Making Ten with a Homemade Ten-Frame

The skill: identify a part and its complement to ten

The target is finding how many of ten positions are occupied and how many are still empty. If six spaces contain one counter each, four spaces are empty. Six occupied spaces and four empty spaces make ten positions. A child can count or recognize each part, then verify the total.

The empty spaces are positions in the frame, not physical counters. If you ask how many counters are present, the answer is six. If you ask how many more counters would fill the frame under the one-per-space rule, the answer is four. Distinguishing these questions keeps the model accurate.

A pair such as six and four can be recorded as 6 + 4 = 10. The child can first use gestures, counters, or a verbal explanation. The written equation becomes useful when it describes the visible arrangement rather than replacing it.

Make a frame and choose counters

Draw a rectangle on paper and divide it into two rows of five roughly equal boxes. Count the boxes together: five in the top row and five in the bottom row, ten in all. Make each box large enough to hold one counter without covering its neighbors.

Gather ten suitable counters, such as large paper circles or blocks. The counters should be distinct and safe for the child using them. A pencil and paper are enough for the frame; a ruler can help make straight lines but is not necessary for the math.

Place the frame on a stable surface. If small objects are difficult to handle, use larger pieces or let the child point to boxes while an adult moves the counters. A child can indicate counts through speech, gesture, eye gaze, or a communication device.

Fill the frame and verify ten positions

Begin with an empty frame. Place one counter in each box, filling the top row from left to right and then the bottom row. Count each counter once as it is placed. This conventional arrangement makes the rows easier to compare, though the mathematical count does not depend on a particular direction.

1. Point to the five spaces in the first row and count them.
2. Point to the five spaces in the second row and count them. Together the frame has ten spaces.
3. Place one counter in each space, counting from one to ten as the counters are added.
4. Check that no box is empty and no box contains two counters. The full frame has ten counters under the agreed rule.
5. Remove four counters and set them beside the frame. Count the six still inside and the four empty spaces.
6. Put one counter back in each empty space and verify that the frame is full again: six and four make ten.

A conversation about the empty spaces

Adult: ‘How many counters are in the frame?’ Child counts six. Adult: ‘How many empty spaces do you see?’ Child counts four. Adult: ‘What would happen if we put one counter in each empty space?’ Together they add four and count ten counters in the full frame.

Child: ‘There were already ten counters.’ Adult: ‘The frame has ten spaces. Before we filled the empty ones, six of those spaces had counters. Let's point to the counters and then to the empty spaces.’ Both counts describe different parts of the same frame.

If the child counts the empty spaces as counters, use a gesture to distinguish an object from a place where an object could go. A paper outline or colored marker can indicate a space without becoming another counter in the collection.

Show six and four in two ways

Fill the top row with five counters and place one counter in the first box of the bottom row. Six spaces are occupied and four bottom-row spaces remain empty. Ask the child to count the empty spaces and predict how many counters would complete the frame.

Now arrange six counters in different boxes, leaving four empty spaces scattered. The counts remain six and four, but the empty spaces may be harder to see as a group. Count each empty box once or use a marker to track them. The layout changes the visual difficulty, not the relationship between six and four.

Restore the familiar left-to-right arrangement if it helps. A structured layout is a useful support, not proof that the child has memorized the complement. The same count can be verified in several arrangements when the one-counter-per-space rule is preserved.

Find a partner for a chosen number

Choose a number from zero through ten and place that many counters in the frame, one per box. For seven, fill five spaces in the top row and two in the bottom. Count the empty spaces: three. The partner of seven that completes this frame is three.

Add three counters, one to each empty box, and count or verify that all ten positions are filled. Record 7 + 3 = 10 if using equations. The check connects the number of empty positions to the number of additional counters needed under the agreed rule.

Try another starting number, such as eight. The frame has two empty positions, so eight and two fill ten. Do not change the starting count while asking for its partner without telling the child; the question depends on the arrangement that is actually visible.

Explore zero and ten

An empty ten-frame has zero counters and ten empty spaces. It takes ten counters, one per space, to fill it. Place a 0 card beside the occupied count and a 10 card beside the empty count if numeral cards are being used.

A full frame has ten counters and zero empty spaces. No additional counter is needed to fill it. The pairs 0 and 10 and 10 and 0 both make ten, with the roles of occupied and empty positions reversed.

The empty frame still has ten boxes; zero describes the number of counters in it. The full frame still has boxes, but none are empty. State whether the child is counting counters, occupied positions, or empty positions so the answers remain distinct.

Record all pairs systematically

Start with zero counters and ten empty spaces. Add one counter at a time, recording the number occupied and the number empty after each addition. The pairs are 0 and 10, 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5, 6 and 4, 7 and 3, 8 and 2, 9 and 1, and 10 and 0.

The first count increases by one at each step and the empty count decreases by one. This happens because a counter fills one previously empty position; the number of positions in the frame stays ten. Point to the changed box when comparing two consecutive rows in the record.

If the record treats occupied and empty counts as distinct roles, the reversed pairs are separate states. If a later task lists unordered pairs of addends, decide whether to include reversals separately. The underlying sum remains ten in either interpretation.

Check a missing-number equation

Write 6 + __ = 10 beside a frame containing six counters. Ask what the blank represents: the number of additional counters needed to fill the four empty spaces. Count those spaces, place four counters, and write 4 in the blank.

For 3 + __ = 10, clear the frame, place three counters, and count the seven empty spaces. If the child selects 6, add six counters and inspect the frame: one space remains empty, so six does not complete this arrangement to ten. Add the seventh and check again.

The equation is meaningful because the frame contains ten positions and each holds at most one counter. If a counter is placed between boxes or two share one box, fix the representation before using empty spaces to infer a missing number.

Distinguish the total from the remaining spaces

With eight counters in the frame, ask three separate questions: ‘How many counters are present?’ ‘How many spaces are empty?’ and ‘How many spaces are in the frame altogether?’ The answers are eight, two, and ten. Point to the relevant part for each question.

A child might answer ten to the first question because the frame has ten boxes. Acknowledge that there are ten positions, then count only the counters. A child might answer two to the first question because two spaces are empty; distinguish the empty places from the objects.

This language precision is especially useful when the frame is full. Ten counters occupy ten positions, while zero positions are empty. Two of the counts happen to be equal, but they refer to different things.

Compare two representations of the same pair

Show a frame with four counters and six empty spaces. Beside it, place four blocks on one mat and six blocks on another. In the frame, the second part is a set of empty positions; on the mats, both parts consist of physical blocks. Each representation can show the numerical pair four and six.

Count the two mat groups together to verify ten blocks. Then fill the six empty frame spaces and verify ten counters. Explain the difference in what is being counted before the frame is filled: there are four counters in the frame, not ten counters already present.

A drawing can also show four shaded boxes and six unshaded boxes. The shading represents one partition of the ten positions. The child can match each version to 4 + 6 = 10 while describing whether the parts are objects, occupied positions, or empty positions.

Change one part and predict the other

Place five counters in the top row and one in the bottom row, leaving four empty spaces. Ask what will happen to both counts if one counter is added to an empty space. The occupied count rises to seven and the empty count falls to three.

Make the move and count to check. No box was added to the frame; one position changed from empty to occupied. The two counts still add to ten because they partition the same ten positions.

Remove a counter and ask the reverse question. The occupied count decreases by one and the empty count increases by one. If the child predicts six occupied and three empty, inspect the frame: one position has been omitted from the two counts. Recount both sets of positions.

Make a frame story

Imagine ten seats on a small bus, represented by the ten boxes. Six toy passengers occupy six seats, one per seat. Ask how many seats are empty. Four are empty, so four more passengers could fill the remaining seats under this model.

Place four additional figures in the empty boxes and verify that all ten seats are occupied. Ask how many passengers are on the bus now. The answer is ten in this model, provided each figure represents one passenger and each box represents one seat.

If a passenger leaves, one seat becomes empty. The physical story gives meaning to the occupied and empty counts. It does not require the child to assume that every real bus has ten seats or that all passengers always occupy a separate seat.

Make the task easier

Start with a full frame and remove a small, visible group of two counters. Count the empty spaces and return the same two counters. The child can point to each empty box while an adult counts or moves the pieces.

Use a large frame with strong lines and counters that contrast with the paper. Keep the five-box rows clear. A child who needs assistance with manipulation can direct an adult to place one counter in each chosen space.

If counting to ten is not yet reliable, count the boxes together and use the full arrangement as a reference. Focus first on the specific pair eight and two. Add other starting amounts when the child wants to explore them.

Add challenge without hiding the model's rules

Show a frame with a chosen number of counters, allow the child to examine it, and then cover part of the frame. State the total number of positions and which portion is hidden. Ask for a prediction about the covered spaces only if the visible information is sufficient to determine it; reveal the frame to check.

Alternatively, provide a record of ten pairs with one row missing and ask which occupied count was skipped. Use the frame to build and verify the missing pair. The challenge is noticing a pattern and confirming it with the model.

For a writing extension, ask the child to write both 4 + 6 = 10 and 6 + 4 = 10 for the same pair of group sizes. Explain whether the order records occupied versus empty roles or simply reverses two addends. Keep that meaning clear in the accompanying drawing.

Check a frame with a placement error

Put six counters on a frame but place two in one box and leave five boxes empty. Ask whether counting empty spaces alone tells how many more counters would make ten counters. The one-per-box rule has been broken, so the usual filled-versus-empty comparison no longer matches the number of counters.

Move the extra counter into an empty box. Now six boxes are occupied and four are empty. Four additional counters can fill the remaining positions, producing ten counters. The repair shows why the representation's rules matter.

A counter partly outside the frame can also make its status unclear. Agree whether it is in a box before counting. The child can reposition it or ask the adult to do so. Clear placement avoids treating a visual ambiguity as a mistake in arithmetic.

Use a partner to explain the pair

One person places counters in the frame and asks the other how many more will fill it. The partner counts or recognizes the empty spaces, names the number, and adds counters to check. Swap roles and choose another starting count.

If the partner offers a different number, add that many counters one per empty box and inspect what remains. Too few leaves empty boxes; too many cannot fit under the one-per-box rule. Count the needed spaces again and revise the answer.

Both participants can use the frame and numeral cards. The goal is to explain a relationship supported by the visible positions, not to guess a hidden number or answer within a time limit.

Keep a useful record

Draw a small frame, shade six boxes, and leave four unshaded. Write ‘6 occupied, 4 empty’ and, if useful, 6 + 4 = 10. Another row can show seven and three. The child can dictate the labels or point while an adult draws.

When looking back, ask what the shaded and unshaded boxes represent. If both are simply called counters, the record may be misleading. Add a key or draw actual counters in the occupied spaces to make the distinction explicit.

A note about the response might say: ‘Counted four empty spaces when six counters were visible, then filled them to check.’ Include whether the frame and adult prompts were available. A single correct pair does not establish knowledge of every combination.

Use the five-space rows as a reference

Fill the first row and leave the second row empty. There are five counters and five empty positions, so five and five make ten. Point to the complete top row as one group and the empty bottom row as the other. Count the positions in each row if the child wants to check.

Add two counters to the bottom row. The frame now shows five and two counters, seven in all, with three positions empty. The complete first row can help a child organize the count, but they may still count each counter if that is the most reliable method for them.

If counters are scattered across both rows, a row may no longer be full even though seven positions are occupied. Rearrange the same seven counters into a full row and two additional spaces, or count the original arrangement directly. Neither layout changes the number of counters; the full row is a helpful visual structure rather than a requirement for seven.

Compare two frames without combining their spaces

Prepare two separate ten-frames. Put six counters in the first and eight in the second. Count the empty spaces: four in the first and two in the second. Each frame has ten positions, but the frame with more occupied positions has fewer empty positions.

Ask how many counters must be added to each frame to fill it. The answers are four and two. Add the counters separately, one per empty box, and verify both frames. Do not treat the two frames as a single ten-position frame; together they contain twenty positions.

If the child says that eight needs more because eight is greater than six, compare the actual empty spaces. The question asks about the additional counters needed, not which frame already contains more. The visible layouts provide evidence for both answers.

Find and repair an incorrect record

Show a frame with three counters and seven empty positions next to the equation 3 + 6 = 10. Ask whether the equation matches the picture. Place six additional counters into six empty positions: one space remains empty, and the frame contains nine counters. One more counter is needed to reach ten.

Replace 6 with 7 and check again using the original three-counter arrangement. The corrected equation is 3 + 7 = 10. If the child prefers, count the seven empty spaces before adding counters. The correction is grounded in a specific mismatch between the equation and the model.

For a second example, show 8 + 2 = 10 beside a frame with seven counters and three empty spaces. The equation is true, but it does not describe that particular picture. The child can change the picture to eight and two or change the equation to 7 + 3 = 10. Checking both truth and correspondence helps distinguish two different issues.

Continue with parts and counting

For a smaller whole using two groups of objects, see Making Five in Different Ways. For counting objects once each, try One-to-One Counting with Snack-Time Supplies. The ten-frame adds a fixed set of positions to that counting practice.

Numbers and counting offers an optional digital continuation. The homemade frame and suitable counters are enough for the activity. Keep the model available whenever it helps the child explain a pair.