Wetland Pattern Detectives β€” Maths Helper

Wetland Pattern Detectives

Your maths helper β€” stuck? Scroll to the "Remember" cards. Ready? Pick your sheet.

Remember 1 β€” The = sign means BALANCE 🐦

Same picture as on the board β€” the pΕ«keko seesaw.

Big idea: = does NOT mean "the answer is coming". It means both sides weigh the same β€” like two pΕ«keko on a seesaw.
8 + 4 7 + 5 = = 12 = 12 Both sides make 12 β€” the seesaw balances
Try it: 8 + 4 = ☐ + 5. The left makes 12, so the right MUST also make 12. ☐ + 5 = 12, so ☐ = 7. (It's NOT 12 β€” that would tip the seesaw!)

Remember 2 β€” Patterns have a STEP rule and a SHORTCUT rule 🌿

Same picture as on the board β€” the growing harakeke.

Big idea: The step rule tells you what changes each time. The shortcut rule lets you jump straight to Year 10 without drawing.
Yr 1 Β· 3 +2 Yr 2 Β· 5 +2 Yr 3 Β· 7 ... Year 10? 10 Γ— 2 + 1 = 21 the SHORTCUT
  1. Step rule: +2 leaves each year.
  2. Shortcut rule: leaves = year Γ— 2 + 1. Test on Year 3: 3 Γ— 2 + 1 = 7 βœ“
  3. Jump to Year 10: 10 Γ— 2 + 1 = 21 leaves. No drawing!
Detective check: always TEST your shortcut on a year you know before trusting it.

Remember 3 β€” Function machines have a SECRET RULE πŸ’§

Same picture as on the board β€” the wetland filter.

Big idea: Something goes IN, the machine's secret rule changes it, something comes OUT. Your job: crack the rule.
IN 3 dirty SECRET RULE + 4 the filter OUT 7 clean
  1. Ask: "How do I get from IN to OUT?" Try + or Γ— first.
  2. Check your rule on EVERY clue β€” a rule that only fits once isn't the rule.
  3. Some machines do TWO things, like Γ— 2 then + 1.
Running it backwards: if OUT is 12 and the rule is + 4, do the opposite: 12 βˆ’ 4 = 8 went IN.

Remember 4 β€” Missing numbers: do the OPPOSITE to undo 🐟

Same picture as on the board β€” the eel that grew over summer.

Big idea: Every operation has an opposite that undoes it. + undoes βˆ’, βˆ’ undoes +, Γ— undoes Γ·, Γ· undoes Γ—.
35 cm + ☐ 52 cm
Worked example: 35 + ☐ = 52.  Undo + with βˆ’:  52 βˆ’ 35 = 17 cm.  Check: 35 + 17 = 52 βœ“
Another: 3 Γ— ☐ = 15.  Undo Γ— with Γ·:  15 Γ· 3 = 5.  Check: 3 Γ— 5 = 15 βœ“
Always CHECK: put your answer back in the box. If both sides balance, you've cracked it.

Remember 5 β€” Meet n, the number we don't know 🌳

Same picture as on the board β€” the kahikatea (NZ's tallest tree).

Big idea: Mathematicians got tired of drawing boxes. They use the letter n instead. Same maths, fancier costume.
leaves = year Γ— 2 + 1 becomes... leaves = n Γ— 2 + 1 where n = the year (any year!)
Try it: if n = 5, leaves = 5 Γ— 2 + 1 = 11.  If n = 100, leaves = 100 Γ— 2 + 1 = 201.
Solving with n: n + 5 = 14. Undo the + 5: 14 βˆ’ 5 = 9. So n = 9.

🟒 Sheet 1 β€” Detective in Training

Everyone starts here. Show your working in your book.

A. Balance it (Remember card 1)

  1. 5 + 3 = ☐
  2. 6 + 4 = ☐ + 2
  3. 9 + 1 = ☐ + 5
  4. 7 + 7 = ☐ + 4

B. Continue the patterns (Remember card 2)

  1. Raupō reeds: 2, 4, 6, 8, ☐, ☐  What is the step rule?
  2. PΕ«keko chicks: 3, 6, 9, 12, ☐, ☐  What is the step rule?
  3. Eel lengths in cm: 10, 15, 20, 25, ☐, ☐
  4. Draw the next harakeke: Year 1 has 3 leaves, Year 2 has 5, Year 3 has 7. Draw Year 4. How many leaves?

C. Crack the machines (Remember card 3)

  1. Machine rule is + 3. Complete:
    IN251020
    OUT5☐☐☐
  2. Crack the secret rule:
    IN14710
    OUT691215
    The rule is: ________

D. Missing number mysteries (Remember card 4)

  1. 10 pΕ«keko at the wetland. Some fly away. 6 are left. 10 βˆ’ ☐ = 6
  2. A baby eel is 12 cm. It grows to 20 cm. 12 + ☐ = 20
  3. ☐ + 5 = 14
  4. ☐ βˆ’ 4 = 9 (careful β€” undo the take-away!)

Finished? Tap βœ“ Check answers, fix any mistakes, then head up to Sheet 2! 🟑

🟑 Sheet 2 β€” Pattern Detective

For detectives who finished Sheet 1. Two-step thinking starts here.

A. Trickier balance

  1. 15 βˆ’ 5 = ☐ + 4
  2. 12 + 8 = ☐ + 13
  3. 3 Γ— 4 = ☐ + 2
  4. 18 βˆ’ ☐ = 6 + 5 (work out the right side first)

B. The shortcut rule

A kahikatea seedling pattern: Year 1 = 4 branches, Year 2 = 7, Year 3 = 10.

  1. Copy and complete:
    Year1234510
    Branches4710☐☐☐
  2. What is the step rule?
  3. What is the shortcut rule? (year Γ— ? + ?) Test it on Year 3!
  4. How many branches in Year 20?

C. Two-step machines β€” and backwards!

  1. Crack this two-step rule:
    IN23510
    OUT571121
    The rule is: ________
  2. Same machine: what comes OUT if 8 goes IN?
  3. Same machine running backwards: 15 came OUT. What went IN? (undo the steps in reverse order)
  4. Machine rule is Γ· 2. 18 went IN β€” what comes OUT? 7 came OUT β€” what went IN?

D. Missing numbers with Γ— and Γ·

  1. Four matuku nests each hold the same number of eggs β€” 20 eggs altogether. 4 Γ— ☐ = 20
  2. ☐ Γ— 5 = 45
  3. 30 ÷ ☐ = 6
  4. A tuna swims the same distance every day. After 6 days it has swum 42 km. How far each day? Write it as a missing number sentence first.

Done? Tap βœ“ Check answers, learn from any slip-ups, then the red sheet awaits, detective... πŸ”΄

πŸ”΄ Sheet 3 β€” Master Detective: Meet n

Advanced challenge. Mathematicians got tired of drawing boxes β€” they write n instead. n just means "the number we don't know yet". So ☐ + 5 = 14 becomes n + 5 = 14. Same puzzle, fancier costume.

A. Solve for n (undo it, then CHECK)

  1. n + 9 = 23
  2. n βˆ’ 7 = 15
  3. 4 Γ— n = 36
  4. n Γ· 3 = 8
  5. 2 Γ— n + 1 = 17 (two steps β€” undo the +1 first)

B. Write the rule with n

  1. The harakeke rule was "leaves = year Γ— 2 + 1". If the year is n, write the rule using n.
  2. A machine's rule is "Γ— 3 then βˆ’ 2". If n goes IN, what comes OUT? (write it with n)
  3. Use your rule from question 2: what comes OUT when n = 10? When n = 100?
  4. Same machine: 28 came OUT. Write the puzzle with n, then solve it.

C. Reasoning like a mathematician

  1. The harakeke pattern goes 3, 5, 7, 9, ... Could a plant in this pattern EVER have exactly 50 leaves? Explain how you know without listing every number.
  2. Balance puzzle β€” both boxes hide the SAME number: ☐ + ☐ + 4 = ☐ + 10
  3. Wetland maths: NZ once had about 2,500,000 hectares of wetland. Only 10% is left. How many hectares is that? How many hectares were lost?

D. Design your own (the real test of a master)

  1. Invent a two-step machine rule. Make an IN/OUT table with 4 clues and ONE missing OUT. Swap with a partner β€” can they crack it?
  2. Invent a wetland growing pattern (a plant or animal population). Give the first 3 terms and challenge your partner to find the shortcut rule and term number 10.

Done? Tap βœ“ Check answers β€” you've cracked the case, Master Detective! πŸ•΅οΈ