The Same Amount
How can different arrangements have the same number?
The story unfolds
The mathematics beneath the story
Compare small collections by matching, counting, and composing. Separate number from arrangement, object size, or spread. PK children can show “the same number”; formal use of the equal sign is not a prerequisite.
A learning breakpoint to notice
“More spread out” can be mistaken for “more.” Ask what was added or removed and use one-to-one matching. Counters should be compared by number, not by their physical mass or size.
See the structure
Teacher investigation
Make two groups of five counters in different arrangements. Match them. Move one group farther apart. Has the number changed? What evidence supports your answer?
Try it before revealing the solution. Name every quantity and unit. Accept several valid representations.
Study a completed solution
Five counters match five counters with none left over. One side may be arranged as three and two, while the other is four and one. Each side still has five counters.
Compare approaches
Counting gives five on both sides. Pairing objects gives a second argument without needing number symbols. A drawing can preserve the pairs after the objects are put away.
Analyze an incorrect response
“The longer row has more counters.” Length is not the quantity being compared. Match each counter to one in the other group; if none remain unmatched, the groups have the same number.
Identify the first invalid inference. Ask what evidence would help the learner revise it.
Complete the reasoning
One collection has four counters. Another has three counters. Add __ counter to make the same number.
Check after trying
One counter. Both collections then contain four.
Transfer independently
Make a group of six objects. Ask a partner to make the same number using a different arrangement. Explain how you know the numbers match.
Compare your reasoning
Any two collections of six, with a count or one-to-one match showing equality of number.
Pause and reflect
Which unit must students keep track of? What would an accurate answer conceal? Choose a different context or representation to test whether the idea transfers.
Bring it to your classroom
PK–K · Printable student practice and a separate teacher key. Use the developmental guidance in the key to select tasks.