Objectives for the lesson

Transcription

1 Foundations of Mathematical Reasoning Student Pages 14.C, Blood alcohol content Lesson 14 Part C, Blood alcohol content Theme: Risk Assessment Blood alcohol content (BAC) is a measurement of how much alcohol is in someone s blood. It is usually measured as a percentage, so a BAC of 0.3% is three-tenths of 1%. That is, there are 3 grams of alcohol for every 1,000 grams of blood. A BAC of 0.05% impairs reasoning and the ability to concentrate. A BAC of 0.30% can lead to a blackout, shortness of breath, and loss of bladder control. In most states, the legal limit for driving is a BAC of 0.08%.1 BAC is usually determined by a breathalyzer, urinalysis, or blood test. However, Swedish physician E.M.P. Widmark developed the following equation for estimating an individual s BAC. This formula is widely used by forensic scientists:2 " 2.84iN % B = 0.015it + $ ' # W ig & Credit: istockphoto Objectives for the lesson You will understand that: o The location of a variable is important to its effect on the size of an expression. You will be able to: o Explicitly write out order of operations to evaluate a given formula. The variables in the formula are defined as: B = percentage of BAC N = number of standard drinks (N should be at least 1) (A standard drink is one 12-ounce beer, one 5-ounce glass of wine, or one 1.5ounce shot of liquor.) W = weight in pounds g = gender constant, 0.68 for men and 0.55 for women t = number of hours since the first drink 1 Blood alcohol content. In Wikipedia. Retrieved June 7, 2014, from Blood_alcohol_content. Note that different countries measure BAC in different ways involving mass and volume. 2 Gullberg, R. G. (2007, August). Estimating the uncertainty associated with Widmark s equation as commonly applied in forensic toxicology. Paper presented at the T2007 Conference. Retrieved June 7, 2014, from The Charles A. Dana Center at Version 2.0 (2014) 1

2 Student Pages 14.C, Blood alcohol content 1) Look at the right side of the equation. How do the variables and their location make sense in calculating BAC? For example, based on their locations, which variables will make BAC larger? Which will make it smaller? 2) Consider the case of a male student who has five beers and weighs 180 pounds. Simplify the equation as much as possible for this case. What variables are still unknown in the equation? 3) Using your simplified equation, find the estimated BAC for this student one, three, and five hours after his first drink. What patterns do you notice in the data? 4) Record the sequence of steps you took to get from time to BAC. Be specific. For example, did you multiply, add, subtract, etc.? What values and in what order? 2

3 Overview and student objectives Lesson 14, Part C Blood Alcohol Content Foundations of Mathematical Reasoning Suggested Instructor Notes 14.C, Blood alcohol content Overview In this lesson, students investigate the use of variables in mathematical equations through the examination of the formula for blood alcohol content (BAC). Students create and work with a simplified form of the formula and then analyze the sequence of steps needed to determine the BAC. This formula is more extensive than the one in Lesson 14, Parts A and B, as this new simplified equation involves multiplication and addition. This analysis leads to Part D, in which students undo those steps they are given a specific BAC value and will solve for the elapsed time. The time allotment for Lesson 14, Parts C and D is fluid. You may spend more or less time in the simplifying and the solving stages, but the two parts together should take about 50 minutes. Lesson Length: 25 minutes Prior Lesson: Lesson 14, Part B, Target Weight Next Lesson: Lesson 14, Part D, Balancing Blood Alcohol (25 minutes) Constructive Perseverance Level: 3 Theme: Risk Assessment Outcomes: A1, A2, A3 Goal: Problem Solving Objectives Students will understand that: The location of a variable is important to its effect on the size of an expression. Students will be able to: Explicitly write out order of operations to evaluate a given formula. Suggested resources and preparation Materials and technology Computer, projector, document camera Preview Assignment 14.CD Student Pages for Lesson 14, Part C Practice Assignment 14.CD 3

4 Suggested Instructor Notes 14.C, Blood alcohol content Prerequisite assumptions Before beginning this lesson, students should be able to: Substitute a value for a variable in a mathematical expression or formula, then simplify. Verbalize the sequence of steps involved in evaluating an expression. Making connections This lesson: Connects back to simplifying expressions using the order of operations, and solving equations involving multiplication and division. Connects forward to solving more complex equations. Background context In the Preview Assignment for this lesson, students evaluated whether equality was maintained by equivalent operations to both sides of an equation. In addition, they began preparation for future lessons by plotting points on the coordinate plane. Suggested instructional plan Frame the lesson (6 minutes) Think-Pair- Share Class Discussion Distribute or display the Student Pages for this lesson. Introduce the equation given in the student handout. Ask students to think about how the variables given in the equation would impact the BAC and then share with a neighbor. Take student responses to the introduction. Carefully discuss why each variable and its location might be important to the BAC calculation. For example, students saw in the previous activities that higher weight led to higher BMI. In this case, weight is in the denominator, so higher weights will result in lower BAC values. The variables B, N, W, g, and t change depending on the person. Ensure that the students understand that the units of B are a percentage. Therefore, 0.08 is not 8%, but eight-hundredths of a percent. Notes on the Widmark Equation: This equation has been simplified from the one found in the reference. 4

5 Suggested Instructor Notes 14.C, Blood alcohol content Lesson activities (15 minutes) The numbers and 2.84 are constants based on the average person. The value is the average rate of elimination of alcohol, and the value 2.84 is a conversion factor between weight (in pounds) of an individual, density of alcohol in a standard drink, and density of water in an average person. Many high-end beers have twice as much alcohol as the standard beer, which assumes 4% alcohol. Transition to the lesson activities by briefly discussing the Objectives for the lesson. Questions 2 4 Group Work Question 2 mimics the process from Lesson 14, Parts A and B in which students created and explored simplified equations. Remind them that they are holding some variables (N, W, and g) fixed as a way of further investigating the model. The simplified equation is B = t Now there are only two variables remaining in the equation. Question 3 Note that the BAC after 1 hour (0.101) is above the legal limit for driving and (after 3 hours) is still high. (0.05 is the legal limit in Canada and in May 2013, the U.S. National Transportation Safety Board recommended that states lower their limits to 0.05.) Group Work Point out that students solved for B when they substituted in t. That is, B was the only unknown left in the equation. Students should notice that BAC decreases over time. Students might try substituting in more hours. If they try to substitute 8 hours or more (actually, for t > 7.7) for this simplified equation, the BAC is negative. o Talk about how to interpret this value. After a certain point, the BAC is 0 because the alcohol has been metabolized by the liver, even if the equation gives a negative number for BAC. Question 4 As seen in Lesson 14, Parts A and B, this question lays the foundation for students to know how to solve for t when B is known. Ensure that students are being explicit about the order of operations and remind them how to determine this from the equation. It is important that they see the relationship between the equation and the steps. Also, reinforce that the steps are based on the order of operations. 5

6 Suggested Instructor Notes 14.C, Blood alcohol content Guiding Questions In an equation with two operations [use an example from the lesson], how do you decide which operation to do first? (It can be useful to demonstrate that the result is different if the work is done in a different order.) How do you decide which operation to use? What if negatives are involved? Why do you have to perform the same operation on both sides of the equation? How can you check if a solution to an equation is correct? Wrap-up/transition (4 minutes) Wrap-up In this lesson, students were introduced to the idea of creating equivalent expressions based on mathematical rules. Each step of solving an equation creates a new equivalent form of the original equation. Students must understand that the rules that govern solving equations apply to all equations. Have students refer back to the Objectives for the lesson and check the ones they recognize from the activity. Alternatively, they may check objectives throughout the lesson. Transition This example continues in the next lesson as students further examine how to solve for variables when given the BAC. o What if a forensic scientist tells you someone s BAC level. Can you determine how much time has passed since the person s first drink? Suggested assessment, assignments, and reflections Give Practice Assignment 14.CD. Give the Preview Assignments, if any, for the lesson activities you plan to complete in the next class meeting. 6

7 Foundations of Mathematical Reasoning Suggested Instructor Notes 14.C, Blood alcohol content Lesson 14 Part C, Blood alcohol content Theme: Risk Assessment ANSWERS Blood alcohol content (BAC) is a measurement of how much alcohol is in someone s blood. It is usually measured as a percentage, so a BAC of 0.3% is three-tenths of 1%. That is, there are 3 grams of alcohol for every 1,000 grams of blood. A BAC of 0.05% impairs reasoning and the ability to concentrate. A BAC of 0.30% can lead to a blackout, shortness of breath, and loss of bladder control. In most states, the legal limit for driving is a BAC of 0.08%.1 BAC is usually determined by a breathalyzer, urinalysis, or blood test. However, Swedish physician E.M.P. Widmark developed the following equation for estimating an individual s BAC. This formula is widely used by forensic scientists:2 " 2.84iN % B = 0.015it + $ ' # W ig & Credit: istockphoto Objectives for the lesson You will understand that: o The location of a variable is important to its effect on the size of an expression. You will be able to: o Explicitly write out order of operations to evaluate a given formula. The variables in the formula are defined as: B = percentage of BAC N = number of standard drinks (N should be at least 1) (A standard drink is one 12-ounce beer, one 5-ounce glass of wine, or one 1.5ounce shot of liquor.) W = weight in pounds g = gender constant, 0.68 for men and 0.55 for women t = number of hours since the first drink 1 Blood alcohol content. In Wikipedia. Retrieved June 7, 2014, from Blood_alcohol_content. Note that different countries measure BAC in different ways involving mass and volume. 2 Gullberg, R. G. (2007, August). Estimating the uncertainty associated with Widmark s equation as commonly applied in forensic toxicology. Paper presented at the T2007 Conference. Retrieved June 7, 2014, from The Charles A. Dana Center at Version 2.0 (2014) 7

8 Suggested Instructor Notes 14.C, Blood alcohol content 1) Look at the right side of the equation. How do the variables and their location make sense in calculating BAC? For example, based on their locations, which variables will make BAC larger? Which will make it smaller? Answers will vary. Sample answer: The time is multiplied by a negative, so it will reduce the BAC as time passes. Number of drinks is in the numerator, so that will increase the BAC, and larger weights in the denominator will decrease BAC. 2) Consider the case of a male student who has five beers and weighs 180 pounds. Simplify the equation as much as possible for this case. What variables are still unknown in the equation? Answer: B = t Time and blood alcohol level are the only two variables. 3) Using your simplified equation, find the estimated BAC for this student one, three, and five hours after his first drink. What patterns do you notice in the data? Answer: When t = 1, BAC = 0.101; when t = 3, BAC = 0.071; and when t = 5, BAC = ) Record the sequence of steps you took to get from time to BAC. Be specific. For example, did you multiply, add, subtract, etc.? What values and in what order? Answers will vary. Sample answers: First, replace t with the given hours in the equation (i.e., plug in the value for t). Second, multiply the hours by Third, add The result is the BAC. 8

9 Foundations of Mathematical Reasoning Student Pages 14.D, Balancing blood alcohol Lesson 14 Part D, Balancing blood alcohol Theme: Risk Assessment Recall Widmark s blood alcohol formula: " 2.84iN % B = 0.015it + $ ' # W ig & The variables in the formula are defined as: B = percentage of BAC N = number of standard drinks (N should be at least 1) (A standard drink is one 12-ounce beer, one 5-ounce glass of wine, or one 1.5ounce shot of liquor.) W = weight in pounds g = gender constant, 0.68 for men and 0.55 for women t = number of hours since the first drink Find the simplified formula for the 180-pound male who drank five beers. What sequence of steps did you take to determine his BAC after 3 hours? Remember to be specific about operations used, values, and order. Credit: istockphoto Objectives for the lesson You will understand that: o Addition/subtraction and multiplication/division are inverse operations. o Solving for a variable includes isolating it by undoing the operations in an expression. You will be able to: o Solve for a variable in an equation. o Explicitly write out the order of operations to solve a given equation. 1) How long will it take for this student s BAC to be 0.08, the legal limit? How long will it take for the alcohol to be completely metabolized, resulting in a BAC of 0.0? The Charles A. Dana Center at Version 2.0 (2014) 9

10 Student Pages 14.D, Balancing blood alcohol 2) A female student, weighing 110 pounds, plans on going home in two hours. Part A: Determine this student s simplified formula. Part B: If she has one glass of wine now, what will her BAC be when she leaves to go home? Part C: What if she has three glasses of wine? Part D: Calculate an estimate of how many drinks she can have and keep her BAC less than

11 Overview and student objectives Lesson 14, Part D Balancing Blood Alcohol Foundations of Mathematical Reasoning Suggested Instructor Notes 14.D, Balancing blood alcohol Overview In this lesson, students continue their investigation of mathematical equations, using the BAC formula. In the previous lesson (Part C), students created and worked with a simplified form of the formula and then analyzed the sequence of steps needed to determine the BAC. Here, students undo those steps they are given a specific BAC value and will solve for the elapsed time. Solving requires subtraction and division (in Lesson 14, Part B, solving required only multiplication and division). The time allotment for Parts C and D is flexible. You may spend more or less time in the simplifying and the solving stages, but the two parts together should take about 50 minutes. Lesson Length: 25 minutes Prior Lesson: Lesson 14, Part C, Blood Alcohol Content Next Lesson: Lesson 15, Part A, Proportional Reasoning in Art (25 minutes) Constructive Perseverance Level: 3 Theme: Risk Assessment Outcomes: A1, A2, A3 Goal: Problem Solving Objectives Students will understand that: Addition/subtraction and multiplication/division are inverse operations. Solving for a variable includes isolating it by undoing the operations in an expression. Students will be able to: Solve for a variable in an equation. Explicitly write out the order of operations to solve a given equation. Suggested resources and preparation Materials and technology Computer, projector, document camera Preview Assignment 14.CD Student Pages for Lesson 14, Part D Practice Assignment 14.CD 11

12 Suggested Instructor Notes 14.D, Balancing blood alcohol Prerequisite assumptions Before beginning this lesson, students should be able to: Understand the use of variables in mathematical equations. Understand that an equation is a statement of equality. Substitute a value for a variable in a mathematical expression or formula, then simplify. Verbalize the sequence of steps involved in evaluating an expression. Making connections This lesson: Connects back to simplifying expressions using the order of operations. Connects forward to solving more complex equations. Background context In the Preview Assignment for this lesson, students evaluated whether equality was maintained by equivalent operations to both sides of an equation. In addition, they began preparation for future lessons by reading points from and plotting points on the coordinate plane. Suggested instructional plan Frame the lesson (2 minutes) Class Discussion Distribute or display the Student Pages for this lesson. If this is a continuation of the same class period as Lesson 14, Part C, simply continue with the case of the 180-pound male student who drank five beers. If this is a new class meeting, direct students to find their materials from the previous class. Confirm that all students have the same simplified formula. You may wish to ask the students for their sequence of steps and document them on the board. Transition to the lesson activities by briefly discussing the Objectives for the lesson. 12

13 Lesson activities (20 minutes) Questions 1 and 2 Foundations of Mathematical Reasoning Suggested Instructor Notes 14.D, Balancing blood alcohol Group Work Note: Question 2 is an extension question for students who finish question 1 quickly. You may not wish to wait for all students to reach this question before beginning the class debrief. Let students struggle with question 1. Some will plug in hours until they come close to 0.08 and 0.0, but they will not get those BAC values exactly. Direct them back to the equation, asking them what is unknown in this case. Ask them to look at their algorithm above and discuss how finding the hours when given the BAC is really working backward through their algorithm and undoing each step. To get students started, you may need to ask what undoes addition and what undoes multiplication. Introduce the term inverse operation in talking about undoing. You can ask them to write out the undoing steps such as: o First, start with the known value of BAC for B in equation. o Second, subtract o Third, divide by o The result is t. Guiding Questions How do you decide which operation to use? In an equation with two operations [use an example from the lesson], how do you decide which operation to do first? (It can be useful to demonstrate that the result is different if the work is done in a different order.) What if the variable is multiplied by a negative? Why do you have to perform the same operation on both sides of the equation? How can you check if a solution to an equation is correct? Although some students may have solved similar equations algebraically in the past, you should explicitly and carefully model how this undoing works, emphasizing the importance of keeping both sides of the equation balanced using the B = 0.08 case. Then give students a chance to work through the case of B = 0 on their own before moving to the next question. A demonstration on the board that links their algorithm with the algebraic operations might look like the table below. You may wish to use a bald equation like y = 3x + 8, find x when = 14, then have students share their BAC work on the document camera. 13

14 Suggested Instructor Notes 14.D, Balancing blood alcohol Undoing Algorithm Step 1: Replace given BAC for B. Step 2: Subtract Step 3: Divide by The result is t. Looks Like with the Equation 0.08 = i t = i t and simplified: = i t = i t or 2.4 = t t = 2.4 hours Notes Substitute B into the equation. To maintain equality, any operation you do on one side must also be done to the other side of the equation. Again, divide both sides of the equation by to maintain a balanced and equal equation. By convention, you rewrite so that the variable is on the left of the equation. Undoing (inverse operations) and balancing equations are two key ideas that students can use to solve for an unknown variable nested in an equation. It is also important for students to realize that undoing is directly related to the order of operations in essence, performing the order of operations backwards. One strategy in deciding what to do first in solving an equation is to first put in a value for the variable and simplify as modeled in this lesson. The steps in simplifying (the order of operations) are reversed for solving. Make sure to remind students that 2.4 hours is not 2 hours and 4 minutes (or 40 minutes). If there is time, you could ask them to convert 2.4 hours into 2 hours and 24 minutes. Ask students to check the reasonableness of their answers with the numbers they got in the going forward model above. Ample time should be spent on question 1. For students ready to move on, direct them to question 2. For students who need more practice, stick with the above model and provide them varying values for B. 14

15 Suggested Instructor Notes 14.D, Balancing blood alcohol Question 2 Wrap-up/transition (3 minutes) Point out how, in this case, you have held t fixed and let N be the variable. Students should notice that if they fix N and let time increase, BAC goes down (as seen above). If they fix t and let N increase, BAC goes up (as seen in this case). Ask why this makes sense. Finally, if needed, remind students that adding a negative 0.03 is equivalent to subtracting positive 0.03, which might help them in the undoing process in Part (b). Solving for N in this equation is a bit trickier than before because the unknown is in the second half of the equation and has been multiplied and divided by different numbers. If students get stuck, remind them about order of operations and encourage them to write out the steps going from N to B and then back from B to N. Some students may recognize that they can divide the coefficient of N to get rid of the fraction. This is fine, but it is important that you discuss how to solve the equation by undoing the division. Common error: Watch for students who add the unlike terms in the equation B = N. Wrap-up In this lesson, students were introduced to the idea of creating equivalent expressions based on mathematical rules. Each step of solving an equation creates a new equivalent form of the original equation. Students must understand that the rules that govern solving equations apply to all equations. Have students refer back to the Objectives for the lesson and check the ones they recognize from the activity. Alternatively, they may check objectives throughout the lesson. Suggested assessment, assignments, and reflections Give Practice Assignment 14.CD. Give the Preview Assignments, if any, for the lesson activities you plan to complete in the next class meeting. 15

16 Suggested Instructor Notes 14.D, Balancing blood alcohol 16

17 Foundations of Mathematical Reasoning Suggested Instructor Notes 14.D, Balancing blood alcohol Lesson 14 Part D, Balancing blood alcohol Theme: Risk Assessment ANSWERS Recall Widmark s blood alcohol formula: " 2.84iN % B = 0.015it + $ ' # W ig & The variables in the formula are defined as: B = percentage of BAC N = number of standard drinks (N should be at least 1) (A standard drink is one 12-ounce beer, one 5-ounce glass of wine, or one 1.5ounce shot of liquor.) W = weight in pounds g = gender constant, 0.68 for men and 0.55 for women t = number of hours since the first drink Find the simplified formula for the 180-pound male who drank five beers. What sequence of steps did you take to determine his BAC after 3 hours? Remember to be specific about operations used, values, and order. Credit: istockphoto Answers will vary. Sample answer: Simplified formula: B = t Steps: First, replace t with 3 hours in the equation (i.e., plug in 3 for t). Second, multiply 3 by Third, add The result is the BAC. The Charles A. Dana Center at Version 2.0 (2014) 17

18 Suggested Instructor Notes 14.D, Balancing blood alcohol Objectives for the lesson You will understand that: o Addition/subtraction and multiplication/division are inverse operations. o Solving for a variable includes isolating it by undoing the operations in an expression. You will be able to: o Solve for a variable in an equation. o Explicitly write out the order of operations to solve a given equation. 1) How long will it take for this student s BAC to be 0.08, the legal limit? How long will it take for the alcohol to be completely metabolized, resulting in a BAC of 0.0? Answer: t = 2.4 hours for BAC = 0.08 and t = 7.73 hours for BAC = ) A female student, weighing 110 pounds, plans on going home in two hours. Part A: Determine this student s simplified formula. Answer: Part B: If she has one glass of wine now, what will her BAC be when she leaves to go home? Answer: When N = 1, B = Part C: What if she has three glasses of wine? Answer: When N = 3, B = Part D: Calculate an estimate of how many drinks she can have and keep her BAC less than Answer: When B = 0.08, N = 2.34 or 2 drinks. 18