Rearranging a sum so it can be done in your head

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What to assume they know

How many numbers have a given number of digits, and why digit sums behave: how many digits a number has, and how digit counts behave
Addition and subtraction of numbers up to six digits
Multiplication as repeated addition, and multiplication tables to ten
Counting the parts of a shape sequence produces a number sequence: counting the parts of a shape arrangement to produce a number

What they should be able to do

Reach a stated total by combining given numbers, reusing them as needed
Show more than one combination that reaches the same total
Decide whether a stated total can be reached at all, and justify the decision
Use subtraction as well as addition to reach a target more cheaply
Explain why adding then taking away can beat adding four times
Produce examples for stated digit-count scenarios, and identify which scenarios are impossible
Classify a statement about digit counts as always, sometimes or never true, with a reason
Find the total of a patterned arrangement by grouping rather than by counting one at a time
Read a diagram's shape as an instruction about how to group
Build an arrangement of your own that adds up to a chosen total

Where it usually goes wrong

"Mental maths means doing the same sum faster in your head." It means doing a different, shorter sum that has the same answer. 38,800 by column addition and 38,800 as 25,000 + 400 + 400 + 13,000 are not the same amount of work.
"There is one right way to make a target." 28,000 and 63,000 can each be made more than one way. The figure invites several routes and the class should compare them rather than converge on a single answer.
"If you can make 2,000 you can make 1,000." You cannot. The two smallest parts are 400 and 1,500, and no number of 400s lands on 1,000. This is the first impossibility argument in the section and it is short enough to do fully.
"Subtraction is a step backwards, so it cannot make things quicker." 39,800 is reached fastest by overshooting to 40,000 and coming back. Overshoot-and-correct is the whole reason the second toolbox exists.
"Adding two 4-digit numbers can give a 6-digit number." It cannot, and the Always, Sometimes, Never block is built to expose exactly this. Two 4-digit numbers are each under 10,000, so the sum is under 20,000. Give the argument, not a failed search.
"To total a picture you count everything in it." That is what §3.9 exists to refuse. The arrangement is the hint; the rows of figure (a) and the two blocks of figure (c) are placed as they are on purpose.
"The dotted grids are decoration." They are sums too — of dots. They are the hardest of the six because you must decide what the repeated part is before you can count anything.

Questions to check understanding

Reach a stated total from a given set of parts, using each as often as wanted
Show that a stated total cannot be reached, with a reason
Fill in a target using both addition and subtraction
Write an example fitting a stated digit-count scenario, or explain why none exists
Classify a digit-count statement as always, sometimes or never true and justify it
Total a patterned arrangement, with the grouping shown as working
Design an arrangement that sums to a stated number — the chapter-end block asks exactly this for a number between 210 and 390 (p.72)
Items combining the two: reach a target both by column arithmetic and by grouping, and compare the effort

Examples worth working on the board

The mental-math figure (§3.8, p.65). A middle column of five numbers — 25,000, 400, 13,000, 1,500 and 60,000 — with four targets down the left (38,800, 28,000, 61,600, 31,000) and four down the right (3,400, 63,000, 19,500, 20,900). Arrows run from the middle to the targets. Middle numbers may be used as often as wanted. Two routes are printed:
- 38,800 = 25,000 + 400 taken twice + 13,000
- 3,400 = 1,500 + 1,500 + 400 The arrows are drawn and do not extract; the numbers do.
The Math Talk questions (§3.8, p.66). Can 1,000 be made, and why not? What about 14,000, 15,000 and 16,000 — the book says these can be, and asks how. Then: which whole thousands are impossible?
Adding and Subtracting (§3.8, p.66). Six boxes — 40,000, 7,000, 300, 1,500, 12,000 and 800 — and five targets: 39,800 (worked as 40,000 minus 800 plus 300 plus 300), then 45,000, 5,900, 17,500 and 21,400 left blank.
Digits and Operations (§3.8, p.66). Two printed examples: 12,350 + 24,545 = 36,895, and 48,952 − 24,547 = 24,405. Then a grid of ten scenarios — five additions and five subtractions — each naming the digit counts of the two inputs and of the answer, to be satisfied by an example or shown impossible.
Always, Sometimes, Never (§3.8, p.67). Five statements, (a) to (e), about the digit count of a sum or difference. Each must be classified and the classification argued. Note: the interesting ones are the two that are never true, because those need an argument about the largest and smallest numbers of a given length, not an example.
§3.9 figure (a) (p.67). Five rows, alternating: four 40s, then five 50s, then four 40s, then five 50s, then four 40s. That is twelve 40s and ten 50s. The grouping the picture is pushing is by row.
§3.9 figure (c) (p.68). A block of four rows of eight 32s, sitting on a shape made of four rows of four 64s — three at the left and one at the right, with a gap between. That is thirty-two 32s and sixteen 64s. The two blocks turn out to be worth the same, which is the observation the arrangement is engineered to produce.
§3.9 figure (b) (p.67). An 8 × 8 grid of dice-style tiles. Each tile shows either a single dot or a five-dot face. The five-dot tiles are darker and sit in four two-by-two blocks near the corners of the inner area and one two-by-two block in the middle, twenty tiles in all; the other forty-four carry one dot. What is being summed here is dots, not printed numbers.
§3.9 figure (d) (p.68). A second dice-style grid, in purple and red, mixing more than one face value. It is entirely artwork. I could not read it reliably enough to state its composition and this brief gives none — see Notes.
§3.9 figure (e) (p.68). A large hexagon tiled with smaller hexagons, the numbers 15, 25 and 35 arranged around each, with an outer band in a second colour. The labels crowd at shared vertices; this brief states no composition for it either.
§3.9 figure (f) (p.68). A bullseye of discs: one 1000 at the centre, then a ring of four 500s, then a ring of eight 250s, then an outer ring of 125s. The value halves at each ring outward, and the first two rings double in count — but the outer ring does not carry sixteen. Counted on the printed page of page 68 it holds eighteen discs, so the four rings do not have equal subtotals and an explanation that assumes they do will state a wrong total.
Build your own (chapter-end Figure it Out, p.72, question 7). Choose a number between 210 and 390 and design an arrangement in the style of §3.9 that sums to it. This is the exercise that proves the student has understood grouping rather than counting.

Figures to have open

The §3.8 opening figure with its middle column, its eight targets and the two printed routes (p.65). The arrows are the mechanism and must be drawable one at a time.
The six-box Adding and Subtracting panel with its five targets (p.66), four of them blank.
The ten-cell scenario grid from Digits and Operations (p.66), blank.
§3.9 figures (a) and (c) (pp.67–68) redrawn to scale, since the grouping argument is geometric.
§3.9 figures (b), (d), (e) and (f) (pp.67–68). All four are artwork carrying their numbers inside the drawing. They must be redrawn from the printed page rather than reconstructed from the extracted text, which loses most of them — see Notes.
No photograph or dataset is required.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 3 "Number Play": §3.8 Mental Math, pp.65–67, and §3.9 Playing with Number Patterns, pp.67–68
The opening figure and the two worked routes, p.65
The Math Talk questions about 1,000 and the three fourteen-to-sixteen thousands, p.66
The bold sub-headings Adding and Subtracting and Digits and Operations, p.66
The Figure it Out block: the scenario grid on p.66 and Always, Sometimes, Never? on p.67
The six patterned figures (a) to (f), pp.67–68
The chapter-end Figure it Out block, p.72, question 7, which asks the student to build a §3.9-style arrangement
Forward link: §3.11, pp.69–71 (Estimation: when an approximate answer is the right answer), where the same regrouping habit is used to estimate rather than to compute exactly