the floor plan. Using one foot rulers, show that it takes 9 rulers to cover the length of the floor. Using the foldable ruler, review that there are 12 inches in a foot. Convert the length and width of the floor from feet to inches by inputting the correct number to complete the multiplication problem (___ft x 12 inches) using a calculator or computer program. Place manipulative tiles down the length of one side of the taped floor plan and count how many tiles are needed to cover the length of that side. Repeat for one side the width of the floor. Multiply the two numbers to determine the number of tiles needed to cover the floor or use a computer program to input the tiles and input the numbers into a multiplication problem to solve the problem. Important Note for Communicators Considered Pre Symbolic: The number load may need to be reduced.

Lesson 3: Closure – 15 minutes

A. Revisit/Review Lesson and Objectives Remind students that they were to make decisions about units and scales that are appropriate for problem solving situations involving mathematics or across disciplines or contexts, and: 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. Solve problems requiring calculations that involve different units of measure within a measurement system. 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: Share 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, students refer to the lesson objectives they recorded or collected in their mathematics journals or in 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, students may grab the tactile cue for area to state, "I have learned that the area is all the space measured within a figure." Students should refer back to the photographs and/or tactile representations of examples of real-life situations when these concepts are used to share when they could use these new skills. For example, students could touch the tactile representations for area and tabletop to state, "I can use area to determine if an object will fit on my desk."

B. Exit Assessment 1. Students solve a third problem using models and paper and pencil. For Example, How many 6 in square tiles are needed to cover a 3 ft. by 5 ½ ft. counter top? Discuss the results showing more than one strategy. Multiple means of representation: Allow students to have a written copy of the problem, drawn models of the situation, and/or conversion formulas as needed/requested. Multiple means of expression: Students may draw or use manipulatives to model solutions or use the computer. Multiple means of engagement: Create situations that include areas of interest to students.

Additional Considerations for Emerging Readers and Emerging Communicators

* Use the same supports as used in the practice section.

Lesson 3 Resources

UDL_HS_Math_Lesson_3_Resources.pdf

Return to Lesson 2 Proceed to Lesson 4