The purpose of this activity is for students to learn about prime numbers and composite numbers. Students are given a set of cards with rectangles on them. They sort the rectangles by area and then attempt to draw an additional rectangle for each category. They notice that some areas can be represented by more than one rectangle and some areas can only be represented by one rectangle.

During the Activity Synthesis, highlight that the side lengths of each rectangle represent one factor pair (each pair of side lengths should be used only once), and that the area of each rectangle represents a multiple of each side length. Students learn that a number with only one factor pair—1 and the number itself—is a prime number, and a number with more than one factor pair is a composite number.

• Invite 2–3 students to share the rectangles they added to each category.
• “¿Por qué pudieron hacer más rectángulos para algunas áreas y no para otras?” // “Why were you able to create more rectangles for some areas and not others?” (Some of the numbers had more factor pairs. For some numbers, there was only one possible factor pair.)
• Repeat student reasoning. “Solo se puede hacer un rectángulo para el área de 7. Los números como este se llaman números primos. Un número primo solo tiene una pareja de factores: 1 y él mismo” // “Only one rectangle can be made for the area of 7. Numbers like 7 are called prime numbers. Prime numbers have only one factor pair: 1 and itself.”
• “Los números como el 15, que tienen más de una pareja de factores, se llaman números compuestos” // “Numbers like 15 that have more than one factor pair are called composite numbers.”
• “¿Con qué otros números compuestos trabajaron? ¿Cómo saben que son compuestos?” // “What other composite numbers did you work with? How do you know they are composite?” (Twenty-four is a composite number because I can make 2 rows of 12 or 4 rows of 6. Eighteen is composite because it has factor pairs of 2 and 9 and 3 and 6.)

In this activity, students use the area of rectangles to find all factor pairs of a given whole number and decide if the number is prime or composite. The Activity Synthesis focuses on finding all possible rectangles for a given area as a strategy to find all the factor pairs of a number. Students may notice that they do not need to find all possible rectangles to determine whether a number is prime or composite.

• Invite 3–4 groups to share their strategy for finding the number of rectangles for a given area.
• “¿Cómo se relaciona el número de parejas de factores con el número de rectángulos?” // “How does the number of factor pairs relate to the number of rectangles?” (The side lengths of each rectangle is a factor pair. So, finding all the rectangles would give us all the factor pairs. Finding all the factor pairs of the number would tell us how many rectangles have that number for their area.)
• “¿Cuáles son todos los números primos de nuestra lista? ¿Cómo sabemos que son primos?” // “What are all of the prime numbers in our list? How do we know they are prime?” (2, 11, 23, 31. They each only have one factor pair, 1 and the number itself.)
• “¿Qué observan sobre los números primos?” // “What do you notice about the prime numbers?” (They are odd numbers except the number 2.)
• “¿Cuál es el número primo más pequeño de nuestra colección? ¿Es éste el número primo más pequeño que hay?” // “What is the smallest prime number in our set? Is it the smallest prime number?” (2. I don’t know. Is 1 a prime number?)
• Display a rectangle with an area of 1 square unit.
• “¿Cuánto miden los lados de un rectángulo que tiene un área de 1 unidad cuadrada?” // “What are the side lengths of a rectangle with an area of 1 square unit?” (1 and 1)
• “Como 1 solo tiene 1 factor, no tiene ninguna pareja de factores, así que no es ni primo ni compuesto” // “Since 1 only has 1 factor, it doesn’t have any factor pairs, so it is neither prime nor composite.”
• “¿Cuáles son todos los números compuestos de nuestra colección? ¿Cómo sabemos que no son primos?” // “What are all the composite numbers in our set? How do we know they are not prime?” (10, 48, 21, 60, 32, 42, 56. They each have more than 1 factor pair.)