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Preparing for Problem Solving and Revisiting Freckleham

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Frecklehammer 1Although Teaching Children Mathematics is building a great collection of rich tasks on this blog, we thought it might be a good time to step back and examine one way to support students as they tackle the tasks. One way to do so is with the Preparing for Problem Solving Interview.  

In our previous post, we discussed the Preparing for Problem Solving Interview as a way to attend to students’ mathematical thinking. In this follow-up blog, we share some of our take-aways from conducting the Preparing for Problem Solving Interview. We then showcase solution strategies to the People of Freckleham task and discuss how students’ interview answers can be used to support implementation of the task.

We observed patterns in our fourth-grade students’ responses that honored procedures, tips, or tricks versus those with responses that honored processes and problem-solving ideas. For example, Maya said that she “looks for words that tell me what to do. Like when it says  ‘altogether,’ that means add.” Her use of word clues is of limited value. Maya would likely benefit from opportunities to solve problems that contain no such clues or from problems that include the phrase altogether but do not involve additive reasoning.

In response to being asked for ways that Cameron would try to help another student solve a problem, he said he would rewrite the problem using a context familiar to the student, using his or her name. Cameron’s ability to re-contextualize problem-solving situations is an important problem-solving practice that he could share with others in the class. Below is a summary of some of the responses we received and the kinds of actions that could be taken.

Student       Reponse  Action to be taken 
Maya Said she uses certain words to tell her which operation to use: “I look for words that tell me what to do. Like when its says, ‘altogether,’ that means to add.” Purposely composing and presenting problems where the phrase altogether does not involve additive reasoning will show Maya that clue words do not always result in the same operation being used. Presenting problems where word clues are absent will force Maya to chose an operation on the basis of what makes sense in the context.
Ella

Cameron

Sophia
Talked about finding a context to fit the problem. Ella said, “Some problems are like the Windows and Tower problems. You have to find a pattern.”  Cameron referred to a measurement problem for which he created a context that was familiar to him.  When given problems that are not in context, it helps students to think of a context in which the numbers and operations make sense. This is a strategy that others could benefit from. Building students’ fluency with creating problem contexts can be accomplished through problem-posing activities.
Beau Said he could check his answer with a neighbor to see if it was right. Asked if there was anything else he could do, he said no.  Beau would benefit from using multiple representations in problem situations, alongside opportunities to see relationships among operations (e.g., inverse operations) that would allow him opportunities to use various representations and operations to ensure accuracy.
Jamiel Focused his responses on the end product (answer). Asked if there was anything he could do to see if he was on the right track while working through the problem, he said no.  The idea that mathematicians are checking their progress throughout the problem-solving experience is a new idea to many. Asking questions throughout problem solving to look for comprehension supports students as they self-monitor their work. This is similar to the strategy of monitoring for understanding while reading. 
Danielle When asked about strategies, said, “Counting on fingers,” but she was reluctant to share beyond that. Honoring all strategies when having whole-group conversations is going to be important for Danielle to feel as though her strategies have merit—even those strategies that we hope she will move away from. Moving away from finger counting necessitates access to more efficient methods, likely methods that her peers are using. Having open discussions of those strategies and making connections between them is a critical part of our follow-up discussion in problem solving. 

Frecklehammer 2Implementing the People of Freckleham Task  

The Freckleham task from our previous blog entry provides an interesting demonstration of students’ problem-solving strategies. First, the task has no word clues that students like Maya can rely on. By drawing all the Frecklehammers first, students like Beau are scaffolded into representing the problem situation in productive ways. One student’s representation of the Frecklehammers looked like this: 

Student Work 1 from blog post 11/24/14Whether students draw Frecklehammers, use ordered pairs, or create some other representation, a systematic structure to accounting for all Frecklehammers is critical. We see this structure in both students’ solutions above. We often ask the question, “How do you know you have listed all the possible Frecklehammers?” Students who systemically list all possibilities can provide sound responses. In reference to the problem-solving interviews, Jamiel benefits from considering this intermediate step in problem-solving situations. Additionally, Beau benefits from the opportunity to see other students represent Frecklehammers in multiple ways.

Another useful representation, particularly when considering how many times the greeting was said at the town meeting, involves the use of a table. Presenting the Frecklehammers in this manner allows for an opportunity to account for the number of statements, 54 in all. 

Freckles      1 2 1 2 2 2 3 3 3
Hairs 1 2 3 1 2 3 1 2 3
"I have more freckles" 0 0 0 3 3 3 6 6 6
"I have more hairs" 0 3 6 0 3 6 0 3 6
Student Work 2 from blog post 11/24/14

Similarly, one student drew upon the ordered pairs representation to arrive at the same solution: 

 Students like Danielle may be tempted to account for the number of statements said by counting one by one. Encourage these students to look for   patterns in solutions. For instance, after students determine the number of statements said for, “I have more freckles than you,” they could be encouraged to consider how the solution to freckles could be used to determine the solution to hairs.

Students similar to Ella, Cameron, and Sophia draw on contexts to make sense of problem-solving situations. As such, they may benefit from acting out the problem or personalizing it in ways that allow for even greater identification. Creating space in classroom discussions for students to consider alternate contexts, strategies, and representations is essential in furthering students’ abilities to see themselves as mathematicians capable of solving the sorts of challenging mathematics tasks that deepen and extend mathematical proficiency. The use of the interview protocol allowed us to better design tasks and scaffold students’ work. We hope that this protocol provides others with insights into students’ views of problem solving and ultimately informs the implementation of problem-solving tasks.

Your Turn 

We want to hear from you. If you are an NCTM member, log in and post your comments. Alternatively, anyone may share his or her thoughts on Twitter @TCM_at_NCTM using #TCMtalk.

 


Thomas E. Hodges
 
Thomas E. Hodges is an assistant professor of mathematics education at the University of South Carolina. He teaches field-based mathematics methods courses, capitalizing on opportunities for preservice teachers, teacher educators, classroom teachers, and elementary students to learn with and from one another. He published on the field-based design in NCTM’s 2014 Annual Perspectives in Mathematics Education and regularly contributes manuscripts to Teaching Children MathematicsMathematics Teaching in the Middle School, and Mathematics Teacher.  
 Malisa Johnson  Malisa Johnson teaches a self-contained fourth-grade class at Oak Pointe Elementary School in Irmo, South Carolina. In her thirteenth year of teaching, Johnson often hosts mathematics methods courses in her classroom and collaborates with university faculty and other classroom teachers on mathematics education publications. She is interested in productive discourse and students’ use of representations in mathematics classrooms. 
 Krysten Fandrich  Kyrsten Fandrich is a Master of Arts in Teaching candidate at the University of South Carolina, completing her internship experience in Johnson’s classroom. She is interested in learning alongside her fourth graders through careful attention to students’ mathematical thinking. 

 

Preparing for Problem Solving

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Highly effective teachers know their students well. They can attend to students’ mathematical thinking, and then design instruction that capitalizes on what students know and are able to do. Our collaborative efforts often center on contextualized tasks, or story problems, where students have opportunities to think about and reason through problem-solving situations. Yet we know first hand how challenging this work can be.

The contexts and design of problem-solving tasks isn’t taken lightly. We’ve found there is much more than the mathematics content to attend to. As such, we designed the Preparing for Problem-Solving Interview, which has helped us unpack how students approach problem-solving situations. The protocol mirrors the Burke Reading Inventory, which provides insight into students’ beliefs about the reading process.

Try asking the following questions of your students, one-on-one, and see what they have to say about problem-solving. If you cannot ask all your students, try asking the questions of a few students you are particularly curious about.

  1. Tell me the name of a student in our class or another class that you think is a good mathematician.
  2. What do you think makes her or him a good mathematician?
  3. Do you think he or she ever struggles to solve a story problem? If so, what do you think he or she would do?
  4. When you are solving a story problem, what are some different ways you can find an answer?
  5. If you tried a way that didn’t work, what might you do next?
  6. If you get an answer that you think might be incorrect, what would you do next?
  7. If you were helping another student solve a problem, what might you do to help?
  8. Do you think you are a good mathematician? Why do you think that?

Students’ responses provide insight into their engagement with the Standards for Mathematical Practice (SMPs) (CCSSI 2010). In particular, how students make sense of problem situations (SMP 1) and their willingness to draw upon a variety of representations (SMP 2 and 4) and tools (SMP 5) to solve problems are often visible when asking these questions. Furthermore, we are able to observe how students position themselves academically in relation to others and how they come to value the knowledge that others bring to classroom learning experiences.

When asked about strategies, students often talk about using calculators, counting on fingers, asking other group members, using manipulatives, and decomposing large numbers. Some students talk about contextualizing problem situations, creating their own stories to fit problem scenarios.

What do your students say? Post responses from your students in the comments section. In our next post, we’ll provide more details on our own students’ responses and how we have designed problem-solving experiences out of these responses to deepen and enrich students’ mathematical proficiency.

A Task to Consider 

As a follow-up to the interview, trying having students solve this derivation of the classic People of Freckleham problem (Treffers and Vonk 1987). Think about what students said and did during the interview to support their work and discussion of the task.

Frecklehammer 

The people of Freckleham are interesting creatures. Every Frecklehammer is different from the other and has at least one freckle and one hair but no more than three freckles and three hairs.

Make a list of all of the different Frecklehammers.

The mayor of Freckleham decided to improve the manners of his townsfolk. He issued an order:

When two Frecklehammers meet, the one with the most hairs or freckles will greet the other and say, “I have more __________ than you have.” A Frecklehammer might say, “I have more freckles than you have,” or a Frecklehammer might say, “I have more hairs than you have.” Or a Frecklehammer might not be able to say anything at all.

At a town meeting of all of the Frecklehammers, the greeting “I have more _________ than you have” was heard many times. How many times?

Your Turn 

We want to hear from you. Try the Problem-Solving Interview and/or the Freckelham task; then come back to this blog and post your comments. You may also share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk.

Citation 

Treffers, Adrian, and Vonk, H. 1987. Three Dimensions: A Model of Goal and Theory Description in Mathematics Instruction—The Wiskobas Project. Dordrecht: Reidel.

 


Thomas E. Hodges
Thomas E. Hodges is an assistant professor of mathematics education at the University of South Carolina. He teaches field-based mathematics methods courses, capitalizing on opportunities for preservice teachers, teacher educators, classroom teachers, and elementary students to learn with and from one another. He published on the field-based design in NCTM’s 2014 Annual Perspectives in Mathematics Education and regularly contributes manuscripts to Teaching Children MathematicsMathematics Teaching in the Middle School, and Mathematics Teacher.  
 Malisa Johnson Malisa Johnson teaches a self-contained fourth-grade class at Oak Pointe Elementary School in Irmo, South Carolina. In her thirteenth year of teaching, Johnson often hosts mathematics methods courses in her classroom and collaborates with university faculty and other classroom teachers on mathematics education publications. She is interested in productive discourse and students’ use of representations in mathematics classrooms. 
 Krysten Fandrich Kyrsten Fandrich is a Master of Arts in Teaching candidate at the University of South Carolina, completing her internship experience in Johnson’s classroom. She is interested in learning alongside her fourth graders through careful attention to students’ mathematical thinking. 

 

Addition and Subtraction Fluency through Games

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In the November 2014 issue of Teaching Children Mathematics, authors Jennifer Bay-Williams and Gina Kling shared a collection of fun games that can be used to develop students’ fluency with addition and subtraction facts. They explained that (1) games should be selected on the basis of a developmental progression from counting to mastery and (2) teachers must emphasize reasoning strategies as students play:

Games support strategy development only when the use of reasoning strategies is explicitly built into the games and reinforced through student-teacher and student-student interactions. (p. 246)

Basic fact fluency can be encouraged through conversations both during the game as students share how they determined their answers, as well as in pregame or postgame discussions. For example, at the end of the game time, you might pull the class back together, pose a challenging fact, such as 8 + 7, and invite children to describe strategies they used during the game to solve this fact. Discussing strategies puts the “meaning” into meaningful practice.

Number Cube      Dice 

Example games from “Enriching Addition and Subtraction Fact Mastery throughGames 

Game: Roll and Total  

Goal of Game: To encourage children to move from counting all to counting on

Game in Brief: Students have one numeral die and one dot die. They find the sum by starting with the numeral die and adding on the quantity from the dot die.

Image for game Salute 

Game: Salute!

Goal of Game: To encourage children to practice reasoning strategies and see relationships between addition and subtraction

Game in Brief: In small groups of three, two students lift a numeral card to their forehead without looking at it. The student without a card says the sum. On the basis of what the players see on the other person’s forehead, the pair with cards figures out what is on their own forehead.

See the article for more games, ideas for differentiating, and suggestions for involving families. Also consider how teachers can assess student fluency while students play games (rather than using timed tests). See “Assessing Basic Facts Fluency” in the April 2014 issue of Teaching Children Mathematics for assessment tools to use as students play games. 

Let’s Talk 

This space is dedicated to continuing this conversation. Please share your ideas with respect to these questions:

 

  • What strategy-focused games do you and your students like? What do students like about them?
  • What questions do you typically ask students as they play facts games?
  • How might we help families focus on reasoning strategies and games as they help their children master basic facts?
  • How might we make strategy-focused games a primary way we develop, practice, and assess mastery of basic facts?

 

We want to hear from you! Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk.

 


 

Jennifer M. Bay-Williams, j.baywilliams@louisville.edu, University of Louisville, teaches courses and offers professional development on effective mathematics teaching for all students. Gina Kling, gina.garza-kling@wmich.edu, teaches elementary mathematics education courses at Western Michigan University and is a curriculum developer for Everyday Mathematics.  

More on the Story of Gauss

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Welcome back! I hope you and your students had the opportunity to explore finding the sum of a series of consecutive numbers. This problem can easily be adapted for any grade level and can offer opportunities for good classroom discourse. At the first-grade level, students in one classroom were asked to find the sum of the first five numbers: 1 + 2 + 3 + 4 + 5.

One first grader rearranged the order of the numbers and added to find a sum of 15:

(1 + 5) + (2 + 4) + 3

Another student added the numbers in the order they were given:

    (1 + 2) =         3

    (3 + 3) =         6

    (6 + 4) =       10

  (10 + 5) =       15

The teacher asked the class to compare the two different approaches that their classmates had used. The discussion provided a great opportunity for students to explore the associative (grouping) property. The teacher demonstrated the problem by showing the class one red Unifix® cube, two blue Unifix cubes, three green Unifix cubes, four yellow Unifix cubes, and five white Unifix cubes. She asked students to think about how they could put the cubes together to find the total sum of cubes. As students shared such strategies as putting the red and white cubes together, grouping the blue and yellow cubes, and then adding the green cube, their teacher helped them make the visual connection to the two different student strategies.

At the third-grade level, some students used a hundred chart to find the sum of the first fifty numbers. First, they found the sum of each row:  

  Hundred Chart    
      1    + 2 + …    + 9 + 10  =   55,

     11 + 12 + …  + 19 + 20  = 155,

     21 + 22 + …  + 19  + 30  = 255,

     31 + 32 + …  + 19 + 40 = 355,

     41 + 42 + …  + 49  + 50 = 455.

Then, they added 55 + 155 + 255 + 355 + 455 = 1275. Students noticed the pattern for the sum of each row increasing by 100 (a fine example of the eighth of the Common Core State Standards for Mathematical Practice, SMP 8: Look for and express regularity in repeated reasoning). When students were asked to justify why the sum of each row increased by 100, several students stated that the numbers in each row had a value that was ten more than the number in the row above it. Thus, 10 × 10 = 100. This showed the teacher that they were connecting their knowledge of place value to the hundred chart.

When the teacher asked students to find the sum of the first 100 numbers, some of them used a calculator and added (500 + 600 + 700 + 800 + 900) + (55 × 5) = 3775 to the 1275 (the sum of the first fifty numbers) for a total sum of 5050. Other students added 555 + 655 + 755 + 855 + 955 = 3775 to the sum of the first fifty numbers (1275) for a total of 5050. The use of the hundred chart to look for patterns with the sums of each row helped students reason quantitatively and make mathematical connections with patterns and groupings of numbers.

In both examples, we see how students from different grade levels used their knowledge of mathematical properties to find the sum of a series of the first n consecutive integers. Problems and tasks such as these give students opportunities to use their knowledge of basic operations, the commutative and associative properties, and different strategies to solve problems as well as to share and defend their approaches. They are engaged in such Standards for Mathematical Practices as SMP 1: Make sense of problems and persevere; SMP 2: Reason abstractly and quantitatively; SMP 3: Construct viable arguments and critique the reasoning of others; SMP 4: Modeling with mathematics; SMP 5: Use appropriate tools strategically; SMP 7: Look for and make use of structure; and SMP 8: Look for and express regularity in repeated reasoning.

The historical connection to Carl Friedrich Gauss (1777–1855) offers students an opportunity to see the richness of mathematics over time. Having students solve the summation of the series of integers from one to n before sharing Gauss’s approach may help students appreciate his contribution to mathematics. Students can see how Gauss engaged in SMP 7: Make use of the structure of the problem to find a shortcut to a solution.


Wilburne-Jane-100x140.jpgJane M. Wilburne is an associate professor of mathematics education at Penn State Harrisburg. She teaches content and methods courses for both elementary and secondary mathematics teachers as well as graduate mathematics education courses. She is a co-author of Cowboys Count, Monkeys Measure, and Princesses Problem Solve: Building Early Math Skills Through Storybooks (Brookes Publishing 2011) and has published numerous manuscripts in Teaching Children Mathematics, among other journals. Jane began serving as a member of the Teaching Children Mathematics Editorial Panel in May 2014, and her term will continue through April 2017. 

 

The Story of Gauss

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Carl Friedrich GaussI love the story of Carl Friedrich Gauss—who, as an elementary student in the late 1700s, amazed his teacher with how quickly he found the sum of the integers from 1 to 100 to be 5,050. Gauss recognized he had fifty pairs of numbers when he added the first and last number in the series, the second and second-last number in the series, and so on. For example: (1 + 100), (2 + 99), (3 + 98), . . . , and each pair has a sum of 101.

Fifty Pairs of Numbers 

50 pairs × 101 (the sum of each pair) = 5,050.

Another way to represent the problem could be to list the integers from 1 to 100 and write the same list in reverse order below the first list.

Another representation of fifty pairs of numbers 

This gives us 100 addends of 101 for 10,100. Because the list of numbers from 1 to 100 was doubled, we need to divide the total by 2, giving us a sum of 5,050.

This representation of the way Gauss solved the problem may help students explore the connection to the algebraic generalized form for finding the sum of a series of consecutive numbers: n(+ 1)/2.

Problems that involve the sum of a series of integers can be adapted for different elementary grade levels.

Students in the primary grades can explore ways to find the sum of the first five or ten counting numbers. Challenge older elementary students to find the sum of the first twenty or thirty counting numbers. What properties of operations might they use to help them find the sum? What strategies might they use to find the sum? What manipulatives would be best to use to help students find the sum? How can teachers facilitate students’ learning to help them make connections to Gauss’s approach without directly showing them his strategy? Please share your ideas and strategies as well as samples of how your students have found the sums. 

Let’s change the problem to find the sum of a series of even or odd numbers. What is the sum of the first 20 even numbers? What is the sum of the first 30 odd numbers? What is the sum of the first 100 odd numbers?

I have found that students often misinterpret these problems. For example, some students find the sum of the even numbers up to 20 instead of the sum of the first 20 even numbers. Problems such as these can engage students in the Standards for Mathematical Practice (SMP) (CCSSI 2010), such as “Attend to precision” (SMP  6) and “Make sense of problems (SMP 1).

For Grade 6 students, opportunities to explore other questions exist, such as the following:

  • What is the sum of the integers from –10 to +10?
  • What is the sum of the series of integers from 24 to 78?

What extensions related to finding the sum of an arithmetic series have you used with your students? Have fun sharing The Story of Gauss task with them. We look forward to your input. NCTM members are welcome to log in and add comments. Comments can also be Tweeted to @TCM_at_NCTM using the hashtag #TCMtalk. Comments can also be emailed to TCM@nctm.org.


Wilburne-Jane-100x140.jpgJane M. Wilburne is an associate professor of mathematics education at Penn State Harrisburg. She teaches content and methods courses for both elementary and secondary mathematics teachers as well as graduate mathematics education courses. She is a co-author of Cowboys Count, Monkeys Measure, and Princesses Problem Solve: Building Early Math Skills Through Storybooks (Brookes Publishing 2011) and has published numerous manuscripts in Teaching Children Mathematics, among other journals. Jane began serving as a member of the Teaching Children Mathematics Editorial Panel in May 2014, and her term will continue through April 2017. 

 

What Is the Largest Number You Cannot Make? Part 2

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TCM-Blog-Chicken-150x154.jpgIf you have not had a chance to engage your students in the What Is the Largest Number You Cannot Make? problem, you can find the task here. What interesting patterns did your students find? What strategies did they use? Please post a comment and share with others. We enjoy learning about how these tasks are used in classrooms.

Some students start the problem by making a list of the numbers from 1–20. Then they list the multiples of 4 and multiples of 7 and make combinations of both sets of multiples. As they combine sets, they check off the total number of nuggets from the list of numbers from 1–20. Students soon realize they needed to extend the list to 40. For example, 3 sets of 4 is 12, 4 sets of 7 is 28, so 12 + 28 = 40 nuggets total.

Other students might list the numbers from 1–30 and start with 1, then 2, and so on to determine whether they can buy that number of nuggets. For example, they cannot buy 10 nuggets; but they can buy 11 nuggets (pack 4 + pack 7), and they can buy 12 nuggets (3 sets of 4 nuggets). They cannot buy 13, but they can buy 14 nuggets (two packs of 7 nuggets), and so on. Students often continue with this approach, sometimes arguing why they could or could not buy a particular number of nuggets. For example, Josh could not figure out how to buy 22 nuggets, but Andre explained that if you could buy 11 nuggets, you could buy 22 nuggets by doubling the number of packs to make nuggets (2 packs 4 + 2 packs 7 = 22 nuggets).

Did your students prefer one of these strategies? Which other strategies did your students use?

In a fifth-grade classroom, the teacher asked her students how many successful combinations of nuggets they could buy before they determined they could buy any number of nuggets beyond it. Sharise shared that once she had four numbers in a row, she was able to make any number of nuggets by just adding four to some of the number of nuggets she could buy. She shared that she could buy 18 nuggets (1 pack of 4 nuggets + 2 packs of 7 nuggets), 19 nuggets (3 packs of 4 nuggets + 1 pack of 7 nuggets), 20 nuggets (5 packs of 4 nuggets) and 21 nuggets (3 packs of 7 nuggets). Therefore, by adding one more pack of 4 nuggets to 18 nuggets, you could buy 22 nuggets. Adding one more pack of 4 nuggets to 19 nuggets would allow you to buy 23 nuggets, and so on. She used repeated reasoning to see that she could buy every number of nuggets beyond 17 using this approach.

Which other mathematical practices were your students engaged in when working on this problem?

One extension to the problem would be to ask, “How many packs of 4 and 7 nuggets would you need to have 98 nuggets?” “How many different ways could you purchase 98 nuggets?”

What other extensions can you suggest?

Students could create their own similar problems. As long as the two numbers (a, b) you select for the packs of nuggets are relatively prime (they have no common factors other than 1), you can find the solution by 

(a * b) – (a + b). Now you can create your own “Greatest Number You Cannot Buy” problems.


Wilburne-Jane-100x140.jpgJane M. Wilburne is an associate professor of mathematics education at Penn State Harrisburg. She teaches content and methods courses for both elementary and secondary mathematics teachers as well as graduate mathematics education courses. She is a co-author of Cowboys Count, Monkeys Measure, and Princesses Problem Solve: Building Early Math Skills Through Storybooks (Brookes Publishing 2011) and has published numerous manuscripts in Teaching Children Mathematics, among other journals. Jane began serving as a member of the Teaching Children Mathematics Editorial Panel in May 2014, and her term will continue through April 2017. 

 

What Is the Largest Number You Cannot Make?

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TCM-Blog-Chicken-150x154.jpgAn interesting problem that I have used with elementary school students, classroom teachers, and preservice teachers involves opportunities to engage in various problem-solving strategies. The most important step in this problem is Understanding the problem. It offers students the chance to reason and think critically about what the problem is asking them to find and the meaning of the problem. The problem can be adapted in context and quantity to meet the needs of students from primary grades to upper middle level grades.

A fast food restaurant sells chicken nuggets in packs of 4 and 7. What is the largest number of nuggets you cannot buy? How do you know this is the largest number you cannot buy? 

The word cannot is the key term in the problem. In many situations where I have presented this problem, students are confused by the meaning of the problem. I usually have to start them off by asking them if they could buy only 1 nugget (no), 2 nuggets (no), 3 nuggets (no), 4 nuggets (yes: 1 pack of 4), 5 or 6 nuggets (no), 7 nuggets (yes: 1 pack of 7), 8 nuggets (yes: 2 packs of 4 nuggets), and so on.

In most cases, students make a list of consecutive numbers and try different addition combinations of 4 and 7 to see if they can make the number of nuggets. Some students use colored chips to represent the packs of nuggets. For example, red chips represent packs of 4 nuggets, blue chips represent packs of 7 nuggets. Two red chips and one blue chip would represent 4 + 4 + 7 = 15 nuggets. Students get lots of practice adding combinations of multiples of 4 and multiples of 7 because addition can be used to solve the problem. Students in primary grades can engage in a similar problem with smaller numbers of nuggets in each package.

Try the problem and see what you get. Then try to create another problem using a different context and different numbers. For example, what is the largest number of pencils you could not purchase if pencils came in packages of 5 and 8? Do you see any patterns with respect to the solution and what types of numbers work best in the context of the problem?


Wilburne-Jane-100x140.jpgJane M. Wilburne is an associate professor of mathematics education at Penn State Harrisburg. She teaches content and methods courses for both elementary and secondary mathematics teachers as well as graduate mathematics education courses. She is a co-author of Cowboys Count, Monkeys Measure, and Princesses Problem Solve: Building Early Math Skills Through Storybooks (Brookes Publishing 2011) and has published numerous manuscripts in Teaching Children Mathematics, among other journals. Jane began serving as a member of the Teaching Children Mathematics Editorial Panel in May 2014, and her term will continue through April 2017. 

 

Frogs and Worms, a Second Look

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How did your students do with the Frog problem and the Worm problem? When I have used these problems in the past, typically students have quickly decontextualized them, representing the problems in some way and finding a solution. Below are some common responses. Both these solution processes are straight­forward and mathematically correct. In fact, the students providing these solutions have done a nice job of decontextualizing the problem.

FrogThe Frog problem

  • 5 meters is 500 cm. The total race is 1,000 cm.

  • Frog 1 jumps 80 cm every 5 seconds.

1000 cm ÷ 80 cm per jump = 12.5 jumps 

12.5 jumps ´ 5 seconds per jump = 62.5 seconds 

  • Frog 2 jumps 15 cm every second.

1000 cm ÷ 15 cm per jump = 66.67 jumps

Each jump takes one second, so 66.67 seconds 

  • Therefore, Frog 1 wins.

WormThe Worm problem

  • Each day, the worm has a net gain of 1 foot.

  • If he gains 1 foot per day, he will take 12 days to get to the top of the 12-foot wall.

A Closer Look 

I will usually have one or two students who are quick to say, “Wait a minute!” These wait-a-minute students have noticed something that other students have not, and what they have noticed resulted from them having contextualized the problem. Let’s take a closer look at the Frog problem.

Do the frogs really travel 1000 cm to complete the race? The wait-a-minute students say no. They argue that if that were the case, the frogs would have to stop in mid-air at the 500 cm mark and reverse their path—an impossibility. Instead, the frogs have to complete their jump over the 500 cm and then turn around to go back to the starting line. As a result, how far does each frog actually travel? Does this change which frog wins the race?

Similarly, consider the Worm problem. The wait-a-minute students argue that at some point, the worm makes it to the top of the wall and does not continue sliding up and down.

Wait a minute! Keeping your eye on the problem, or contextualizing, seems to be important.

I hope that these two problems gave you and your students an interesting way of thinking about the importance of decontextualizing and contextualizing. In the Comments section below, please share how your students handled the problem.
 


 Angela Barlow, Middle Tennessee State UniversityAngela T. Barlow is a Professor of Mathematics Education and Director of the Mathematics and Science Education Ph.D. program at Middle Tennessee State University. During the past fifteen years, she has taught content and methods courses for both elementary and secondary mathematics teachers. She has published numerous manuscripts in Teaching Children Mathematics, among other journals, and currently serves as the editor for the NCSM Journal of Mathematics Education Leadership

Frogs and Worms

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With school starting, many of us are focusing on the need to support students’ engagement in the Standards for Mathematical Practice (SMP). Regardless of whether your state has adopted the Common Core State Standards, the SMP represent processes and proficiencies that we all want to develop in our students. Within these standards, decontextualize and contextualize represent two unfamiliar terms for many of us. Here, I offer two problems to help you and your students think about the processes embodied in these terms.

First, the Frog Race problem: 

FrogTwo frogs have a race. One frog makes a jump of 80 centimeters once every five seconds. The other frog makes a jump of 15 centimeters every second. The rules of the race require that the frogs must cross a line 5 meters from the start line and then return to the start line to complete the race. Which frog wins the race?  (NCTM 1994) 

This problem is appropriate for upper elementary school students. For those in the lower grades, consider the Worm problem: 

WormA worm is at the bottom of a 12-foot wall. Every day it crawls up 3 feet, but at night it slips down 2 feet. How many days does it take the worm to get to the top of the wall? (Herr and Johnson 2001)

As students work to solve either of these problems, drawing a diagram may be an appropriate initial strategy. After that, students may move toward using symbols to represent and solve the problem. These symbols will be manipulated without considering the problem. That is, students will be decontextualizing the problem.

The richness of these problems, however, comes from contextualizing—that is, pausing during the process of working with the symbols to look back at how the symbols connect to the original problem. For both the Frog Race problem and the Worm problem, this process of “keeping an eye on” the problem is key to finding the solutions.

I encourage you to solve both of these problems and consider using them with your students. And be sure to decontextualize and contextualize—the results may surprise you.

You are invited to share your thoughts and comments here or via Twitter @TCM_at_NCTM.  I’d also like to see samples of student work. I’ll be back in a couple of weeks with my reflections on the Frog and Worm tasks.

References 

Herr, Ted, and Ken Johnson. 2001. Problem Solving Strategies: Crossing the River with Dogs and Other Mathematical Adventures. 2nd ed. Emeryville, CA: Key Curriculum Press.

National Council of Teachers of Mathematics (NCTM). 1994. “Menu of Problems.” Mathematics Teaching in the Middle School 1 (November-December): 223. http://www.nctm.org/publications/article.aspx?id=37609


 Angela Barlow, Middle Tennessee State UniversityAngela T. Barlow is a Professor of Mathematics Education and Director of the Mathematics and Science Education Ph.D. program at Middle Tennessee State University. During the past fifteen years, she has taught content and methods courses for both elementary and secondary mathematics teachers. She has published numerous manuscripts in Teaching Children Mathematics, among other journals, and currently serves as the editor for the NCSM Journal of Mathematics Education Leadership

Reflecting on the Counterfeit Bill Problem

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Clip art of cartoon billsI hope that you and your students or colleagues enjoyed discussing the Counterfeit Bill problem. I suspect that a variety of solutions were offered, including these:

$40—The shoe owner gave $20 to the grocer and $20 (counterfeit) to the FBI.

$55—The shoe owner gave $15 to the customer, $20 to the grocer, and $20 (counterfeit) to the FBI.

To think about whether these solutions are correct, let’s start by trying the act-it-out strategy.

Although the shoe-store owner is not able to make change for the $20, you can assume that he has some money in his cashbox (just like you can assume that other pairs of shoes are in the store). Let’s say, for the sake of argument, that he has $40 in his cashbox and that he has the slippers on his sales rack. 

Transaction 1
The customer hands the fake $20 to the shoe-store owner. The shoe-store owner hands the fake $20 to the grocer. The grocer hands the shoe-store owner four $5 bills. The shoe-store owner hands the customer $15 and the slippers.
Result of transaction 1: The shoe-store owner has $45 in his cashbox.

Transaction 2
The shoe-store owner gives the grocer $20 in exchange for the fake $20.
Result of transaction 2: The shoe-store owner has $25 in his cashbox and a fake $20.

Transaction 3
The shoe-store owner gives the fake $20 to the FBI.
Result of transaction 3: The shoe-store owner has $25 in his cashbox.

Compare
The shoe-store owner had $40 and a pair of slippers to start with, and then he ended with $25. He lost $15 and a pair of slippers—or $20, if you assume the value of the slippers is $5.

If you don’t believe me, act it out!

Alternatively, you might use the strategy look at the problem from a different view. With this in mind, consider the following argument. By handing the shoe-store owner a counterfeit bill, what did the customer receive free of charge? That’s right, $15 and a pair of shoes. So the shoe-store owner lost what was taken from him: $15 and a pair of shoes.  

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I have used this problem in a variety of settings, and it is always interesting that students expect me to tell them who is right. If you tried this problem in your classroom, I suspect the same thing was true with your students. Not telling them, though, supports students in making sense of the problem, listening to one another, and considering others’ justifications. And by doing so, this problem supports the establishment of your classroom norms. I hope you’ll share your students’ experiences of the Counterfeit Bill problem with us.

Did you or your students use a different strategy? You are welcome to share photos or work samples. We hope to hear from you.


Angela Barlow, Middle Tennessee State UniversityAngela T. Barlow is a Professor of Mathematics Education and Director of the Mathematics and Science Education Ph.D. program at Middle Tennessee State University. During the past fifteen years, she has taught content and methods courses for both elementary and secondary mathematics teachers. She has published numerous manuscripts in Teaching Children Mathematics, among other journals, and currently serves as the editor for the NCSM Journal of Mathematics Education Leadership

The Counterfeit Bill Problem

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Photo of stack of $20 billsI am often asked what the best way is to start the school year. My answer is always, “With a problem, of course!” Not just any problem will do, though, as I want a problem that will spark discussion by eliciting a variety of solutions and/or solution strategies. One problem that I have found to be particularly fun is the Counterfeit Bill problem (Sobel and Maletsky 1999). 

A customer enters a store and purchases a pair of slippers for $5, paying for the purchase with a $20 bill. The merchant, unable to make change, asks the grocer next door to change the bill. The merchant then gives the customer the slippers and $15 change. After the customer leaves, the grocer discovers that the $20 bill is counterfeit and demands that the shoe-store owner make good for it. The shoe-store owner does so, and by law is obligated to turn the counterfeit bill over to the FBI. How much does the shoe-store owner lose in this transaction? 

In the past, I have asked students to work collaboratively in groups to solve this problem and represent their work on poster paper. The mathematics in the problem is limited to addition and subtraction, thus allowing engagement of a wide range of students in terms of both grade level (grades 3 and up) and ability level. The power of the problem lies in its ability to support students in recognizing the need to understand the problem rather than rushing to compute with numbers and to elicit the act-it-out strategy, a strategy often forgotten as students get older. In using this problem, typically three or four different solutions surface, which selected groups can then present for discussion. In doing so, students engage in justifying their solution processes to the class. Of equal importance, however, are critically listening to and critiquing the arguments of others, which are necessary for the class to move forward in agreeing on the solution. By engaging in these processes, students are able to begin establishing classroom norms that will support their mathematical adventure.

Try the Counterfeit Bill problem. Here’s a hint: Two problem-solving strategies that you might find useful are act it out and look at the problem from a different view. Note that for younger students, the problem may be modified to involve a $10 counterfeit bill, and you may want to provide counters or other manipulatives to support student engagement with the problem.

Reference 

Sobel, Max A., and Evan M. Maletsky. 1999. Teaching Mathematics: A Sourcebook of Aids, Activities, and Strategies. 3rd ed. Boston, MA: Allyn and Bacon.

 


Angela Barlow, Middle Tennessee State UniversityAngela T. Barlow is a Professor of Mathematics Education and Director of the Mathematics and Science Education Ph.D. program at Middle Tennessee State University. During the past fifteen years, she has taught content and methods courses for both elementary and secondary mathematics teachers. She has published numerous manuscripts in Teaching Children Mathematics, among other journals, and currently serves as the editor for the NCSM Journal of Mathematics Education Leadership

Answering the Question, “When Is Halving Not Halving?”

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It’s late in the school year, but I hope you had a chance to try out the perimeter and area comparison problem with your students. If not, you might try it early next year.

Recall that students were going to start with a rectangle and then make another one with half the area. The task was to see what fraction of the old perimeter the new perimeter could be.

I tried this problem in a sixth-grade class and had a really interesting experience. The regular classroom teacher had four students who normally did not stay with the others for math; they usually went to a special education classroom. We asked if, for this problem, the students could stay and, in fact, each of them was successful in creating the second rectangle and in realizing that the new perimeter was not only not half the old perimeter but also that it was greater than half.

Some students figured out that if the cutting in half was on a line of symmetry, the new perimeter had to be more than half of, but less than all of the old perimeter. That’s because even if you cut the length (or width) in half, you retain the full width (or length) of the original.

For example, in the two rectangles below, note that the length of the top edge and bottom edge do not change when the shape is cut on a line of symmetry; however, the lengths of the left and right sides (shown in blue on the original figure) are half the length of the new figure (shown in green on the halved figure).

      Figure 1 - comparison of edge lengths 

fig. 1                    Perimeter = 2 full red + 2 full blue       Perimeter = 2 full red + 2 half-blues

 

Some students realized that if you have a long rectangle or a tall and skinny oneand cut it in half by making it even skinnier, you end up keeping most of the perimeter, so the fraction of the old perimeter and that of the new perimeter turn out to be really close to 1.

Compare the change in perimeter on the left versus on the right. It is much less of a change on the left.

Figure 2: Graphic comparing change of perimeter on left vs.right
 

fig. 2 

Keeping the same perimeter or even increasing the perimeter actually is possible, but only if the original shape is not cut on a line of symmetry. For example, a 1 ´ 6 rectangle has half the area of a 3 ´ 4 rectangle but the same perimeter. And a rectangle that is 6 ´ 5 with a perimeter of 22 can be halved in area by using a rectangle that is 1 ´ 15. But the new perimeter is 32, even bigger than the original perimeter.

How did your students respond to the task? 


Marian SmallMarian Small is the former dean of education at the University of New Brunswick, where she taught mathematics and math education courses to elementary and secondary school teachers. She has been involved as an NCTM writer on the Navigations series, has served on the editorial panel of a recent NCTM yearbook, and has served as the NCTM representative on the MathCounts writing team. She has written many professional resources including Good Questions: Great Ways to Differentiate Mathematics Instruction (2012), Eyes on Math (2012), and Uncomplicating Fractions to Meet Common Core Standards in Math, K–7 (2013), all co-published by NCTM.

13 Rules that Expire

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In the August 2014 issue of Teaching Children Mathematics, authors Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty initiated an important conversation in the elementary mathematics education community. We are dedicating this discussion space as a place where that conversation can continue.

In their article, “13 Rules That Expire,” the authors point out thirteen math rules commonly taught in the elementary grades that no longer hold true in later grades; in fact, these rules “expire.” For example—

Rule 1: When you multiply a number by ten, just add a zero to the end of the number.  

This rule is often taught when students are learning to multiply a whole number times ten. However, the rule is not true when multiplying decimals (e.g., 0.25 × 10 = 2.5, not 0.250). Although the statement may reflect a regular pattern that students identify with whole numbers, it is not generalizable to other types of numbers. Expiration date: Grade 5 (5.NBT.2).

See the article for the other rules.

Using the comment section that follows this blog post, submit additional instances of “rules that expire” or expired language that the article does not address. If you would like to share an example, please use the format of the article (as listed below):

  1. State the rule that teachers share with students.
  2. Explain the rule.
  3. Discuss how students inappropriately overgeneralize it.
  4. Provide counterexamples, noting when the rule is untrue.
  5. State the “expiration date” or the point when the rule begins to fall apart for many learners. Give the expiration date in terms of grade levels as well as CCSSM content standards in which the rule no longer “always” works.

If you submit an example of expired language that was not in the article, include “What is stated” and “What should be stated” as shown in the table below (for additional examples, see table 1 in the published article.

Expired mathematical language and suggested alternatives 
What is stated      What should be stated 
Using the words borrowing or carrying when subtracting or adding, respectively Use trading or regrouping to indicate the actual action of trading or exchanging one place-value unit for another unit.
Using the phrase ___ out of ___ to describe a fraction—for example, one out of seven to describe 1/7 Use the fraction and the attribute. For example, say the length of the string. The out of language often causes students to think a part is being subtracted from the whole amount (Philipp, Cabral, and Schappelle 2005).
Using the phrase reducing fractions  Use simplifying fractions. The language of reducing gives students the incorrect impression that the fraction is getting smaller or being reduced in size.

 


 

Karen S. Karp, University of Louisville  Karen S. Karp, karen@louisville.edu, a professor of math education at the University of Louisville in Kentucky, is a past member of the NCTM Board of Directors and a former president of the Association of Mathematics Teacher Educators. Her current scholarly work focuses on teaching math to students with disabilities.  
Sarah B. Bush, Bellarmine University  Sarah B. Bush, sbush@bellarmine.edu, an assistant professor of math education at Bellarmine University in Louisville, Kentucky, is a former middle-grades math teacher who is interested in relevant and engaging middle-grades math activities.  
Barbara J. Dougherty, University of Missouri  Barbara J. Dougherty is the Richard Miller Endowed Chair for Mathematics Education at the University of Missouri. She is a past member of the NCTM Board of Directors and is a co-author of conceptual assessments for progress monitoring in algebra and an iPad® applet (MOTO) for K–grade 2 students to improve counting and computation skills.    

 


 

When Is Halving Not Halving?

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Exploring the relationship (or lack of relationship) between perimeter and area is interesting for students—even for simple shapes like rectangles. For example, if you cut a rectangle’s area in half, do you also cut the perimeter in half?

Using the rectangles shown below, it is easy to see that the figure on the left was cut in half to create the figure on the right. When we measure the area, the rectangle on the left is 16 units and the area of the smaller rectangle is 8 units—exactly half of the original rectangle. However,  it turns out that the perimeter of the figure on the left was not cut in half when the new rectangle was created. In fact, the new perimeter is a full 3/4 of the old perimeter.

Figure 1 - Halving Horizontally

Is the new perimeter always 3/4 of the old one? Let’s try a different rectangle. This time, let’s cut it vertically instead of horizontally.

Figure 2 - Halving Vertically

Once again, the area is halved, but the perimeter changes from 16 units to only 10 units. This time, the ratio of new perimeter: old perimeter, is not 1/2 and also not 3/4. Instead, it is 5/8.

You could provide students with square tiles with which to build rectangles, or they could explore the challenge using geoboards and geobands. Alternatively, students might digitally access squares they can put together to make rectangles using the Patch Tool at http://illuminations.nctm.org/Activity.aspx?id=3577, the Shape Tool at http://illuminations.nctm.org/Activity.aspx?id=3587, or the virtual geoboard available at the National Library of Virtual Manipulatives website.

Encourage students to then explore exactly what fractions of the old perimeter the new perimeter could be if a rectangle’s area is cut in half.

Could it be 2/3?

Could it be 1/3?

Could it be 5/6?

Could it be really close to 1?

Could it be really close to 0?

Is it ever 1/2?

Alternatively, if time is limited, ask students to determine the dimensions of rectangles with specific new perimeter: old perimeter ratios, such as 5/6 or 2/3.

Have your students try the problem and see how it goes for them. If you are already on summer break, you could challenge you own children or neighborhood children to explore the task. I welcome you to share your students’ experiences with us.



Marian SmallMarian Small is the former dean of education at the University of New Brunswick, where she taught mathematics and math education courses to elementary and secondary school teachers. She has been involved as an NCTM writer on the Navigations series, has served on the editorial panel of a recent NCTM yearbook, and has served as the NCTM representative on the MathCounts writing team. She has written many professional resources including Good Questions: Great Ways to Differentiate Mathematics Instruction (2012), Eyes on Math (2012), and Uncomplicating Fractions to Meet Common Core Standards in Math, K–7 (2013), all co-published by NCTM.

Reflecting on the Build a Number Problem

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I hope you have had a chance to try the Build a Number problem with your students. I had lots of fun with it when I tried it with some third- and fourth-grade students.

Recall that students were going to use base-ten blocks to build a number that has—

  •  twice as many ten rods as hundred flats, and
  • one-fourth as many ten rods as unit blocks.

We heard from a couple of you who saw the potential of this task for differentiation. It was a great idea to suggest using the blocks to represent decimals as well.

Someone asked about modifications for kindergarten or grade 1. One possibility, more likely for grade 1, is to propose only one condition; for example, there are twice as many ones as tens.

Most students start with the hundred flats. Once they put out 1 hundred flat, they realize that they need 2 ten rods and then 8 units to go with it, i.e., 128. You might ask students if the problem could be solved by starting with the rods or units first. Of course, it can, but it may take more experimenting. For example, if students start with 1 unit block, that won’t work. They might start with 4 unit blocks and 1 rod. But then they realize that, oops, there wouldn’t be a flat; they have to go back to the beginning to start with 8 unit blocks. It might be interesting for students to realize that sometimes the order in which you solve a problem matters, but not always.

After students get to 128, many just stop there, but they can easily be encouraged to look for more numbers. You might ask them to use 2 flats, and they soon see they would need 2 flats, 4 rods, and 16 units. Some students will wonder whether it is “legal” to have 16 units. Of course, it is, but the number will need to be written as 256, not 2 4 16.

At this point, most students just keep adding 1 flat, 2 rods, and 8 units, realizing that each time, they get a new correct answer. Once a few of these numbers are created, there is usually some excitement when students notice that the numbers are 128 apart; if you keep adding 128, you get more and more answers, namely 128, 256, 384, 512, 640, 768, 896, . . . . A class of older students might notice that these numbers are, in fact, multiples of 128.

It would be worth exploring why no other numbers are possible. In effect, if there have to be 2 rods and 8 units for every flat, any increase in a number has to come as a package of 128.

In one classroom, a student told me that there was actually a number lower than 128—namely 0. He said 0 flats, 0 rods, and 0 units does the trick, and so it does. That was quite insightful. It would be equally interesting to ask if there is a greatest number. There is not, because 128 could continue to be added.

An alternative where there were three times as many blocks as ten rods was also offered in the earlier post. Here the values are all multiples of 126. The problem isn’t really simpler; it’s just the language of saying “three times as many” instead of “one-fourth as many.”

What did you like about this task? What would you have changed?


Marian SmallMarian Small is the former dean of education at the University of New Brunswick, where she taught mathematics and math education courses to elementary and secondary school teachers. She has been involved as an NCTM writer on the Navigations series, has served on the editorial panel of a recent NCTM yearbook, and has served as the NCTM representative on the MathCounts writing team. She has written many professional resources including Good Questions: Great Ways to Differentiate Mathematics Instruction (2012), Eyes on Math (2012), and Uncomplicating Fractions to Meet Common Core Standards in Math, K–7 (2013), all co-published by NCTM.

Build a Number Problem

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In a lot of school districts in my region, there is an emphasis on building proportional reasoning even before it is formally introduced in the curriculum.

A problem I used recently is the one I’ve proposed here. You would provide students with base-ten blocks, including hundred flats, ten rods, and unit blocks; or students could use virtual base-ten blocks.

Build a number [Note: You might limit the value to numbers under 1000] where all of the following are true: 

Children using Base-Ten Blocks 
  • There are twice as many ten rods as hundred flats.
  • There are one-fourth as many ten rods as one blocks.
  • What could the number be?

A simpler variation that might better suit some students is created by changing the second condition:

  • There are three times as many one blocks as ten rods.

Whichever version you use, good things happen. Students think about how to represent three-digit numbers and how to regroup and name them using an equivalent form; for example, they realize that 256 can be represented with 2 flats, 4 rods, and 16 ones—and not just 2 flats, 5 rods, and 6 ones. If the value is limited, for example, to 1000, they start realizing why 8 flats is too many: because 8 flats + 16 rods + 64 ones is too much.

Students will think about what terms like twice as much or one-fourth as much mean. In particular, they will realize that another way to say that one number is one-fourth as much as another is to say that the second number is four times the first. You might lead in to the task with the following questions:

  • You model a number with 6 ones and some tens. What could it be?
  • You model a number with 15 ones and some tens. What could it be?

I encourage you to try one of the variations with your class and tell us how it went. Encourage students to work through the problem with a partner. As you discuss solutions, include questions like these:

  • How many rods did you use before you traded? Why not 3 or 5 or 7?
  • How many ones did you use before you traded? Why were these the only possibilities?
  • Which did you choose first—the number of flats, rods, or units? Did you have to?
  • What was the least number you could have had?
  • What did you notice about the possible solutions?
  • Were there numbers you tried to get and just couldn’t? What were they?

I’ve presented this activity recently to a third-grade class, and students were fully engaged. Let me know how it goes for you. I look forward to hearing your experiences, and we’ll talk about the Build a Number problem in a couple of weeks.


Marian SmallMarian Small is the former dean of education at the University of New Brunswick, where she taught mathematics and math education courses to elementary and secondary school teachers. She has been involved as an NCTM writer on the Navigations series, has served on the editorial panel of a recent NCTM yearbook, and has served as the NCTM representative on the MathCounts writing team. She has written many professional resources including Good Questions: Great Ways to Differentiate Mathematics Instruction (2012), Eyes on Math (2012), and Uncomplicating Fractions to Meet Common Core Standards in Math, K–7 (2013), all co-published by NCTM.

Reflecting on the Pondering Patterns Problem

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Greetings! Over the past few months, it has been great fun sharing some of my favorite “Math Tasks to Talk About” with you and becoming a blogger in the process. The plan for the TCM blog is for a series of guest bloggers to continue adding to this rich collection as they share and discuss their favorite tasks, so I now need to step aside and make room for the next person. I hope you’ve enjoyed the tasks I’ve shared with you, and I’d certainly be delighted if you’d share your thoughts by commenting on the blog!

So, how did you and your students respond to the Pondering Patterns task? Let’s start by looking at the four patterns talked about in the first paragraph and how students might go about finding a specified term in each pattern. The example I used was finding the fifteenth term, so let’s begin with that. The first pattern was the one generated by the Handshake problem: 1, 3, 6, 10, 15, 21, 28, . . . . Most elementary school students would likely find the fifteenth term here by just continuing the pattern out fifteen numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120. But if you recall the discussion from that task, students might also recognize that the fifteenth term would be the total number of handshakes for sixteen people shaking hands, or 15 + 14 + 13 + 12 + . . . = 120, and some students might be able to generate a formula for the nth term of the pattern, n(n + 1)/2. So, for the fifteenth term, it would be (15 × 16)/2 = 240/2 = 120.

By the way, if students were sticking with “handshake reasoning,” the formula would be slightly different—the fifteenth term would be the number of handshakes for sixteen people, or (16 × 15)/2, but the result would be the same.

For the pattern generated by the  How Many Squares on a Checkerboard? Task: 1, 4, 9, 16, 25, 36, . . . , with very little prompting or scaffolding, most students would be able to recognize that for any specific term, the number would just be that term multiplied by itself, so the fifteenth term would be 152 = 225. For the arithmetic progression 1, 4, 7, 10, 13, 16, 19, again most students would just continue the pattern of adding three until they got to the fifteenth term. But, depending on the grade level and how much time the teacher wanted to spend on a discussion of arithmetic progression patterns, students could be led to determine the formula for the nth term of such a progression, namely an = a1 + (n – 1)d, where an is the nth term, a1 is the first term, and d is the common difference (in this case, three). So the fifteenth term of the progression above would be 1 + (14 × 3) = 43. Because geometric progression patterns increase in value so quickly, for elementary students, one would likely not ask for a term as large as the fifteenth term (finding the fifth or sixth term would be more appropriate), but for those of you determined to find out, in the pattern given: 2, 10, 50, 250, the nth term of the progression could be found with the formula an = a1 × rn – 1 where an is the nth term, a1 is the first term, and r is the common ratio (in our example, five). So the fifteenth term would be 2 × 514 (or a very big number!).

OK, let’s move on to the far less complicated (but perhaps more sneaky) patterns given in the last task.

Complete the following pattern:

5 -----> 4

36--> 8

11---> 1

53---> 7

942---> 14

18---> ?

49---> ?

371---> ?

This is a great example for showing how it’s possible to sometimes overthink patterns, as students and teachers try all manner of combinations of operations to get from the first number to the second one, before “stepping back” and realizing that the second number is just one less than the sum of the digits of the first number, so 18 --> 8 ; 49 --> 12 ; and 371 --> 10.

Study the numbers below, and continue the pattern by listing the next five numbers in the sequence:

1, 1, 2, 2, 8, 10, 3, 27, 30, 4, 64, 68, 5, __, __, __, __, __

This pattern becomes more obvious as you look “further in” to it, seeing 2 followed by 8; 3 followed by 27; and 4 followed by 64. The pattern is in groups of three: a number, that number cubed (raised to the third power), and then the sum of the number and the number cubed. So after 4, 4= 64 , and 4 + 64 = 68, we would have 5, 5125, and 5 + 125 = 130; 6, 216 (63), 222 as the next five numbers in the pattern.

Study the numbers below, and continue the pattern by listing the next five numbers in the sequence:

13, 4, 17, 8, 25, 7, 32, 5, 37, 10, 47, 11, 58, ___, ___, ___, ___, ___

This pattern has a starting number, followed by the number which is the sum of the digits of that number, followed by the sum of the two numbers. It continues by finding the sum of the digits of that number, adding the two again, and so on. So the next five numbers in the pattern would be 13 (sum of the digits of 58); 71 (sum of 58 and 13); 8 (sum of the digits of 71); 79 (71 + 8); 16 (sum of the digits of 79).

I hope that you and your students had some fun with these somewhat unusual patterns. Do you have a pattern to share? Any and all comments regarding these patterns and others you might want to talk about are welcome! I’ll be around to respond to your comments, and then it will be, “Ralph has left the blogosphere!” 


RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

Pondering Patterns

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I hope you’ve been enjoying TCM’s “Math Tasks to Talk About.” From those who understand a lot more about how these things work, I gather the blog is getting a good number of visits, which is really nice to hear, but not too many readers are taking that next step and commenting on the task or the discussion of it. Since I’m now “clear proof” that you don’t have to know anything about blogging to participate in a blog, I’m hoping that folks will realize that all you have to do to comment on the blog is log in as an NCTM member. I look forward to hearing from some of you!

For this next math task, I’m going to venture away from the “classic problems” of the last two tasks, but stick with something they had in common—namely, looking for patterns. This is such a powerful problem-solving strategy that it warrants a lot of attention. The Handshake Problem, when exploring the number of handshakes for different-size groups, generated a pattern of 1, 3, 6, 10, 15, 21, 28, . . . . The Squares on a Checkerboard task generated a pattern of 1, 4, 9, 16, 25, 36, . . . in its solution. These patterns are a bit more complex than something like an arithmetic progression pattern such as 1, 4 ,7 , 10, 13, 16, 19, . . . where the next term in the pattern can be found by adding a specific number (in this case 3) to the number before it, or a geometric progression pattern like 2, 10, 50, 250, . .  . where the next term in the pattern can be found by multiplying the number before it by a specific number (in this case 5). However, one thing all the above patterns have in common is that there is a way of determining what a particular term in the pattern (say the fifteenth term) will be.

Since this will be the last Math Tasks to Talk About problem that I will pose, I’m going to make it a “biggie” by challenging you and your students to ponder  several possible tasks with the same theme—patterning. The discussion above has already provided one such task, namely just asking students to find out what the fifteenth term (or whatever number term you choose, depending on the grade level of your students) in each of the above patterns would be.

Here are some other patterns to ponder, but I should warn you that you’ll have to “think outside the box” when you consider these patterns. Unlike the previous patterns, there’s not necessarily any way to determine a particular term in the pattern. You just have to look at the numbers already in the pattern, and use what you see to find the next number:

Pattern 1  

Complete the following pattern:

5 ----->4

36--> 8

11---> 1

53---> 7

942---> 14

18---> ?

49---> ?

371---> ?

 

Pattern 2 

Study the numbers below, and continue the pattern by listing the next five numbers in the sequence:

1, 1, 2, 2, 8, 10, 3, 27, 30, 4, 64, 68, 5, __, __, __, __, __

 

Pattern 3 

Study the numbers below, and continue the pattern by listing the next five numbers in the sequence:

13, 4, 17, 8, 25, 7, 32, 5, 37, 10, 47, 11, 58, ___, ___, ___, ___, ___.

 

I hope you and your students will have some fun with these patterns, and I look forward to your thoughts and comments. I’ll be back in a couple of weeks with my reflections on these pattern tasks.

 

RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

Reflecting on the How Many Squares on a Checkerboard? Problem

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So, how did things go in your classroom with the How Many Squares on a Checkerboard task? I’m told that we’re still getting a good number of visits to the blog, but few visitors are taking the next step and leaving a comment. The hope is that the blog will become an interactive way for members to share thoughts, comments, ideas, and so on. Please realize that a comment need not be long—a simple “Tried it with my class—they really liked it” is great. Maybe your students solved the task in a really interesting way that you hadn’t thought of before; maybe you had to remove roadblocks before students got going on the task. We would love to hear from you!

All right, so let’s have a look at the How Many Squares on a Checkerboard task and some approaches to solving it. A common strategy is to start with a simpler problem:

How many squares on a 1 × 1 square? 1 

How many squares on a 2 × 2 square? 5
(four 1 × 1 squares and one 2 × 2 square)

How many squares on a 3 × 3 square? 14

For this last one, solvers must see that the board has not only squares of different sizes but also overlapping squares, so a 3 × 3 square has 9 (nine) 1 × 1 squares; 4 different 2 × 2 squares (overlapping, as demonstrated in the checkerboard examples below; and 1 (one) 3 × 3 square), and so on, until eventually arriving at the following solution:

CBoard01-200x192.jpg 

 CBoard02-200x601.jpg 

 checkerboard 

How many squares on an 8 × 8 checkerboard? 204 

64  1 × 1 squares    49  2 × 2 squares 
36  3 × 3 squares    25  4 × 4 squares 
16  5 × 5 squares    9  6 × 6 squares 
4  7 × 7 squares    1  8 × 8 square 

Strategies for identifying and extending patterns, drawing diagrams, making a table, and so on soon come into play. Often the final result for the Checkerboard problem is presented in a table like the one below.

Number of Squares on Various Boards 

Students and teachers readily identify patterns that emerge. The problem can then be extended to determine the total number of squares on any size square board, with the corresponding algebra being introduced as appropriate.

In the higher grades, an extension might be to find the total number of rectangles that can be found on an 8× 8 checkerboard. (Warning: This is not at all trivial!)

So what did you think of the Checkerboard task? How did your students respond to the task? What strategies did they use? Did they encounter any stumbling blocks? How did the student resolve the stumbling blocks? You are welcome to share class photos or student work samples. Hope to hear from you!

RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

How Many Squares on a Checkerboard?

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Now that I’m an official blogger (with two blogs posts under my belt), I found selecting the next problem to be a real dilemma. I have decided to post another “classic” problem.

How many squares are on a standard (8 x 8) checkerboard? 

checkerboard 

As with the Handshake problem, the appeal of this problem (and what probably makes these problems classics) is its accessibility to students across many grade levels, the variety of problem-solving strategies that can be brought to bear in its solution, and the large number of variations/extensions. The simplicity in stating and setting up the problem is also part of its appeal.

A word of caution when introducing this task: Often students see this problem as somewhat trivial, counting just the 64 small squares; some go an extra step and realize that the whole board is also a square, for a total of 65. So, realize that students (or teachers) might need some prompting to recognize that the board also has 2 x 2 squares, 3 x 3 squares, and so on.

So, there you have it. Go ahead and have some fun with this task!

I was gratified to see the response to the launching of the TCM Blog. The site had lots of visits and a few comments. I’m hoping that for this post, we’ll get even more visits, and that more of you who visit the site will take the extra step to post a comment/question/random thought/whatever drawn either from your own experience/reflections or from introducing the problem in your classroom.

As with the first task, I’ll be back in a couple of weeks to post solutions/thoughts/extensions/variations to the task. I hope to hear from you soon and that you’re enjoying “Math Tasks to Talk About.”

RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

Reflecting on The Handshake Problem

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Well, I’ve now been officially initiated into the blogosphere (is that actually a word?) I really appreciated those who took the time to comment on the first task, and I am sincerely hoping that this blog entry, the discussion of the task, encourages more discussion/comments.

So, how’d you do with the Handshake task? If you missed it, here’s the link.

As you know, I love this problem! It’s overflowing with the variety of problem-solving strategies that can be brought to bear in its solution—act it out, draw a diagram, look for a pattern, solve a simpler problem, make an organized list, make a table, use logical reasoning, . . . .

Young students (and math-anxious teachers) can use (and combine) the strategies of acting the problem out, solving a simpler problem, and looking for a pattern, as they build up to the given problem. For example, have 2 students act it out—1 handshake

3 students—3 handshakes

4 students—6 handshakes

5 students—10 handshakes

6 students—15 handshakes

I always find it interesting when students act out the problem: They often go from random hand shaking, which they discover is very hard to count, to organized hand shaking (lining themselves up and going down the line), and finally (what we hope for) to the realization that when an extra person joins the group, they don’t have to repeat all the handshakes that came before, but rather just add on how many handshakes the new person has to do. Therefore, the 7th person would have to do 6 handshakes, 15 + 6 = 21, so 7 students—21 handshakes. Often at this point (if not before), the pattern-seeking students will see that as the number of people goes up by 1, the number of handshakes goes up by first 2, then 3, then 4, then 5, and so on. Continuing, they arrive at the solution:

8 students—21 + 7 = 28 handshakes

9 students—28 + 8 = 36 handshakes

10 students—36 + 9 = 45 handshakes 

Older students (and teachers) will tackle the whole problem with a combination of organized listing and looking for a pattern. They mentally or physically line up 10 people and reason that the 10th person will go down the line and shake hands with 9 people; the next person will go down the line and shake hands with 8 people; the next one, 7; the next one, 6, and so on. So the total number of handshakes is 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 handshakes.

This usually leads to the discovery of a generalization: for 20 people, add 19 + 18 + 17 + … + 1. The generalization still takes a lot of computation to find the total number of handshakes, but it’s certainly within the capabilities of students in elementary school. A wonderful YouTube video shows two grade 3 girls solving the problem in this manner, using linking cubes as an aid, for the number of handshakes for 35 people! How’s that for “perseverance in problem solving”?

With teachers and older students, if no one has yet suggested this problem-solving strategy, I like to demonstrate using logical reasoning and throw in my incredibly well-reasoned (but incorrect!) hypothesis:

Well, if I’m one of the 10 people at the party, I shake hands with 9 other people. So does every other person at the party. Since there are 10 of us, and we each shake hands with 9 other people, it’s obvious that the answer is 10 × 9 = 90 handshakes. But this doesn’t match the answer arrived at by using the other methods. What’s wrong?

After some pondering, they realize that I’ve broken one of the ground rules and have counted every handshake twice, so the correct answer would be 90 ÷ 2 = 45. This discovery will then lead to the algebraic generalization that for any group of n people, the number of handshakes will be

n(n – 1)/2,

which matches the formula for finding the sum of all the numbers up to (but not including) a given number. The process is a nice, concrete way of showing why that formula makes sense: n people would each shake (n – 1) hands, but that counts every handshake twice, so we have to divide by 2. Thus,  

n(n – 1)/2.

The handshake problem has many variations in presentation. A way of incorporating the problem into a history context is effectively shown on NCTM’s Illuminations website, which discusses the tradition of the Supreme Court Justices all shaking hands with one another before each session. Then follows the Handshake problem, which asks how many handshakes that scenario would take.

The Illuminations website also has an applet that draws a diagram, along with creating a chart, for the number of handshakes for 2 people up to 12 people.

And, for students in grade 6 and beyond, Illuminations has a nice extension/connection between the Handshake problem and triangular numbers.

OK—so there you have it. I hope you’ll agree with me that this is indeed a “Math Task To Talk About.” Maybe you have some other interesting connections to the Handshake problem, great ways that your students thought about it, or thought-provoking activities that build on it. Please share!

RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

The Handshake Problem

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kidshandshakeI have the honor of being the “inaugural blogger” for the new Teaching Children Mathematics (TCM) blog, “Math Tasks to Talk About.” Now, to be clear, what I know about blogging could fit in a thimble with plenty of room still left for your finger. However, the talented staff at NCTM can take whatever I submit and magically make it blog-worthy, so here goes!

My absolute favorite math task to talk about is a classic known as the Handshake problem. Alternatively, you may know it as the How Do You Do? problem or the Meet and Greet problem or one of more than at least a dozen different names. No matter what you call it, this problem is my favorite because it can be easily made accessible and interesting to students at all levels, from first grade through high school!

All right—here’s the problem:

Ten [or however many you want] people are at a party, and you want everyone to meet (shake hands with) everyone else at the party. How many handshakes will it take?  

For students in the early grades, simply reduce the number of people who are shaking hands; and for students in the upper grades, move to a generalization for a large number of people shaking hands.

The other reason I find the problem appealing is that the conditions are few and easy to understand. In exploring the problem, students will discover two things: (1) you wouldn’t shake hands with yourself, and (2) when two people shake hands, it counts as one handshake, not two.

OK—that’s it! Solve away! Clearly, the only way this will become a robust and interesting blog is if there is interaction. Give a version of the problem to your class, talk about the different ways your students approach solving it, and talk about your own strategies/reasoning as you think about the problem. Afterward, come back here and post a comment about how it went. You are also welcome to share sample student work and photos. We need your help to make “Math Tasks to Talk About” a rich problem-solving resource.

I’ll be back in two weeks as a follow-up to this post with solutions and ruminations about the problem. In the meantime, I look forward to hearing from you!

 

RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

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