I’ve been doing research on the Trends in International Mathematics and Science Study (TIMSS) lately and came across this sample 8th grade question (source):
PQRSTU is a regular hexagon. What is the measure of the angle QUS?
A. 30º
B. 60º
C. 90º
D. 120º
All the other questions (including the 4th grade ones) seem reasonable (easier than I expected, even) but is this one really an 8th grade question? Even my current geometry students would have difficulty. I don’t recall ever getting anywhere past area and perimeter in middle school; is the TIMSS expecting a full geometry class by 8th grade, so algebra would be taken 7th grade? I imagine the majority of students could still solve this problem by eyeballing the figure and guessing, but that hardly seems the intent of the question.
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The Number Warrior 
Anecdote: I was taught http://en.wikipedia.org/wiki/Inscribed_angle_theorem before I had any hope of thinking about it apart from “oh okay here’s a rule I’ve memorized let’s apply it”.
I extrapolate from that to assume everyone’s taught it mega-early. 🙂 If so, the hexagon problem is a fine little problem of recognizing which memorized rule to apply.
del rettangolo abcd della figura a lato si sa che: ^
AE = EF = FD = AD = 18CM,
One doesn’t need to calculate anything to solve this problem, nor is it necessarily a matter of vague eyeballing. Simply recognize that, because of the symmetries of a regular hexagon, QUS is an equilateral triangle, and remember the fact (taught long before 8th grade, I’m sure) that the interior angle in an equilateral triangle is 60 degrees.
Simply recognize that, because of the symmetries of a regular hexagon, QUS is an equilateral triangle
I guess my question straight out then is: do 8th graders know enough to do this?
(I am also happy accepting it’s simply meant to be a boundry-of-possible-skill problem, which is fine; it’s just this problem was the only case of the samples I looked at that left me wondering.)
I teach high school Geometry to primarily 9th and 10th grade students. The problem is a bit challenging, but not out of the realm for something the students in my class could conquer. Knowing about interior angles of a regular polygon (check) and a bit about types of triangles (check) makes this one of those “bring a few ideas typically taught in isolation together” problems.
Granted after a proper geometry class one can do this, but do they know this in 8th grade?
U.S. students, in both grades 4 and 8, have consistently performed least well in geometry and measurement when compared against themselves in other areas: numbers, data & statistics (we excel!), and algebra (1995, 1999, 2003, 200).
As part of a small video study during TIMSS 1995, one of the public release videos from the U.S. included a geometry lesson, where, toward the end, the teacher gives the students the formula for finding the sum of the angles of regular polygons. By contrast, in other countries, students are asked to generalize a way of finding the sum of the angles of any regular polygon, knowing the sum of the angles of a triangle and a quadrilateral.
D. Eppstein’s simple but elegant solution requires that students know: 1. what is a regular polygon, 2. definitions of isosceles and equilateral trianges, 3. sum of the angles of a triangle.
Geometry begins seriously in grade 4 and is taught at every grade after this in most other countries.
Re < Granted after a proper geometry class one can do this, but do they know this in 8th grade?
They should.
They should.
Granted.
Will you grant, then, that the US curriculum needs a major shift to hit this by 8th grade?
I think we’re still feeling the backlash of the anti-Geometry sentiments in math ed from the 60s. Just for reference, my students come in not knowing what an acute or obtuse angle is, let alone a regular polygon or the number of degrees in a hexagon or triangle.
The problem above is solvable after chapter 5 in the US high school Discovering Geometry textbook.
Re < Will you grant, then, that the US curriculum needs a major shift to hit this by 8th grade?
Yes, particularly since you say:
"my students come in not knowing what an acute or obtuse angle is, let alone a regular polygon or the number of degrees in a hexagon or triangle."
Their lack of knowledge may not be due to the fact they never were taught acute/obtuse angles, but that they didn't remember/learn it?
A 7th grader, when asked to solve 0.87 x 10, gave an answer of 0.870. What he remembered from elementary school was that when you multiply by 10, you add a zero.
Perhaps we need to be continually assessing what students understand from what we say to them?
Re < The problem above is solvable after chapter 5 in the US high school Discovering Geometry textbook.
This would be grade 9 for "advanced" students and grade 10 for regular students?
the angles in a regular polygon: not learned.
acute / obtuse: sometimes learned, but it is as you point out forgotten.
This would be grade 9 for “advanced” students and grade 10 for regular students?
Yes.
My comments aren’t going to be constructive or helpful, but I have to say this problem should not belong in the 8th grade curriculum. To provide a comparison with the 8th grade math (and science) syllabus in India, say, one may take a look at this product, for instance.
Re < this problem should not belong in the 8th grade curriculum.
Where would you suggest it belongs?
The product is for what grade(s)?
As I said, the product is for 8th grade students.
I am sorry I forgot to answer your other question. The problem should not belong to a curriculum higher than the 6th. (That’s my opinion, of course, but its based on personal experience.)
were the the line QS to be drawn in, it would be almost trivial (5th grade?). Without the line, it seems quite a lot harder.
I would expect my 8th graders to do be able to do it, but I have an advanced class. I think you would be hard pressed to find a “regular” 8th grade class in CA that would expect their students to do a problem like this one.
Not for average students in Lafayette Indiana in my opinion.
I distinctly remember being taught about the fact that the sum of angles in a triangle equals 180° in 4th, maybe 5th grade. Basic properties of isosceles triangles are also taught before 8th grade (at least here in Italy), and AFAIR also properties about regular polygons.
It’s trivial to see that PQU is isosceles, but I have to admit that the subsequent necessary adding and subtraction of angles necessary to solve the problem are not really fit for eight graders. Still, I’d keep the problem in for extra points 😉
Re < subsequent necessary adding and subtraction of angles necessary to solve the problem are not really fit for eight graders.
There is no need to add/subtract angles, if one sees that one has an equilateral triangle, right?
Yep, that’s a faster way I didn’t see.
PQU, STU and QRS are congruent since two sides and one angle are congruent, but I can’t remember when congruency conditions are taught.
@agos: no need to consider triangle PQU. Imagine the line QS is drawn. QUS is now an _equilateral_ triangle (because it’s vertices are evenly spaced around the same circle as the hexagon).
I do think most 8th graders know that the angles of an equilateral triangle are each 60 degrees, But I think only a few would have the insight to draw the right line to make that all clear.
my students come in not knowing what an acute or obtuse angle is, let alone a regular polygon or the number of degrees in a hexagon or triangle.
I don’t remember what grade level you teach. But I do know that the school where I volunteer, kids are taught acute/obtuse from about 5th grade on. They may not remember into high school, but they certainly are exposed many times. They also learn the definition of a regular polygon. 6th graders learn how many degrees are in a triangle and a quadrilateral, and that is something they do appear to remember. We also hit the pythagorean theorem and similar triangles (great practice for proportion problems). The only middle schoolers who would learn how many degrees in a hexagon would be honors-level students. But as noted above, knowing how many degrees in a hexagon is not required to solve this problem.
AZ Grade 7 Standards: Describe the relationship between the number of sides in a regular polygon and the sum of its interior angles.
MA Grade 7 Standards: Analyze, apply, and explain the relationship between the number of sides and the
sums of the interior angle measures of polygons.
Real students, other than eyeballing – unlikely.
I think the problem is a nice assessment of how well students are able to use/put together what they have learned in earlier grades. Several of you indicate your students in grade 8 won’t be able to solve the problem now. The problem perhaps shows us where we should go and what we should do across the grades so they will be able to solve such problems?
Here’s an offer. If you send me an email message (pwangiverson@gmail.com) with your mailing address (no PO box, please, in case I use UPS), I will be happy to send you a copy of Elementary Geometry for Teachers by Tom Parker and Scott Baldridge, which is aligned with the scope and sequence of Primary Mathematics, U.S. edition (Singapore). Some of the homework assignments come from the PM texts and New Elementary Mathematics I text (equivalent to our grade 7).
Yesterday I did a survey of my (relatively sharp) geometry students. After discovering that a triangle has 180º I asked how many had heard it before. 3 hands went up (out of 30).
The upside is the Common Core standards (which I will be posting about fairly soon) present an opportunity to reboot the US teaching of geometry.
Here is a Singapore 2009 Primary School Leaving Exam (PSLE) question (Grade 6 students took the exam last week):
Jim bought some chocolates and gave half of it to Ken. Ken bought some sweets and gave half of it to Jim. Jim ate 12 sweets and Ken ate 18 chocolates. The ratio of Jim’s sweets to chocolates became 1:7 and the ratio of Ken’s sweets to chocolates became 1:4. How many sweets did Ken buy?
Patsy
Oddly enough, I was about to post about that. Some of the Signapore bloggers boggled. There’s also a geometry question which seems questionable.
What’s the geometry question?
Thanks.
Just to toss in another data point (re: the comment that this is a grade 6 problem in India) here is a textbook from India online:
http://ncertbooks.prashanthellina.com/class_6.Mathematics.Mathematics/index.html
Nothing seems to indicate the students would be able to solve the problem above. Grade 7 however, does give enough background:
http://ncertbooks.prashanthellina.com/class_7.Mathematics.Mathmatics/index.html
Just so everyone is aware, the grade 6 mathematics textbook that Jason is referring to is an NCERT textbook. Science and Math textbooks published by NCERT generally tend to be of a “lower” level when compared to CBSE and ICSE textbooks. It wouldn’t be an exaggeration at all if one said that an NCERT grade 7 (science or math) textbook would be equivalent to an ICSE grade 6 textbook. My earlier comment about the geometry problem being that of grade 6 level was based on the ICSE “benchmark”. I guess I should have qualified my statement earlier; my apologies for not doing the same earlier.
For a comparison between ICSE and CBSE (as well as NCERT), here’s mom’s perspective.
Thank you, that was most edifying.
DEL RETTANGOLO ABCD DELLA FIGURA A LATO SI SA CHE
AE = EF = FD = AD =18CM;
The segment joining the point on the circle to the centre is called——————
a) diameter
b) radius
with explaination
Every 8th grader should be able to eyeball the picture and get the 60 degree answer. The problem would be much harder if it weren’t multiple-guess with such a huge difference between the answers.
This is a tricky question and I definitely think it’s for 9th graders. After all, most 9th graders take Geometry. You have to know a certain formula. The formula for determining all of the angles added up in a polygon is 180(n-2) with n being the number of sides. Then to figure out the degrees of each angle of a regular polygon, you must divide that number by the number of sides it has. So, 180(6-2)= 720. 720/6= 120. Since this is a regular polygon, angles SUT and QUP are congruent and when added together are equal to angle QUS. So, 120/2= 60.
I believe that most good 8th-grade math students would be able to answer this question correctly; if not “by sight”, then at least after a little thought. As recently appointed Math Coordinator at my private school, I am currently in the process of “de-celerating” our curriculum. Currently (and for one more year of “grandfathering”) I teach an 8th-grade Honors Geometry course–a complete and rigorous curriculum. However, I believe that this is more age-appropiate when taught in 9th grade (still an “accelerated” track.) Just because students “can” handle advanced math at a younger age does not necessarily mean that they “should”.
Hi, I’m an Eighth Grader in Indonesia (Southeast Asia) and we had this in grade 6. Yes, Algebra was taught to us in Grade 6 and by Grade 8 we had studied how to factorize quadratic equations 😀 This is quite a simple question, if I may say, since even if the average 8th grader could not see the segment QS, then he or she would still be able to find the sum of interior angles of a hexagon, then PUQ is an isosceles triangle, they will find eventually that angle UQR and USR are right angles. After finding that angle QRS is 120° by dividing the interior angle sum of a regular hexagon by 6, then they would be able to find angle QUS by subtracting all above values from 360° to get 60 °.
I did pre algebra in 6th grade Algebra in 7th and Geometry in 8th. I live in California.
Yeah, there’s a fair number of people who do hit Geometry in 8th; I’ve got some of them this year. The question is are we expecting _everyone_ to (alternately, they are able to answer the question above without having a full geometry class).