A Pyramid of Crunchkins

Transcription

1 Pictured below you will see Nestle Crunchkins stacked in a triangular pyramid. Each layer is in the shape of an equilateral triangle, and the top layer is a single Nestle Crunchkin. How many Nestle Crunchkins are in a pyramid 4 layers high? How many Nestle Crunchkins are in a pyramid 5 layers high? How many Nestle Crunchkins are in a pyramid 10 layers high? 1 of 12

2 Suggested Grade Span 3 5 Grade(s) in Which Task Was Piloted 5 Task Pictured below you will see Nestle Crunchkins stacked in a triangular pyramid. Each layer is in the shape of an equilateral triangle, and the top layer is a single Nestle Crunchkin. How many Nestle Crunchkins are in a pyramid 4 layers high? How many Nestle Crunchkins are in a pyramid 5 layers high? How many Nestle Crunchkins are in a pyramid 10 layers high? Alternative Versions of Task More Accessible Version: (Use the same pyramid image that appears in the original version of the task.) Pictured below you will see Nestle Crunchkins stacked in a triangular pyramid. Each layer is in the shape of an equilateral triangle, and the top layer is a single Nestle Crunchkin. How many Nestle Crunchkins are in a pyramid 3 layers high? How many Nestle Crunchkins are in a pyramid 4 layers high? How many Nestle Crunchkins are in a pyramid 5 layers high? 2 of 12

3 Teacher Note: See page 6 of the PDF to print a complete worksheet with graphics. More Challenging Version: (Use the same pyramid image that appears in the original version of the task.) Pictured below you will see Nestle Crunchkins stacked in a triangular pyramid. Each layer is in the shape of an equilateral triangle, and the top layer is a single Nestle Crunchkin. How many Nestle Crunchkins are in a pyramid 5 layers high? How many Nestle Crunchkins are in a pyramid 10 layers high? How many Nestle Crunchkins are in a pyramid n layers high? Teacher Note: See page 7 of the PDF to print a complete worksheet with graphics. NCTM Content Standards and Evidence Algebra Standards for Grades 3 5: Instructional programs from pre-kindergarten through grade 12 should enable all students to Understand patterns, relations and functions. NCTM Evidence A: Describe, extend and make generalizations about geometric and numeric patterns. Exemplars Task-Specific Evidence A: This task requires students to describe the number of Crunchkins in each layer of the triangular pyramid, to analyze how each layer increases and to extend the pattern to 10 layers. NCTM Evidence B: Represent and analyze patterns and functions, using words, tables and graphs. Exemplars Task-Specific Evidence B: This task requires students to represent and analyze the number of Crunchkins in each layer of the pyramid. Use mathematical models to represent and understand quantitative relationships. NCTM Evidence: Model problem situations with objects and use representations such as graphs, tables and equations to draw conclusions. Exemplars Task-Specific Evidence: In order to solve this task, many students create models of pyramids using objects and then represent their models in tables in order to analyze the data and to draw conclusions. 3 of 12

4 Time/Context/Qualifiers/Tip(s) From Piloting Teacher This task turned out to be a medium-length task for my students, as many needed to create a model to understand the task. Students often are not given the opportunity to visualize the inside of three-dimensional objects. This task requires students to do so in order to achieve a solution. It also requires students to find a pattern to solve the problem in that it is very time consuming creating a model that is 10 layers high. To make my own model to motivate students in solving the task, I used Nestle Crunchkins (foilwrapped chocolate candies) and hot-glued a pyramid of them together. I used a digital camera to take a photo to embed in the task. I recommend that you create your own model for students to look at and hold. You can also create the pyramid with other objects that may fit a unit you are studying, such as using moth balls to make a model of a snowball pyramid. Many of my students used acorns to build their models! It is often a challenge to get students to document strategies when creating models or acting out solutions. Having a digital camera in the classroom available for students to use to record their work in progress not only motivates students but is also a wonderful way to document their thought processes without being time consuming or laborious. After competing the task, students can go back and label their photos to communicate to their audience their thought processes, reasoning and proof. There is a wonderful opportunity when presenting the task to teach students geometry vocabulary such as face, vertex, equilateral triangle and triangular pyramid. To print a copy of the more accessible version of the task with the graphic image, refer to page 6. To print a copy of the more challenging version of the task with the graphic image, refer to page 7. Links This task could link to any unit you are doing simply by changing the medium used in making the pyramid. Other objects that could be used are round macaroni (for a unit on Italy) or different types of cereal balls, such as Cap'n Crunch (for a unit on pirates). You could also link this task to a unit on ancient pyramids! Common Strategies Used to Solve This Task Most students will need to either create a model or make a diagram to understand the different levels of the pyramid. From there, students will need to make some sort of chart to organize the information, to look for patterns, to extend them and to draw conclusions. Possible Solutions Nestle Crunchkins pyramid 4 layers high = 20 Nestle Crunchkins pyramid 5 layers high = 35 Nestle Crunchkins pyramid 10 layers high = of 12

5 More Accessible Version Solution: Nestle Crunchkins pyramid 3 layers high = 10 Nestle Crunchkins pyramid 4 layers high = 20 Nestle Crunchkins pyramid 5 layers high = 35 More Challenging Version Solution: Nestle Crunchkins pyramid 5 layers high = 35 Nestle Crunchkins pyramid 10 layers high = 220 Nestle Crunchkins pyramid n layers high = [n(n + 1)]/2 + {(n 1)[(n 1) + 1]}/2 + {(n 2)[(n 2) + 1]}/2... Task-Specific Assessment Notes General Notes In order to be most successful with this task, students will need to create a mathematical representation. Clearly labeled ones will assist in documentation and will affect the level of communication with the audience. This task lends itself well to students identifying patterns and underlying math concepts such as that of triangular numbers. Therefore, there are many opportunities for students to make connections. Novice The Novice will have no strategy for understanding the contents of the pyramid, seeing it as a hollow figure. No problem-solving strategies will be engaged, and no math language will be used to communicate. Apprentice The Apprentice s approach will lack organization, leading to an incorrect solution. If students don t organize their work, they will not be able to see the pattern or achieve a correct solution, as it is important, even when creating models, to document the number of items in each layer along the way. Practitioner The Practitioner will have an organized approach that will lead to a correct solution. Expert The Expert will not need to draw or build every layer of the pyramid but s/he will recognize and use patterns to arrive at a correct solution or will make a mathematically relevant comments or observations. 5 of 12

6 More Accessible Version Worksheet Pictured below you will see Nestle Crunchkins stacked in a triangular pyramid. Each layer is in the shape of an equilateral triangle, and the top layer is a single Nestle Crunchkin. How many Nestle Crunchkins are in a pyramid 3 layers high? How many Nestle Crunchkins are in a pyramid 4 layers high? How many Nestle Crunchkins are in a pyramid 5 layers high? 6 of 12

7 More Challenging Version Worksheet Pictured below you will see Nestle Crunchkins stacked in a triangular pyramid. Each layer is in the shape of an equilateral triangle, and the top layer is a single Nestle Crunchkin. How many Nestle Crunchkins are in a pyramid 5 layers high? How many Nestle Crunchkins are in a pyramid 10 layers high? How many Nestle Crunchkins are in a pyramid n layers high? 7 of 12

8 Novice 8 of 12

9 Apprentice 9 of 12

10 Practitioner 10 of 12

11 Practitioner 11 of 12

12 Expert 12 of 12