Ganita Prakash, period by period — fitted to your timetable.
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What Does Your Number Say About Your Neighbours?
Numbers can Tell us Things · Inductive
The power of this opening unit lies in noticing that structural constraints, not arithmetic, decide which sequences are possible — students who jump to 'try all cases' often miss the logical short-cut. A common slip is forgetting that the two end-children each have only one neighbour, so they can say at most '1', making '2' impossible at the ends. Figure it Out Q6 (sequence 0,1,2,1,0) is a rewarding case to close on because it is achievable and the construction is instructive.
0–8Math Talk, section 3.1 p.55 ("List five different situations in which numbers are used....") — students call out uses of numbers from daily life; teacher charts responses and steers toward the idea that numbers can encode relational information, not just counts.
8–20Teacher sets up the core scenario: a row of five children of different heights, each announcing how many immediate neighbours are taller. Students work in pairs to act out several short height orderings at their desks using finger-heights, recording the announced sequence. Through these concrete cases, the class begins to see that position (end vs. middle) and relative height together constrain which numbers are possible.
20–32Figure it Out Q5, section 3.1 p.56 ("For a group of 5 children, is the sequence 1, 1, 1, 1, 1 possible?") and Figure it Out Q6, section 3.1 p.56 ("Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?") — students attempt each question in writing, then share justifications with a partner; teacher circulates and probes reasoning.
32–40Whole-class discussion: pairs report their justifications for Q5 and Q6. Teacher draws out the structural argument — why 1,1,1,1,1 forces two adjacent children to be mutually taller, a contradiction — and affirms that the reasoning pattern (constraint from position + uniqueness of heights) is the key tool.
Homework
How would you rearrange the five children so that the maximum number of children say '2'? (**Figure it Out Q7, section 3.1 p.56**)
Figure it Out Q5, section 3.1 p.56 — For a group of 5 children, is the sequence 1, 1, 1, 1, 1 possible?
Five children of different heights stand in a line. Each child announces how many of their immediate neighbours are taller than themselves. Which of the following sequences of announcements is impossible?
- A 0, 1, 0, 1, 0
- B 0, 1, 2, 1, 0
- C 1, 1, 1, 1, 1
- D 1, 2, 1, 2, 1
C. 1, 1, 1, 1, 1
What each wrong option reveals
- A The student may not have checked all positions; 0,1,0,1,0 is achievable with a height pattern like short-tall-short-tall-short.
- B The student may have confused the achievable symmetric sequence 0,1,2,1,0 (tallest in the middle) with an impossible one.
- D The student may have overlooked that the end positions in 1,2,1,2,1 would require end-children to have two taller neighbours, which is impossible since end-children have only one neighbour — however, option D is also impossible, making this a valid alternative answer the teacher should probe.
Also saved as 7 periods or 10 periods.
- 2From Height Sequences to Supercells: Spotting the Rule
Numbers can Tell us Things - 3Filling Grids: Engineering Your Own Supercells
Supercells - 4Pinning Large Numbers: Reading and Placing on Scaled Lines
Patterns of Numbers on the Number Line - 5Digit Sums and the Detective Work of Counting Sevens
Playing with Digits - 6Reverse, Add, Reflect: The Logic of Palindromes
Pretty Palindromic Patterns - 7Running the Kaprekar Process and Finding Palindromic Times
The Magic Number of Kaprekar - 8Calendar Puzzles and Spider-Web Mental Arithmetic
Clock and Calendar Numbers - 9Always, Sometimes, Never — and Summing Patterned Arrays
Mental Math - 10Why Does the Shortcut Work? And Will Collatz Always Reach 1?
Playing with Number Patterns - 11Estimation Sense and Finding the Winning Move
Simple Estimation - 12Four Problems, One Chapter
Synthesis
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