Use counting, ten, and related facts to add and subtract

Learning goal: Add and subtract within twenty using meaningful strategies, understand subtraction as finding a missing addend, and develop fluency within ten over repeated varied practice. A strategy should explain the quantity relationship rather than supply a memorized chant with no model.

Relate counting to addition. To find 7 + 2, start at seven and take two steps forward: eight, nine. The starting seven is not the first added step. Counting on avoids recounting the entire original group. To find 9 − 2 by counting back, take two decreases: eight, seven. Count the steps, not the number of labels spoken including the start.

Use ten as an organizing amount. For 8 + 6, split six into two and four. Eight plus two makes ten, and four more makes fourteen. The six is still fully represented as two plus four; no counters vanish during the split. Write 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14 and connect every step to a model.

Subtract through ten similarly. For 13 − 4, remove three to reach ten, then remove the remaining one. The result is nine. All four removed counters are accounted for as three and one. An error such as stopping at ten means the second part of the removal was not completed.

Subtraction can also ask for a missing addition part. For 12 − 8, think eight plus what makes twelve. Count up four or use the related fact 8 + 4 = 12, so the difference is four. This does not reverse the meaning of subtraction; it uses the relationship between the whole and its parts.

Use a known near-double when helpful. Six plus seven can be six plus six plus one: twelve plus one is thirteen. The extra one belongs to the seven group after six of it was matched with the first six. Explain why the adjusted addend still represents the original seven.

Fluency within ten develops with accurate, efficient use of known relationships. Offer brief mixed practice across several days and ask occasionally for a strategy explanation. Do not use one easy fact, one fast response or a stressful timer as proof of fluency across the whole range.

Worked example 1: Count on accurately. To solve 6 + 3, begin at six and mark three forward steps: seven, eight, nine. A learner who answers eight may have counted six itself as the first added step. Show the starting quantity separately from each increase.

Worked example 2: Missing addend. To solve 15 − 9, ask nine plus what makes fifteen. One more reaches ten; five more reaches fifteen. Six were added, so the difference is six. Check with 9 + 6 = 15.

Worked example 3: Make an easier sum. For 9 + 5, move one from the five to complete ten with nine. Four remains from the five, so 10 + 4 = 14. The grouping changed, not the total number of counters.

Guided practice: Model a count-on fact, a make-ten sum, a through-ten subtraction and a missing-addend subtraction. Have children choose between strategies and justify a reasonable choice. A method need not be the adult’s favorite if it is accurate and efficient for the learner.

Independent evidence: Solve varied within-twenty problems with explanations, then sample addition and subtraction facts within ten in mixed order over multiple occasions. Record accuracy, prompts and strategy use. Include zero and related facts so recognition of patterns can develop across the full studied range.

Feedback: If a split addend changes the total, use counters to verify that its new parts reconstruct the original amount. If a child confuses steps with endpoints, physically mark the starting number and count movements. If a known fact is uncertain, return to a model and revisit it later. Extend by explaining two strategies for one problem and identifying when each feels useful.