Add three groups and choose a useful grouping

Learning goal: Solve stories with three whole-number addends and totals no greater than twenty, and use addition properties as meaningful strategies. Children need not memorize the formal names commutative and associative, but they should explain why changing order or grouping preserves the sum when the same objects are counted exactly once.

A class collects four red counters, six blue counters and three yellow counters. The question asks how many counters there are altogether. Record 4 + 6 + 3 = □ and show three distinct parts. A drawing need not show details such as the color or shape of each counter, but it should show the mathematics in the problem: three separate groups and how many are in each. That rule applies wherever drawings are mentioned in the standards. Combine four and six first because they make ten, then add three to make thirteen. The total includes all three groups, not just the first easy pair.

Change the order to 3 + 4 + 6. The same thirteen counters are present, so the sum is unchanged. This order change works for addition; it does not mean that reversing a subtraction expression always preserves its value. Compare 5 − 2 with 2 − 5. Taking two counters from five leaves three, but five counters cannot be taken from a group of only two, so 2 − 5 does not name the same amount as 5 − 2. Switching the order is safe for addition, not for subtraction.

Grouping means choosing which part of an addition is combined first. In 2 + 6 + 4, six and four make ten, then two more make twelve. Combining two and six first also works: eight plus four is twelve. Both methods count the same total, but the ten combination may be easier to reason about.

The diagram shows four, six and three counters as separate parts, then a statement about the total. Each part should be counted once. If a picture includes an additional row labeled total, that row is a second representation of the already combined collection, not a fourth addend. This lesson’s picture avoids that ambiguity by showing the three original groups only.

Use a known addition fact in reverse order: if 8 + 3 = 11 is established, 3 + 8 also equals eleven. Demonstrate with two groups before relying on symbolic rearrangement. The strategy is justified by the collection, not by a trick that changes any numbers in any equation. Keep every addend and ask which grouping makes the next step easier. The order property belongs to addition, not to subtraction, but the link between addition and subtraction lets it help you subtract as well. Because 8 + 3 = 11 and 3 + 8 = 11 name the same two parts of eleven, removing the three leaves eight and removing the eight leaves three: 11 − 3 = 8 and 11 − 8 = 3. Show this with eleven counters split into a group of eight and a group of three. Taking away either group leaves the other, so one addition fact, read in either order, solves two subtractions.

Worked example 1: Three-color collection. Four red, six blue and three yellow counters make thirteen. First make ten from four and six; then count three more. Record the intermediate step: 4 + 6 + 3 = 10 + 3 = 13.

Worked example 2: A useful regrouping. Seven children choose one activity, two choose another and three choose a third. Group seven and three to make ten, then add two: twelve in all. The arrangement changes the order of addition, not the children’s actual choices.

Worked example 3: Catch an omitted part. A learner solves 5 + 5 + 4 by answering ten. Ten is the sum of the first two groups, but the four still needs to be included. Count all three original parts and write fourteen as the final total.

Guided practice: Give partners three small groups whose total is at most twenty. Ask each partner to choose a first pair, explain the choice and find the full sum. Compare two correct strategies for the same collection. Include an example with no pair that makes ten so the child does not think every problem is designed that way.

Independent tasks: Solve 2 + 6 + 4, 7 + 3 + 5 and 4 + 4 + 3 with a drawing or objects. Create a three-group story, write an equation with a box for its total, and explain why reversing two addition parts does not change the amount. Keep the overall total within twenty.

Assessment and feedback: Credit inclusion of all three addends, an accurate total and a valid grouping/order explanation. If rearrangement causes double counting, physically move the same counters instead of drawing a fresh untracked set. If the child overgeneralizes to subtraction, return to concrete actions and distinguish addition’s properties from other operations. Extend by identifying more than one useful strategy for a given three-part sum.