PARCC Grade 6 Mathematics Lesson 18: Performance-Based Assessment Expressions and Equations Reasoning and Solving Rationale Students develop understanding of expressions, inequalities, and equations as representative of real-world situations. The students will write inequalities and equations to interpret real-world problems. Goals To understand that inequalities have multiple solutions To create a visual representation of an inequality To express a real-world problem as a mathematical equation Objectives Standards Students will interpret real-world problems. Students will write and solve equations. Students will evaluate an expression to express exponential growth. Students will explain their reasoning. Students will experience Mathematical Practices 1, 2, 3, 4, 5, 6 and 7. 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents. 6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are nonnegative rational numbers. 6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Materials Performance-Based Assessment Tasks-There are two separate tasks: o Task # 1 Type III o Task # 2 Type II Class folder labeled Lesson 18: Performance-Based Assessment Expressions and Equations (At the end of the lesson, place the class papers in the folder. If students used graph paper, please have students attach the graph paper to their assessments.) Procedures for the Performance-Based Assessment Be sure all students have a pencil, scrap paper, and an assessment. Provide a reasonable amount of time in which most students can complete the task. Lesson 18: PBA Expressions and Equations Page 1
Teacher & Teachers Aide Observations During the Assessment Be sure to circulate the classroom and monitor students while they are completing the assessment. Which students are using their time wisely? Which students seem alert with good posture and energy? Which students are spending too much time on one particular task? Which students seem to be making small mathematical errors that could easily be fixed with a mini-lesson? Assessment or Check for Understanding Follow-Up Use the rubric to score the students responses. Look for patterns of error. Ask students for their feedback on the Performance-Based Assessment. Make note of their comments on content and rigor. PARCC Technology Tips PARCC equation editors are provided as the answer boxes for responses that include math, utilizing special mathematical functions. In this lesson, a written response is needed and the Open Response Equation Editor is provided. It is possible to respond with a combination of words and math. It is suggested that letters, numbers, and punctuation symbols from the standard keyboard be used. Any part of the response that indicates mathematical processes can be described using the function keys provided in the Open Response Equation Editor. When the pointer (curser) hovers over a function key, the name of the function appears as a pop-up; the question mark function represents the unknown in an equation. It is NOT possible to create diagrams, models and/or step-by-step solution processes in a vertical format (such as solving with an algorithm). Lesson 18: PBA Expressions and Equations Page 2
PARCC Grade 6 Mathematics Lesson 18: Performance-Based Assessment Expressions and Equations Reasoning and Solving Task #1 (PARCC 6.D.1) Part A Greg is a swimmer. He and his teammates pay entry fees when they compete in swim meets. The fee for swimmers 12 years of age and under is $4 per event. The fee for swimmers over the age of 12 is $5 per event. The team pays a total of $500 for entries in a meet. Write and solve an equation to find the number of 12 years of age and under entries if the team paid $400 for over the age of 12 entries. Let n represent the number of 12 and under swimmers. Enter your equation and solution in the space provided. Lesson 18: PBA Expressions and Equations Page 3
Part B In order to qualify for an important meet, Greg needs to improve his time. He cannot compete unless his time in the 50-meter freestyle is 35 seconds or less. Write an inequality to express this requirement. Enter ONLY the inequality. Represent possible solutions to the inequality on the number line. 5 10 15 20 25 30 35 40 45 50 55 Lesson 18: PBA Expressions and Equations Page 4
PARCC Grade 6 Mathematics Lesson 18: Performance-Based Assessment Expressions and Equations Reasoning and Solving Task #2 (PARCC 6.C.6) The pool that Greg s team uses for practice requires daily maintenance to keep the water clean and safe to swim in. If the correct chemicals are not added on schedule, bacteria grow rapidly. The number of bacteria doubles each day. With regular maintenance, 3 are present. How many bacteria would there be on each day without maintenance? Enter your answers in the boxes. Day 0 (regular maintenance) is 3 Day 1 (no maintenance) Day 2 (no maintenance) Day 3 (no maintenance) Day 4 (no maintenance) Greg wrote an expression to represent the bacteria growth: 3 x 2 n million bacteria. n represents the number of days without maintenance. Is he correct? Explain your reasoning. Enter your answer and explanation in the space provided. Lesson 18: PBA Expressions and Equations Page 5
Task # _1_ Part A Score Description Student response includes each of the following 2 elements: Correct equation Correct answer, 25 swimmers Sample Student Response: 2 400 + 4n = 500 4n = 100 n = 25 swimmers (or other valid equation) 1 Student Response includes 1 of the 2 elements. 0 Student Response is incorrect or irrelevant. Task # _1_ Part B Score Description Student response includes each of the following 2 elements: Correct inequality, t < 35 or 35 > t Correct representation of less than 35 on the number line 2 Sample Student Response: t < 35 5 10 15 20 25 30 35 40 45 50 55 Open circle around 35 and a solid line going back to the beginning of the number line 1 Student Response includes 1 of the 2 elements. 0 Student Response is incorrect or irrelevant. Lesson 18: PBA Expressions and Equations Page 6
Task # _2_ Score Description Student response includes each of the following 3 elements: Correct answers for days 1-4: 6, 12, 24, 48 Correct answer, yes Valid explanation to support the determination that Greg s expression is correct. Sample Student Response: 3 6 million, 12 million, 24 million, and 48 Yes, Greg s expression is correct. When the number of bacteria doubles each day, the result is equivalent to the expression that Greg wrote. Doubling is the same as multiplying by 2 so increasing the exponent of the base 2 to reflect consecutive days is the same as multiplying by one more 2. 3 x 2 1 = 6 (day 1) 3 x 2 2 = 12 (day 2) 3 x 2 3 = 24 (day 3) 2 Student Response includes 2 of the 3 elements. 1 Student Response includes 1 of the 3 elements. 0 Student Response is incorrect or irrelevant. Lesson 18: PBA Expressions and Equations Page 7