Hi, Parents,
Throughout this year, we will tackle some challenging problems in math. Currently, I am working on some visual growth patterns with the scholars. These problems are wonderful for getting students to discuss how they visualize problems and lead to creating equations to describe the patterns they find. Scholars are pushed to find figures in the pattern that would be tedious to draw out. This leads them to look for more efficient strategies. In early stages, scholars often try to “multiply” to find the area of a larger figure. They may not recognize why multiplying figure 10 by 10 will not give them the area of figure 100. I often ask them to check smaller figures. Does doubling figure 2 give you figure 4?
We are embracing the idea that “Mistakes and challenges are the best time for your brain!”
These problems also encourage scholars to “attend to precision” which is one of the common core mathematical practices. If scholars do not accurately portray the figures, I ask them to check to see if they are following the pattern that they identified.
I also love these problems because they are very accessible for all students. Scholars who are not ready to write equations, can still identify patterns and create other figures in the sequence. As scholars share how they see the patterns grow with others, they start to understand how some visualizations lead more easily to developing equations that describe the pattern than others.
I am including some of the student representations of the patterns that they saw as they worked with the “Growing, Growing, Growing…” problem, along with an explanation of how their visualization can lead to an algebraic solution.
In the visualization at the right, a scholar showed how she saw the pattern growing. Each new color accurately shows the tiles added to the previous figures.
A key step in the development of describing these figures with equations is when scholars learn to use the figure number, and not just the previous figure, to find the next figures in the sequence.
Throughout this year, we will tackle some challenging problems in math. Currently, I am working on some visual growth patterns with the scholars. These problems are wonderful for getting students to discuss how they visualize problems and lead to creating equations to describe the patterns they find. Scholars are pushed to find figures in the pattern that would be tedious to draw out. This leads them to look for more efficient strategies. In early stages, scholars often try to “multiply” to find the area of a larger figure. They may not recognize why multiplying figure 10 by 10 will not give them the area of figure 100. I often ask them to check smaller figures. Does doubling figure 2 give you figure 4?
We are embracing the idea that “Mistakes and challenges are the best time for your brain!”
These problems also encourage scholars to “attend to precision” which is one of the common core mathematical practices. If scholars do not accurately portray the figures, I ask them to check to see if they are following the pattern that they identified.
I also love these problems because they are very accessible for all students. Scholars who are not ready to write equations, can still identify patterns and create other figures in the sequence. As scholars share how they see the patterns grow with others, they start to understand how some visualizations lead more easily to developing equations that describe the pattern than others.
I am including some of the student representations of the patterns that they saw as they worked with the “Growing, Growing, Growing…” problem, along with an explanation of how their visualization can lead to an algebraic solution.
In the visualization at the right, a scholar showed how she saw the pattern growing. Each new color accurately shows the tiles added to the previous figures.
A key step in the development of describing these figures with equations is when scholars learn to use the figure number, and not just the previous figure, to find the next figures in the sequence.
This scholar broke her pattern into areas that can easily lead to the development of an equation. Although it is not the “most efficient,” it leads to a very valid solution. The left column is the figure number (x) plus 2, The middle rectangle is (x-1)(x+1). The low rectangle on the right (students often call it the “tail”) is x-1. This leads to the expression (x+2) + (x-1)(x+1) + (x-1) which will give you the total number of tiles in any figure. |
Here a scholar is seeing patterns that will lead them to an efficient algebraic strategy. He saw a square in the middle with an area of x squared. The bottom rectangle is 2x-1. If you add the single square on the top to the bottom rectangle, you get 2x. So, to compute the area of any figure in this sequence, you can use x squared + 2x. |
Sometimes scholars recognize that you could break off the “tail” and move it.
As your scholar works on these types of problems, encourage them to think about how they might break the figure apart to make it simpler. If they are not ready for the algebraic equations yet, don’t push them to get there. This understanding will come as they share strategies with their classmates. Please don’t feel the need to solve the problem for them or show them how you would do it. We want to value their math understanding and let them focus on how they see the problem.
An added layer to this mathematical understanding is the work we are doing with writing in math class. Scholars are working to clearly explain and support their thinking. This is a struggle for many, since they are not used to writing about math. As the year progresses, I think you will see how much more clearly they think about and explain the way they solve their math problems.
I hope to hold a parent meeting about math soon. I hope you will join me for that. Thanks for reading this lengthy math letter! I truly enjoy working with your children to help them learn about how math helps us understand our world!
Warmly,
Mrs. Campbell
- “I can take the sticking-part out and put it on the blank space on the top row to make a big rectangle. We can multiply the number of rows by the number of columns to get the number of tiles.”
As your scholar works on these types of problems, encourage them to think about how they might break the figure apart to make it simpler. If they are not ready for the algebraic equations yet, don’t push them to get there. This understanding will come as they share strategies with their classmates. Please don’t feel the need to solve the problem for them or show them how you would do it. We want to value their math understanding and let them focus on how they see the problem.
An added layer to this mathematical understanding is the work we are doing with writing in math class. Scholars are working to clearly explain and support their thinking. This is a struggle for many, since they are not used to writing about math. As the year progresses, I think you will see how much more clearly they think about and explain the way they solve their math problems.
I hope to hold a parent meeting about math soon. I hope you will join me for that. Thanks for reading this lengthy math letter! I truly enjoy working with your children to help them learn about how math helps us understand our world!
Warmly,
Mrs. Campbell