Lesson 14

The purpose of this Number Talk is to elicit strategies and understandings students have for multiplying by 5. These understandings help students develop fluency and will be helpful later in this lesson when students represent and solve a problem involving groups of 5.

When students reason why as one factor increases by 1, the product increases by 5, they are looking for and expressing the regularity they notice in the expressions (MP8).

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “¿Qué patrón reconocen al observar todos los problemas? ¿Por qué ocurre esto?” // “What pattern do you see as you look at all of the problems? Why is that happening?” (The factor that isn’t 5 goes up by 1 each time. The products increase by 5. We added another group of 5.)

The purpose of this activity is for students to relate equations to multiplication situations and diagrams using a symbol for the unknown number. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). Students explain their matches to their peers and revise their language for precision and clarity when they describe how the numbers and symbols in the equations match the representations (MP3, MP6). In the synthesis, students explain the meaning of the factors and products, and what a symbol in an equation represents.

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

• Groups of 2
• Display: \(4\times5=?\)
• “¿Cuál podría ser el significado de esta ecuación?” // “What might this equation mean?” (There are 4 groups of 5. There's a number in the equation that we don't know. We don't know the total.)
• 2 minutes: partner discussion 
• Share responses. 
• “Se pueden usar diferentes símbolos para representar el número desconocido en una ecuación. Algunos símbolos comunes son los signos de interrogación, los espacios en blanco y los cuadros” // “Different symbols can be used to represent the unknown number in an equation. Some that are common are question marks, blank spaces, and boxes.”
• “Por ejemplo, si en la ecuación \(80 = 8\times10\) no conociéramos el producto, podríamos escribir: ? = \(8\times10\)” // “For example, in the equation \(80 = 8\times10\), if we didn’t know the product we could write ? = \(8\times10\).” Display these equations as you explain. 
• “Si no conociéramos uno de los factores, ¿qué ecuación se podría escribir usando un símbolo para representar el número desconocido?” // “If we didn’t know one of the factors, what is an equation you could write using a symbol for the unknown number?” (\(80 = \underline{\hspace{1 cm}} \times10, 80 = ? \times10\))
• Distribute one set of pre-cut cards to each group of students.

• “Este grupo de tarjetas incluye ecuaciones, situaciones y diagramas. Emparejen cada ecuación con una situación o con un diagrama. Con su compañero, justifiquen sus elecciones” // “This set of cards includes equations, situations, and diagrams. Match each equation to a situation or diagram. Work with your partner to justify your choices.”
• 5-7 minutes: partner work time
• Monitor for students who explain the meaning of the factors and the product, specifically that the symbol is for a missing number that represents a missing amount in the diagram or situation.

• Have students share their matches and how they know those cards go together.
• Choose 1-2 equations (at least one with a missing factor) and ask:
  • “¿Qué representa cada número en la ecuación?” // “What does each number in the equation represent?”
  • “¿Qué representa cada signo de interrogación (o espacio o cuadro)?” // “What does each question mark (or blank or box) represent?”
  • “¿Cómo podemos descifrar qué número debe ir en el espacio para que la ecuación sea verdadera?” // “How can we figure out which number goes in the blank to make the equation true?”
• Listen for the language students use to describe their matches and the equations, diagrams, and situations clearly and precisely. As needed, ask:
  • “¿Qué quieres decir con ______?” // “What do you mean when you say _____?”
  • “¿El número desconocido era el producto, o era uno de los factores? Explica” // “Was the unknown the product or one of the factors? Explain.”
  • “¿Qué representaba el factor desconocido? Explica” // “What did the unknown factor represent? Explain.”
• Highlight terms such as “factor” and “product.”

The purpose of this activity is for students to write equations for multiplication situations and diagrams using a symbol for the unknown number. When students write an equation to represent a situation, including a symbol for the unknown number, they model a situation with mathematics (MP4).

Students find an unknown factor or unknown product in multiplication problems. In this task, the unknown factor diagrams and situations only include the “how many groups” problem type and the factors 2, 5, and 10. This sets students up to skip-count to find the unknown number.

This problem type will be revisited extensively in future lessons and will be related to division. It is not necessary to make the connection to division now. In the synthesis students explain how the equations they wrote represent the diagram or situation.

• Groups of 2

• “Ahora van a practicar la escritura de sus propias ecuaciones que tengan un símbolo para representar el número desconocido” // “Now you will practice writing your own equations with a symbol for the unknown.”
• 2–3 minutes: independent work time
• “Compartan sus ecuaciones con su pareja. Discutan cómo saben que cada ecuación corresponde al diagrama o a la situación” // “Share your equations with your partner. Discuss how you know each equation matches the diagram or situation.”
• 2–3 minutes: partner discussion
• Have a whole-class discussion focused on how the equations match the different representations.
• Consider asking:
  • “¿Cómo usaron las representaciones para escribir una ecuación que tuviera un símbolo en el lugar del número desconocido?” // “How did you use the representations to write an equation with a symbol for the unknown?” (I looked for what was missing in the diagram. I thought about the situation to figure out if it was the number in each group, the number of groups, or the total that was missing.)
• “Ahora encuentren el número desconocido de cada ecuación y escriban una nueva ecuación que incluya la solución” // “Now find the missing number in each equation and write a new equation that includes the solution.”
• 3–5 minutes: partner work time

If students write an equation without a symbol for the unknown number, consider asking:

• “¿De qué manera tu ecuación representa el diagrama (o la situación)?” // “How does your equation represent the diagram (or situation)?”
• “¿Cómo podrías mostrar con un símbolo el número que no se conocía?” // “How could you show the number that was unknown with a symbol?”

• “¿Qué estrategias usaron para encontrar los números desconocidos?” // “What strategies did you use to find the unknown numbers?” (I counted by 2 to get the total. I counted by different numbers until I found a number that gave me the product.)
• “¿Cómo cambió cada ecuación luego de encontrar el número desconocido?” // “How did each equation change as you found the unknown number?” (The symbol was replaced with the number.)