Vertical angles, as you will recall from high school Geometry, are the opposite angles that appear whenever two lines cross. Despite the name, vertical angles need not be vertically aligned. The image to the left has two pairs of vertical angles.
My students are studying vertical angles, so I challenged them with the following question.
How many pairs of vertical angles are in the diagram below?
Some students answered "three pairs," while others said "six." I asked students to justify their answers by listing the pairs of vertical angles. Students quickly found the three pairs below.
I asked, "Did anyone find a fourth pair?" An intrepid student raised his hand. "Angles C X F and D X E." These vertical angles are harder to see, because of the extra line that runs through them. Students responded to his discovery with "ohhhh!!!" and a well-deserved round of applause. The final two pairs of vertical angles were then easy to find, bringing the total to six pairs of vertical angles in the diagram.
Student imagination and communication are on display in the following activity that I give annually to my Geometry class at the beginning of the year.
How do you get from
using only these three rules?
One solution is
Now imagine that there is a fourth rule, but you don't know what it is. All you know is that your friend used Rule 4 to get from
Students came up with several suggestions for what the mystery rule could be.
How do we know what Rule 4 actually is?
We need more data to test our three hypotheses!
In my math class students learn to apply math they know to new situations and to communicate about the process. As an educator, I am constantly creating new ways to teach these skills.
By initially taking away numbers from the problem, I force students to use language to describe their solution. I am careful to check that they use precise language and that their work in part 2 reflects their response from part 1.
Has the work you've done ahead of time prepared you to answer this final question?
Learning Vietnamese is like playing the popular party game Code Names in reverse. In Code Names, players link seemingly unrelated items with a single-word clue. In Vietnamese, I encounter seemingly unrelated compound words which share a common component. I have to figure out with the common link is.
In Code Names a player may need to think of a single word to link the items “bank”, “fortune”, and “story”. The clue “teller” would be a good link for these three items, invoking the compound words “bank teller”, “fortune teller”, and “story teller”.
In unrelated situations around Vietnam, I encountered the compound words “quý khách” (customer), “xe khách” (tour bus), and “khách sạn” (hotel). So, what concept does the word “khách” represent on its own? Click here for the answer.
Another example of seemingly unrelated compound words revealed itself only gradually. I first learned the word “mật ong” for “honey”. “Ong” means “bee”, so I guessed “mật” was another word for “food”.
Later, I learned the word “mật gấu” for bear bile (extracted from live bears for medicinal use). “Gấu” means “bear”, so perhaps “mật” is better translated as “animal by-product”.
Finally, I encountered “mật mía” for “molasses” or "syrup". “Mía” means “sugarcane”. Keeping all three of “mật ong”, “mật gấu”, and “mật mía” in mind, I conclude that “mật” is closest to “(sweet) thing that comes from a plant or animal by some process”.
With that in mind, I propose “mật ngô” and “mật me” be added to the Vietnamese lexicon for “corn starch” and “tamarind pulp”, respectively. My Vietnamese friends will have to tell me how far off I am!
I can now count "conference workshop presenter" among my many roles as a mathematics educator. This weekend I had the privilege to present on Sources of Authority at the Vietnam Tech Conference in Hanoi. My presentation was about techniques to empower students to check and verify an answer to a math problem without asking the teacher. For my presentation I pulled from topics I've discussed earlier on my blog.
Pen & Pencil - students use contrasting writing implements to show their corrections
Math Mosaic - students work individual efforts into a group creation
Students Reverse - students complete problems which are secretly the answer key to a partner's problems
Side-Side-Side Construction - students test their geometric construction against the authority of Geogebra software
Thanks to all the people who attended my presentation. You can find the handouts below.
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Hanoi, Vietnam. Your comments are always welcome.