Success With Open-Ended Questions. Alabama Reading and Mathematics Test (ARMT). Understanding, Designing, Teaching, and Assessing Open-ended Responses in Math. Why Use Open-Ended Questions? Characteristics of Open-Ended Questions Creating Open-Ended Questions
Success With Open-Ended Questions Alabama Reading and Mathematics Test (ARMT)
Understanding, Designing, Teaching, and Assessing Open-ended Responses in Math • WhyUse Open-Ended Questions? • Characteristicsof Open-Ended Questions • CreatingOpen-Ended Questions • How to AnswerOpen-Ended Questions • Rubrics SampleFor Open-Ended Questions • ARMT Open-Ended Questions and instructionalstrategies
Why Use Open-Ended Questions? • Traditionally: questions required single number answers • What if? Forgot what 8 x 6 is equal to but remembers what 5 x 6 is equal to • How could a student use this fact to figure out 8 x 6?
5 x 6 = 30 I need 3 more 6’s to make a total of 8 groups of 6 – that’s 18 30 + 18 = 48 5 x 6 = 30 Using my fingers I can start at 30 and count out six fingers 3 times. That gives me 48 Two students with different approaches
Johnny wants to plant a garden in the shape of a rectangle. Using the figure below, how would he divide the garden so that 50% of the garden is peas, 25% is beans, 15% is corn, and 10% is carrots.
Two Possible Solutions! (at least) 25% 15% 10% 50 % 50% 25% 15% 10%
Why Use Open-Ended Questions? You can insert the Ma and Pa Kettle Explain Math movie clip here. The web site is: http://facstaff.bloomu.edu/fdangelo/ (Scroll to the bottom of the page and you will see the underlined area to click in order to obtain this clip.)
Use Open-Ended Questions to: • Promote individuality - Not all students are alike - some are creative, some think “inside the box”, some need step-by-step instructions. • Allow students to approach problem-solving however they choose. • Address when to use certain mathematical procedures rather than how to use them. • Reveal “holes” in the student’s understanding. When a student stops communicating (whether written or spoken) that’s where he/she stops understanding. • Develop confidence in the ability to do math, increase the ability to reason and solve problems, give a better understanding of why certain problems work the way they do and foster creativity.
Characteristics of Open-Ended Questions Open-ended questions should: • Involve significant thought • Elicit a range of responses • Require communication • Be clearly stated • Lend themselves to a scoring rubric
Situational Awareness Scenario: • You are driving in a car at a constant speed. On your left side is a valley and on your right side is a fire engine traveling at the same speed as you. In front of you is a galloping pig which is the same size as your car and you cannot overtake it. Behind you is a helicopter flying at ground level. Both the giant pig and the helicopter are also traveling at the same speed as you. • Question: • What must you do to safely get out of this highly dangerous situation? • Answer: Get off the kid’s merry-go-round. You’re drunk!
Mathematics Open-Ended Format • Question (Item Stem) • No answer choices • Problem may have multiple parts or steps that require student reasoning • Student either: • Shows all work • Writes a narrative with answer(s) and explanation(s) of how the problem was solved discussing ALL steps Valued at three points each
FYI about ARMT • Standards are tested because they are expected to be mastered. • Bullets are not tested unless directly stated in the standard, however, they must be taught. The bullets are considered to be introduced in that grade. • On gridded responses: 4th grade has no fraction bar or percent sign 5th grade has no percent sign 6th grade has no fraction bar This is because there are no questions on the gridded response portion of the test that require these symbols.
Alabama Reading and Mathematics Test Grade 3 Mathematics
Multiple-choice and gridded questions are one point each. Open-ended questions are three points each.
Number of Open-Ended Questions on ARMT Per Grade Level Grade 3 - One open-ended item per Standards 2, 6, 7, and 9 - (12 points) • Standard 2 – Solve addition and subtraction problems, including word problems, involving two-and three-digit numbers with and without regrouping. • Standard 6 – Use coins to make change up to $1.00. • Standard 7 – Complete a given numeric or geometric pattern. • Standard 9 – Specify locations on a coordinate grid by using horizontal and vertical movements.
Grade 4 One open-ended item per Standards 6, 7, 15, and 17 - (12 points) Standard 6 – Solve problems, including word problems, that involve addition and subtraction of four-digit numbers with and without regrouping. Standard 7 – Solve problems, including word problems, involving the basic operations of multiplication and division on whole numbers through two-digit multipliers and one-digit divisors. Standard 15 – Represent categorical data using tables and graphs, including bar graphs, line graphs, and line plots. Standard 17 – Represent numerical data using tables and graphs including bar graphs and line graphs.
Grade 5 One open-ended item per Standards 2 and 3; Two open-ended items per Standard 14 - (12 points) Standard 2 – Solve problems involving basic operations on whole numbers, including addition and subtraction of seven-digit numbers, multiplication with two-digit multipliers, and division with two-digit divisors. Standard 3 – Solve word problems that involve decimals, fractions, or money. Standard 14 – Analyze data collected from a survey or experiment to distinguish between what the data show and what might account for the results.
Grade 6 Two Open-ended items per Standard 2; One open-ended item per Standards 7 and 10 - (12 points) Standard 2 – Solve problems involving decimals, percents, fractions, and proportions. Standard 7 – Solve problems involving perimeter and area of parallelograms and rectangles. Standard 10 – Interpret information from bar graphs, line graphs, and circle graphs.
Grade 7 One open-ended item per Standards 8, 11, 12, and 13 - (12 points) Standard 8 – Recognize geometric relationships among two-dimensional and three-dimensional objects. Standard 11 – Solve problems involving ratios or rates, using proportional reasoning. Standard 12 – Determine measures of central tendency (mean, median, and mode) and the range using a given set of data or graphs, including histograms, frequency tables, and stem-and-leaf plots. Standard 13 – Determine the probability of a compound event.
Grade 8 Two open-ended items per Standard 4; One open-ended item per Standards 7, 11, and 13 – (15 points) Standard 4 – Graph linear relations by plotting points or by using the slope and y-intercept. Standard 7 – Solve problems using the Pythagorean Theorem. Standard 11 – Determine the surface area and volume of rectangular prisms, cylinders, and pyramids. Standard 13 – Interpret data from populations using given and collected data.
Original Closed-Ended Item Which of the following numbers are prime? 7, 57,67,117 Round 37.67 to the nearest 10th. Find the LCM of 18 and 24. Revised Open-Ended Item Fred thinks that 57 and 67 are prime because they both end in 7, which is a prime number. Dick says he is wrong. Who is correct and why? Generate three different numbers that when rounded to the nearest 10th give 37.7. Why can’t 48 be the LCM of 18 and 24? Creating Open-Ended Questions
Closed- Ended Find the value of n if 3 x n = 12 Open-Ended Solve the riddle using the clues and numbers in the table. 1. It is not 3 x 4. 2. It is not > 56. 3. It does not equal 4 tens. 4. It does not equal 2 x 7. 5. It is not the missing number in 3 x n = 12. The number is ______. Creating Open-Ended Questions continued
ARMT MATHEMATICS • Look at the spinners. Yellow Yellow Yellow Blue Blue Red Red Blue Red Spinner A Spinner B Spinner C Julie, Greg, and Lori each used a different spinner to record the results of 40 spins.
Look at the spinners. Yellow Yellow Yellow Blue Red Blue Red Blue Red Spinner A Spinner B Spinner C • This table shows Julie’s results. Julie’s Spinner Results Which spinner did Julie most like use? Show your work or explain how you know. • This table shows Greg’s results. Greg’s Spinner Results Which spinner did Greg most likely use? Show your work or explain how you know.
Look at the spinners. Yellow Yellow Yellow Blue Red Blue Red Blue Red Spinner A Spinner B Spinner C Lori used the remaining spinner. Make a table to show the most likely results of Lori’s 40 spins. Explain your reasoning.
MoreOpen-Ended Examples: Ask students to create a situation or an example that satisfies certain conditions. Make a 4-digit even number using the following digits: 3 6 7 1 5 Give an example of an event that has a probability of 0 Draw a rectangle and label the sides so that the perimeter is between 19 and 20 Ask students to explain who is correct and why Ask students to justify their answers or explain why their answer is true
Activity for Converting Closed-Ended Questions to Open-Ended • How many sides does a quadrilateral have? A shape has: exactly four angles exactly 4 sides sides that are all the same length a. What shape could this be? b. If you added two more sides what geometric shape would you have? • Frank has 3 coins. Each coin has a different value. The total value of the coins is 40¢. What types of coins does Frank have? a. What types of coins does Frank have? Use a dollar sign and a decimal point to write your answer. b. Use numbers, words, or pictures to show your work or explain how you know.
Conversions continued • Write an equivalent fraction for 2/6. Kanetra and Susan each bought a pizza of the same size. Kanetra’s pizza was cut into 3 equal slices. She ate 1 slice. Susan’s pizza was cut into 6 equal slices. She ate 2 slices. a. Explain in words, numbers, or pictures how Kanetra did or did not eat the same amount of pizza as Susan. • Find the area of the rectangle in square feet. Jonathon wants to make a garden with an area of 24 square feet. The garden must be rectangular. a. On the grid provided, draw 2 different rectangular gardens Jonathon could make. The perimeter of each garden must be different. b. What are some possible dimensions of a rectangular garden whose area and perimeter are the same? Explain your answer by using words, pictures, or numbers. 8 3
Conversions continued • Using the numbers below, identify the next number in the pattern. 24, 20, 16, 12, 8 Nannette wrote the number pattern shown below: 24, 20, 16, 12, 8 • What could be the rule for Nannette’s pattern? • Use the rule you wrote in part (a) to write the next number in Nannette’s pattern. 24, 20, 16, 12, 8, ______
Things to Consider Before Answering an Open-Ended Mathematics Question • Read the ENTIRE question critically: What is the question asking me to do? Straight computation Multiple steps (scaffolding) Probability • Plan: What do I do? Equation Discussion Equation and discussion Draw graph with explanation • What do I know that can help me answer this question? • How can I make this answer better?
Using a Rubric to Score Student Responses A scoring rubric is helpful in several ways: • Helps focus on what students know and can do rather than what they do not know and cannot do. • Helps teachers keep grading consistent. • Allows for partial credit. • If ask students to evaluate previous students’ responses using the same scoring rubric you will use they will stand a better chance of understanding expectations and levels of responses. • OPEN-ENDED ITEMS ON ARMT ARE SCORED BY INDIVIDUALS USING ITEM SPECIFIC RUBRICS AND SCORE-POINT ANCHOR PAPERS.
Sample Scoring Rubric • 3 Answer is complete, logical, and accurate • 2 Answer has accurate logic and some corresponding answers or contains accurate logic only • 1 Answer reflects partial logic and one or more answers • 0 Answer is not correct (blank, insufficient, off-topic, rewrites problem, foreign language, illegible, refusals, off-task)
Practice Scoring Open-Ended – 4th Grade This problem requires you to show your work and/or explain your reasoning. You may use drawings, words, and/or numbers in your answer. Your answer should be written so that another person could read it and understand your reasoning. It is important that you show all your work. A group of students were surveyed to see in which small country they would prefer to live. The table below shows the results of the survey. Use the information from the table to make and label a bar graph. Show all your work and/or explain your reasoning in the space provided in the answer document.
Practice – Continued – 6th Grade This problem requires you to show your work and/or explain your reasoning. You may use drawings, words, and/or numbers in your answer. Your answer should be written so that another person could read it and understand your reasoning. It is important that you show all your work. The junior and senior football teams at Strawberry High School washed cars for an athletic fund-raising project. The junior high players washed 54 cars in 1 ½ hours. The senior high players washed 70 cars in 2 hours. a. How many cars did the junior high players wash per hour? b. How many cars did the senior high players wash per hour? c. At the same rate of washing cars, about how long should it take both teams working together to wash a total of 175 cars? Show all your work and/or explain your reasoning for each part in the space provided in the answer document.