Math Memos

# The Goats and Chickens Problem

• the variety of possible solution methods: arithmetic, algebraic, pictorial, mathematical, and representational;
• the humor inherent in the problem;
• the potential fun for students and their joy in working it out;
• the gender switch on Farmer Montague;
• seeing the different ways students draw the chickens and goats;
• being surprised at who solves the problem and how;
• seeing the light come on in students’ eyes when they arrive at the solution.

This problem pushes students to think outside the box. It is non-discriminatory in that I can, have, and did give this and a similar problem to students learning at the most basic level and those taking the HSE exam in the same week.

# The Ice Cream Combinations Problem

With this problem, I hope to observe how students will organize data and if they are able to recognize patterns. I also wanted to see what counting techniques the students might use to find all of the possible outcomes. Some of my students have experience developing harts and graphs, while others are new to these concepts. I was interested in seeing which students would figure out different solutions and help others to formulate solutions if they were having difficulty. My class is composed of students with math grade equivalents between 5.0 to 11.0 on the TABE.

# The Sum and Difference Problem

I like this problem because I think it’s a good one for introducing problem-solving strategies with a new influx of students that had just begun my class. Because the computational skills required to successfully solve the problem are low, I felt this would remove an anxiety barrier some of my newer students may present with. Additionally, I had just received a few new students who have very low TABE reading levels and much higher math scores. I felt this “less wordy” problem would allow us to focus more on developing the problem-solving skills, without overwhelming them at the same time with a longer story problem. For my students who have been with me longer, many still struggle with organizing their work, recognizing patterns, or persisting long enough to make their guesses count. This problem could provide room for growth for everyone.

# The Arch Problem

I like the arch problem because it starts as a visual model, and then challenges students to develop their algebraic thinking skills. It’s also a problem that allows students of all ability levels to be successful. Lower-level students may be able to simply sketch some of the figures but might be truly challenged by some of the questions that show up further in the handout. Higher-level students will be challenged by the bonus questions, and I can even ask them to go a step further and create an equation that they could use to calculate the number of squares in any figure. The problem also asks for written explanation, so students will have to explain their mathematical thinking verbally–not just with numbers or pictures. This is something that students often struggle with, but being able to articulate mathematical ideas in writing is a skill that can help them to make their learning more concrete and to emphasize key vocabulary.

# The Multiples of Nine Problem

I recently discovered this problem, and I really like it for a number of reasons. First, it requires a little bit of vocabulary in order to get started. Students will have to know what a multiple is, they will have to know what digits are—and more specifically, how digits can differ from numbers—and they’ll have to understand the difference between even and odd numbers. I also like how nonintimidating it looks at first glance. “How hard could it be to find a multiple of 9 that has only even digits? I shouldn’t have to count up very far.” Because the problem doesn’t look lengthy or challenging, it comes as a surprise when the correct answer is actually the 32nd multiple of nine. I anticipate a lot of students writing out 9, 18, 27, 36, 45, 54, etc, and then getting frustrated or giving up when they don’t get to the answer fairly quickly.