Goal: Each group of students will complete a poster of all the possible arrays for the numbers one through ten, labeling the dimensions of each. Students will utilize their poster as a model for multiplication and explore prime and composite numbers, with their work focusing on finding factors of numbers, and recognizing prime numbers as numbers with only one pair of factors.

Level: 3rd or 4th grade

Materials: One-centimeter graph paper

scissors

glue or rubber cement

tape

poster board (one piece per group)

fine-tipped markers

Interlocking cubes

Procedure: Students will be placed in heterogeneous groups of 3-4 students.

The teacher will begin by talking about groups of people that need to sit down. These groups often need an arrangement of chairs to fit a specific space. The class can then take turns naming places where this might happen (Examples include: movie theater, school cafeteria, meeting, assembly, band concert).

The teacher then picks a number (example: 20) and asks the students to describe different rectangular arrays for the chairs. The teacher can draw and label the arrangements/arrays on the board or overhead as the students name them. It is important to include every possible array for the number. This model can remain visible for the students during their group activity.

Discuss with students how each multiplication pair is related: 1x20, 20x1, 4x5, 5x4, 2x10, 10x2. Ask students how they would count the squares to see that they have a total of 20. They will volunteer many inventive ways of counting. Emphasize counting by groups as a lead into multiplication.

Each group of students is then instructed to make a poster of all possible arrays for the numbers one through ten. Students will cut the arrays from graph paper. Each group will be responsible for deciding how they will arrange and glue the arrays on the poster in an "orderly" fashion. Students are instructed to label each array, including any prime numbered arrays. Students may use the interlocking cubes to help them find the arrays before cutting graph paper.

The teacher will circulate around the room, talking with the groups and

asking them:

• How will you know when you have found all the arrays for a number?

• Have you found any numbers that have only one array?

• How did you count your array to know that the total was correct?

• What method are you using to organize your arrays?

• Can you find anything true about odd or even numbers?

• Did you find any prime numbers?

End the activity by having each group briefly share their poster. Each group member must speak and be a part of the explanation and discuss their discoveries.

This should include classmates asking any questions they may have of the group presenters. This should also include a class discussion about prime numbers, which have only two factors- one and itself. Numbers with two or more factors are called composite numbers. Students should also notice odd and even number of factors and discuss any patterns they recognized.

Post the posters in the room or hallway for further viewing.

Heterogeneous Groups: Students are grouped to include high, medium, and low ability students including special needs students. Static and dynamic norms will be factors in determining the quality and productivity of the group, actively influencing interpersonal relationships.

Positive Interdependence: The students must work cooperatively to complete the task. Members have reciprocal influence over each other. Interpersonal helpfulness is essential. All members need to help. Students are developing interpersonal and conflict- resolution skills, working to gain peer group support of ideas, depending on each group member to accomplish the task. The group goal is to work productively, and all students need to feel that they belong and that they contributed. There must be an exchange of information to determine arrangement, process, and job description of the individual members of the group. There will also be interpersonal contacts during the whole group discussion (collective orientation). A sense of connectedness, trust, and inter- responsibility contribute to a more positive interdependence within the groups.

The circular interpersonal process is evident as the individual members of the group develop working expectations of how to accomplish the task. The attitudes and expectations that the individuals hold of themselves and others in the group will be key factors in deciding who does what in the group. 

Face to Face Interactions: Each group member is physically seated at the same table and must interact to make decisions about the project. Elements of cooperation , or working together to achieve a common goal, supporting each other and helping each other, are dependent on this face-to-face interaction and communication.

 Social Skills: Students bring a unique set of characteristics to each group. The personalities and self-concepts and especially how they perceive others can have an impact on getting the job done. The skills and competencies of the individuals also affect the progress of the group. They may determine who does what task within the group. Gender socialization will also influence the group dynamics. In this development of social skills, they must focus on the collective well-being of the group, reaching agreement together and cooperatively. They must learn to be accepting of others’ opinions and ideas, skills, and attributes which will help them build communitiesoutside the classroom ( Sapon-Shevin, 1999, p.31).

Individual accountability: Task completion requires that all members participate. The members are interchangeable in the task-group goals. This project has a high degree of "groupness" in its goals. Individuals could work in parallel situations- (Schmuck, 2001, p,32) task-individual. Mary could do the 2’s, Beth could do the 3’s, and so on. Social-emotional group discussion takes place in the end when they share their group expectations. Each individual should be accountable for speaking during the presentation.

Evaluation: This activity would not necessarily have to require a "group grade".

"Finn wrote that expectations are evaluations" ( Schmuck. 2001, p.165,). In this case, the teacher has communicated certain expectations for academic performance.

The teacher is providing feedback by circulating around the room during the project time. The teacher’s expectations of the final product would include:

• Have the students included all possible arrays for each number?

• Did the students label each array? labeled correctly?

• Are the required elements arranged in an organized manner?

• Did the students identify all prime numbers?

The teacher can also evaluate the students’ understanding of the activity during the group discussion.

• Did each student in the group speak during the group presentation?

• Did each student in the group demonstrate a basic understanding of the concepts?

* This lesson was adapted from INVESTIGATIONS in Numbers, Data, and Space, Arrays and Shares, TERC, Cambridge, Massachusetts.