# The Broken Eggs

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## 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.

## Opening

10 minutes

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.

## Investigation

30 minutes

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.

## Closing

20 minutes

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 in 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.

## Citations

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