# Tasks, Units & Student Work

## 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 B (Performance at Standard )

Students at this level still struggle with giving good verbal descriptions of what they see. The biggest difficulties were in finding generalizable rules rather than recursive rules. 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.

Students need opportunities to compare and contrast strategies and to evaluate their usefulness. So during class discussions it is important not to stop after several students have shared how they solved the problems. Teachers need to ask students to think about and reflect on the strategies. "Which one is easier to use? Why? Which one would be most useful for finding the 1000th term? Why? What is the same about both formulas or strategies? What is different? How can we be sure that they will always give the same solution?"

Students need more opportunities to develop justification and make convincing arguments. They should have frequent opportunities to question, to critique, and to improve the arguments of others. "Does this convince you? Why or Why not? What would make it more convincing? Why do you disagree? How could you convince the person to change their mind?"

### Student C (Performance below Standard)

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.