Add numeracy modulo lesson (fixes #397)

csunplugged/topics/content/en/structure.yaml

@@ -2,6 +2,7 @@ topics:
   - binary-numbers
   - error-detection-and-correction
   - sorting-networks
+  - unplugged-programming
 
 programming-challenges-structure: programming-challenges-structure.yaml
 curriculum-areas: curriculum-areas.yaml

csunplugged/topics/content/en/unplugged-programming/numeracy/lessons/8-10/modulo.md

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+# The Modulo operator Unplugged
+
+{panel type="note" title="Note from the authors"}
+
+This lesson is primarily on a maths topic rather than Computational Thinking,
+but the modulo operator is so widely used in digital systems that we've
+provided this lesson plan to teach it explicitly rather than introduce it in
+each of the several topics that use it.
+The "mod" operation (short for modulo) is used a lot in the error correction
+and detection lessons; in the Unplugged activities it also comes up in
+searching (hashing), and it applies to angles in turtle based languages like
+Scratch and Logo.
+Students may already be familiar with the idea, since it is essentially the
+same as the "remainder" after division; it's also referred to as "clock
+arithmetic" because it is how we can add time.
+
+{panel end}
+
+## Key questions
+
+-   Which month comes first, January or December?
+    What is the month after December?
+-   What is the weekday that comes after Friday?
+    What is the weekday that is 4 days before Friday?
+-   If you arrive at a car park at 5pm (17:00 if you use 24 hour time),
+    and pay for 8 hours of parking, when will it expire?
+
+### Potential answers could include:
+
+-   With the months, each month is both before and after every other month!
+    If it's January, then there's a December just before it, but there's also
+    one coming up in 11 months.
+-   With the days, again, Monday comes after Friday, but it's also before
+    Friday.
+-   The car parking ticket expires at 1am (1:00).
+    When you add 8 to 5 (or 17), you get 1!
+
+## Lesson starter
+
+Print the modulo clock resource with modulo 10 out for the students, or draw
+it on the board (This sequence of questions could also be run using the train
+or ferris wheel model; see the next section).
+
+{image file-path="img/topics/modulo-10-clock.png" alt="A clock face with the numbers 0 to 9 around inside."}
+
+Ask what is unusual about this "clock" (it has only 10 points on the dial,
+and it has 0 at the top instead of 10).
+Explain that although it's a little odd, it's actually very useful.
+
+## Lesson activities
+
+Ask which number is 2 hours (or 2 ticks) after 3?
+(The answer is 5 i.e. just count 2 units clockwise).
+If students have their own copy of the clock, they could put a counter on
+the 3, and move it 2 places.
+
+Which is 3 hours after 9? (2) We'll refer to this as 9+3.
+
+What is 2+10 (i.e. 10 ticks after 2)?
+(It is 2, since adding 10 goes right around the clock).
+What is 2+20?
+(it's also 2; you go around the clock twice).
+What is 5+100?
+(5, since going a multiple of 10 gets you back to the same place).
+
+Now compare the following sums on the clock with the normal sum:
+
+-   8+3 (1, compared with 11).
+-   9+8+7 (4, compared with 24).
+-   6+6+6 (8, compared with 18).
+
+Ask the students if they can see the pattern; with sufficient examples they
+will realise that it keeps the last digit of the sum e.g. the last digit of
+11 is 1, of 24 is 4, and of 18 is 8.
+If they have done the product code activity, they may recognise this as the
+function that we use for working out the check digit in product codes.
+
+The last calculation above is 3 times 6 (you could start at 0 and
+count 6 ticks 3 times).
+
+Ask what 4 times 6 might be on the clock (it's 4, which is
+the last digit of 24).
+How about 8 times 5? (it's 0, since it's the same as the number 40).
+
+This strange kind of clock arithmetic that we're doing has a name:
+it's the *modulo*.
+The clock here has 10 ticks on it, so the correct way to write
+a calculation is: `(8+3) modulo 10`.
+
+This means add the 8 and 3, but reset to zero every time you would have
+got to 10.
+
+(This function is also called the "remainder" after division, which for
+positive numbers is the same as the modulo.)
+
+### Ferris wheels and train routes
+
+The above exercise could also be done with the train or ferris wheel handouts,
+which also cycle every 10 steps.
+Students can use counters to track their movement around the stations.
+We have provided resources for train routes above.
+
+{image file-path="img/topics/modulo-train-stations-tracks-circular.png" alt="A circle of 10 train stations connected by train tracks."}
+
+### Time to rewind
+
+The modulo counting is like a circular number line.
+As well as going forward, let's experiment with going backwards
+(i.e. subtraction, or adding negative numbers).
+On the clock it would be winding the time back; on the train and ferris
+wheel it's going in reverse.
+
+Ask the class what they think 2 - 4 is on this system (i.e. 4 steps before 2).
+(This takes you to location 8).
+
+How about 2 - 10? (It's still 2).
+Or 2-100?
+(We're still going to end up at 2).
+
+What is 0 - 7? (3)
+
+But what about -8 i.e. negative 8?
+(That's 8 units before 0, which is 4.)
+Other numbers that land at 4 are -18, -28, and so on.
+
+(This idea comes up with calculating some check digits; for several systems
+we need to ask "what do you add to x to get to 0 in modulo 10";
+the short way to calculate that in a computer program is "(0-x) mod 10".)
+
+### Other sized clocks
+
+Now have the students imagine (or draw) a clock with 11 ticks on it,
+numbered from 0 to 10.
+
+What is 9+3 on this clock (it's 1).
+
+What is 7+11? (7, since 11 goes right around the clock).
+
+You can experiment with other sums, and other clock sizes.
+Modulo 11 is mentioned because it is used by ISBN-10 numbers.
+
+If you get the students to consider modulo 12 or 24, they should see that it
+relates to normal clocks (12-hour or 24-hour clocks respectively).
+For example, (8+5) modulo 12 is the same as working out what the time is 5
+hours after 8 o'clock; or (22+4) modulo 24 is the same as working out the
+time 4 hours after 22:00.
+In the modulo system, the number at the top of the clock would be 0 rather
+than 12 or 24, but all the other values work as expected.
+
+An interesting one to experiment with is modulo 360 i.e. a clock with 360
+ticks, numbered from 0 to 359.
+This kind of calculation happens when we add angles, since turning 360 degrees
+ends up in the same place as the starting position.
+
+Have the students stand up and face the front.
+Now ask them to turn right 90 degrees (a right angle).
+Now have them turn another 90 degrees.
+Ask what angle they've turned in total (180 degrees).
+
+Continuing on, have them turn 90 degrees twice more.
+They should now be facing the front.
+How far have they turned in total? (90+90+90+90 = 360 degrees)
+Students may be familiar with the idea that a "360" represents a full rotation!
+
+Now ask which way they would be facing if the turned another 360 degrees?
+(It would be facing the front again.)
+What if they turn 360 degrees 10 times (i.e. 3600 degrees)?
+(Still facing the front).
+
+Using similar questions, help the students to realise that the total of the
+angles they turn modulo 360 gives the final result.
+For example, turning 3690 degrees to the right is the same as turning 90
+degrees to the right (and the latter is a lot easier!)
+Doing rotations is just another kind of modulo arithmetic.
+
+Doing these calculations in modulo 360 relates to programming in turtle-based
+languages such as Scratch and Logo, where sprites turn through angles.
+Turning left 90 degrees six times is the same as turning 540 degrees,
+but that puts the sprite in the same direction as it would be if it had
+turned 180 degrees.
+Likewise, turning 3600 (or 360, or 720) degrees is the same as turning
+0 degrees!
+And 365 degrees is the same as 5 degrees.
+Turning -10 degrees is the same as 350 degrees.
+
+Going to another extreme, ask the students to consider a clock with only
+two marks on it.
+What would they be? (They would be 0 and 1).
+
+{image file-path="img/topics/modulo-2-clock.png" alt="A clock face with the number 0 at the top (12 o'clock position) and number 1 at the bottom (6 o'clock position)."}
+
+In this modulo 2 system, what is 1+1? (0).
+
+What is 1+1+1? (1).
+
+Count up through the numbers.
+For example, 4 is 4 ticks past 0, which gets you back to 0.
+Moving 5 ticks gets you to 1, 6 gets you back to 0.
+With some hints, students should realise that a number modulo 2 tells you
+if it is odd or even.
+This is used for the parity code!
+If you count black as 1 and white as 0, then each line with even
+parity adds up to 0 modulo 2.
+There's a parity error if it adds up to 1 modulo 2; and the extra
+bit that was added can be thought of as the value needed to make the
+line add up to 0, which is the same idea that is used for most of
+the other checksums, just with a different modulo!
+
+### Does the order matter?
+
+Going back to the modulo 10 clock, ask students if the number 7+8 is
+different to the number 8+7.
+(They aren't, since they both are 15 ticks after 0).
+If necessary, use more examples, so that students recognise that this new
+way of adding isn't affected by the order.
+
+Likewise, have them experiment with applying the mod function at each
+step of adding three numbers, compared with adding them first.
+For example, 7+8+6 gets to 1 on the clock.
+Now have them consider 7+8 (which gets to 5), and then add the 6 to it.
+It turns out that you can apply the modulo at each step, or just
+at the end; both give the same result.
+
+This means that you can add numbers up quickly by grouping them.
+For example, suppose you're adding up: 3 + 7 + 2 + 9 + 1 + 3
+
+You could work out the total (25), which is 5 in modulo 10.
+Or you could add up the digits in pairs.
+In this example, the first two digits add up to 0 mod 10: 3 + 7 = 0
+
+This means that the 3 and the 7 cancel each other out - they won't
+affect the total.
+There's also a 9+1 in the sum, so that will also make no
+difference to the total.
+That leaves just 2 and 3 to add up, which come to 5 for the grand total.
+
+By grouping numbers that add to 10 (or 20 etc.), you can usually add up
+several digits quite quickly in your head.
+This isn't useful on computers, but means you could possibly add up
+a product code checksum without even using pencil and paper!
+
+## Plugging it in
+
+The "modulo" or "mod" function is easily tested in most programming languages.
+
+In Scratch, you can just choose the "mod" block from the Operators, type in
+two numbers and double click it to see the result:
+
+{image file-path="img/programming/scratch-mod-example.png" alt="An image of the mod Scratch blocks."}
+
+In several programming languages, the `%` sign is used to represent "modulo",
+which is a very confusing choice!
+You would type `37 % 10` to do the above calculation in Python, Java, C, C++,
+C#, and JavaScript.
+Some other languages use `mod`, including spreadsheets - for example, you
+can type `=mod(37,10)` in a spreadsheet formula.
+
+You could project a program onto the screen, and ask students to predict
+the value when you type in two numbers.
+For example, in Scratch you could put `scratch:((8)+(5))mod(12)` into the
+Scratch work area and double click it once students make their predictions.
+In Python you would use `(8+5)%12`.
+Now try `8+4` instead of `8+5`. `8+12`? `8+24`? `8+25`?
+
+### Fun facts
+
+The months of the year are modulo 12; 4 months after November is March,
+13 months after January is February, and so on (if you want to use the
+modulo function, you'd need to number the months from 0 to 11, which is
+a little unconventional).
+
+{image file-path="img/topics/modulo-keyboard.png" alt="A keyboard with keys labelled A to G, with the D keys highlighted."}
+
+The naming of musical notes is modulo 7; if you start at the note "D" on a
+piano, and count up 7 white notes, you'll end up back on "D",
+which is the same note an octave higher.
+This raises the interesting fact that there are 7 different notes in an octave;
+it gets its name because if you count the notes at the start and end
+("D" in this case, there are 8 notes).
+
+Actually, there are really 12 different notes in an octave if you count the
+black keys (if you go 12 semitones up from the note "D", you'll end
+up at D again), so in this sense the 12 different notes on a piano are a
+lot like the 12 months or the 12 ticks on a 12-hour clock;
+but it would be really confusing if we referred to the notes
+as "January, February,...".
+
+The cyclic number relationship comes up when playing notes in computer
+programs, which often use a digital representation of piano notes called
+MIDI (for example, these numbers are used in Scratch).
+In this system, middle C is note number 60, C# is 61, and D is 62.
+So an octave above 62 is 62+12 = 74.
+Every 12th number corresponds to the same note name.
+This will be explained in more detail in the activity on programming
+musical scales, which is under development.
+
+### Applying what we have just learnt
+
+Mathematical concepts are part of our everyday life.
+Modular arithmetic (which uses the modulo function) is the formal term
+for clock arithmetic.
+As we expose students to the technical terms of things that we use every
+day, such as when learning to tell the time, the months of the year,
+and during our music lessons, the mathematical links become more obvious.
+The modulo function is also closely related to finding the remainder.
+
+### Lesson reflection
+
+-   Who were the students who were able to make the connections quickly?
+-   Were there any students who found this a logical way to calculate the
+    remainder of a number?
+-   What surprised you and your students when learning about Modulo arithmetic?
+
+## Seeing the Computational Thinking connections
+
+This topic is primarily a maths topic, which supports later activities for
+computational thinking.
+The connections with Computational Thinking are therefore more indirect,
+since CT isn't the main purpose of the topic, and so we haven't
+provided the usual list of connections.
+
+Nevertheless, there are some connections worth noting.
+One connection with computational thinking is an example of **abstraction**:
+sequences like "Monday, Tuesday, Wednesday..." or "January, February,
+March..." or "C, C#, D, D#, E, F..." can be represented by a
+cycle of numbers (0, 1, 2, 3...).
+There are also a lot of **patterns** for students to explore; for example,
+using modulo 10, the result is always the last digit, and using modulo 2
+the pattern for ascending numbers is 0, 1, 0, 1, 0, 1..., that is, it
+alternates for even and odd numbers, and also corresponds to the right-hand
+bit of the binary representation.

csunplugged/topics/content/en/unplugged-programming/numeracy/numeracy.md

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+# Numeracy unit plan
+
+Placeholder text.

csunplugged/topics/content/en/unplugged-programming/numeracy/numeracy.yaml

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+modulo:
+  min-age: 8
+  max-age: 10
+  number: 3
+  duration: 30
+  generated-resources:
+    modulo-clock:
+      description: Clock face with blank, 2, and 10 digit combinations.
+    train-stations:
+      description: Train stations with circular or twisted tracks.
+    piano-keys:
+      description: Piano keys with optional highlighted keys.
+  classroom-resources:
+    - Counters to mark positions.

csunplugged/topics/content/en/unplugged-programming/unplugged-programming.md

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+# Unplugged programming
+
+Placeholder text.

csunplugged/topics/content/en/unplugged-programming/unplugged-programming.yaml

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+unit-plans:
+    - numeracy/numeracy.yaml