Wow! Five weeks of school have flown by!
In Math, we have been working to improve our math communication and explain our reasoning. We learned that “Mistakes are Valuable.” Mistakes and challenges are the best time for your brain to learn. Although we usually prefer not to struggle, struggle and mistakes help our math brains grow. We are learning the most when we are struggling at the edge of our understanding.
The Towers problem that we worked on last week was a great example of this concept. My goal with this problem was to give scholars a problem that would not present an easy solution, but that they would need to think deeply about. Scholars started by building the model in class using blocks. Scholars were encouraged to record their model on graph paper in cross-sections.
In Math, we have been working to improve our math communication and explain our reasoning. We learned that “Mistakes are Valuable.” Mistakes and challenges are the best time for your brain to learn. Although we usually prefer not to struggle, struggle and mistakes help our math brains grow. We are learning the most when we are struggling at the edge of our understanding.
The Towers problem that we worked on last week was a great example of this concept. My goal with this problem was to give scholars a problem that would not present an easy solution, but that they would need to think deeply about. Scholars started by building the model in class using blocks. Scholars were encouraged to record their model on graph paper in cross-sections.
Scholars had two homeschool days (equivalent to 2.5 hours of math class) to think and write about this problem. (Many parents were struggling alongside their children. Way to go, parents! I hope that you were able to share with each other that this problem made you think!) Some scholars reached out to ask clarifying questions. So exciting for me!! I encourage scholars to do this more, as it is so much better than being “stuck” because you don’t understand!
Since scholars were only given the 5th case of the pattern, they had to make inferences about the other cases. They had to infer and support a pattern that they saw. Scholars saw the pattern in several different ways, depending on the view that made the most sense to them. Some scholars saw the cross-sections in View 2 of the problem more clearly: | Other students saw View 3 more clearly. Either view led to identifying a pattern that would allow students to make a conjecture about the height of Case 100. |
Many students assumed that since 100 is 20 times 5, they could find the height of the 100th case by multiplying the height of the 5th case by 20. They concluded that since the height of the 5th Case was 32, the height of the 100th Case would be 32x20= 640. This method has a flaw, since the pattern does not grow in this way. We are learning about using smaller cases to test out our ideas to see if they work. Testing this with Case 2 and Case 4 would have uncovered the error.
Some students noticed that the height of each case grows by 8. If they decided that Case 1 has zero blocks, then they often realized that the height of Case 5 was 8x4=32. 4 is one less than the Case number (5-1). If they tested this with smaller cases, the rule would hold.
One student reasoned that they needed to divide the number of blocks in the 5th Case by 5 to find the number of blocks in the lower cases. This is a great mathematical idea! They showed their ideas using diagrams that followed a mathematical pattern, but did not lead easily to a way to find the height of the 100th Case.
Some students noticed that the height of each case grows by 8. If they decided that Case 1 has zero blocks, then they often realized that the height of Case 5 was 8x4=32. 4 is one less than the Case number (5-1). If they tested this with smaller cases, the rule would hold.
One student reasoned that they needed to divide the number of blocks in the 5th Case by 5 to find the number of blocks in the lower cases. This is a great mathematical idea! They showed their ideas using diagrams that followed a mathematical pattern, but did not lead easily to a way to find the height of the 100th Case.
A few students worked to show their ideas 3-dimensionally. One student even worked with a parent to program a 3 dimensional image of the growth pattern.
Students are learning about a structure to use for writing more clearly about our ideas in math. Students are asked to “restate the problem” in the same way that they restate the question in answering essay questions. This helps the students summarize the information that they are given and identify their tasks they need to complete. Students should answer and explain their reasoning for anything that they are asked to solve. We are working on Claim > Evidence > Reasoning writing in Math. Students state their claim (the answer they are supporting). Then they give evidence to support their claim (usually how they calculated or thought about their answer). Last, they explain their reasoning for their strategy, Scholars were scored on their writing for this assignment and I also looked at their math understanding.
In a reflection, one scholar wrote, “To grow, I will read all of the directions before and after I finish the problem…” What a great plan for future assignments!
I look forward to more challenging learning with our awesome scholars!
In a reflection, one scholar wrote, “To grow, I will read all of the directions before and after I finish the problem…” What a great plan for future assignments!
I look forward to more challenging learning with our awesome scholars!