Single Ten-Frames

The single ten-frame is a rectangular arrangement of two columns (or rows) of five squares as shown below.


The ten-frame is an extension of the dot pattern activity. It provides a spatial organization for the dots that supports children's development of five-referenced, ten-referenced, and doubles-referenced conceptions of numbers up to ten and the development of mental imagery for such numbers. It also supports development of partitions of ten.

This activity involves showing a number on the ten-frame. Using transparent chips, the teacher can create an arrangement of a number ten or under. The students' task is to figure out how many there are. The approach of flashing patterns encourages children to develop mental images and operate on their mental images rather than on the visual material. By arranging the chips to encourage five-referenced, ten-referenced, or doubles-referenced strategies, the teacher can facilitate children's development of number relationships. The ten-frame lends itself to a variety of solution methods. For example, in the first frame shown below, children might think of seven as five (the top row) and two more, or as four (the four on the left) and three more. They might also think of it as three empty boxes so its three less than ten. In the second frame shown below, children might think of it as four (the top row) and three more (the bottom row), as two, two, two and one more (focusing on the columns), or as three and three and one more (using a doubles-references strategy). Discussions in which children explain their strategies provide occasions for children to reflect on their solution methods in light of the methods others describe. Students then have the opportunity to attempt to make sense of solution methods that they might not have used previously themselves without feeling obliged to use them. As a result, children develop alternative solution methods and strategies that are in keeping with their current understandings.

 five-referenced seven  doubles-referenced seven   random seven

Children's descriptions of how they thought about what they saw are productive for discussion. As with the dot pattern activity, the teacher can capitalize on children's descriptions and notate them using standard or pedagogical notation. This further supports number relationships and discourages counting. As with the spatial dot patterns, random arrangements can be used to foster discussions about easy/hard. Easy arrangements are those that can be figured out without counting by ones. In addition, questions about the number of empty squares in the ten-frame encourages the development of partitions of ten.

One productive way to pose the ten-frame tasks is to develop a scenario that children can use as the basis for developing imagery. For example, the ten-frame might be a crate at Earl's Fruit Stand. Earl keeps his pumpkins in the crates. When the crate is full it has ten pumpkins in it. A scenario such as this creates a forum for posing questions in a variety of ways. By thinking about the scenario, children can use their imagery of the situation to help them think about how they might solve the problem. For example, if a ten-frame with seven chips is shown the teacher can ask a variety of questions such as:


The last two questions can contribute to children's development of partitions of ten and filling up the ten strategy.

Double Ten-Frame

The double ten-frame is useful to support children's development of number relationships based on five- and ten -referenced strategies and doubles strategies for numbers up to 20. In addition, it is productive for developing thinking strategies, such as the +1, -1, and compensation strategies and filling up the ten strategy. Comparisons, including more, less, how many more, how many less can also be facilitated by use of the double ten-frame.
The activity involves showing a number in each of two adjacent ten-frames. The children's task is to figure out how many there are. Alternatively, the task can be to figure out how many are missing or how many more (or fewer) there are in one frame than in the other. Posing the tasks using chips of two different colors, one color for each frame, facilitates the children's explanations as well as the posing of tasks. For example, if the following display shows red chips in the upper frame and blue chips in the lower frame, a child might explain that s/he moved one blue chip to the frame with the red chips to make five and five. Because five and five is ten, there are ten chips in all.



Another child might explain that s/he (mentally) moved the four red chips beside the blue chips to fill up the upper ten-frame. So there are ten in all. By capitalizing on solutions such as these which make spontaneous use of thinking strategies the teacher can make these strategies explicit topics of discussion. Furthermore, the teacher can actually move the chips to visually show the compensation or filling up the ten. In this way some children might develop visual imagery for some of the strategies that they hear described verbally.

Typically, children's thinking shifts from thinking about visual items and counting-based solution methods to numerical interpretations based on five- or ten referenced and strategy-based interpretations. Increasingly, children develop number relationships for number combinations up to 20 and come to "just know" the addition facts up to 20. At the same time, those children for whom visual imagery is still necessary have a way to participate in the activity. Thus, the task accommodates the individual needs of the children while at the same time providing opportunities for all of them to advance in their interpretations.

The double ten-frame can be used to support children's understanding of comparisons and inequality by asking questions relating the quantities shown in the two frames. For example, questions based on the above display might include:


Children might think about the visual material to develop answers to these questions. For example, in the double ten-frame shown above some children might think of putting two more red chips in the upper frame to make it "the same" as the lower frame. The teacher can notate such descriptions using standard notation such as 4 + 2 = 6. As noted in the discussion of the spatial flashing (dot patterns) activity, in this way the teacher develops ways of notating naturally. The notation emerges from the children's thinking and is introduced as a means of recording thinking and facilitating communication.
The double ten-frame can be useful to facilitate children's development of thinking strategies by posing tasks in sequence which provide opportunities for children to relate the tasks to each other. For example, consider the following sequence of four double ten-frame tasks.


  task 1  task 2  task 3  task 4

Some children might solve task 2 by relating it to task one using the compensation strategy, that is, they might mentally move one blue chip from the upper frame to the lower frame to make six and six and reason that it is the same problem as task 1. For task 3, some children might relate it to task 2 by noticing that one more blue chip was added and so reason that the number of chips is one more than in task 2. Some children might solve task 4 by relating it to task 3 and reason using the compensation strategy. Others might say that there is one more red chip than in task 2 and reason using the +1 strategy. Still others might use the +1 compensation strategy but relate it to task 1 noticing that there is one more blue chip. Other children might not relate the problems but solve each one independently. However, by posing the tasks in a sequence such as this the teacher can create the opportunity for thinking strategy solutions. Typically, some children will use thinking strategies when tasks are posed in sequence. As the teacher capitalizes on these solutions in whole class discussions, solutions based on relating problems and on number relationships become the norm. As this happens, children begin to spontaneously use thinking strategies by relating the number combinations to some that they already know. The example given above for the problem that showed four in the first frame and six in the second frame is illustrative. A child might solve this task by relating it to the known fact 5 + 5 = 10, by thinking of moving one from the frame with six to the frame with four.