**The final angles problems in 6th grade Everyday Math (first) and 6th grade Singapore Math (second):**

Extra Credit: a. Compare the 1-3 step Everyday Math problems with the 4-6 step Singapore Math problems. Why does only the former request explanations for answers? b. Relate your answer to (a) to the Everyday Math "Time to Reflect" questions that follow this section, in which pride, challenge, and learning are supposed to be made explicit: Which activity in this unit do you believe is an example of your best work? Why do you think so? Which activity in this unit did you find the most challenging? Why? What is something new you learned about geometry in this unit? |

## 4 comments:

It took me a few minutes to solve the first problem. Is this right? I'd like a quick sanity check, because I'll be teaching this to my daughter.

What you need to know:

1.) opposite angles of a rhombus are equal.

2.) The sum of the angles of a triangle is 180 degrees.

And the rest is logic, right?

4.) Facts 1 and 2 imply angles ABD and ADB, which sum to 140 degrees, are each 70 degrees.

5.) Now look at the triangle AED. From Fact 2, we have angle EAB = 180 - 40 - 40 - 70 = 30 degrees.

OK, I think I've got it down.

Yup, that's it! One thing I love about these Singapore geometry problems is how much logic is involved. It's a very nice preparation for the proofs that students later do (or used to do!) in high school level geometry.

Katherine, I like the logic angle too. And I like the way it's not just about memorizing algorithms. Frankly, my memory is lousy, so my strategy is to memorize as few things as possible and derive the rest.

That's why I was so pleased to discover you only need 2 facts to solve the problem. And the 2 facts both involve rock-bottom, non-derivable conventions; namely, the definition of a rhombus and the number of degrees in a line.

When I was teaching calculus in graduate school, I often had students complain about certain problems: "But that's like, logic, not math." I am having a a hard time recalling what the offending problems were. One was like this: you have a triangle with two sides of length 1; think of the area as a function of the angle. First, what if any is the maximum area; second, what if any is the minimum area. There is no minimum area. The most popular answer tends to be 0. Here the student has to understand that the angle is greater than 0 and less than 180. The logic part is recalling that a triangle has three sides and realizing that a line segment does not have three sides; or, three colinear points do not form a triangle.

This same complaint about logic came up in exponential problems and others. This was mostly from students who had had calculus in high school and were taking it again in college. Many of them had an on-going hard time with the fact that college calculus was not the same as their high school class. They wanted to plug and chug.

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