Tasks, Units & Student Work

High School Algebra: Aussie Fir Tree

High School Algebra: Aussie Fir Tree

The Aussie Fir Tree task is a culminating task for a 2-3 week unit on algebra that uses the investigation of growing patterns as a vehicle to teach students to visualize, identify and describe real world mathematical relationships. Students who demonstrate mastery of the unit are able to solve the Aussie Fir Tree task in one class period.

Suggested Use: Review the task and rubric before looking at the student work. Then, look at the student work and click on the red arrows to see an explanation of the student's performance on the task. Scroll down to the bottom of the student work to see suggested instructional next steps. 

Student A (Achieves Standards at a High Level )

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Student B (Performance at Standard )

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Student C (Performance below Standard)

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Grading Criteria

Many students at this level are using recursive rules to extend the pattern. While this is helpful and easy to use for a small number of cases, it is cumbersome and prone to errors when trying to extend the pattern for larger numbers. Students need to see that it is more helpful to find more generalizable rules to solve problems for all cases.

One helpful way to do this is to ask students break down the pattern into simpler parts. "What do you see when you look at the pattern? How can you decompose the shape into parts? How does the tree trunk grow? How could we use algebra to describe how to find the tree trunk for any pattern number?" Breaking down the problem into simpler steps helps students manage the thinking in smaller chunks. But students in this stage need to learn questions to help them progress in their thinking, to develop strategies for finding a generalizable rule.

Students need to be encouraged to give more detail about what they see. So in class the teacher might ask, "That's interesting, can you tell me a bit more? Or where do you see the n in the diagram or the (n+1)?" The more detailed their descriptions usually the easier it is to quantify the ideas symbolically. Students should also be more descriptive in thinking about classes of numbers. In elementary school it is good to notice that numbers are odd or even, but by this grade level students should start to classify numbers as consecutive or consecutive odd numbers, multiples of . . ., powers of . . . , triangular numbers, etc. The types of patterns that students think about should be expanded.

Students need to have experiences thinking about types of linear patterns; those that are proportional and those that are not proportional. Take the work of student 692. The strategy of doubling from case 5 to case 10 works for proportional patterns, but not for patterns with a constant. Students can benefit by looking at two cases at the same time and comparing which one will work by doubling and which one won't. Drawing graphs of the situations can help to clarify this idea.

Student D (Demonstrates Minimal Success)

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Grading Criteria

Students need help learning to describe accurately and precisely what they see as the visual pattern grows. These descriptions become tools that can be used to describe the pattern algebraically.

Students need ways of organizing their thinking, such as making tables to see how the pattern grows numerically. The work done by student 209 can help to lead to a full solution algebraically, but students need discussions about the purpose of the tools. Students at this level don't know or understand the purpose of the tools, so even when they make a table that aren't sure of what they learn from it that can help them make the generalization. A useful teaching devise is self talk. The teacher starts to solve a problem or a student describing a solution goes part way, then the teacher stops and asks, "What do you think comes next?" Students need to start anticipating how the solution progresses. In this case, students might notice that each stage increases by a consecutive even number.

Students need to talk about cases. When making a conjecture, they need to understand that just one example is not sufficient. They need to test their ideas against several cases to see if it at least holds true for all the examples that they have available. At later stages of their development, they will learn that there are never enough examples to prove a case and that the proof lies in the physical geometry of the pattern. However, many students at this level made a generalization, when they had other examples on their paper that disproved their assertions. Students need frequent opportunities to work with patterns, organize the information for them, and make and test conjectures.