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* *Reflection:
The Broken Eggs - Section 2: Investigation

I noticed many opportunities for differentiation during today's lesson. In this year's class, I have many special education students along with many students who have been out of school for an extended period of time. I also seem to have a few students who declare their hatred for math! This problem can be a challenging one for some students as it requires them to practice** SMP 1:** **Make sense of problems and persevere in solving them**. I noticed during this class that many different things were happening simultaneously in order to keep students working on the problem.

- First, I started out with a demonstration using small cubes for the eggs. I think this visualization of the problem helped some students to "make sense of the problem." Some students continued with the manipulatives for a while, while other students moved on to pen and paper.

- Two students came out with some great ideas right at the start, but had trouble incorporating those ideas into their work. One student immediately said, well the answer has to be a multiple of seven. We wrote up this "noticing" for the class in order to help students keep notes on their process. Another student talked about how the answer could not be divisible by 2, 3, 4, 5, or 6. Students were able to take the idea of not being divisible by 2 into account and ruled out any even numbers, but I noticed they were unable to do anything else with this information even though it was useful. Later in the class, I did end up showing them a trick for how to figure out if a number is divisible by 3.

- Students definitely needed help thinking about decimals once they got to larger numbers and were using calculators. The way I approached this issue with most students was to ask them what an example would be of a number that would have a remainder of 1 when divided by 3. Most students could get to the next number, 4, realizing that 1 would be left over. We then talked about what 1/3 looks like as a decimal.

- One student needed help organizing her information and needed a lot of encouragement to stay with the problem. We worked to create a table to organize which numbers fit the divisibility requirements of the problem. Once students realize that there have to be more than 100 eggs, they often get frustrated. For this reason, I think it's good to cut off the work at some point and get students writing about what they have tried so far.

*Opportunities for Differentiation*

*Opportunities for Differentiation*

# The Broken Eggs

Lesson 5 of 14

## Objective: SWBAT to make sense of a word problem and use multiple approaches to justify their answers. SWBAT be able to explain their mathematical thinking in writing.

## Big Idea: Students work on their first math riddle. This lesson gives them the opportunity to solve a problem in different ways and spend some time explaining their own, unique mathematical thinking.

*60 minutes*

#### Opening

*10 min*

The Broken Eggs problem is a low-entry / high ceiling problem that allows students multiple entry points and ways to solve. I first came across the Broken Eggs problem in the IMP Curriculum, but it is based on a well known problem. The IMP version of the problem is on page 6 in their Year 1 textbook. IMP also does a great job of guiding students to write about their problem solving process so you may want to check out their textbook for support around the writing piece as well.

This is the first word problem students will see in this class and it is important that all students feel they have a way in to get started. In the context of a farmer bringing her eggs to market, the problem asks students to find a number(s) whose remainder is 1 when divided by 6, 5, 4, 3, and 2, and no remainder when divided by 7. This problem is derived from a well known problem written by the Hindu mathematician Brahmagupta.

An old woman goes to market and a horse steps on her basket and crushes the eggs. The rider offers to pay for the damages and asks her how many eggs she brought. She does not remember the exact number, but when she had taken them out two at a time, there was one egg left. The same happened when she picked out three, four, five, and six at a time, but when she took them seven at a time, they came out even. What is the smallest number of eggs she could have had?

Begin class by having students read the Broken Eggs problem aloud together. You can let students know they can work in groups or alone on this problem. You can discuss the Write Up portion of the problem later. For now, have students focus on problem solving. You should tell students they will have time at the end of class to summarize in writing what they have worked on so far. Therefore, you should encourage them to keep notes about different strategies that they try as they work.

#### Resources

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#### Investigation

*30 min*

Let students get to work! If they have trouble starting, I like to use manipulatives to illustrate the problem. You can have a bunch of small object (I use chemistry cubes) and have them represent the eggs. You can ask students to choose a number of eggs to start with. See if anyone has ideas about what number might make sense. If students say 8 eggs for example, you might try and see if anyone can rule out the number 8 because it is divisible by 2 and therefore does not have a remainder. You can try to get students to generalize this to all even numbers, but don't tell them, let them come up with that idea. Make a big deal of this generalization. You might stop the class and say "hey everyone, look what John found." Then you might write the generalization on the board as an example of the kind of notes that students will want to make as they work. Point out that these notes will be useful later when they write an explanation of how they solved the problem.

You may also need to help students come up with a way to organize their information. You might suggest some kind of chart where they are testing numbers and what the remainders are. Students may begin to notice a pattern within the chart.

Students may begin to get exasperated when the numbers start to get really big. This is where you can help them with **SMP1: Make sense of problems and perservere in solving them**. You might ask students how they can work with these larger numbers without using manipulatives or doing out long division. Let them generate some ideas. If students suggest working on calculators, make sure they have an understanding of what the remainder of 1 will look like as a decimal. This is a good opportunity for students to share their understanding of how fractions and decimals work. You might have one student explain why, for example, the decimal should be .5 when a number divided by 2 has a remainder of 1.

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#### Closing

*20 min*

Today's closing is a little longer than usual because you want to give time for students to begin writing about their work on this problem so far. You can explain to students that you will frequently be asking them to write about their mathematical thinking. You can tell them that today's problem provides many opportunities for them to share their thinking. Let them know that they will eventually complete a full "write up" for this problem. One part of the write up is called the process. The process section asks them to explain how they approached the problem. Some questions they might answer in the process section include:

- How did you start the problem?
- What did you try? Why?
- Where did you get stuck?
- Who helped you?
- How did you get unstuck?
- What else did you try?

Give students ample time to write about what they have accomplished so far today. While they are writing (in paragraph form), you can circulate around the room and make sure they are using descriptive language. You might show an example of how you started the problem and how you would write up that work.

**Homework**: For homework tonight, tell students they will be writing a Problem Statement for the Broken Eggs problem. A problem statement is simply a paragraph that states what the problem is asking them i** n their own words**. I like to say to students, "Imagine the principal walked into class right now and asked you what you were working on. How would you describe the problem to him if you did not have the paper right in front of you?" Emphasize to students that they should write a full paragraph, not just bullet points.

Note: This problem is not meant to be solved after one class period. You may give students additional class time on another day, or assign pieces of the problem for homework. This class will revisit this problem in a few lessons and spend additional time on math writing to complete the problem write-up.

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Program, I. (2008, June 3). *POW 1: The Broken Eggs*. Retrieved from the Connexions Web site: http://cnx.org/content/m15963/1.3/

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- UNIT 1: Introduction to Algebra: Focus on Problem Solving
- UNIT 2: Multiple Representations: Situations, Tables, Graphs, and Equations
- UNIT 3: Systems of Equations and Inequalities
- UNIT 4: Quadratics!
- UNIT 5: Data and Statistics
- UNIT 6: Arithmetic & Geometric Sequences
- UNIT 7: Functions

- LESSON 1: First Day of Class - The Marshmallow Challenge
- LESSON 2: How Does This Math Class Work? Creating a Positive Classroom Climate
- LESSON 3: Generating Student Discourse
- LESSON 4: What Makes Something a Pattern?
- LESSON 5: The Broken Eggs
- LESSON 6: Introduction to Functions
- LESSON 7: Tables, Words, and Equations
- LESSON 8: A Communication Challenge
- LESSON 9: Mystery Bags Game
- LESSON 10: Writing About Math in the Cafeteria
- LESSON 11: Post-It Note Equations
- LESSON 12: Solving for x
- LESSON 13: Inequalities: True or False?
- LESSON 14: The Great Inequality Debate