Sarah Forst, 8th grade Pre-Algebra (23 May 2014)
Margaret McNamara, 5th grade (4 June 2014)
David Lochbihler, 3rd grade (30 May 2014)

Standards:

Problem Solving Strategies
Algebraic Formulae

Lesson Study Title:

Cathedral Construction

Overview of the lesson:

Students in grades three, five, and eight will work individually and in small groups to solve the Cathedral Construction problem. Exploration will take place in groups of three, with each group receiving a chart paper for recording of pictures, graphs, patterns, and problem solving strategies.

Various aspects to STEAM will be utilized in this Lesson Study:
SCIENCE — the building of cathedrals during the Middle Ages and to the present day will be discussed
TECHNOLOGY — IPad calculators will be available for student use
ENGINEERING — the building of cathedrals during the Middle Ages and to the present day will be discussed
ART — the building of cathedrals during the Middle Ages and to the present day will be discussed
MATHEMATICS — students of varying aptitudes will use problem solving strategies and algebraic knowledge

Here is the Cathedral Construction word problem as originally presented in our Tuesday class:
“To finish a cathedral, four stonemasons and three artists can be hired for 33 guilders for one day. A second option is to hire three stonemasons and four artists for 37 guilders for one day. How much would it cost to hire one stonemason and one artist for one day?”

The three teachers in this Lesson Study are teaching three different grade levels: Grade Three, Grade Five, and Grade Eight Pre-Algebra. Because of this, at the suggestion of his colleagues, the third grade teacher adapted the problem for his students as follows:
“To finish a cathedral, four builders and three artists can be hired by the bishop for 13 dollars for one day. The bishop also may decide to hire three builders and four artists for 15 dollars for one day. How much would it cost to hire one builder and one artist for one day?”

Anticipated student strategies: (BOARD PLAN with Sequencing and connections)

We anticipate the students will utilize these strategies:
➢ Guess & Test
➢ Draw a Picture
➢ Solve a Simpler Problem
➢ Find a Pattern
➢ Algebraic Formulae
Here are some possible solutions for the original word problem:

A. An incorrect Guess & Test:

S = 6 S = 6 S = 6 S = 6 A = 3 A = 3 A = 3 —– Total 33 Guilders

S = 6 S = 6 S = 6 A = 3 A = 3 A = 3 A = 3 —– Total 30 Guilders

B. The correct Guess & Test:

S = 3 S = 3 S = 3 S = 3 A = 7 A = 7 A = 7 —– Total 33 Guilders

S = 3 S = 3 S = 3 A = 7 A = 7 A = 7 A = 7 —– Total 37 Guilders

C. Pictures
FOUR STONEMASONS & THREE ARTISTS

THREE STONEMASONS & FOUR ARTISTS

D. Solve a Simpler Problem & Patterns
STONEMASONS ARTISTS GUILDERS
7 0 21
6 1 25
5 2 29
4 3 33
3 4 37
2 5 41
1 6 45
0 7 49

E. Incorrect Algebraic Formula

4S + 3A = 3S +4A

F. Correct Algebraic Formula

S + S + S + S + A + A + A = 33
S + S + S + A + A + A + A = 37
7S + 7A = 70
Dividing both sides of the equation by 7, 1S + 1A = 10

G. Sample Algebraic Reasoning

S + S + S + A + A + A + A = 37
MINUS
S + S + S + S + A + A + A = 33
EQUALS
A – S = 4 —– This might prompt the students to understand that the artists four dollars more than the stonemasons.

Formative Assessment strategies: (Rubric/checklist)

Here are several items the teachers will consider in assessing the students’ work:
1. Students will begin their work individually before gathering in groups of three.
2. Each student’s written work will be assessed to encourage their critical thinking and reasoning skills.
3. Each small group’s written work will be assessed to encourage their critical thinking and reasoning skills.
4. Students will be encouraged to share their ideas verbally both within their small groups and to the entire class.
5. We will try to make sure each member in a small group of three students can explain the group’s work.

Goals
What are your mathematical goals for the lesson:
1) What do you want students to understand as a result of this lesson? …as a result of this unit?
2) What mathematical processes are you working to develop?
3) How does this lesson contribute to their continuing development as learners?

Prior Knowledge
In what ways does the task build on students’ previous knowledge? What definitions, concepts, or ideas do students need to know in order to begin to work on the task? What questions will you ask to help students access their prior knowledge?
Goals
➢ We want our students at all grade levels to understand there is more than one way to solve a word problem. To this end, the value of multiple solution strategies will be emphasized.
➢ We want our students at all grade levels to understand the importance of critical thinking and reasoning in solving word problems
➢ We want the students to understand several problem solving strategies: Guess & Test; Draw a Picture; Solve a Simpler Problem; Find a Pattern
➢ We want our middle school students and some elementary students to understand the role of functions and algebraic connections

Prior Knowledge
➢ Given the three grade levels in our Lesson Study, the prior knowledge of the students will vary greatly
➢ The third grade students will learn about the different problem solving strategies for the first time
➢ The fifth grade students have used the problem solving strategies in the past but will be unaware of middle school algebraic expressions
➢ The eighth grade students will have experience in the problem solving strategies, but the algebraic expressions may be new to them
➢ Regardless of the grade level, students must be able to read and comprehend the word problem
➢ There are several vocabulary terms to be discussed: guilder, stonemason, artist, cathedral

Possible questions include:
➢ What do we call money here in the USA and in other countries?
➢ What do you know about cathedrals?
➢ What do you know about Michaelangelo and the Sistene Chapel?
➢ How were cathedrals built in the Middle Ages?
➢ What does a stonemason do?
➢ Why do you need an artist when you build a cathedral?
➢ What are some of the problem solving strategies you have used?
Anticipating Student Responses
Identify the ways in which the task can be solved.
• Which of these methods do you think your students will use?
• What misconceptions might students encounter?
• What errors might a student make?

Expectations
What are your expectations for students as they work on and complete this task?
• What resources or tools will students have to use in their work?
• How will the students work — independently, in small groups, or in pairs — to explore this task? How long will they work individually or in small groups/pairs? Will students be partnered in a specific way? If so in what way?
• How will students record and report their work? Anticipating Student Responses
We expect the student responses to vary according to their mathematical age-appropriate aptitude:
➢ Third Grade — Guess & Test, Draw a Picture
➢ Fifth Grade — Guess & Test, Draw a Picture, Solve a Simpler Problem, and Find a Pattern
➢ Eighth Grade — Guess & Test, Draw a Picture, Solve a Simpler Problem, Find a Pattern, and Algebraic Expressions

The third grade teacher is considering simplifying the numbers in the problem:
➢ Each Builder will be paid one dollar
➢ Each Artist will be paid three dollars
➢ 4 Builders + 3 Artists will be paid 13 dollars
➢ 3 Builders + 4 Artists will be paid 15 dollars

Misconceptions:
➢ All workers are paid equally
➢ The first set of workers will be paid differently from the second set of workers

Errors:
➢ The word problem may not be thoroughly read
➢ The word problem may not be accurately understood
➢ Younger students may add incorrectly

Expectations
➢ Begin with independent thinking for two or three minutes
➢ The students will be given manipulatives to allow them to consider different numerical combinations
➢ The students will be encouraged to draw pictures of the stonemasons and the artists
➢ Encourage the students to write all their ideas down on paper, even those that do not lead to the correct answer
➢ We hope the students will avoid frustration and continue working even if the correct answer is not forthcoming
➢ Small heterogeneous groups of three work together
➢ The small groups will use chart paper to showcase their thought processes and possible solution strategies for a large group presentation.

Launch
How will you introduce students to the task so as not to reduce the problem solving aspects of the task(s)?
What will you hear that lets you know students understand the task(s)?

Explore/Monitoring
As students are working independently or in small groups:
• What questions will you ask to focus their thinking?
• What will you see and hear that lets you know how students are thinking about the mathematical ideas?
• What questions will you ask to assess students’ understanding of key mathematical ideas, problem solving strategies, or their representations?
• What questions will you ask to advance students’ understanding of the mathematical ideas?
• What questions will you ask to encourage students to share their thinking with others or to assess their understanding of their peers’ ideas?

How will you ensure that students remain engaged in the task?
• What will you do if a student does not know how to begin to solve the task?
• What will you do if a student finishes the task almost immediately and becomes bored or disruptive?
• What will you do if students focus on non-mathematical aspects of the activity (e.g., spend most of their time making a beautiful poster of their work)?

Whole Class Discussion/Selecting, Sequencing, Connecting
Which solution paths do you want to have shared during the class discussion in order to accomplish the goals for the lesson?
• Which will be shared first, second, etc.? Why?
• In what ways will the order of the solution paths helps students make connections between the strategies and mathematical ideas?
Launch
The students will be asked to share and discuss all the problem solving strategies they have used in the past.

The 3rd grade teacher will introduce the different ways you can go about trying to solve a word problem.
➢ Guess & Test
➢ Draw a Picture
➢ Solve a Simpler Problem
➢ Find a Pattern

The 5th grade teacher will have students write independently for about five minutes on the following focus questions: What strategies have I used in the past that have proven helpful in problem solving? Is it possible to use different methods and achieve the same outcome? How can I make sure I understand what the problem is asking? Students will discuss their answers collaboratively and as a larger group.

Upon receiving the word problem, students will identify any vocabulary they do not understand and use context clues to determine meanings. They will then complete a chart to organize what they know about the problem and what they need to find out. Students will also discuss any assumptions about the problem they may have.

The 8th grade teacher will pose an initial question of the students, “If one person can …., How many will it take to …”. The teacher will provide an example such as “If one person can east 5 cookies, how many people will it take to each 500 cookies?”. Students will be given 3 minutes to evaluate the problem individually. The class will be brought together to share responses, followed by a brainstorming session for math concepts needed to solve such problems.

Upon receiving the word problem, there will be a class discussion to make sure the students have thoroughly read and understood what the problem is asking. Hyperlinks will be built into a slide show to help clarify any vocabulary that is unknown to students.

Students will ask any clarifying questions they think are necessary prior to beginning the task.

Some behavior we hope to see:
➢ Students placing the manipulatives in different piles
➢ Students using their calculators or adding numbers together
➢ Students drawing pictures of stonemasons and artists
➢ Students using symbols to indicate the difference between stonemasons and artists
➢ Students writing possible solutions to the problem
➢ Students attempting more than one solution strategy

Some questions we hope to hear:
➢ May we use any strategy we want to solve the problem?
➢ What do we do if we get stuck?
➢ May we work with a partner?

Explore/Monitoring

To facilitate engagement in the task, we will utilize the following strategies for success:
➢ We will provide individual help to any students stuck and unable to advance on the task.
➢ If a student finishes the task early, we will encourage him or her to try a different problem solving strategy.
➢ If a student finishes the task early, we will encourage them to share their solution with the rest of their group.
➢ If a fifth or eighth grade student successfully employs two different problems solving strategies, we will encourage them to consider solving the problem algebraically
➢ If a third grade student successfully employs two different problems solving strategies, we will provide to them the advanced sheet used in the original Middle Grades Cathedral Construction and see if they can solve this more challenging word problem
➢ If a student turns the assignment into an art project, we will encourage their artistic ability (a part of STEAM) but try to make sure each of the group members is able to share the group’s findings with the entire class.

Whole Class Discussion/Selecting, Sequencing, Connecting
We hope to bring together a variety of different problem solving strategies from the students and thus encourage them to demonstrate their critical thinking and reasoning skills both within their small group and with the entire class. We will try to get a different problem solving strategy from each small group. The simplest strategy (Guess & Test) will be presented first, and from there we will move to more complex problem solving strategies (Draw a Pictures, Solve a Simpler Problem, and Find a Pattern). The use of several algebraic formulae will be presented to the entire class, beginning with the simplest solution and continuing to the most complex. Beginning with the simple and progressing to the more complex, the students will be able to understand there are many ways to solve a word problem. In addition, the students will discover the thinking and reasoning process is more relevant for critical thinking than simply finding the answer. With the student’s permission, incorrect answers also will be presented for large group discussion. The use of incorrect answers may have to be presented anonymously by the teacher depending on student sensitivity. The use of both correct and incorrect answers is essential if the students are to move beyond simply finding an answer and shift to a deeper appreciation of the problem solving process.
Assessment
What will you see and hear that lets you know that students in the class understand the mathematical ideas addressed in the lesson?
Students will reflect on the activity. It would be helpful if each student could write or explain both their favorite problem solving strategy and at least one other strategy presented by one of their classmates.

References: Cathedral Construction. COMPLETE: Center for Outreach in Mathematics Professional Learning and Educational Technology. George Mason University, Fairfax, Virginia (http://completecenter.gmu.edu).

APPENDIX ONE
Third Grade
Cathedral Construction

A similar replication of:
CATHEDRAL CONSTRUCTION

To finish a Cathedral, four Builders and three Artists may be hired by the Bishop for 13 Dollars for one day.

The Bishop also may decide to hire three Builders and four Artists for 15 Dollars for one day.

How much would it cost to hire one Builder and one Artist for one day?

APPENDIX TWO
Third Grade
Guess & Test

Your FIRST Guess & Test MAY Look Like This:
BUILDER = 1 BUILDER = 1 BUILDER = 1 BUILDER = 1
ARTIST = 2 ARTIST = 2 ARTIST = 2
Total = 10 Dollars
Correct Answer = 13 Dollars
Your GUESS is way too LOW.

BUILDER = 1 BUILDER = 1 BUILDER = 1
ARTIST = 2 ARTIST = 2 ARTIST = 2 ARTIST = 2
Total = 11 Dollars
Correct Answer is 15 Dollars
Your GUESS is way too LOW.

Your Second Guess & Test MAY Look Like This:
BUILDER = 2 BUILDER = 2 BUILDER = 2 BUILDER = 2
ARTIST = 3 ARTIST = 3 ARTIST = 3
Total = 17 Dollars
Correct Answer = 13 Dollars
Your GUESS is way too HIGH.

BUILDER = 2 BUILDER = 2 BUILDER = 2
ARTIST = 3 ARTIST = 3 ARTIST = 3 ARTIST = 3
Total = 18 Dollars
Correct Answer = 15 Dollars
Your GUESS is way too HIGH.

Your THIRD Guess & Test MAY Look Like This:
BUILDER = 3 BUILDER = 3 BUILDER = 3 BUILDER = 3
ARTIST = 1 ARTIST = 1 ARTIST = 1
Total = 15 Dollars
Correct Answer = 13 Dollars
Your GUESS is a little too HIGH.

BUILDER = 3 BUILDER = 3 BUILDER = 3
ARTIST = 1 ARTIST = 1 ARTIST = 1 ARTIST = 1
Total = 13 Dollars
Correct Answer = 15 Dollars
Your GUESS is a little too LOW.

APPENDIX THREE
Third Grade
Draw a Picture

FOUR BUILDERS & THREE ARTISTS (Total Salary = $13)

THREE BUILDERS & FOUR ARTISTS (Total Salary = $15)

APPENDIX FOUR
Third Grade
Lesson Plan

I. Launch
The Goal of this lesson is to introduce the rising fourth grade class to Problem Solving Strategies, specifically Guess & Test and Draw a Picture. To introduce the scholars I will be teaching next fall to these strategies, I tried to get them to see “the Big Picture.”

Introductory questions: first, who can recall the last time I taught a single class?; second, was I next year’s 4th grade teacher yet?
I wanted to thank them, because earlier this school year, I taught a Math Lesson to them under the auspices of our principal. At that point in time, I had not been hired yet, and this class lesson was part of the interview problem. The class went very well and included a college Math problem two students successfully solved. I told the students because that class session went so well, I was hired and will become their fourth grader teacher next year, and I thanked them.

I shared with the students this quote from my evaluation from the principal: “MOST IMPORTANT – he taught REASONING SKILLS/CRITICAL THINKING.” I told the students: “Thanks to you, I was hired to teach 4th Grade and asked to teach Math to both 4th and 5th grade next year.” And once a week next school year, I want to dedicate the Math lesson to Problem Solving.

II. Explore/Monitoring
We considered the following two questions as a class. First, what is more important in sports, hustle & attitude or skill & ability? (Most students thought skill & ability). I then gave to them my perspective as a former high school varsity basketball and soccer coach — why hustle & attitude is even more important than skill & ability. I then elicited answers to the questions: what is most important in Math? What is most important in solving problems? I was looking for the fact that Process is more relevant than finding an answer. I then asked, how on the problem sheet will they be able to show Process? After several guesses, one student said, “Show your work!”

➢ Even if your work leads to a wrong answer, that’s okay if you show your work
➢ Really really important to show you work
➢ Also important to be able to explain your work to your friends
➢ Show work on the problem sheet
➢ If at first you don’t succeed, try, try again (hint for Guess & Test)
Our discussion focused on a common mistake older Middle School students make. “What do you think is the one of the most basic or fundamental or simple mistakes in Word Problems.
➢ Not reading the problem
➢ Not clear on what is being asked
I handed the Word Problem to the students. As a student read the problem out loud, we were to discuss what each sentence means. I forgot to ask, but in my written Lesson Plan it would have been helpful to ask, “If you were the teacher, how many Square Pattern Blocks would you put in each bag?”

Three phases of Problem Solving once students received the problem sheet:
➢ Work on your own (teacher walks around and monitors progress while answering questions);
➢ Hand out and discuss Guess & Test sheet to introduce first Problem Solving Strategy, students continue working on the problem (teacher walks around and monitors progress while answering questions);
➢ Hand out and discuss Draw a Picture sheet with bag of Pattern Block Manipulatives to introduce second Problem Solving Strategy, students continue working on the problem (teacher walks around and monitors progress while answering questions).
➢ At this point on my Lesson Plan, I was going to have students begin working in groups of three to discuss their different strategies and possible solutions, and then draw their strategies to share on big sheets of paper with the whole class. Unfortunately I did not monitor my time well, and I completely forgot this part of my gameplan! To the students’ credit, as they were sitting at desks, I could tell by their whispering that they were going over their problems together as they often do with their regular teacher, simply following their usual procedure.

III. Whole Class Discussion: Selecting, Sequencing, Connecting
The time flew by and I did not complete this essential part of the Lesson Plan with the students. As this is not my class, I could not go back later and finish this part of the Lesson Plan with them. Were I their classroom teacher, we would have extended the class, or I would have finished this part of the Lesson Plan with them later. This will be the launch of our 4th Grade Problem Solving next school year and will be completed with them then.

On both Smarter Cookie and as an appendix to our Group Lesson Study, and as follows, here is how I would have selected, sequenced, and connected the students’ work.
A. SELECTING
Of the thirty students, I would have tried to incorporate all their work somehow. For even incorrect answers, the students’ reasoning and thought processes are critical. As a sample of how I would have featured Selection, Sequence, and Connection in the Whole Class Discussion, I grouped and gathered twelve samples of their work, commenting on each of them in “13 of 13 — Cathedral Construction” on Smarter Cookie. A summary of this exercise follows.

I selected twelve works representing a sampling to the students’ thinking. I based my selection on the following factors. First, I wanted to make sure to place some incorrect samples in the mix because, as indicated in our class discussion, process is more relevant than finding the right answer. Second, I wanted to show some evidence of a student really reading and comprehending the problem, because this is an essential “first step” in Problem Solving. Third, I wanted to feature some samples that showed “out-of-the-box” thinking, regardless of whether or not the student solved the problem correctly. I wanted to sequence these early to reinforce our emphasis on being process-driven rather than answer driven. Fourth, I wanted to feature both incorrect and then correct Guess & Test samples, highlighting the simplest and first Problem Solving Strategy. Fifth, I wanted to feature both incorrect and then correct Draw a Picture sample, highlighting an alternative Problem Solving Strategy.

B. SEQUENCING
After selecting twelve works representing a sampling of the students’ thinking, I would have sequenced them as follows:
➢ Picture One — I first placed a Number Sentence equaling 13, one facet of what the students learned in our previous GMU lesson on the Associative and Commutative Properties for the Elementary Grades class.
➢ Picture Two — This picture was sequenced next to emphasize the importance of thoroughly reading and understanding a Word Problem prior to beginning to solve.
➢ Pictures Three & Four — These samples featured a very common mistake. This shows a good attempt at coming up with an answer, albeit it incorrect, but not fully grasping the problem.
➢ Pictures Five & Six — These samples featured a distinct attempt at a solution by combining all Builders and Artists together prior to solving the problem
➢ Pictures Seven & Eight — These were good samples of Guess & Test, our first Problem Solving Strategy.
➢ Pictures Nine, Ten, & Eleven — These were good samples of Draw a Picture, our second Problem Solving Strategy.
➢ Picture Twelve — I sequenced this at the end because it best epitomizes the goal of this lesson. It shows the most expansive reasoning skills and critical thinking, with a wide variety of attempts at solving the problem.

C. CONNECTING
There are a variety of important connections to be highlighted in the student samples:
➢ Pictures Three & Four show similar thinking and probably the most common mistake in solving this problem.
➢ Pictures Seven & Eight both show an attempt to Guess & Test in order to solve the problem.
➢ Pictures Five & Six are connected to Pictures Seven, Eight, and Nine in that all five samples tried to Draw a Picture to solve the problem.
➢ Perhaps the second most important connection: The students using Draw a Picture also employed Guess & Test in reaching a final solution.
➢ Perhaps the most important connection: All the students in each sample tried to show their work, an essential step in solving problems, mathematical thinking, and sharing strategies and solutions with your classmates.

APPENDIX FIVE
Third Grade
Student Work

The third graders Student Samples are attached as Appendix One. Here is a brief summary of the numbered pictures:
1. This student understood a Number Sentence with addition and multiplication but could not grasp the depth of the Word Problem.
2. Notice the Arrows from the paragraphs. This student read and understood the Word Problem exceptionally well. During classroom discussion, by the number of artists and the total payments, a student deduced that an artist earned more than a builder.
3. This student decided to Draw a Picture and solved the first problem by counting by twos, and then giving the last artist one dollar. To her credit, the Word Problem did not state the respective artisans had to be paid equal sums.
4. This student also decided to Draw a Picture and came to the same conclusion.
5. Within the pink rectangle, this student decided to group all the artisans from both professions together. She also realizes the total payment is $28.00. Within our Teacher Small Group at GMU, this was the first step towards several algebraic solutions.
6. This student tried a number line and then decided to Draw a Picture. She also grouped everyone together and realized each Builder earned one dollar while each Artist earned three dollars.
7. This is a great sample of Guess & Test. You may not see clearly on the paper, but behind the correct solution are other guesses.
8. This Guess & Test thought outside the box to consider Decimals rather than Whole Numbers. She arrived at the correct solution on the top left of the page.
9. Here is a student trying to Draw a Picture to solve the problem. There are cross-outs and erasures showing their work.
10. This student decided to Draw a Picture to successfully solve the problem. She tried many numbers and erased a lot. Next school year I will try to get them to show all their work and not erase.
11. This student decided to Draw a Picture to successfully solve the problem. The picture is quite clever, with tools for the Builders and paint brushes for the Artists. The students using Draw a Picture also incorporated Guess & Test to solve the problem.
12. This student used a variety of strategies to solve the problem. She circled some data and demonstrated a very good understanding of the Word Problem. The “1D” meaning “one day” in both the setup and the solution shows this thoughtful understanding.

APPENDIX SIX
Third Grade
Student Follow-up

At the end of the lesson, the teacher gave to the students the actual Middle School Cathedral Construction from our GMU class and encouraged them to try it at home. Here is the follow-up from one student who with her dad successfully completed the advanced Word Problem in several clearly articulated steps: