Fred G. Harwood

Maths teacher, mentor teacher, ed consultant fascinated with discovery learning and inquiry, problem solving, assessment, visualization & making a difference.

[Fred G. Harwood @HarMath]

@howie_hua Correction to this excellent post: Every positive odd number can be written as the sum of two consecutive natural numbers, not the sum of the squares.

[Fred G. Harwood @HarMath]

@pwharris 51 x 16 is one more 16 than 50 x 16. I'd quickly do 50x16 by taking half of 100 x 16 --> 800 + 16 = 816. I could also have used doubling and halving 50 x 16 as 100 x 8.

[Fred G. Harwood @HarMath]

@dmarain @KarenCampe @MatildaSpicer @sonukg4india @mATH_e_matics @RaminKhosravi4 @DonFraser9 @MrValencia24 @dment37 @jamestanton @asitnof @scodiu @NCTM @maanow @RonySarker71 I got a summer man-cold just before the school term started, so I haven't been back...yet. I will give this a go. I filled a page with approaches and loved how rich the organizations could become.

[Fred G. Harwood @HarMath]

@dmarain @KarenCampe @MatildaSpicer @sonukg4india @mATH_e_matics @RaminKhosravi4 @DonFraser9 @MrValencia24 @dment37 @jamestanton @asitnof @scodiu @NCTM @maanow @RonySarker71 It seems to be a very 'directed' inquiry into evens being one more than odds so 10 odds need 10 ones to make them all even. I'd much rather make this more open and 'look' for the class to discover this number theory as just one of many ideas that develop from these patterns.

[Fred G. Harwood @HarMath]

@mrallanmaths @VisMathsBuilder 56's digits don't add to a multiple of 9

[Fred G. Harwood @HarMath]

@YohakuPuzzle 1 - 3 - 6
2 - 10 - 12
4 - 60 - 8 --> puzzle broke by noticing column 1 and 2 needed an odd number {1, 3, 5, 15} and the pair had to be in the same row as all the right hand numbers are even. I haven't checked that this solution is unique.

[Fred G. Harwood @HarMath]

@CAndrewFrank @howie_hua Nice way to draw the golfers into a math class.

[Fred G. Harwood @HarMath]

@howie_hua I like doubles like 35 + 35 so this becomes 35 + 2 + 35 + 3 = 70 + 5 = 75. It could also be 40 – 3 + 40 – 2 = 80 – 5.

[Fred G. Harwood @HarMath]

@pwharris Doubling/halving. If I double one dimension, the other needs to be half to keep the same area. Good visual of this. Another approach is 25x18 = 25x2x9 = 50 x 9

[Fred G. Harwood @HarMath]

@dmarain A lovely inquiry that seems to generate its own furthering questions in response to the "Wait! What? Really?" Extensions: can it work with three different factors? (1/n = 1/a + 1/b + 1/c ) Does the n have to be a good composite number?

[Fred G. Harwood @HarMath]

@pwharris #MathStratChat How many 1/2's are in 9/2? Nine as 9/2 is nine halves.

[Fred G. Harwood @HarMath]

@dmarain @gerryvarty @mATH_e_matics @ATMMathematics @howie_hua @ybgoi @Math @NCTM @sonukg4india @dment37 @jamestanton If 50^2 is 2500 then 51^2 is two 50's and 1 in the corner bigger or 2601, then 52^2 is 103 bigger etc.

[Fred G. Harwood @HarMath]

@dmarain @gerryvarty @mATH_e_matics @ATMMathematics @howie_hua @ybgoi @Math @NCTM @sonukg4india @dment37 @jamestanton Or for 51^2 and 42^2. This problem was a centre piece of my "visualizing mathematics" workshop. The sum of consecutive odd numbers always being a square (when arranged as you've shown) leads to estimating the next square from a known square.

[Fred G. Harwood @HarMath]

@dmarain @mATH_e_matics @howie_hua @jamestanton @dment37 @RaminKhosravi4 @sonukg4india @maanow @NCTM Left X for a while: visually seeing the steps losing all but the highest stack is probably the easiest solving. Simple cases allowed me to do it mentally. Algebraically was the 'messiest'.

[Fred G. Harwood @HarMath]

@pwharris 24524–12875 --> 24649 – 13000 = 11,649
Solved by pushing each up 125 as a constant shift has an identical difference.

[Fred G. Harwood @HarMath]

@mrdardy @dmarain Pt. 2 Now I'm curious what either of your approaches might change by recording the data in this way for a specific rectangle. Would the emerging patterns lead to a generalization?

[Fred G. Harwood @HarMath]

@mrdardy @dmarain When I had extended the squares question on a checkerboard, I asked, "How many rectangles of any size is on an m x n grid?" The students started with simple cases, organized counting and then one had a breakthrough by organizing the data on a grid. A 6x4 has 210 rectangles Pt.1

[Fred G. Harwood @HarMath]

@MickeyCootes @MathGuyTFL This is the correct approach. Those stating PEMDAS need to see that - 4^2 has the 4^2 taking priority and then being multiplied by the -1.
(-4)^2 =16 but -4^2 = -16
Even the platforms AI knows this.

[Fred G. Harwood @HarMath]

@RHSTUDYZONE 5(s^2 – 25) = 0
(s^2 – 25) = 0
(s + 5)(s – 5) = 0 so either
s + 5 = 0 or s – 5 = 0
then s = -5 or 5