When One Whole Block Is Left

Original lesson metadata

code: "EMP-HALV-L1-T1-WK8"
curriculum_id: "halving"
curriculum_name: "Halving"
level_id: "L1"
level_name: "All Ages"
age_line: "around 2 to around 6"
term: 1
week: 8
title: "When One Whole Block Is Left"
slug: "week-8"
aim: "Identify that an odd group leaves one unmatched block when whole blocks must be shared into two equal groups."
blocks_this_week: 5
minutes_per_block: 6
weekly_minutes: 30
blocks: [{"n":1,"title":"Try One Block With Two Empty Mats","minutes":6,"aim":"Recognise that one whole block cannot give a matching block to both mats.","staging":"Offer one central block and invite a move to either mat. Compare the one and zero groups, then return it to the centre. Close by stating that two empty mats still leave the original block unused.","say":"There is only one whole block. Put it on either mat and look at the other. Can it have a partner there?"},{"n":2,"title":"Find the Spare in a Group of Three","minutes":6,"aim":"Identify one unmatched block after one and one are shared from three.","staging":"Set three central blocks, invite alternating moves and keep the last block visible. Read \"One unmatched block means the whole group is not shared equally.\" Close by checking that the original three are not all in the two matched groups.","say":"One block is on each mat. Another block is still waiting. Have we shared the whole group equally?"},{"n":3,"title":"Keep All Five in the Task","minutes":6,"aim":"Recognise that two and two with one remaining do not equally share all five.","staging":"Offer five blocks and let the child form two pairs, keeping the remaining block central. Invite one move of that block to a mat and compare the unequal result, then restore it. Close without removing or splitting any block.","say":"Two pairs match, but one block has no partner. Keep that block in view. What happens if it joins either group?"},{"n":4,"title":"See the Same Limit With Seven","minutes":6,"aim":"Identify one unmatched block after three pairs are formed from seven.","staging":"Distribute seven in alternating turns until three pairs and one central block are visible. Ask whether every starting block is used in the matched groups. Close with the accurate conclusion that the whole seven cannot be shared equally as whole blocks.","say":"You have made three matching pairs. There is still one whole block. Can all seven join two equal groups this way?"},{"n":5,"title":"Explain the Unmatched Block","minutes":6,"aim":"Identify that an odd group leaves one unmatched block when whole blocks must be shared into two equal groups.","staging":"Place nine blocks in the centre for your demonstration; offer three or five for a child needing a smaller turn. Form matched groups with one remaining. Keep that block visible while the child shows why the whole group cannot match. Leave the total unchanged through the explanation, then return every block to the centre.","say":"Show the block that has no partner. We must keep every block whole and use them all. What stops this group matching?"}]
domains: ["M","CL","PD"]
objectives: [{"domain":"M","text":"Identify that an odd group leaves one unmatched block when whole blocks must be shared into two equal groups."},{"domain":"CL","text":"Say that one whole block is left or has no partner."},{"domain":"PD","text":"Keep the unmatched block visible instead of removing it from the task."}]
materials: ["Ten matching large solid blocks from smooth untreated wood offcuts freely given and prepared by someone with suitable tools before the lesson, or an existing borrowed set; each must be safe for the children to handle and free of splinters, residue and loose parts","Two flat pieces of clean reclaimed cardboard, each roomy enough for six chosen blocks in a column with gaps","An old box kept by you for blocks outside the chosen total","The supplied Two Equal Groups Reference on an existing screen or copied onto used paper"]
space: "A stable low table or clear indoor floor holding two flat sharing mats at least fifty centimetres square, a central starting space and a clear view for up to four children. Enlarge either mat if six chosen blocks cannot fit in a column with clear gaps. Keep access around the edges so the active child can reach both mats by changing position without leaning over blocks."
differentiation: {"younger":"Use three blocks, make one matched pair and point to the visible block left in the centre.","older":"Explain the same limit for five, seven or nine without dividing a block or changing the starting total."}
watching_for: "Whether a child correctly notices the unmatched whole block and avoids claiming that equal smaller groups use the full original total."
resources: []
version: 1
date: "2026-09-09"
access: "Free"
publisher: "Empathy Global"
author: "Eric Gonzalez-Payne"
author_url: "https://www.linkedin.com/in/eric-gp/"
copyright: "© 2026 Empathy Global. All rights reserved."
licence: "Free to read. Licensed for use in any setting by holders of an Empathy Global account, free or paid. Not licensed for reproduction, redistribution, resale or use in a setting by anyone else."
source_url: "https://empathy.global/curriculum/halving/"

HALVING · TWO EQUAL GROUPS WITH WHOLE BLOCKS

ALL AGES · around 2 to around 6 · TERM 1, WEEK 8 · 5 × 6


DOMAINS: M, CL, PD
CODE: EMP-HALV-L1-T1-WK8
DAY & DATE:

Objectives

M Identify that an odd group leaves one unmatched block when whole blocks must be shared into two equal groups.
CL Say that one whole block is left or has no partner.
PD Keep the unmatched block visible instead of removing it from the task.

Before the two mat turns

You will want to put the spare away so the two groups look equal. Leave it visible. You need the whole original group in view, including the block without a partner.

One child will put the unmatched block in the storage box and announce that the groups match. Return it to the centre as part of the stated starting group and check all original blocks again. The box is not a way to complete the task.

Two children will pass the last block back and forth between the mats. Let one transfer happen, then compare the sides. The block changes which side has more, but it does not acquire a partner.

How we use the sharing space

Stay beside the sharing surface with up to four children. Give individual turns. Keep unused blocks in your box outside the display.
Use matching large, solid, single piece blocks suitable for the children. Check for smooth surfaces, sound edges and no cracks, splinters, residue, loose fittings or detachable parts. No small counters, coins, beads, seeds, food or broken pieces. Keep blocks out of mouths.
Use a stable low table or clear floor and two flat sharing mats. Leave room for six blocks in each column with clear gaps and a starting group between the mats. Let children move around to reach, without leaning over blocks.
Place one block gently at a time. Keep fingers clear. No throwing, stacking, standing on blocks or carrying piles.
Keep blocks whole. No cutting, tearing, breaking or food sharing. The centre holds the starting group; it is not a third share.
Use every starting block and check every partner. Never add, hide or remove one to make the groups look equal. Keep any unmatched block visible and name the limit.
If anything unfamiliar enters, stop and move back without touching it. Continue in another clear place after the usual response.

Five Short Sharing Turns

Try One Block With Two Empty Mats · 6 minutes

AIM: Recognise that one whole block cannot give a matching block to both mats.

Use one minute to introduce the turn, four minutes for individual attempts with a reset between children, and one minute for the closing check. Offer one central block and invite a move to either mat. Compare the one and zero groups, then return it to the centre. Close by stating that two empty mats still leave the original block unused.

SAY: "There is only one whole block. Put it on either mat and look at the other. Can it have a partner there?"

Find the Spare in a Group of Three · 6 minutes

AIM: Identify one unmatched block after one and one are shared from three.

Set three central blocks, invite alternating moves and keep the last block visible. Read "One unmatched block means the whole group is not shared equally." Close by checking that the original three are not all in the two matched groups.

SAY: "One block is on each mat. Another block is still waiting. Have we shared the whole group equally?"

Keep All Five in the Task · 6 minutes

AIM: Recognise that two and two with one remaining do not equally share all five.

Use one minute to introduce the turn, four minutes for individual attempts with a reset between children, and one minute for the closing check. Offer five blocks and let the child form two pairs, keeping the remaining block central. Invite one move of that block to a mat and compare the unequal result, then restore it. Close without removing or splitting any block.

SAY: "Two pairs match, but one block has no partner. Keep that block in view. What happens if it joins either group?"

See the Same Limit With Seven · 6 minutes

AIM: Identify one unmatched block after three pairs are formed from seven.

Use one minute to introduce the turn, four minutes for individual attempts with a reset between children, and one minute for the closing check. Distribute seven in alternating turns until three pairs and one central block are visible. Ask whether every starting block is used in the matched groups. Close with the accurate conclusion that the whole seven cannot be shared equally as whole blocks.

SAY: "You have made three matching pairs. There is still one whole block. Can all seven join two equal groups this way?"

Explain the Unmatched Block · 6 minutes

AIM: Identify that an odd group leaves one unmatched block when whole blocks must be shared into two equal groups.

Use one minute to introduce the turn, four minutes for individual attempts with a reset between children, and one minute for the closing check. Use nine for the final demonstration and three or five for a child needing a smaller practice turn. Form matching groups with one remaining, then ask for the limit without changing the total. Record the actual total used and the child’s recognition; this is not marked as a completed equal split.

SAY: "Show the block that has no partner. We must keep every block whole and use them all. What stops this group matching?"

If you have more time

After the nine block demonstration in block five, add two minutes to move the unmatched block to each mat in turn and see why neither result is equal.

A small sharing conversation

Ask your child what one unmatched block tells us when all blocks must stay whole. Naming the limit prevents an incomplete share being called equal.

Look for a complete match

The child keeps the odd group's unmatched block visible and says why the whole group cannot be shared equally.

Notes: