The Two Equal Groups Reference
Exact whole block splits
| Starting total | First mat | Second mat | Blocks left in the middle |
|---|---|---|---|
| 2 | 1 | 1 | 0 |
| 4 | 2 | 2 | 0 |
| 6 | 3 | 3 | 0 |
| 8 | 4 | 4 | 0 |
| 10 | 5 | 5 | 0 |
Each row uses every starting block. The two groups match by number, regardless of their spacing. The task does not require fractional symbols or language about cutting an object. Two empty groups also match an empty starting group, but zero is not a required early turn.
Prepare the sharing surface
Use ten matching solid one piece blocks intended for this age group, each at least six centimetres in every dimension, with smooth surfaces and no cracks or detachable fittings. Do not substitute small counters or foods. Use two flat mats at least fifty centimetres square. Before teaching, check that six of the chosen blocks fit on each mat in a column with clear gaps; enlarge the mats if needed. The repair turns begin with as many as six on one side, even though the final equal groups use at most five. Place the mats side by side with a clear central space for the starting group and enough room for the children to see the far ends. The centre is the unshared source group, not a third recipient. Keep access around the edges so the active child can change position to reach both mats without leaning over blocks.
Keep blocks outside the selected total in the ordinary adult storage box, away from the working display. Count the starting total together before any move. Reset between children with that same total; do not leave blocks from the preceding turn on a mat. No mat labels, number cards, timers or extra loose objects are needed.
Share, then check
- Put the stated starting group in the centre with both mats empty.
- Move one block to one mat and one block to the other, continuing in turns until the centre is empty or one unmatched block remains. A child may use another strategy if it preserves the full group and produces a complete check.
- To check, arrange one mat’s blocks in a column. Place one block from the other mat opposite each block in that column. Keep the two groups on their separate mats.
- Touch or point to one pair at a time. Every block must have a partner; equal space or equal row length is not enough.
- Check the centre and the original total. Two matching groups with a block hidden or left in the centre do not complete the original equal sharing task.
Counting each side is useful confirmation. A child who cannot yet say the number sequence reliably can demonstrate matching by arranging and pointing to the pairs. Accept a spoken equal, same number or they match alongside that complete visible check. Do not treat counting words without a corresponding block action as proof.
Repair examples
| Original total | Unequal groups | Repair | Equal result |
|---|---|---|---|
| 4 | 3 and 1 | Move one from the larger group to the smaller | 2 and 2 |
| 6 | 4 and 2 | Move one from the larger group to the smaller | 3 and 3 |
| 8 | 5 and 3 | Move one from the larger group to the smaller | 4 and 4 |
| 10 | 6 and 4 | Move one from the larger group to the smaller | 5 and 5 |
Move a whole block already in the starting group. Adding a new block or removing one from the task changes the total. Rearranging within a mat changes the display but does not change its number.
When the starting number is odd
| Starting total | Matched groups possible | Unmatched block |
|---|---|---|
| 1 | 0 and 0 | 1 |
| 3 | 1 and 1 | 1 |
| 5 | 2 and 2 | 1 |
| 7 | 3 and 3 | 1 |
| 9 | 4 and 4 | 1 |
These rows do not complete an equal split of the original group. Keep the unmatched block visible in the centre. If it is moved to a mat, that side then has one more. Moving it back does not make it disappear from the original total. A correct conclusion is that these whole blocks cannot all be shared into two equal groups. Do not break a block, remove the spare or claim that the smaller matched groups use the whole original group.
The queue outcome is kept exactly. With whole objects and fractions excluded, an even total can be split equally and an odd total cannot. The lessons teach both the successful equal split and recognition of that limit; they do not promise an impossible split for every integer up to ten.
Six minute blocks
Use one minute for the demonstration or prompt, four minutes for short individual turns and one minute for a closing match check. Up to four children share the surface, with a fresh reset for each turn. The five blocks can be spaced across the week. Start with smaller totals for supported practice, keeping the stated final task visible in the record. If a turn needs more time, use the closing minute or the specified extension instead of moving the child’s blocks to finish for them.
Final observation
Across the five final blocks use starting totals 2, 4, 6, 8 and 10. Each child distributes the blocks and checks both sides using one to one pairing, with counting if helpful. Record all blocks used, complete pairs and any adult prompt separately. The final consolidation checks the earlier totals as well as the ten block task. Week 8 records the odd total limit; a visible unmatched block is not marked as an equal split.
Canonical lines
Quote one line exactly once per lesson.
- Give every block a partner on the other side.
- Use all the blocks in the starting group.
- Space can change while the number stays the same.
- Move a block, then check both sides again.
- One unmatched block means the whole group is not shared equally.
- Two equal groups have the same number of blocks.
Factual verification
The even and odd tables were checked by arithmetic: the two mat counts are equal, and their sum plus the displayed unmatched count equals the starting total. Every repair preserves its original total and ends with matching mat counts. These are direct numerical relationships; no research claim, learning guarantee or developmental threshold is asserted.