Lesson 3

The purpose of this Choral Count is to invite students to practice counting by 2 and 5 and notice patterns in the count. These understandings help students develop fluency and will be helpful later when students find factor pairs.

When students predict common multiples for 2 and 5 based on the numbers recorded from the count and what they know about multiplication, they look for and express regularity in repeated reasoning (MP8).

• “Cuenten de 2 en 2, empezando en 0” // “Count by 2, starting at 0.”
• Record as students count.
• Stop counting and recording at 30.
• “Cuenten de 5 en 5, empezando en 0” // “Count by 5, starting at 0.”
• Record as students count.
• Stop counting and recording at 75.

• “¿Qué patrones ven en los conteos individuales?” // “What patterns do you see in the individual counts?”
• 1–2 minutes: quiet think time
• Record responses.
• “¿Qué patrones notan si miran los dos conteos juntos?” // “What patterns do you see between the two counts?”
• 1–2 minutes: quiet think time
• Record responses.

• If it doesn’t come up in the student responses, ask: “¿Cuántos dos se necesitaron para llegar a 10? ¿Cuántos cincos se necesitaron para llegar a 10?” // “How many twos did it take to get to 10? How many fives did it take to get to 10?” (It took 5 twos and 2 fives to get to 10.)
• “Diez es un múltiplo de 2 y de 5. ¿Observan algún otro número que sea múltiplo tanto de 2 como de 5?” // “Ten is a multiple of 2 and 5. Do you notice any other multiples of both 2 and 5?” (20 and 30 are on both lists.)
• 1 minute: partner discussion
• Record responses.
• “Si los conteos siguieran, ¿qué otros números verían que son múltiplos tanto de 2 como de 5?” // “If the counts continue, what other numbers would you see that are multiples of both 2 and 5?” (I think 40 would be the next common multiple because the multiples are going up by 10. I think 100 would be a common multiple because \(2\times50 = 100\) and \(5 \times 20 = 100\).)
• 2 minutes: partner discussion
• Record responses.

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 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.

Here is an image of the cards for reference.

• Create a set of cards from the blackline master for each group of 2. 

• Groups of 2
• Give each group a set of cards from the blackline master.
• “Clasifiquen las tarjetas en categorías de cualquier forma que tengan sentido para ustedes” // “Sort the cards into categories in any way that makes sense to you.”
• 2 minutes: partner work time
• Ask students to share ways in which they sorted.

• “Si no lo han hecho todavía, clasifiquen los rectángulos por área” // “If you did not already, sort the rectangles by their area.”
• 3–5 minutes: partner work time
• Ask students to check their work with another group to make sure the cards in each category match.
• “Ahora, hagan por lo menos un rectángulo para agregar a cada categoría de su clasificación de tarjetas” // “Now, create at least one rectangle to add to each category in your card sort.”
• 3–5 minutes: partner work time
• Observe the rectangles students add to each category. Monitor for students who notice that no new rectangles could be drawn for the area of 7 square units.

• Select 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.)
• Revoice 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 this 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 area of rectangles to find all of the factor pairs of a given whole number and decide if the number is prime or composite. The 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.

• Groups of 2
• Give each group access to inch tiles and grid paper.
• “Si se les diera un número que es el área de un rectángulo, ¿cómo podrían saber cuántos rectángulos se pueden hacer que tengan esa área?” // “If you were given a number that is the area of rectangle, how could you find out how many rectangles with that area can be made?” (Test it out with tiles. Think about factor pairs for the number.)
• 1 minute: partner discussion
• Share and record responses.

• “Completen esta tabla con su compañero. Si quieren, pueden usar fichas de pulgada o papel cuadriculado” // “Work with your partner to complete this table. Inch tiles and grid paper are available if you’d like them.”
• 10 minutes: partner work time
• Monitor for different ways students find the number of rectangles, such as:
  • building the rectangles from inch tiles
  • drawing rectangles on grid paper
  • drawing rectangles freehand
  • listing the factor pairs of the number and knowing that one rectangle corresponds to each pair

• Invite 3–4 groups 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. Or, 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, 23, 31. They each only have one set of side lengths, 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 el número primo más pequeño?” // “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.)