The Common Core Mathematics Standards: Transforming Practice through Team Leadership
Publication Year: 2012
The Standards for Mathematical Practice, according to the CCSS document describe the mathematical “habits of mind” that teachers, at all levels, should develop in their students, and without which the content standards cannot be successfully implemented. Attention to the Mathematical Practices connected with content must be enacted in teaching, which will require professional development. Though the CCSS Mathematical Content Standards differ in detail from other content standards, their form is familiar to teachers: a list of things to know. The Mathematical Practices are not so easily condensed into a lesson or unit, not so easily tested and, generally, not so familiar. Content standards are specified grade by grade and build on each other rather than repeating year after year. The Mathematical Practices are different. Though ...
- Front Matter
- Back Matter
- Subject Index
- Chapter 1: Overcoming Resistance to Change: Four Strategies for Teams
- Chapter 2: Transforming Instruction
- Chapter 3: Promoting Adoption and Avoiding Rejection
- Chapter 4: Focusing on Students Brings Success
- Chapter 5: Attaining the Common Core Practices
- Chapter 6: Visiting a Transforming Classroom
- Chapter 7: Building Support for Collegial Relationships
- Chapter 8: Maintaining Support to Increase Implementation
- Chapter 9: Leading the Way for Change
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Copyright © 2012 by Corwin
All rights reserved. When forms and sample documents are included, their use is authorized only by educators, local school sites, and/or noncommercial or nonprofit entities that have purchased the book. Except for that usage, no part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.
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Library of Congress Cataloging-in-Publication Data
Hull, Ted H.
The common core mathematics standards: transforming practice through team leadership/Ted H. Hull, Ruth Harbin Miles and Don S. Balka.
A Joint Publication with the National Council of Teachers of Mathematics
Includes bibliographical references and index.
ISBN 978-1-4522-2622-4 (pbk.: acid-free paper)
1. Mathematics teachers—Training of. 2. Leadership. 3. Mathematics—Study and teaching. I. Miles, Ruth Harbin. II. Balka, Don S. III. Title.
This book is printed on acid-free paper.
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Improving student success and achievement in mathematics are the goals of this book. The theme is consistent across each of our previous books. In 2009, Corwin released our first book, A Guide to Mathematics Coaching: Processes for Increasing Student Achievement (Hull, Balka, & Harbin Miles), which was designed to assist coaches in directly impacting student performance by working with teachers. Within a few months, our second book was released, A Guide to Mathematics Leadership: Sequencing Instructional Change (Balka, Hull, & Harbin Miles, 2009). We focused on strategies school leaders could use to increase student success in mathematics achievement. In 2010, we released Overcoming Resistance to Change: A Guide for School Leaders and Coaches, and in 2011, Corwin and the National Council of Teachers of Mathematics (NCTM) released our book Visible Thinking in the K–8 Mathematics Classroom. Both books specifically targeted classroom instructional change.
Our series of books is based on a study of the research, a review of the literature, and our professional experiences with a combined total of more than 100 years working with school leaders and teachers to improve mathematics achievement. With both our work and our books, we strive to provide school leaders responsible for mathematics achievement, mathematics leaders, and mathematics teachers a practical, sequential process to establish meaningful, significant improvements in mathematics teaching and learning.
Shortly after the release of our first two books, a never before witnessed phenomenon occurred in our nation. A wide-ranging group of state governors (48 of the 50) met to initiate the creation of common content standards. With work completed, the Common Core State Standards (CCSS) were released in 2010. Now, more than 40 states, the District of Columbia, and U.S. territories have signed on to this initiative as work continues and will continue on the assessment portion for many years.
Yet, with the CCSS, school leaders are facing a significant undertaking in transitioning to the new content standards and Standards for [Page ix]Mathematical Practice. They need more specific help that is directly related to the CCSS than what is contained in our previous books.
With these thoughts in mind, we have written a book for leaders, teachers, and leadership teams that is precise and easy to read, one that selectively pulls ideas from our other books that directly impact leadership concerns and issues. Four different groups of educators need this book:
- Leaders responsible for mathematics such as principals, assistant superintendents, and curriculum directors;
- Mathematics leaders such as coaches, specialists, and coordinators;
- Mathematics teachers; and
- Leadership teams consisting of representatives from the above three groups.
We are recommending this companion book for all educators responsible for mathematics because it assists them in working collaboratively to understand and adopt the mathematical content and practices. More important, we provide a guide, with supporting forms, for successfully leading the implementation of the eight identified Standards for Mathematical Practice for students that are contained in the CCSS.
Corwin is grateful for the contributions of the following reviewers:
D. Allan Bruner, Science and Math Teacher Colton High School Colton, OR
Elizabeth Marquez, Mathematics Assessment Specialist Educational Testing Service (ETS) Princeton, NJ
Edward C. Nolan, Mathematics Supervisor, PreK–12 Montgomery County Public Schools Rockville, MD
Sandra K. Peer, Math Educator Wichita State University Wichita, KS
Lisa Usher-Staats, Response to Instruction and Intervention Expert Los Angeles Unified School District Los Angeles, CA
About the Authors
References[Page 119]2011). Quality tools for the classroom. Beech Grove, IN: Beech Grove High School., & (2009). Guide to mathematics leadership: Sequencing instructional change. Thousand Oaks, CA: Corwin., , & (Common Core State Standards. (2010). Retrieved January 26, 2012, from www.corestandards.org/the–standards2001). Implementing change: Patterns, principles, and potholes. Needham Heights, MA: Allyn and Bacon., & (2009). Guide to mathematics coaching: Processes for increasing student achievement. Thousand Oaks, CA: Corwin., , & (2011a). LCM. Retrieved January 26, 2011, from www.mathleadership.com, , & (2011b). Visible thinking in the K–8 mathematics classroom. Thousand Oaks, CA: Corwin., , & (2010). Overcoming resistance to change: A guide for school leaders and coaches. Pflugerville, TX: Self-Published., , & (2011). Overcoming resistance to change: Why isn't it working? Virginia Mathematics Teacher, 38(1), 36–38., , & (Implementation of the Common Core State Standards. (2010). National Council of Teachers of Mathematics Regional Conference presentation. Retrieved January 26, 2012, from http://www.nctm.org/…/Common_Core_Standards/CCSSM_Grades6–8_120210v.2.ppt2003). Group-Worthy tasks. Creating Caring Schools 60, 6(72–75).(2003). What works in schools: Translating research into action. Alexandria, VA: Association for Supervision and Curriculum Development (ASCD).(National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.National Council of Teachers of Mathematics. (2011). Making it happen: A guide to interpreting and implementing common core state standards for mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.National Research Council. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington DC: National Academy Press.National Research Council. (2004). Engaging schools: fostering high school students' motivation to learn. Washington, DC: National Academy Press.[Page 120] (2006). The learning leader: How to focus school improvement for better results. Alexandria, VA: Association for Supervision and Curriculum Development (ASCD).1995). Diffusion of innovations. New York: The Free Press.(
Appendix: Sample Problems Showing CCSS Content and Practices[Page 121]
It is our hope that teacher teams and leadership teams will find these nine problems useful for initiating conversations about instructional change. The intent of the problem examples is not for teachers to copy and paste the problems into a lesson but to discuss the problems with colleagues while studying the Sequence Chart and Proficiency Matrix. Teacher teams discover that, as students attend to solving challenging problems, better and clearer thinking and reasoning emerge. Provided with appropriate opportunities, students share their reasoning with classroom partners, the teacher, and other students in the classroom. This developmental progress continues until the teachers master a variety of strategies, students attain proficiency on the Standards for Mathematical Practice, collaborative teams function effectively, and leadership teams achieve a critical mass of adopters.CCSS Problem 1Grade Level: 1
In problem 1, teachers are using the strategy of pair-share to provide students additional time to reason and reflect on the problem. By working [Page 122]with a partner and sharing individual understandings, students are better prepared to explain their thinking aloud. By using the strategy of showing thinking, teachers are able to assess student understanding. Teachers are also able to provide ongoing formative assessment as they observe, listen, and talk to students.Directions and Strategies
Discuss the idea of collecting objects, and provide some examples such as stamps, coins, dolls, or stuffed toys. Then ask students to share examples. Next, read the problem to the students.The Problem
Tell the students:
“With your partner, agree how to solve the problem.”
(Use the think-pair-share strategy with a learning partner. Students will individually take a minute to think about a process to solve the problem and then be paired with another student to share ideas together.)
“Be prepared to demonstrate how you solved the problem, and be prepared to explain your thinking.”[Page 123]
(Use Showing Thinking in Classrooms. Students will demonstrate thinking by explaining their selected solution process.)Solution and Discussion
Sam had 19 stickers. (Total Stickers – 7 = 12.)
Move around the classroom; observe and listen to students as the problem is being worked. You may observe students working this problem in a variety of ways. For example:
Students may elect to role-play the problem. In this case, they would gather an uncounted number of objects to represent stickers. One student would be Sam and count out 7 stickers (objects) to his or her partner (Dan). Students then count out 12 objects to the student acting as Sam to demonstrate the remaining stickers. Students recognize they need to combine the 7 stickers with the 12 stickers to find the original amount.
Students may decide to select 12 objects that represent the stickers that Sam had after giving 7 stickers to Dan. The students would then count up from 12 (remaining stickers) the 7 stickers (given stickers). Students may do this with objects representing the stickers by counting out 12, then counting out 7, and then counting the number of objects in the pile. These same strategies can be used by drawing representations of stickers.
Some students may actually set up an equation by selecting a symbol (?, X, or box) to represent the original number of stickers. They then show subtracting 7, and that this expression (? – 7) is equal to 12 (? – 7 = 12). Students then perform the operation of adding 7 to 12 and get the answer of 19.
When they have completed working the problem, call on several student pairs to provide their answers to the problem and their method. (Make sense of problems: Explain their thought processes in solving a problem one way.) Choose three methods to compare. Then, select two students to role-play, and write two other methods on the whiteboard. (Reason abstractly and quantitatively: Reason with models or pictorial representations to solve problems.) As the students enact the problem, point out the numbers contained in the problems, and encourage students to discuss symbolic representations used to indicate the stickers and operations. (Reason abstractly and quantitatively: Translate situations into symbols for solving problems.)CCSS Problem 2Grade Level: 2
In problem 2, students are given a task for which they get to work with partners. The task, while not that challenging, does take collaboration and communication. Students will need to agree on creating and recording numbers. Further, students need to discuss and arrange answers to ensure all possibilities are found. Finally, students will not be allowed to just provide answers. Students will need to explain their reasoning in verifying that all combinations have been found. Students need to learn how to effectively work with partners and small groups.Directions and Strategies
Discuss with students the concept of place value in base ten. Have students identify and name 100s, 10s, and 1s in base ten blocks. Read the problem with the students.The Problem
There are many three-digit numbers that can be made using any combination of the base ten blocks shown. How many can you find?[Page 125]
Tell the students:
“With your partner, select one 100, three 10s, and four 1s and place them on your desk. One partner needs to get a sheet of paper and a pencil. Find and record as many three-digit numbers as you can using the place value blocks.”
(Use Initiating Think-Pair-Share.)
After a reasonable amount of time, when students are engaged and finding answers, ask the students to stop for a moment and share how they are finding answers.
(Use Showing Thinking in Classrooms.)
“Now, class, continue working, but this time, I want you to think about how you will know when you have discovered every possible answer. Organize your work so you can explain your thinking.”
(Use Showing Thinking in Classrooms and Questioning and Wait Time.)Solution and Discussion
There are 20 possible three-digit numbers if teachers allow 0s to be used.
Three-digit numbers require the 100 number block, so for the most part, students are finding combinations with the 10s and 1s. The smallest three-digit number is 100. By counting up, students find:
100; 101; 102; 103; 104
Continuing to count up, students note that they cannot create 105, and so on. The next series begins with 110; 111; 112; 113; 114. Again, students should find they cannot create 115, so the next number series is 120; 121; 122; 123; 124. This is followed by 130; 131; 132; 133; and 134.
Students may try to continue counting up to ensure they are correct but should clearly grasp that, without additional 10s or 1s, no further numbers can be created.[Page 126]CCSS Problem 3Grade Level: 2
In problem 3, students are provided a challenging problem to solve with a partner. In this case, students need to critically think about the problem and carefully read and reread the problem. Strategies need to be discussed and agreed upon as students organize their work and seek a solution. As students learn to work with partners, they find it much easier to persevere in finding a solution.Directions and Strategies
Provide a variety of coins for pairs of students to work with. Using the think-pair-share strategy, ask students to review combinations of coins that add up to $.50. Some students may want to use a 0–100 number chart to display the combinations. (Reason abstractly and quantitatively: Reason with models or pictorial representations to solve the problem.) Have students work with a partner to talk about a process to use with the problem below. (Make sense of problems and persevere in solving them.) Ask students to share their thinking, demonstrate their solutions, and critique the reasoning of others. (Construct viable arguments and critique reasoning of others.)The Problem[Page 127]Solution and Discussion
One strategy students may consider is to construct a table using a guess-and-check model. With this model, record each guess, and determine how close the guess is. A good beginning may be to think of 58¢ as 50 + 8. Chart all guesses to determine the correct solution. It is important to remember that the total of all of the coins is 9. Students can adjust the guesses as they get closer to a solution.CCSS Problem 4Grade Level: 3
In problem 3, students are challenged to find the solution to a realistic situation. The students must then take their understanding and translate it into symbolic form. In either form, students need to explain their reasoning and thinking.[Page 128]Directions and Strategies
Talk about allowances and savings to purchase something students want to buy. Share examples with each other. Help students organize thinking with a T-chart (input/output chart). (Reason abstractly and quantitatively: Reason with models or pictorial representations to solve problems.) For example, a new CD costs $15.00. A student receives a $3.00 allowance weekly. Week 2 allowance added to week 1 is $6.00, and so on. The T-chart can be filled in as follows:
Weeks $ 1 $3.00 2 $6.00 3 $9.00 4 $12.00 5 $15.00
Have students work with partners to explain their thought processes in solving a problem and representing it in several ways. (Make sense of problems: Explain their thought processes in solving a problem and representing it in several ways.) Ask the partners to use appropriate vocabulary and explain to each other why their solutions are correct. Then, allow students to share their thinking, demonstrate their solutions, and critique the reasoning of other children in the classroom. (Construct viable arguments and critique reasoning of others.)The ProblemSolution and Discussion
A T-chart (input/output) is easy to complete. Depending on the amount of money needed, an equation may be an easy way to determine [Page 129]the solution to the problem, for example, N × $5.00 = $45.00 or $45 ÷ N = $5.00. If the amount needed is $95.00, a T-chart may take time to complete; however, writing an equation with a symbol for the unknown number to represent the problem is a quick way to find a solution.
Weeks $ 1 $5.00 2 $10.00 3 $15.00 4 ? 5 ? 6 $30.00 7 ? 8 $40.00 9 ?CCSS Problem 5Grade Level: 3
In problem 5, students are working independently to find solutions to the problems. Conversations and thinking abound when students are then provided the opportunity to explain how they answered the various problems.Directions and Strategies
Review and discuss procedures for adding two 2-digit numbers. Distribute a problem sheet similar to the one shown. If possible, distribute number tiles with the digits 3, 4, 5, and 6. Have students read the directions.The Problem
“Use your number tiles or paper and pencil to complete the number sentences.”
“Record your answers.”
“Be prepared to tell me how you solved the problem, not just that the two numbers add up to 108 or they add up to 90. I want you to explain why your answer is correct.”
(Use Showing Thinking in Classrooms.)Possible Solutions and Discussion
45 + 63 = 108 64 + 35 = 99 36 + 54 = 90 45 + 36 = 81 34 + 65 = 99 54 – 36 = 18 65 – 34 = 31 63 – 45 = 18
[Page 131]If students are using number tiles, observe how students place the tiles on the sheet. Focusing on the 1s digit in the sum or difference aids in finding the necessary numbers. To obtain a three-digit sum, a 4 and 6 or 5 and 6 must be in the 10s place. The only way to have 8 as a 1s digit is with 3 and 5. This information leads to 45 + 63. The only way to have 0 as a 1s digit is with 6 and 4. Regrouping must take place. This information leads to 36 + 54. The only way to have 1 as a 1s digit in addition is with 5 and 6. This leads to 45 + 36. There are two ways to get 9 as a 1s digit, 3 and 6 or 4 and 5. This leads to 64 + 35 or 65 + 34. For subtraction, there are three ways to get 1 as a 1s digit 6 – 5, 5 – 4, and 4 – 3. However, only 6 and 3 provide a 10s digit of 3. There are two ways, using regrouping, to get an 8 as a 1s digit, 13 – 5 and 14 – 6. Both cases work in getting a difference of 18.
In moving about the classroom, observe students writing digits in the squares if a pencil is used or placing tiles in the squares. Do they start on the first number sentence because it has a three-digit sum? Did they start on the second number sentence because both digits are in the sum?
Call on students to discuss their thinking. In the first problem, did students consider which digits would produce a three-digit sum? In the second number sentence, did students have different solutions?
To obtain a difference of 18, students need to consider patterns from the basic subtraction table: 13 – 5 = 8 and 14 – 6 = 8.
After discussing student solutions, pose a similar problem using four different consecutive digits such as 5, 6, 7, and 8. Ask students to find the possible sums and differences using the four digits.CCSS Problem 6Grade Level: 4
In problem 6, students need to carefully read the problem for necessary information. They must then decide how to approach the problem so a reasonable solution can be found and they can explain their thinking. Fractions cause students much confusion, especially when the fractions are not in any context for understanding. Once again, teachers are offered the opportunity to assess student knowledge and understanding as they observe, listen, and talk to students.Directions and Strategies
Allow students to talk to each other about a process they can use to solve the problem below. (Make sense of problems and persevere in solving them.) Remind students that organizing thinking helps problem solvers visualize solutions. (Reason abstractly and quantitatively: Reason with models or pictorial representations to solve problems.) Charting or making a table is one way to organize the data given in the problem. Tell students to be prepared to demonstrate their thinking and critique the reasoning of other students. (Critique the reasoning of others: Explain other students' solutions and identify strengths and weaknesses of the solutions and construct viable arguments: Explain their own thinking and thinking of others with accurate vocabulary.)[Page 133]The ProblemSolution and Discussion
Start by using a chart to record how far each person runs in 1 hour. Then, work backward to determine the time spent in hours by dividing each time in half. Using the chart, compute the time for hours.
Time Me My Mom hour 2 3 1 hour 4 6 hours 6 9
In hour, I will cover 6 miles. I need a 3-mile head start to finish with my mom.[Page 134]CCSS Problem 7Grade Level: 5
In problem 7, students are again working with fractions within a context. They must organize what is given in the problem and then organize what is to be found. Students must come to agreement concerning how the information is to be displayed as well as an approach to solving the problem.Directions and Strategies
Provide paper and pencils or manipulative models for students to use in solving the problem.
To solve the problem below, students will need to start with the answer of 2 cookies and should be encouraged to draw pictures as they work backward to find the original amount of cookies. (Reason abstractly and quantitatively: Reason with models or pictorial representations to solve problems.)
Have students partner to read the problem and then decide on a strategy and approach to use in solving the problem. (Make sense of problems and persevere in solving them.) Students will not only share the steps they used but will also explain their reasoning and thinking processes to each other and to the class using pictorial representations. (Reason abstractly and quantitatively: Reason with models or representations to solve problems.) Students should be prepared to justify and explain why their solutions are correct and also critique the reasoning of other students. (Attend to precision: Incorporate appropriate vocabulary and symbols in communicating their reasoning and solution to others.)[Page 135]The ProblemSolution and Discussion
Because you know the amount remaining is 2 cookies, how can you determine how many cookies each teacher and principal ate? Try drawing a model to illustrate the problem. Use the appropriate vocabulary.
In the final drawing, 2 cookies are left.
Students should be able to explain that the music teacher ate 2 cookies because the same amount is left.
Our principal ate 4 times as much as the music teacher. This gives a total of 12 cookies, which is of the total cookies. If 12 cookies are of the total, then 4 cookies are of the total.[Page 136]
The solution is 16 cookies to start with.CCSS Problem 8Grade Level: 7
In problem 8, students are working in small groups to solve a challenging problem. Students will need to carefully engage in the mathematics and try several approaches before unlocking all of the possibilities. Through questions and prompts, this problem allows teachers an opportunity to push students' thinking to the advanced degree of proficiency in the practices.
These sample problems highlight instructional strategies and the degrees of proficiency. The intent of the samples is to demonstrate to teachers and leaders that change can be easily initiated by following the sequence of strategy implementation. As students learn to collaborate and gain confidence in sharing aloud their thinking, teachers find it easier to provide students with more challenging problems. This steady cycle of improvement has students attaining the mathematical practices.[Page 137]Directions and Strategies
Have students work in groups of three. Present the problem to students, and provide time for them to read it.
- “In your group, agree how to solve the problem.”
- “Be prepared to demonstrate how you solved the problem, and be prepared to explain your thinking and the mathematics that follows that thinking.”
(Use Grouping and Engaging Problems, Allowing Students to Struggle, and Encouraging Reasoning.)The ProblemSolutions and Discussion
There is one solution: 360 jellybeans
While moving around the classroom, listen to students discuss their strategies for solving the problem. Initial attempts might involve just adding 1, 12, and 8 to the appropriate numbers and not considering the direction of the differences.
348 + 1 = 349 359 + 8 = 367 368 + 12 = 380 348 + 8 = 356 359 + 12 = 371 368 + 1 = 369 348 + 8 = 356 359 + 1 = 360 368 + 12 = 380 348 + 12 = 360 359 + 8 = 367 368 + 1 = 369 348 + 12 = 360 359 + 1 = 360 368 + 8 = 376
None of these attempts provide the needed result: All three answers must be the same.
The information in the statement of the problems does not indicate whether the guesses were higher or lower than the jellybean total. In other words, the numbers 1, 12, and 8 represent absolute values; they may be positive or negative values. Questions might need to be posed at this point [Page 138]in the problem-solving process for students to understand that different cases must be considered using the three differences. Adding or subtracting the three numbers 1, 12, and 8 must produce the same number of jellybeans.
Some students, however, might be able to determine the necessary condition by focusing on the last result in the list above. The first two sums are the same (360). By subtracting 8 from 368, rather than adding 8, the difference is also 360. The guesses for the first two students were too low, and the guess for the third student was too high.
Other students might begin listing various arrangements of 1, 12, and 8, considering positive and negative possibilities:
- All 3 positive: 1, 12, 8
- All 3 negative: –1, –12, –8
- 2 negative: –1, –12, 8; –1, 12, –8; 1, –12, –8
- 1 negative: –1, 12, 8; 1, –12, 8; 1, 12, –8
Only 1, 12, and –8 produce the needed result.
When students have finished solving the problem, have various groups explain how they obtained their solutions. Discuss the idea of absolute value in reference to the positive and negative differences.[Page 139]CCSS Problem 9Grade Level: High School
In problem 9, students are working throughout the eight Standards for Mathematical Practice as well as most of the degrees of proficiency. This problem also provides the classroom teacher the opportunity to effectively utilize all of the recommended strategies from the Sequence Chart. This type of problem provides students and teachers rich opportunities for discussions, reasoning, and justifying. Students are able to compare various solution strategies and approaches. Finally, students learn to persist in finding a solution by learning from prior solution attempts that provided useful information but fell short of finding a solution.Directions and Strategies
This problem has multiple entry points and involves both algebra and geometry. For algebra, rates of change are important in establishing relationships. If a generalized solution is sought as an extension to the problem, then equations with three variables (angle measure, minutes, hours) are involved. Multiple answers appear. Geometrically, students are dealing with angle measure, a topic that first appears in the CCSS at Grade 4 and then again at Grade 7 in a different context.
Have students work in pairs or groups of three. Present the problem to students, and provide time for them to read it.The Problem
[Page 140]Tell students:
- “With your partner (s), establish a plan to solve this problem for an angle of 65°.”
- “Be prepared to explain your thinking and the mathematics that follows.”
- “If you and your partner(s) find an answer, try a different angle measure.”
(Use Grouping and Engaging Problems, Allowing Students to Struggle, and Encouraging Reasoning.)
Teachers monitor the small groups as they work. They are also careful to use wait time. Teachers also allow students to share their solutions, their thinking, and their problem-solving approaches.Solution and Discussion
There are two solutions: 4:10 and 7:50.
Many students will immediately think about special angle measures on a clock (3:00—90°, 9:00—90°, 6:00—180°). As an entry point, even this information is important because it suggests that there might be more than one time. They will also draw clock faces, attempting to model a 65° angle. This activity in itself will often lead to their reasoning that there is more than one time. (Model with mathematics: Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem.)
Another entry point for students is to consider the number of degrees between each of the numbers 1 through 12 on a clock. With 360° in a circle and 12 numbers on the clock, then the angle between any consecutive numbers is 360/12 = 30°.
With 30° as the angle measure and 5 minutes between each number, then the angle measure between two consecutive minutes is 30/5 = 6°. (Reason abstractly and quantitatively: Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations.)
At this point, many students will move away from the original task and attempt to determine angles for particular times. For example, consider the angle formed at 3:10.
The hour hand has moved toward 4. What is the angle between 3 and the new location at 3:10? It has moved (10/60)(30) = 5°. The minute hand is on 2. Therefore, the angle formed is 30° + 5° = 35°.
In fact, whenever the time is 10 minutes after the hour, the hour hand has moved 5°. Similarly, when the time is 15 minutes after the hour, the [Page 141]hour hand has moved (15/60)(30) = 7.5°. (Construct viable arguments, and critique the reasoning of others; Compare and contrast various solution strategies, and explain reasoning of others.)
Although this attempt is fruitful for particular times and requires significant reasoning on the part of students, it does not provide an immediate solution to the original problem.
Students may now reason that there are rates of change important in determining the angles. Consider the rate of change of the angle in degrees per minute. The hour hand moves 360° in 12 hours or 720 minutes. Therefore, it changes at a rate of 360/720 = 0.5° per minute. The minute hand rotates through 360° in 60 minutes. Therefore, it changes at a rate of 360/60 = 6° per minute.
If a clock is on the hour, then the angle measure is a multiple of 30°. For example, at 2:00, the angle is 2(30) = 60°; at 5:00, 5(30) = 150°. If the clock is not on the hour, then the hour hand moves in multiples of 30° plus a part of 30°. That part is determined by the product of the portion of the hour and 30°. For example, at 2:15, the angle formed by the hour hand and 12:00 is 2(30) + 15/60(30) = 60 + 7.5 = 67.5. This generalizes to 30H + (M/60)(30) = 30H + M/2 = (60H + M)/2.
The rate of change of the minute hand is 6° per minute. So, the angle measured clockwise from 12:00 is 6M, where M is the number of minutes.
The angle ß between the two hands can be found using the formula:
where H is the number of hours and M is the number of minutes.
If ß = 65°, then 65° = |(½)(60H – 11M)|, or 130 = |60H – 11M|.
So, 130 = 60H – 11M or – 130 = 60H – 11M.
If H = 4 and M = 10, then ß = 65°. Also, if H = 7 and M = 50, then ß = 65°.