Near Doubles through One Extra Block

The skill: use a related double and adjust

A near double such as 3 + 4 has addends that differ by one. The smaller double is 3 + 3 = 6. Adding one more to that total gives 3 + 4 = 7.

The larger double also works: 4 + 4 = 8. One of those eight blocks must be removed to match 3 + 4, giving seven. The adjustment must correspond to the change made to the groups.

The child can count all seven blocks to verify either method. The strategy is useful only when the related double is known or can be checked. It is not necessary to memorize a fact before exploring the physical arrangement.

Prepare two groups of blocks

Gather eight suitable blocks or paper squares and mark two spaces on a table. Put three blocks in each space and keep the remaining two outside. A stable flat row is as effective as a tower for comparing group counts.

Count each starting group and the six blocks together. Confirm that the model shows 3 + 3 = 6 before adding an extra block. A pencil and paper can record the equations, but the child may first use objects and gestures.

If moving pieces is difficult, the child can point to a destination and an adult can place the block. Use materials appropriate for the child and make the collection boundary clear. The two spare blocks are not part of the initial six.

Build three plus four from three plus three

Place three blocks in each space. Say: ‘We have three and three, which make six. What happens if one more block joins the second group?’

1. Count the first group as three and the second group as three.
2. Count all six blocks or use the known fact 3 + 3 = 6.
3. Move one spare block into the second group. Its count changes from three to four.
4. Add one to the previous total of six to obtain seven.
5. Count all seven blocks once to check. Match three in the first group with three in the second and identify the extra block.
6. Record 3 + 3 = 6 and 3 + 4 = 7, showing which block accounts for the difference.

A conversation about the extra block

Adult: ‘Three and three make six. Now the second group has one more block. What is the new total?’ Child: ‘Six.’ Adult: ‘Six described the groups before the extra block joined. Let's count the new block as one more.’

Point to the original six and then the added block. Count on from six to seven, or count all seven from one. Adult: ‘The extra block changes the second group from three to four and the whole from six to seven.’

If the child says eight, compare the number of added blocks with the change in the total. Only one block was added to the six-block collection. Recount the actual objects before revising the answer.

See the matching part and the one extra

Place three blocks in one row and four in a parallel row. Pair three blocks across the rows. One block in the second row remains unmatched. The matched groups show 3 + 3; the leftover is the extra one.

Count the matched six and then the extra block to get seven. The extra is part of the four-block group and part of the whole; it is not an additional eighth block outside the problem.

If the child pairs one block with two partners, rearrange so every matched block appears in only one pair. The one-to-one comparison makes the difference between the groups visible.

Use the larger double instead

Build four blocks in each of two groups and count eight. Remove one block from the first group. The groups are now three and four, and seven blocks remain in the counted area.

Record 4 + 4 = 8 and 8 − 1 = 7. The removed block is outside the collection being counted. This is a second route to the same 3 + 4 model.

A child may prefer the smaller double plus one or the larger double minus one. Both are valid if the adjustment matches the action. Counting all seven is an available check for either route.

Compare the two routes

For the smaller-double route, start with six blocks in groups of three and three, then add one. For the larger-double route, start with eight blocks in groups of four and four, then remove one. Both produce groups of three and four and a total of seven.

The intermediate collections differ: one begins with six, the other with eight. Do not add and remove the same extra block simultaneously without clarifying which route is being shown. Reset the materials between demonstrations.

Ask the child to point to the adjustment in each story. One block joins the smaller double; one block leaves the larger double. The final arrangement can be compared directly.

Check the double before using it

If a child thinks 3 + 3 equals five, build two groups of three and count the six blocks. Correct the starting fact before adding one. An incorrect double leads to an incorrect adjusted total even if the child adds one consistently.

A child who does not know the double can count all blocks or use a table of checked examples. The strategy does not require unsupported recall. Once 3 + 3 is established, the extra block can be used to reason about 3 + 4.

Keep the checked model visible. A fact card can record 3 + 3 = 6, but it should match the physical groups. The card is a reference rather than an additional block.

Try two plus three

Build two blocks in each group. Count four and record 2 + 2 = 4. Add one block to one group, producing two and three. The total rises from four to five.

Pair two blocks across the groups and identify the one extra. Count all five to check. A child can point to the matched part and the extra while an adult writes 2 + 3 = 5.

The larger-double route begins at 3 + 3 = 6 and removes one, also giving five. Choose the route that the child can explain with the objects. There is no requirement to use both in the first round.

Try four plus five

Build two groups of four and count eight. Add one block to the second group. Four and five contain nine blocks altogether: 4 + 5 = 9.

If nine objects are difficult to track, keep the first eight in matched pairs and move the extra block into a clearly marked space. Count the whole once as a check or use a number strip from eight to nine.

A matching total does not alone prove that the groups are four and five. Count each group as well as the whole. Three and six also total nine but are not the intended near-double arrangement.

Understand the two addends and the whole

In 3 + 4 = 7, the first addend describes one group, the second describes another, and seven describes all objects in both groups. The extra block is included in the group of four.

If the child answers four when asked for the total, acknowledge that four is one part. Gesture around both groups and count all seven. The visible parts help distinguish a correct part count from the requested whole.

If the child says that 3 + 4 is a double, compare the group counts. The addends differ by one. It is a near double, while 3 + 3 and 4 + 4 are related doubles.

Reverse the groups

Place four blocks first and three second. The collection still contains seven blocks. Match three across the groups and identify the extra block in the first group this time.

The smaller-double route remains 3 + 3 = 6, plus the one extra. The order of the displayed groups changes, but the matched part and the total do not. Record 4 + 3 = 7 if useful.

Do not confuse a change in order with an additional object. No block was added when the two groups exchanged positions. Count the whole if the child wants to verify.

Use a number strip for the adjustment

Mark 6 on a number strip as the checked total of 3 + 3. Move one interval to 7 for the extra block. The movement represents the change in total after the block joins.

For the larger-double route, mark 8 and move one interval back to 7 after removing a block. The starting marker is not itself a move. Compare each movement with the physical action.

A number strip supports the adjustment but does not show why 3 + 3 is six unless that fact has been established. Keep the blocks or a checked fact card available for the starting double.

Draw the matched groups

Draw three circles in one row and four in another. Connect three pairs across the rows and circle the extra object. Count the six matched circles plus one extra to get seven.

Write 3 + 3 = 6 and then 6 + 1 = 7 beside the drawing if useful. The circled extra is one of the four circles in the second row, not a new mark to add again.

If the drawing shows five circles in the second row, correct the model or explain that it represents a different problem. A drawing must match the addends before it can support the calculation.

Make the first example easier

Begin with one block in each group. Count two, then add one to the second group. One and two make three. The child can see the matching pair and the extra block.

The adult can move pieces while the child indicates when to add the extra. Use large shapes and a clear boundary. A verbal explanation is optional if the child can point to the matched blocks and the extra.

Keep the starting double visible in a drawing or rebuild it before checking. Add a larger example only when the child wants another comparison. Counting all three remains a valid way to confirm the result.

Add challenge with a missing group

Show a group of three and say that the other group has one more. Ask how many blocks should be in the second group and how many there will be altogether. Build four and count seven.

For another problem, state that two groups differ by one and total seven. With whole-number groups, three and four satisfy the conditions. Build and check them; do not infer a unique answer if a different set of allowed quantities or rules is introduced.

The child can explain the result through matching pairs and the extra block. The adult may write the equation after the arrangement is checked. The missing-group task is different from being shown both addends at the start.

Play a partner comparison game

One partner builds a double; the other adds or removes one block to create a near double and describes the new total. Both count the groups and the whole. Swap roles after a checked example.

If the partner adds one block to each group, the new arrangement is another double, not a near double. For example, 3 + 3 becomes 4 + 4, and the total increases by two. Count the actual additions before naming the result.

A child can direct a partner's movements and use a fact card or number strip. The game is about explaining the relationship between the arrangements, not rapid recall.

Predict and verify the next total

Build two groups of two and verify the total four. Before adding an extra block, ask what the total will be when one group has three. The child may predict five, choose a card, or ask to count the resulting blocks.

Add one block to the second group and check. Two matched pairs account for four blocks and the extra makes five. If the prediction was six, identify how many blocks were actually added to the original four.

Reset the model before trying a different double. A prediction is checked against a particular starting group and action; changing either one changes the question.

Distinguish a near double from an ordinary double

Show 2 + 2 and 2 + 3 using two sets of rows. In the first, both groups have two, so the expression is a double. In the second, the groups differ by one, so the expression is a near double.

Count the totals as four and five. The difference between the totals is one because one block was added to one group. If one block had been added to each group, the second model would be 3 + 3 and the total would increase by two.

Ask the child to identify the equal part and the extra in the near-double model. A point or a line connecting pairs can show the relationship. The written labels are useful only when they match the actual groups.

Use a known total without treating it as a group

In the three-and-four example, six is the total of the matched three-and-three portion. It is not a third group of six blocks to place beside the other two groups. Adding the extra block changes that total to seven.

If a child puts six new blocks on the table after building three and three, separate the additional pieces and explain that the numeral 6 names the six blocks already present. Return to the original collection before adding the single extra.

A numeral card can sit beside the matched groups as a label. When the extra block joins, replace the total label with 7. The label changes to describe the count; it does not add objects itself.

Compare a near double with a larger difference

Build three blocks and four blocks. The groups differ by one, and 3 + 3 plus one gives seven. Now build three and five. The groups differ by two, so 3 + 3 plus two gives eight.

Pair three across the second arrangement and count the two unmatched blocks. The original double still helps, but the adjustment must match the actual difference. Adding only one to six would not account for both extra blocks.

The expression 3 + 5 is not a near double under the one-difference definition used here. It is still a valid addition and can be solved with a related double plus two. Label the rule before classifying the example.

Check a true equation against the blocks

Place groups of three and four beside the equation 2 + 3 = 5. The equation is true, but it does not describe the visible groups. Count each group and compare the addends.

Replace the equation with 3 + 4 = 7, or rebuild the groups as two and three if the intended equation is 2 + 3 = 5. Both repairs are possible; they change different parts of the setup.

The child can point to the mismatch while an adult writes. Check the total after the repair. A correct arithmetic statement is not automatically a correct description of every arrangement.

Represent both routes on paper

Draw three circles in each of two rows and add a fourth circle to the second row. Label the matched six and the extra one. Write 3 + 3 = 6 and 6 + 1 = 7.

On a separate drawing, make two rows of four and cross out one circle in the first row. Label the starting eight and the seven remaining. Write 4 + 4 = 8 and 8 − 1 = 7.

The two drawings show separate routes to the same final group sizes. Do not add all the marks across both drawings; they are representations of alternative calculations. A child may use arrows or dictate the labels instead of writing equations.

Explain the result to a listener

Ask the child to show a partner the three-and-four groups. The partner identifies the three matched pairs and the one extra block. Together they count six in the matched portion and seven altogether.

If the partner asks why six is not the final answer, point to the extra block already included in the four-group. The six is the related double's total before that block is added. The final total includes it once.

The child can point, move the extra, or direct an adult. The listener's questions should refer to the visible model and allow time to check. A memorized explanation is not required for the relationship to be demonstrated.

Revisit the original model

Return to three blocks in each group and verify six. Add one block to the second group and ask the child to identify the changed group and the new total. Use whichever route or count the child prefers.

If the model is rebuilt after pieces were moved, count each group to ensure it really shows three and four. A missing or duplicated piece would represent a different problem. End after the result has been checked or save the materials for another round.

Choose a helpful related double

For 5 + 6, a child who knows 5 + 5 = 10 may add one to get eleven. Another child may know 6 + 6 = 12 and subtract one. Build five and six with objects or drawings and verify the result using either route.

Ask which fact the child used and how the adjustment matches the model. If the chosen double is not known, count its two groups or count the near-double collection directly. A strategy is useful when its starting fact and adjustment can be checked.

Keep a table of previously verified doubles available if it helps. The child can locate 5 + 5 and then add the extra object. Using a reference is a supported strategy, not evidence that the fact was recalled independently.

Check the direction of the adjustment

Starting from 4 + 4, remove one block to make 3 + 4. The total must decrease from eight to seven. Starting from 3 + 3, add one to make the same final groups, so the total increases from six to seven.

If the child adds one to the larger double and obtains nine, show the action represented by nine: one block was added to eight. That creates a nine-block collection rather than the intended seven-block model. Reset and perform the removal.

Observe and continue

A useful note might say: ‘Built 3 + 3, added one block to make 3 + 4, and counted seven’ or ‘Used 4 + 4 and removed one with a visible model.’ Include which double and support were available.

One correct near double does not establish recall of every fact. One error does not establish a persistent difficulty. If questions continue, share specific examples with a teacher or appropriate professional.

For the underlying equal-group model, see Doubles Facts with Matching Towers. For adding objects to a known quantity, try Counting On from a Known Number. Numbers and counting offers an optional digital continuation.