gerard mcn
@germinalmaths.bsky.social
I teach maths in a Scottish state-sector secondary school.
Fourths or quarters?
December 17, 2025 at 9:18 PM
Fourths or quarters?
I’d guess there’s a strategy mark for obtaining and using the length of the common side. Whats obvious to us experts won’t be obvious to exam candidates many of whom, I suspect, wouldn’t know where to start. Possibly a couple of marks for finding any side length at all.
December 13, 2025 at 8:10 PM
I’d guess there’s a strategy mark for obtaining and using the length of the common side. Whats obvious to us experts won’t be obvious to exam candidates many of whom, I suspect, wouldn’t know where to start. Possibly a couple of marks for finding any side length at all.
1/3 is 7 times 1/21
So
(1/21)/(1/3)=1/7=1/(21/3)
1/3 will go into 8/21 8 times as often as it goes into 1/21
So
(8/21)/(1/3)=8(1/7) = 8(1/(21/3)
4/3 will go into 8/21 4 times fewer than 1/3 will
So
(8/21)/(4/3)=(8/4)(1/7)=(8/4)(1/(21/3))=(8/4)/(21/3)
So
(1/21)/(1/3)=1/7=1/(21/3)
1/3 will go into 8/21 8 times as often as it goes into 1/21
So
(8/21)/(1/3)=8(1/7) = 8(1/(21/3)
4/3 will go into 8/21 4 times fewer than 1/3 will
So
(8/21)/(4/3)=(8/4)(1/7)=(8/4)(1/(21/3))=(8/4)/(21/3)
December 12, 2025 at 9:00 PM
1/3 is 7 times 1/21
So
(1/21)/(1/3)=1/7=1/(21/3)
1/3 will go into 8/21 8 times as often as it goes into 1/21
So
(8/21)/(1/3)=8(1/7) = 8(1/(21/3)
4/3 will go into 8/21 4 times fewer than 1/3 will
So
(8/21)/(4/3)=(8/4)(1/7)=(8/4)(1/(21/3))=(8/4)/(21/3)
So
(1/21)/(1/3)=1/7=1/(21/3)
1/3 will go into 8/21 8 times as often as it goes into 1/21
So
(8/21)/(1/3)=8(1/7) = 8(1/(21/3)
4/3 will go into 8/21 4 times fewer than 1/3 will
So
(8/21)/(4/3)=(8/4)(1/7)=(8/4)(1/(21/3))=(8/4)/(21/3)
I’ve been mis-pronouncing your name (in my head) - sorry, and I now know better.
I’ll wager you get some wrong versions of your surname, too.
People commonly get both my first name and surname wrong; some persist in doing so even after I’ve enlightened them.
I’ll wager you get some wrong versions of your surname, too.
People commonly get both my first name and surname wrong; some persist in doing so even after I’ve enlightened them.
December 8, 2025 at 5:38 PM
I’ve been mis-pronouncing your name (in my head) - sorry, and I now know better.
I’ll wager you get some wrong versions of your surname, too.
People commonly get both my first name and surname wrong; some persist in doing so even after I’ve enlightened them.
I’ll wager you get some wrong versions of your surname, too.
People commonly get both my first name and surname wrong; some persist in doing so even after I’ve enlightened them.
From no clue to, wait … , I get this; this makes a lot of sense
in jig time.
in jig time.
December 5, 2025 at 8:38 PM
From no clue to, wait … , I get this; this makes a lot of sense
in jig time.
in jig time.
Noticing that both have the same answer, we can conclude that
2/3 x 2&1/2 = 1&2/3
since 44 is 2/3 of 66.
2/3 x 2&1/2 = 1&2/3
since 44 is 2/3 of 66.
December 5, 2025 at 6:17 PM
Noticing that both have the same answer, we can conclude that
2/3 x 2&1/2 = 1&2/3
since 44 is 2/3 of 66.
2/3 x 2&1/2 = 1&2/3
since 44 is 2/3 of 66.
Answer 110
First instinct: 5 x 1/3 of 66
Next: 66 + 2x 1/3 of 66
Further thought: 2 x 66 - 1/3 of 66.
Further further thought: 3 x 5/3 = 5; 22 x 5
First instinct: 5 x 1/3 of 66
Next: 66 + 2x 1/3 of 66
Further thought: 2 x 66 - 1/3 of 66.
Further further thought: 3 x 5/3 = 5; 22 x 5
December 5, 2025 at 6:14 PM
Answer 110
First instinct: 5 x 1/3 of 66
Next: 66 + 2x 1/3 of 66
Further thought: 2 x 66 - 1/3 of 66.
Further further thought: 3 x 5/3 = 5; 22 x 5
First instinct: 5 x 1/3 of 66
Next: 66 + 2x 1/3 of 66
Further thought: 2 x 66 - 1/3 of 66.
Further further thought: 3 x 5/3 = 5; 22 x 5
My diagram does include the four letters A-S-T-C. I call it the quadrant diagram and use the words/abbreviations All, sin, tan, cos. I don’t see any value in replacing these with other words (eg the All Students Take Calculus example). I avoid CAST as, sequentially, it starts with the wrong letter.
December 3, 2025 at 9:03 PM
My diagram does include the four letters A-S-T-C. I call it the quadrant diagram and use the words/abbreviations All, sin, tan, cos. I don’t see any value in replacing these with other words (eg the All Students Take Calculus example). I avoid CAST as, sequentially, it starts with the wrong letter.
Completely agree. Thanks for that reminder.
December 3, 2025 at 8:28 PM
Completely agree. Thanks for that reminder.
It may, of course, be the teaching that’s at fault, so the “depending on how it’s taught” qualifier could apply regardless of which approach is used.
December 3, 2025 at 8:26 PM
It may, of course, be the teaching that’s at fault, so the “depending on how it’s taught” qualifier could apply regardless of which approach is used.
Some pupils, including some of the stronger ones, have struggled to understand the graphical approach, causing me to doubt my choice. One pupil sought help elsewhere and was shown the CAST method (her words), which she said she finds more useful.
December 3, 2025 at 8:25 PM
Some pupils, including some of the stronger ones, have struggled to understand the graphical approach, causing me to doubt my choice. One pupil sought help elsewhere and was shown the CAST method (her words), which she said she finds more useful.
I taught solving trig equations graphically for the first time very recently. I’ve previously always used the quadrants diagram (CAST, but I avoid that, or any other, acronym). I felt it this change was the right thing to do at the time.
December 3, 2025 at 8:25 PM
I taught solving trig equations graphically for the first time very recently. I’ve previously always used the quadrants diagram (CAST, but I avoid that, or any other, acronym). I felt it this change was the right thing to do at the time.
From “Sciencia”, published by Wooden Books. Or the shorter QED, by the same publisher.
November 30, 2025 at 12:22 PM
From “Sciencia”, published by Wooden Books. Or the shorter QED, by the same publisher.
Very nice. Thank you!
November 28, 2025 at 9:18 PM
Very nice. Thank you!
Very elegant!
I was, in fact, thinking initially of a quantitative (Pythagorean) approach- hence the choice of the 3-4-5 r.a.t.
If understand your argument, the difference in perimeter is
3 - sqrt5?
I was, in fact, thinking initially of a quantitative (Pythagorean) approach- hence the choice of the 3-4-5 r.a.t.
If understand your argument, the difference in perimeter is
3 - sqrt5?
November 28, 2025 at 9:17 PM
Very elegant!
I was, in fact, thinking initially of a quantitative (Pythagorean) approach- hence the choice of the 3-4-5 r.a.t.
If understand your argument, the difference in perimeter is
3 - sqrt5?
I was, in fact, thinking initially of a quantitative (Pythagorean) approach- hence the choice of the 3-4-5 r.a.t.
If understand your argument, the difference in perimeter is
3 - sqrt5?
Same area (not perceptually obvious) but different perimeters, which can increase without limit.
Conceptually hard?
I would say so.
Conceptually hard?
I would say so.
November 27, 2025 at 5:55 PM
Same area (not perceptually obvious) but different perimeters, which can increase without limit.
Conceptually hard?
I would say so.
Conceptually hard?
I would say so.
These may help convince you of the utility of my original suggestion regarding triangles. They’ve helped convince me!
November 26, 2025 at 9:38 PM
These may help convince you of the utility of my original suggestion regarding triangles. They’ve helped convince me!
Rhombus and kite.
November 26, 2025 at 9:34 PM
Rhombus and kite.
Similar approach for parallelograms
November 26, 2025 at 9:29 PM
Similar approach for parallelograms
These images - with thanks to @studymaths.bsky.social (geoboard on Mathsbot.com) - show that a rectangle where one side is the mean of the two parallel sides and the other the height of the trapezium - may be of use.
November 26, 2025 at 9:19 PM
These images - with thanks to @studymaths.bsky.social (geoboard on Mathsbot.com) - show that a rectangle where one side is the mean of the two parallel sides and the other the height of the trapezium - may be of use.