Take numbers apart in more than one way

Learning goal: Decompose any studied number up to ten into two parts in several ways and record the parts without changing the total. Use a plate divided into two regions, ten counters and paper. Work with smaller totals first, then extend to ten over several sessions.

Put five counters on the plate. Move two to the left region and three to the right. Count the whole collection and both parts. Say “five is made of two and three.” The regions organize the collection; the line between them does not create another counter. Record a drawing or 5 = 2 + 3. The equal sign connects two names for the same amount: the total and its two parts.

Now move one counter from the right region to the left without taking anything off the plate. There are three on the left and two on the right, still five altogether. Move another to make four and one. Ask why the total stayed five: only the arrangement changed. This is the central idea, so always verify that the two recorded parts add to the stated total.

The illustration shows three different views of five counters. Each row is a separate example; the rows are not meant to be added together. The color or outlined regions distinguish the two parts. Read one row at a time: one and four, two and three, or five and zero. An empty region can be a part with zero counters. Zero does not mean that the recording is unfinished.

Develop an organized way to find partners. Begin with none on the left and all five on the right. Move one counter at a time from right to left. The left amounts become 0, 1, 2, 3, 4, 5 while the right amounts become 5, 4, 3, 2, 1, 0. A learner does not need to memorize this list immediately. The movement explains why increasing one part by one requires decreasing the other by one to preserve the same total.

Some decompositions reverse the same two part sizes. Two and three and three and two are different left-right arrangements, but both show five. Acknowledge that relationship without demanding formal terminology. Ask whether two and four could be another decomposition of five. Count the proposed parts to find six, so it cannot show the same total.

Worked example 1: Partners for six. Place six counters on a two-part mat. Record 1 and 5, then 2 and 4, then 3 and 3. Check each by counting the whole collection. Six can be decomposed in more than one way; “three and three” is not the only answer because the two parts need not be equal.

Worked example 2: A hidden part. Show seven counters, cover four, and leave three visible. The visible part is three, but the total is still seven. Uncover the four to verify “seven is three and four.” Repeat with a different covered amount. A cover changes what can be seen, not the amount present.

Worked example 3: Repair a record. A child draws two counters in one region and five in the other but labels the total six. Count all the counters: seven. Either change the total label to seven or remove one counter to make a decomposition of six. Explain which quantity was supposed to remain fixed.

Guided practice: Choose a total of four, then eight. For each, make three different two-part arrangements and draw them. Ask a partner to check the parts and total. Keep counters on the mat throughout so rearranging is not confused with adding new objects.

Independent tasks: Show three decompositions of nine; make a two-part model of ten with one part empty; and decide whether four and three can represent eight. Require a drawing, object model or spoken explanation as well as part numbers. For a hidden-part game, begin by showing the total before covering anything.

Assessment: Credit a record only when both parts and the total agree. Note whether the learner can generate more than one arrangement and explain why the total is unchanged. If a child makes a fresh collection in every step and loses the fixed total, keep one physical collection and move its counters across the divider. End by asking, “What changes when we move a counter across the line, and what stays the same?”