Number Detectives: Primes, Powers and Codes

You need: 20–30 square tiles or multilink cubes per pair · Grid paper · Recording table

1. Pick a number from 1 to 30 and count out that many tiles.
2. Make a rectangle with every tile used. Record its size, like 3 × 4.
3. Find every different rectangle (a 3 × 4 turned around is the same one).
4. Repeat for every number to 30 and colour the numbers that make only one rectangle.
5. Look at your coloured numbers. What do they have in common?

How you'll know: You can explain that a prime number has exactly two factors, 1 and itself, and show it with tiles.

Go further: Which numbers make a square as one of their rectangles? What do you notice about how many rectangles they have?

You need: A hundreds board (1–100) per student · Four coloured pencils

1. Cross out 1. It is not prime.
2. Circle 2. Cross out every other multiple of 2 in one colour.
3. Circle 3. Cross out every multiple of 3 you have not already crossed out, in a new colour.
4. Do the same for 5 and 7.
5. Circle every number left. Count them and check with a partner.
6. Why did you only need to go up to 7? (Hint: 7 × 7 = 49 and 11 × 11 = 121.)

How you'll know: Your board shows 25 primes below 100, and you can explain why no multiple of 2, 3, 5 or 7 is left.

Go further: Use divisibility rules to check whether 91, 119 and 221 are prime. Which one is the trickiest?

You need: Grid paper · Counters in two colours · Problem cards

1. Sausages come in packs of 8, buns in packs of 12. List multiples of 8 and of 12. Circle the first number in both lists (the LCM).
2. Check with counters: how many packs of each do you need?
3. Now the HCF: 24 kapa haka performers and 36 visitors are split into groups of equal size, and every group has only performers or only visitors. What is the largest group size?
4. Draw factor trees for 24 and 36. Find the HCF and LCM from the primes they share.
5. Write your own HCF or LCM problem about a real school event.

How you'll know: You can solve both problems two ways (lists and factor trees) and get the same answer.

Go further: Two buses leave the school stop together, one every 12 minutes and one every 20. When do they next leave together?

You need: Multilink cubes · A sheet of A4 paper · Calculator · Table template

1. Build squares with side 1, 2, 3 and 4. Record the area as a power: 3 × 3 = 3².
2. Build cubes with side 1, 2 and 3. Record the volume: 2 × 2 × 2 = 2³.
3. Fill in the square root and cube root column: √16 = 4, ∛27 = 3.
4. Fold a sheet of paper in half, then again. Record the layers each time: 2, 4, 8, 16…
5. Predict the layers after 10 folds and check on a calculator (2¹⁰).

How you'll know: Your table shows the square, cube and root for each side length, and you can say why doubling gets big so quickly.

Go further: A sheet of paper is about 0.1 mm thick. How thick would it be after 20 folds? Compare it with a real NZ building or mountain.

You need: Two long strips of paper or masking tape for the corridor · Fact cards · Marker pens · Metre ruler

1. Tape a vertical temperature line from −30°C to 40°C and a height line with sea level at 0.
2. Place fact cards: Ranfurly −25.6°C, Scott Base in winter, a Central Otago summer day, your town today, freezing point 0°C.
3. On the height line add: a diver at −18 m, the school at its height above sea level, a local hill, and a note that Aoraki/Mt Cook (3,724 m) and the Kermadec Trench (about −10,000 m) are too far to fit.
4. Use the line to find differences: count up to zero, then past it.
5. Write three puzzle questions for the escape room from your line.

How you'll know: You can find the difference between a positive and a negative number by counting through zero and show it on the line.

Go further: Scale the height line so Aoraki/Mt Cook and the Kermadec Trench both fit on the corridor wall. What scale did you use?

You need: Two or three different calculators (including a phone) · Whiteboards · GEMA card

1. Type 3 + 4 × 2 into each calculator. Record what each shows.
2. Talk: which answer follows GEMA? Grouped (brackets), Exponents, Multiply and divide, Add and subtract, left to right.
3. Four 4s: make every number from 0 to 10 using exactly four 4s, for example (4 + 4) ÷ (4 + 4) = 1.
4. Swap with a partner. They check each answer step by step using GEMA.
5. Write a lock clue whose answer changes if someone ignores the order of operations.

How you'll know: Your partner can follow every one of your Four 4s answers and get the same result.

Go further: Using √ and powers is allowed. Can you make 11 to 20 with four 4s?

You need: Cipher wheel template (two card circles) · Split pin · Scissors · Coded messages · Tally chart

1. Cut out the two circles and join them at the centre with a split pin.
2. Set a shift (for example A → D) and encode a short message. Swap and decode.
3. Now take a long message with an unknown shift. Tally how often each letter appears.
4. The most common letter in English is usually E. Work out the shift from your tally and test it.
5. Try the same idea with a message in te reo Māori. Which letters are most common? Why does the alphabet you use change the pattern?

How you'll know: You can decode a message without being told the shift, and explain how the letter counts gave it away.

Safety: Cut the card carefully and keep split pins away from mouths.

Go further: A Caesar cipher only has 25 possible shifts. How many keys would a code have if every letter could be swapped for any other letter? (Hint: 26 × 25 × 24 × …)

You need: Combination padlock, lock box or zip bag with a paper lock card · Puzzle card template · Hint card · Answer key · Stopwatch

1. Choose your lock's maths: primes, factors, powers, integers, order of operations or a cipher.
2. Write the puzzle so its answer is the lock's combination (for example a 3-digit number).
3. Solve it yourselves two different ways and write the answer key with a short proof.
4. Give it to a test group. Record: solved, solved with hint, stuck, or found a different answer.
5. Fix any problem and test with a new group.

How you'll know: A new group solves your lock in under 8 minutes, gets the answer you planned, and can explain the maths.

Safety: Never lock doors, cupboards or people. A teacher keeps a list of every combination.

Go further: Link two locks so the answer to one becomes part of the next. Check the chain still has only one path.