These are invented everyday-domain paper scenarios, not real sales, admissions, pupil data or observed success. Use them for modelling/guided teaching, separately from first-response E/F. Each solution is a teaching reference; a child still has to make their own mathematical decisions.
M01 · Day 31 · Library-card inventory
Problem: Twelve pretend card marks need a count that can be checked.
Work it through: Make six pairs, audit two marks in each, and count 2,4,6,8,10,12.
Supported conclusion: The last 12 counts all objects. Six is the number of groups, not the object total. Moving the groups farther apart changes no quantity.
If this mistake appears: If “six cards” is claimed, open one pair to show two cards and link two pairs to 4.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M02 · Day 31 · Game-set audit
Problem: Sixteen game-piece marks move from a row into a ring.
Work it through: Make eight full pairs. Count 2,4,6,8,10,12,14,16; match every original mark to one ring mark.
Supported conclusion: Sixteen objects remain, in eight pairs. A mark drawn twice would be a new extra representation, not an extra original piece.
If this mistake appears: If the ring is said to contain more because it spreads out, match pieces one-to-one before recounting.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M03 · Day 32 · Decoration order changes
Problem: Seven full pairs of decorations are present and one more full pair is added.
Work it through: Count seven pairs to 14, then add two actual new marks. Count the eighth pair at 16.
Supported conclusion: The quantity changes 14 to 16. A single added mark would instead make 15 and leave an incomplete pair.
If this mistake appears: A predicted 16 is not an observed addition; retain which two marks were actually added.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M04 · Day 32 · Footprint diagram
Problem: Ten pairs face different directions on a paper footprint sketch.
Work it through: Audit each pair, track only completed groups and count 2,4,6,8,10,12,14,16,18,20.
Supported conclusion: There are 20 objects, not 10. Facing direction and visual spacing do not change the represented quantities. No body movement is required.
If this mistake appears: If one pair is skipped, the last claimed 18 covers only nine groups; point to the remaining pair and recount a new arrangement.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M05 · Day 33 · Growing picture strip
Problem: A strip shows successive stages of 2,4,6,8 squares; every next stage adds one pair.
Work it through: Identify the two new squares each time. Make a next stage with ten squares and label 10.
Supported conclusion: Next 10 matches the stated increase. The growing quantity is different from a repeating colour/shape unit.
If this mistake appears: A label 10 beside eight squares is only a claim; add the two actual new marks or revise the label.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M06 · Day 33 · Numbered trail
Problem: The first checkpoint is 12; each next checkpoint is two more.
Work it through: Retain the twelve base and add pairs:12,14,16,18,20,22.
Supported conclusion: Next after 20 is 22. The given start is included; counting a new pair from 2 would describe that pair alone.
If this mistake appears: If 21 is chosen, model one-object versus two-object changes in a different small case before a new trail.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M07 · Day 34 · Unequal label rack
Problem: A32-mark rack has sixteen alleged pair boxes, with one containing 3 and one 1; the others contain 2.
Work it through: Check those two groups. Move one mark from 3 to 1; sixteen true pairs now count 2 through 32.
Supported conclusion: The repair keeps 32 objects: fourteen groups of 2 plus 3 plus 1 already total 32, but equal-pair skip counting was unjustified before membership repair.
If this mistake appears: Changing only the printed “pair” label does not change group membership. Actual directed movement/drawing is separately observed.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M08 · Day 34 · Display equivalence
Problem: Twenty-four display marks are represented in pairs and then as tens/singles.
Work it through: Twelve pairs give 2 through 24. Replace boundaries with two full tens and four singles:20+4=24.
Supported conclusion: Both representations name the same 24 objects. They do not make 48 unless they are explicitly two different collections.
If this mistake appears: Retaining two diagrams is allowed; counting the same object set through both as extra objects is a duplicate error.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M09 · Day 35 · Museum-card spacing
Problem: Twenty marks are rearranged into widely spaced groups.
Work it through: Make ten verified pairs and count 2,4,6,8,10,12,14,16,18,20; map each earlier mark to one new position.
Supported conclusion: The object total remains 20. A bigger drawing area is not a larger quantity.
If this mistake appears: If the quantity changes, inspect whether any actual mark was added/removed rather than infer from spacing.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M10 · Day 35 · Counting-machine report
Problem: A sketch has thirteen verified pairs; its report says “13 objects”.
Work it through: Count 2,4,6,8,10,12,14,16,18,20,22,24,26. Annotate 13 groups and 26 objects as different facts.
Supported conclusion: Twenty-six objects is the corrected report. Thirteen correctly names the pairs.
If this mistake appears: Open one pair, then two pairs as 2/4; do not merely tell the child to double a number.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M11 · Day 36 · Bookmark print plan
Problem: Five bookmark panels each contain exactly five marks.
Work it through: Audit members, then count 5,10,15,20,25. Match the final count to all five panels.
Supported conclusion: Five panels represent 25 objects. Frame capacity alone would not guarantee a full group.
If this mistake appears: If an empty slot is ignored, count actual marks and decide whether to refill or honestly relabel.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M12 · Day 36 · Pretend kit with a gap
Problem: Three intended five-panels contain 5,5,4 actual decoration marks.
Work it through: Actual count is 5+5+4=14. Either add one mark for 15 or label the existing collection 14.
Supported conclusion: Both repairs are legitimate with their purpose stated. The unmodified collection has 14, not 15.
If this mistake appears: Blind 5,10,15 does not audit the third unit. Save first response, then examine a different incomplete panel.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M13 · Day 37 · Paper-ticket stages
Problem: Marks grow 5,10,15,20 and the next stage adds another five.
Work it through: Show the new five marks; next stage contains 25.
Supported conclusion: Next 25 is supported by the stated five-object addition. These are invented tickets, not actual admissions.
If this mistake appears: A copied twenty-mark image with a new 25 label is inconsistent; a paper prediction alone is recorded as a prediction.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M14 · Day 37 · Page-marker stages
Problem: Start with ten marks and add five at every stage.
Work it through: Retain the base:10,15,20,25,30. Show five newly added marks in any transition.
Supported conclusion: Next after 25 is 30. The start need not be 5 for the increase to be 5.
If this mistake appears: If the base is discarded, map the original ten and latest five before trying a new start.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M15 · Day 38 · Catalogue panels
Problem: Eight verified five-panels show a paper catalogue collection.
Work it through: Count 5,10,15,20,25,30,35,40. Pair them into four tens using the same marks:10,20,30,40.
Supported conclusion: Forty objects in both forms. Eight versus four describes different group counts;5 versus 10 describes unit size.
If this mistake appears: The new ten boundary replaces two five boundaries. Do not add a new set of ten objects while claiming a pure regrouping.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M16 · Day 38 · Certificate display
Problem: Sixty certificate symbols are to be checked as fives or tens.
Work it through: Twelve fives count 5 through 60; six tens count 10,20,30,40,50,60.
Supported conclusion: Both end at 60. Fewer steps do not mean fewer actual symbols; both strategies are supported.
If this mistake appears: Reporting 6 as the total names groups only. Audit one ten and count the first two groups as 10/20.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M17 · Day 39 · Large paper display
Problem: A teacher-prepared ordinary field has 120 actual unlabelled marks. Child grouping is still to be chosen.
Work it through: One valid account is twelve audited tens:10,20,30,40,50,60,70,80,90,100,110,120. Check no mark omitted/reused.
Supported conclusion: 120 is justified only if the complete actual field is included. The target range/teacher preparation is not a completed child response. Other complete equal-group accounts can be valid.
If this mistake appears: A smaller or stopped count is useful partial work. Continue later or teach coverage on a new smaller set.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M18 · Day 39 · Kit-counter choice
Problem: Fifty marks can be grouped as five tens or ten fives.
Work it through: Ten fives:5,10,15,20,25,30,35,40,45,50. Five tens:10,20,30,40,50.
Supported conclusion: Same 50 objects, different unit sizes/group counts. A full account and check matter, not speed.
If this mistake appears: When converting, match the one original collection to both representations rather than make an unsupported 100 total.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M19 · Day 40 · Mural count claim
Problem: Eight purported ten-groups have member counts 10,10,10,10,10,10,10,9.
Work it through: Seven full tens count 70; nine more make 79. Add one actual mark for 80 or relabel the unmodified plan 79.
Supported conclusion: Actual 79 before repair;80 only after one new mark. A prediction about adding is separate from an actual added mark.
If this mistake appears: Count the near-full unit before relying on eight “ten” labels.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
M20 · Day 40 · Game-score marker design
Problem: One hundred marks need two equivalent count accounts.
Work it through: Ten tens count 10 through 100. Replace boundaries with twenty fives and count 5 through 100.
Supported conclusion: Both 100; the old/new diagrams can coexist as representations but are not automatically 200 original objects.
If this mistake appears: If a doubled total appears, teach replacement with a different small collection and retain the first interpretation.
Access/ownership: Large marks, selected numerals, tactile forms or learner-directed adult operation can show the mathematical decisions. Record any supplied grouping/hint; copied answer text is not independent counting.
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