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
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And once a class makes this discovery, how many verifications will they need to do to believe their hypothesis?
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Attached is my first thought for younger students (middle schoolers who know what a power of 2 is)...
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2025^2 and in 2026, we'd need to add 2x2025+1 to this last number to create the next consecutive square
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For those with 99, If you are going to solve it by sequential patterns, you need to monitor the first number (odd, odd, odd, odd, even). This then hints at some missing sequential patterns like below. Better to generalize on a row: e.g. shown in 2 ways in the table below:
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Visuals are great. I use the same image for division too. Is it 6 ÷ into 2 groups of 3 or 6 ÷ into 2 groups of 3. Visuals can teach a deeper conceptual understanding. Is doubling and halving understood? If I double the number of groups, the groups will be half the size etc.
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Discovered (uncovered) by a Gr. 11 student. Find a power of 10 that your divisor goes into with a small remainder. Ex: 10000 ÷ 17 =588 r 4 so break of 4-digit bundles
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I like Dotty 6's for buddy classes pairing early primary with intermediates. Littles have a great chance of beating the bigs while learning lots of friendly facts to 6. For older classes, I like 10000 (Farkle) as it involves applied probability and risk/reward strategies
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I like to do something similar. I have them reverse engineer how some students were counting from their mathematical expression. Here is a partial list. The shapes are repeated so when they talk to others, they might see/hear an alternate solution. Ss learn formal writing too.
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My take is with Dave Hewitt's Grid Algebra: What happens when you move left and right, up and down in different rows and columns of a multiplication table? What journey can you go on to move efficiently from one 'cell' to another cell? Can you reverse the operations?
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I meant I like 3/7 over 3÷7. My rationale sometimes not distributing comes from the attachment below. I teach order of operations in a hierarchy so it is useful for more than just arithmetic but equation solving and power laws et al.
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A few that my Ss discovered. The 3-digit & bigger method is generalizable easily. Note 100÷7=14 r 2 with the 2 being the base of power of the group multipliers. 10 ÷ 7 = 1 r 3 with 3 being the base for single digits. Ex: 14 = 3x1 + 4 = 7. 100÷19=5R5 use 2-digit & 5 as base
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I drew a line to the top left and bottom right vertices. We know that the two missing acute angles must add to 30° from the large triangle. So the missing angle is 150° as triangles add to 180°
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square root of 5184 is 72 so 72 odds are involved.
This pattern can be shown in simple cases:
Number of odds: 1 2 3 4 5 <-- (1+last)/2
Last Odd #: 1 3 5 7 9 <-- 2n – 1
Total sum: 1 4 9 16 25 <-- n^2
or through this diagram:
This pattern can be shown in simple cases:
Number of odds: 1 2 3 4 5 <-- (1+last)/2
Last Odd #: 1 3 5 7 9 <-- 2n – 1
Total sum: 1 4 9 16 25 <-- n^2
or through this diagram:
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Also on this topic: 2, 5, 10 all use the same test.
4, 25, 100 use same test.
3, 9 us the same test.
11 (my classes have found about 8 different tests)
Composites of co-primes use tests of each prime.
Visualization below
4, 25, 100 use same test.
3, 9 us the same test.
11 (my classes have found about 8 different tests)
Composites of co-primes use tests of each prime.
Visualization below
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THE divisibility rule for 7? How about this one (which can be universalized). Units & 10's digits are x1, 1000's & 100's x2, 100000 &10000's x4 etc.
105 has 2x1 + 5 = 7; 1001 has 10x2+1 =21;
13145 has 1x4+31x2+45 = 4+62+45 = 1011 not a factor of 7 since 10x2+11=31.
105 has 2x1 + 5 = 7; 1001 has 10x2+1 =21;
13145 has 1x4+31x2+45 = 4+62+45 = 1011 not a factor of 7 since 10x2+11=31.
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This is why I prefer a hierarchal system like the attached.
This way "powers", including the 3+4 are done first
This way "powers", including the 3+4 are done first
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Brilliant problem! Lots of patterns, a semi-broken pattern, patterns that urge verifying, divisibility rule and more. . .
I love when there is a break in obvious patterns, 11 sets of 1 _ _ _ 1; 11 sets of 2 _ _ _ 2; . . . leading to 9 sets of 11 or 99... but 100?
I love when there is a break in obvious patterns, 11 sets of 1 _ _ _ 1; 11 sets of 2 _ _ _ 2; . . . leading to 9 sets of 11 or 99... but 100?
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I do a lot of this through problem strings, and chunked activities that have related families of fraction/decimals explored. #fractiontalks and Singapore Bar models also add to these understandings when you can have both decimals and fractions (& percents) mixed.
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11 x 9 = 99 --> 9+9 = 18 --> 1+8 = 9
A multiple of 9 will always have digits adding to a multiple of 9. See the explanation attached
A multiple of 9 will always have digits adding to a multiple of 9. See the explanation attached
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The 3/8 was derived by adding two lines to the top right half. Seeing 4 identical triangles making up 1/2 of the square.
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I found many students couldn't read a ruler but could tell you that 3/4 of a rectangle was shaded. I'd do a few of these and then rub out the top line of each rectangle. This was to shepherd their knowing of one thing to the concept of 3/4 of the way across.
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I agree another constraint needs to be added. If a specific sequence is sought, tell us what the 83rd term is (for example).
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The patterns of 9 & 1 or 1 & 9 endings since the others have a multiple of 5 if the last digits add to 4 or 6 ending. This cuts the data explored. I see teams of 3, one with twin prime list; one with calculator; 1 recorder. Data table, data reorganized.
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Yes I would and one of my Gr. 11's was so excited to take his explorations of patterns for 13 to extend it into an explanation for a universal test. I called it the Funston Factor Finder.
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Agree with the observation that our typewriters created the ambiguity that modern software eliminates:
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Your clear board would help make 4th figure in the pattern. How do we overlap to create the maximum regions. I asked "12 equally spaced points on a circle, how many chords are formed?" Then casually added "how many line segments are formed by these intersecting chords?" Wow!
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Ex: Seed any rectangle with a number, an expression, a radical etc. Now figure out all the other rectangle's lengths based on the seed value. Percents, decimals, percent, integers, algebraic expressions etc.
Shows equivalency, decomposition, composition, proportional reasoning...
Shows equivalency, decomposition, composition, proportional reasoning...
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I pulled a series of 12 questions from ChalkDust 2016 Advent calendar for my middle school math club. They had to solve the 12 problems convert their answers to Base 3, then colour a grid with their answers. We loved this challenge.
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Simple cases from 1/10 – 1/1000; 1/100 – 1/10000; etc
Showed the numerator is 10^2–1 and the denominator is the second fractions. This leads to a consistent 2 nines in each representation and 2024 – 2 zeroes.
Showed the numerator is 10^2–1 and the denominator is the second fractions. This leads to a consistent 2 nines in each representation and 2024 – 2 zeroes.
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I had to work out each one and factor them individually to arrive at 5/197. Cleverly designed . . . or is famous?
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My gr. 8 to 12's all used scientific calculators in my area and many of my Gr. 10-12's used graphing calculators. We use dynamic software systems like GSP, Desmos and Geogebra for these grades too. I prefer a hierarchal system like below:
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Perfect square progressions being consecutive odd numbers apart. Your graphic will look great in the "Taking Math Of the Page" workshop I'm giving soon.
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Since the majority of shapes are parallelogram-based, A = bh. Triangles are half parallelograms, trapezoids (trapeziums) are average base x height. This one idea is one of the 5 formula's needed. The 4 others are circle/sphere. All else is concept-based. Notice the parallelism
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I've only had time to do multiple bases in math clubs. Too much lost time in regular classes. I love pattern blocks as a tool for exposing different bases. And my favourite @pgliljedahl Gold Chain problem...
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#MathStratChat I really enjoyed the artist's solution to the sum of the first n odd numbers. This gave us an amazing ability to square numbers from a known target. E.g. 51^2 would be 50^2+ 2x50 + 1 or 2500+101 and 52^2 would be 103 bigger than 2601.
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#MathStratChat My students broke my ingrained paradigms when they pushed my shortcut for squaring numbers ending in 5 with an extended inquiry that exposed the power of student-generated algorithms and much more . . .
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Good use of colour in this visual "speaking the mathematics". I feel these also provide the richness of alternatives as others have shown. How about reverse engineering the math to "see" their organizations?
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Barrie Bennett wrote an excellent book on this: "Beyond Monet". Below is an except that moved me.
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My students and I discovered this in connecting the diagonal formulae of Pascal's Pyramid to the growing pattern. This is the equivalent of the sum of triangular numbers.
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Continuing as an intermediate student: Wondered if a pattern of the 3's, 6's, 1's exist and is predictable: Data below shows why triangular continues. My pre-breakfast brain isn't seeing a predictable pattern! Parity?
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Interesting. My gut told me zero, decided to grid it out like Ss might to utilize patterns with the sum of digits ÷ by 9 test for divisibility.
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Thinking Base 10 Blocks, 4 goes into all powers of 10 bigger than 10, so we only need to check the last two digits of a number for ÷ by 4. And 3 & 9 go into one less than all powers of 10, thus adding the digits to see if ÷ by 3 or 9.
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#EdCampBCCC Initial thoughts:
Proficiency levels are like plants; a seed just rooting = emerging (Initial); Sprouting seed = developing (Partial); Developing fruit = Proficient (Capable/Solid); Harvesting plant = Extending (Sophisticated) due to reproducing, feeding others etc.
Proficiency levels are like plants; a seed just rooting = emerging (Initial); Sprouting seed = developing (Partial); Developing fruit = Proficient (Capable/Solid); Harvesting plant = Extending (Sophisticated) due to reproducing, feeding others etc.
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I gave ÷ 7 test as homework to a Gr. 9 class. One year I forgot to tell them to work with four digits & bigger. Two different Ss returned with this one. They'd started with 105 and saw 2x1+5. They then had to continue expanding their idea to find this. A gr. 11 universalized it.
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I did it inductively by having them gather lists of 3, 4, 5-digit multiples of 11 and look for patterns and see if they were extendable. E.g. 132 has 13-2 = 11 and 32+1 = 11. Does this work in 4-digits? 5?
Or look at base10 blocks & look for divisibility in the pieces
Or look at base10 blocks & look for divisibility in the pieces
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Thankful for you & this PLC online. Attached are my reflections on thankfulness being fundamental to unity. I shared it yesterday in church. (Sorry it is not prettified and festive like Rose's posts are.)
May we all be unified in purpose with our diverse views & uniqueness
May we all be unified in purpose with our diverse views & uniqueness
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#MTBoS
I was challenged by a lit leader to make a spine poem of my math bookshelf. I've added many titles since, but thought I'd challenge others as I was challenged...
I was challenged by a lit leader to make a spine poem of my math bookshelf. I've added many titles since, but thought I'd challenge others as I was challenged...
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#elemmathchat
I like Q6 but often will ask for different ways of viewing the same data. Here are some examples:
I like Q6 but often will ask for different ways of viewing the same data. Here are some examples:
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A Gr. 11 universalized this method while looking for a test for 13
You can change the length of string (and prove it all with Modular math, but I love that it was discovered by grassroots number theory & kids looking for patterns then testing them.)
You can change the length of string (and prove it all with Modular math, but I love that it was discovered by grassroots number theory & kids looking for patterns then testing them.)
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I prefer tutoring small groups rather than 1 on 1. This gives more opportunities for reasoning & problem-solving together. Have you followed any of the #MathStratChat threads? I also use less Qs but ask for more methods. Below is an example:
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Another fascinating development on this happened as described below. This was a defining moment early in my career, teaching me the power of discovery-based, student driven learning!
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We're doing it again, David. Here is the puzzle I created this morning, after seeing @YohakuPuzzle 's morning puzzle on perfect squares
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And when the younger grades are studying primes, problems like this bring the whole shebang to life. Below is an example that I'd use during these times.
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Notice: Dave should have said the first p primes instead of n.
"Pesky p prime product problem" has a certain ring to it.
Noticed I shouldn't have started on scrap paper
Patterns of triangular no.s; sums of triangular no.s; increasing & decreasing complexity in counts
No d) yet
"Pesky p prime product problem" has a certain ring to it.
Noticed I shouldn't have started on scrap paper
Patterns of triangular no.s; sums of triangular no.s; increasing & decreasing complexity in counts
No d) yet
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And the relationship to fractions over
10^k+1 like 7/11; 17/101; 117/1001 etc.
I love chunking them in families and having Ss have AHA moments.
10^k+1 like 7/11; 17/101; 117/1001 etc.
I love chunking them in families and having Ss have AHA moments.
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#MTBoS
I prefer charting it in a vertical hierarchy so that equation. solving and power laws are present. (hits of log laws as well)
I prefer charting it in a vertical hierarchy so that equation. solving and power laws are present. (hits of log laws as well)
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In pursuit of the solution, Ss will find that line symmetry (esp. vert. or horiz.) won't work with an odd number. Good thing to learn. They'll try diagonal lines looking for odd egg to straddle that line.
Finally, pt. symmetry possible if odd egg is the focus. #MTBoS
Finally, pt. symmetry possible if odd egg is the focus. #MTBoS
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-2, -1, 0, 1, 2
See the tweet with the patterns picture. I think younger students will find the second solution this way without quadratics.
See the tweet with the patterns picture. I think younger students will find the second solution this way without quadratics.
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Forgot to add the patterns that emerge with younger students (arithmetic approach and not algebraic)
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Have you seen nerdle.com?
mathller.com?
And if you want to learn to speak Canadian, canucklegame.github.io/canuckle/
This last could lead to creating a math based wordle style database.
Don't forget these all flowed from Mastermind
mathller.com?
And if you want to learn to speak Canadian, canucklegame.github.io/canuckle/
This last could lead to creating a math based wordle style database.
Don't forget these all flowed from Mastermind
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I sneak modular math at them as they try to do #thinkingclassroom with this problem in middle school. Often a class will have one group 'see' how close to 13k is each number to find their mates.
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#BCEdChat
If reconciliation starts within ourselves, then we need to examine our attitudes and acts stemming from our biases. We may not be committing genocide but are we falling short on the bottom layers that support exclusion & other negative behaviours?
If reconciliation starts within ourselves, then we need to examine our attitudes and acts stemming from our biases. We may not be committing genocide but are we falling short on the bottom layers that support exclusion & other negative behaviours?
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#BCEdChat Slowchat answers:
I can't share about 'my school' (semi-retired), but I can share some details from a powerful sermon by Gray Poehnell this morning. I'll share these in several tweets. The call was for us to be ambassadors in the ministry of reconciliation
I can't share about 'my school' (semi-retired), but I can share some details from a powerful sermon by Gray Poehnell this morning. I'll share these in several tweets. The call was for us to be ambassadors in the ministry of reconciliation
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Great pondering. . . for d) found this site I didn't know about:
gimaths.com/MathPages/Numb…
Mistake in their intro but beautiful visual tools
gimaths.com/MathPages/Numb…
Mistake in their intro but beautiful visual tools
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#MTBoS Great story about collaboration & imagination in moving Wordle to math worlds in nerdlegame.com
Mastermind -ing expressions to reveal thinking skills.
Mastermind -ing expressions to reveal thinking skills.
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