# TK: Mathematics

Build number sense and mathematical explanation through objects, pictures, movement, conversation, and progressively more demanding problems.

## Lessons and pupil materials

- [Count and show how many](/agent/docs/tk-lesson-count)
- [Compare collections](/agent/docs/tk-lesson-compare)
- [Put parts together](/agent/docs/tk-lesson-join)
- [Take away and explain](/agent/docs/tk-lesson-subtract)
- [Find the hidden part](/agent/docs/tk-lesson-missing-part)
- [Build with shapes](/agent/docs/tk-lesson-shapes)
- [Find the repeating part](/agent/docs/tk-lesson-patterns)
- [Measure and show data](/agent/docs/tk-lesson-measure)

## Learning progression

The core follows developmental progressions in quantity and operations, classification and pattern, measurement, and geometry. The IES early-math guide supports progression-based teaching; its evidence ratings differ by recommendation. Do not present every activity below as experimentally proven.

| Idea | Begin with | Connect to | Advanced application |
|---|---|---|---|
| Quantity | Notice small groups; match objects to recipients | Count each object once and understand the final number as the total | Count a fresh arrangement and explain why spacing does not change quantity |
| Number representation | Match a spoken quantity to a collection | Connect collections, fingers, drawings, and numerals | Make a requested collection and explain the representation |
| Comparison | Pair objects to compare amounts | Explain more, fewer, equal, and a difference | Find what must change to make groups equal |
| Parts and wholes | Put together and separate visible collections | Represent a story with objects and drawings | Solve a missing-part story and check it |
| Shape and space | Explore, match, build, turn, and describe | Compare defining features and relative position | Explain why a rotated shape keeps its identity |
| Pattern and classification | Sort by a stated attribute; copy a repeating structure | Identify the repeating unit and change the representation | Create another pattern with the same structure |
| Measurement and data | Compare directly; organize observations | Use equal units without gaps or overlaps | Explain a comparison or a classroom data display |

## Original lesson: enough places at the table

**Materials:** toy guests, place mats, and safe counters representing supplies.

Set a table for a chosen group of guests. Ask “How can we make sure each guest has a place?” Let children distribute, count, or pair. Add a guest and ask what must change. Use a small collection the child can manage. Example quantities are teaching materials, not readiness cutoffs.

**Look for:** one-to-one correspondence, a meaningful total, and a reason for changing the supply. If a child recites numbers but skips objects, move each counted object to a new place while modeling one word per object. Extend by hiding some supplies and asking what is missing, then reveal to check.

## Original lesson: the same collection moved

Count a collection, spread it out, and ask whether anything was added or removed. Let the child check with matching or recounting. Do not simply announce that the quantities are equal. Ask the child to make an arrangement that looks different but has the same amount.

**Look for:** reasoning about the collection, not a memorized conservation answer. Extend through grouping and a drawing. If the child changes the answer with spacing, pair the rearranged objects against the original positions or a second equal collection.

## Original lesson: a pattern someone else can continue

Make a repeating pattern using movements or objects. Ask what part repeats. The child continues it and then represents the same structure in a different medium, such as clap–tap becoming red–blue. Those pairs are examples, not an exhaustive pattern curriculum.

**Look for:** recognition of the repeating unit. If the child copies only the last object, mark the unit and say it as a whole. Extend by asking whether a proposed continuation fits and why. Do not confuse reciting a number sequence with understanding a repeating pattern.

## Original lesson: measure the path

Build a route between model buildings. Lay identical units along it and compare with another route. Ask what would happen if units overlapped or differed in size. Invite a drawing that would let another child understand the comparison.

**Look for:** consistent units and a valid explanation. A larger numeral from smaller units does not necessarily mean a longer path. Extend by predicting, checking, and explaining that discrepancy. Include shape composition and location language as children design routes.

## Through the year

Start with meaningful collections, pairing, and spatial play. Connect quantities to symbols as understanding grows. Revisit earlier ideas while introducing parts and wholes, pattern structure, and comparison. In investigations, organize observations into simple displays. In community design, use number and measurement to solve an actual supply or space problem. At transition, use changed objects and arrangements to check transfer.

## Kindergarten extension targets

For children with secure prerequisites, use the actual kindergarten standards as an advanced destination:

- **K.CC.5:** count arranged collections up to 20 and scattered collections up to 10, and make a requested collection from 1–20.
- **K.OA.5:** develop fluent addition and subtraction within 5 through understood relationships.

The numerical ranges above come from those standards. They are not TK admission rules or evidence of a 99th percentile. Use explanations, drawings, and new problems to check understanding. Extend beyond a mastered range by following the next relevant standard and the child's demonstrated knowledge, not by arbitrarily increasing worksheet digits. Shape composition, reasoning, and mathematical language remain part of advanced work.

## Evidence and response

Save a collection drawing, the child's explanation, and a later problem using different objects. Record counting strategy, correspondence errors, numeral confusion, and the help given. If a child is accurate but cannot explain in English, invite action, another language, or a picture. If a procedure works only in the rehearsed arrangement, reteach the underlying idea. No speed test or pass percentage is required.

## Publication details

| Field | Value |
|---|---|
| Material | Teaching guide |
| Learners | California TK, ages 4–5 |
| Setting | Full classroom |
| Pacing | Teacher selects from observed prerequisites |
| Author | Arcomm |
| Edition | First edition, 2026-09-08 |
| Format | Markdown |
| Evidence status | Original synthesis; not outcome-validated |

## Sources and dates

- [CDE: Mathematics](https://www.cde.ca.gov/sp/cd/re/documents/ptklfmathdomain.pdf): current PTKLF edition. Retrieved 2026-09-08T03:44:00.628834+00:00.
- [IES What Works Clearinghouse: Teaching Math to Young Children](https://ies.ed.gov/ncee/wwc/PracticeGuide/18): 2013 practice guide. Retrieved 2026-09-08T03:44:00.628834+00:00.
- [CDE: Kindergarten Counting, K.CC.5](https://www2.cde.ca.gov/cacs/id/web/262): kindergarten standard. Retrieved 2026-09-08T03:44:00.628834+00:00.
- [CDE: Kindergarten Operations, K.OA.5](https://www2.cde.ca.gov/cacs/id/web/269): kindergarten standard. Retrieved 2026-09-08T03:44:00.628834+00:00.

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## Revisions

[Revision history](/agent/docs/tk-mathematics/history)