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 describing the pattern than others and they start to see how patterns can be defined algebraically.
I am including some of the student representations of the patterns that they saw as they worked with the “Bridges” problem, along with an explanation of how their visualization can lead to an algebraic solution.
Scholars usually see clearly how the bridge designs change and that each bridge adds 3 pegs (dots) to the previous bridge design.
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 describing the pattern than others and they start to see how patterns can be defined algebraically.
I am including some of the student representations of the patterns that they saw as they worked with the “Bridges” problem, along with an explanation of how their visualization can lead to an algebraic solution.
Scholars usually see clearly how the bridge designs change and that each bridge adds 3 pegs (dots) to the previous bridge design.
We use the “Patterns” thinking tool to help us focus on looking for and recognizing patterns in math. Scholars virtually all see that each side of the bridge increases by one peg from the previous design.
This scholar drew her bridges in a careful organized way to show their growth. Most students did not indicate how they visualized the pattern. A few who did explain, usually say that they saw a peg being added to the bottom of each vertical side and another peg “in the middle” of the horizontal side. Most students did not use the words vertical or horizontal. Since scholars saw that each bridge design increased by 3 from the last, most of them used some form of “adding on” or “skip counting” to find the pegs in bridge 10.
This scholar drew her bridges in a careful organized way to show their growth. Most students did not indicate how they visualized the pattern. A few who did explain, usually say that they saw a peg being added to the bottom of each vertical side and another peg “in the middle” of the horizontal side. Most students did not use the words vertical or horizontal. Since scholars saw that each bridge design increased by 3 from the last, most of them used some form of “adding on” or “skip counting” to find the pegs in bridge 10.
In the early stages of equation modeling, students often use different variables to describe how they see the figure.
One scholar wrote his thinking this way:
A + A + B
In this case, the scholar was looking at the vertical and horizontal sides this way:
One scholar wrote his thinking this way:
A + A + B
In this case, the scholar was looking at the vertical and horizontal sides this way:
Another scholar wrote her thinking this way. She is close to using the variables we use most often in algebra.
A few scholars realized that if you look at the horizontal row at the top, the pegs left on the sides are equal to the top row, so you could multiply 3 times the number of pegs in the top row. The next step is to recognize that the pegs on each side equals the figure number +1. This leads to the algebraic solution of 3(x+1).
Another scholar wrote his thinking this way:
3 x A + 3 = B
This scholar recognized that the first figure had an “extra” three pegs and that you needed to add those three pegs into each subsequent bridge. In this case, A is the figure number and B is the total number of pegs in the bridge design.
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
Another scholar wrote his thinking this way:
3 x A + 3 = B
This scholar recognized that the first figure had an “extra” three pegs and that you needed to add those three pegs into each subsequent bridge. In this case, A is the figure number and B is the total number of pegs in the bridge design.
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