Lesson 12

The purpose of this Number Talk is to elicit strategies and understandings students have for adding two numbers when one number is close to a multiple of ten. These understandings help students develop fluency and will be helpful later in this lesson when students add the area of parts of a figure to determine the area of the whole figure.

When students use the fact that one number is close to 10 to find the sum, they look for and make use of structure (MP7).

• 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.

• “¿En qué se parecen los números de las cuatro expresiones?” // “How are the numbers in the four expressions alike?” (The first number has two digits and is 1 away from a whole ten. The second number is a single-digit number.)
• “¿Cómo les ayudaron estas características a encontrar cada suma?” // “How did these features help you find each sum?” (We could add the second number to the whole ten closest to the first number and then take away 1.)
• Consider asking:
  • “¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _______ ’s reasoning in a different way?”
  • “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
  • “¿Alguien pensó en el problema de otra forma?” // “Did anyone approach the problem in a different way?”
  • “¿Alguien quiere agregar algo a la estrategia de ____?” // “Does anyone want to add on to____’s strategy?”

The purpose of this activity is for students to learn that area is additive. Students decompose a rectangle into two smaller ones and find the sum of their areas in order to find the area of the whole rectangle. They can find the area of the two smaller rectangles by counting or by multiplying the side lengths.

When students consider how to decompose a larger rectangle into smaller ones to facilitate the process of finding area, they look for and make use of structure (MP7).

• Groups of 2
• “Van a responder algunas preguntas sobre plantar vegetales y flores en esta huerta. Piensen durante un minuto en los vegetales y las flores que les gustaría plantar en un espacio de una huerta” // “You are going to answer some questions about planting vegetables and flowers in this garden. Take a minute to think about the vegetables and flowers you would plant in a garden space.”
• 1 minute: quiet think time
• Share responses.

• “Ahora trabajen con su pareja en el primer problema” // “Now work with your partner on the first problem.”
• 2–3 minutes: partner work time
• Monitor for students who found the area by:
  • using counting strategies
  • multiplying the side lengths of the entire rectangle
  • adding the areas of the smaller rectangles
• Select previously identified students to share how they found the area.
• “¿Cómo saben que al sumar el área de las partes se obtiene lo mismo que al encontrar el área de toda la huerta?” // “How do you know that adding the area of the parts is the same as finding the area of the whole garden?” (The part covered with vegetables and the part covered with flowers make up the whole garden. If I add the area of both parts, I get the same number if I counted all the squares in the whole garden.)
• “Ahora van a diseñar su propia huerta. Asegúrense de explicar cómo encontraron el área de cada parte de su huerta y el área de toda la huerta” // “Now you are going to design your own garden. Be sure to explain the area of each part of your garden and the area of the whole garden.”
• 5 minutes: independent work time
• Monitor for students who create rectangular gardens and gardens composed of rectangles.

If students count one by one to find the area of their gardens, consider asking:

• “Dime cómo encontraste el área de tu huerta” // “Tell me how you found the area of your garden.”
• “¿Cómo podrías usar la multiplicación para encontrar el área de tu huerta?” // “How could you use multiplication to find the area of your garden?”

• Select 2–3 students who created a rectangular garden that is decomposed into rectangles to share.
• Consider asking:
  • “¿Cuántas unidades cuadradas cubre cada parte de la huerta?” // “How many square units does each part of the garden cover?”
  • “¿Cuántas unidades cuadradas cubre la huerta?” // “How many square units does the garden cover?”
  • “¿Qué ecuación representaría la manera como encontraron el área del rectángulo?” // “What equation would represent how you found the area of the rectangle?”

The purpose of this activity is for students to find the area of a figure by decomposing it into two non-overlapping rectangles. The synthesis should emphasize different strategies and also encourage students to directly link expressions and the use of parentheses to the way they decomposed the figure. If students drew gardens in the shape of the image in the launch, display those drawings as well during the notice and wonder.

Some students may partition diagonally to split the figure into what looks like 2 symmetrical parts, or cut the figure up into more than 2 parts. These are both acceptable ways of finding the area. Ask students who partition diagonally to find the area in the way they partitioned, but then encourage them to find a second way that has partitions on one of the grid lines. As students look through each others' work, they discuss how the representations are the same and different and can defend different points of view (MP3).

When students notice that the smaller parts of the figure can be added to find the total area of the figure they are looking for and making use of structure (MP7).

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

• Groups of 2
• Display the image of the gridded figure.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Students may notice: It looks like 2 rectangles. It looks like a big rectangle with a chunk missing. There are squares. Students may wonder: What is this shape called? Could we find the area of the shape? How would we find the area?)
• 1 minute: quiet think time
• Share responses.
• “Para esta no tenemos un nombre, como un cuadrado o un triángulo. Por esta razón, la llamaremos ‘figura’ mientras trabajamos con ella en esta actividad. Esta palabra será útil para describir otras figuras para las que no tenemos nombres” // “This isn’t a shape that we have a name for like a square or triangle. Because of this, we’ll call it a ‘figure’ as we work with it in this activity. This word will be helpful in describing other shapes that we don’t have a name for.”
• “Hablen con su pareja sobre diferentes maneras de encontrar el área de esta figura” // “Talk with your partner about different ways you could find the area of this figure.”
• 1 minute: partner discussion

• “Hay muchas maneras de encontrar el área de esta figura. Tómense un tiempo para encontrar el área. Vamos a compartir estas maneras de encontrar el área con la clase, así que incluyan detalles como sombreado, notas y marcas para ayudar a los demás a entender cómo pensaron” // “There are many ways to find the area of this figure. Take some time to find the area. We are going to share these with the class, so you may want to include details such as shading, notes, and labels to help others understand your thinking.”
• 5 minutes: independent work time

If students add numbers that indicate they tried to find the area by adding the areas of rectangles that overlap, consider asking:

• “Dime, ¿cómo partiste la figura en rectángulos para encontrar el área?” // “Tell me about how you broke the figure apart into rectangles to find the area.”
• “¿Cómo influiría superponer los rectángulos en el número de cuadrados necesarios para cubrir la figura?” // “How would overlapping the rectangles affect the number of squares it would take to cover the figure?”

MLR7 Compare and Connect

• Have students display their work.
• 5–7 minutes: gallery walk
• “¿En qué se parecen y en qué son diferentes las formas como otros estudiantes encontraron el área de la figura?” // “What is the same and what is different about how other students found the area of the figure?” (Some students broke the rectangle apart vertically and some students broke it apart horizontally. Others imagined a missing part that wasn’t there. They all found the same area. They all broke the rectangle into smaller parts and then added the parts to find the area.)
• 30 seconds quiet think time
• 1 minute: partner discussion
• Display an expression that reflects how students found the area, such as:

  \(\displaystyle 4 \times 5 + 5 \times 10\)
• “¿Cómo muestra esta expresión la forma de encontrar el área?” // “How does this expression show how to find the area?” (The \(4 \times 5\) represents the rectangle on the top of the figure if we break it into 2 rectangles. The \(5 \times 10\) represents the bottom rectangle.)
• Add parentheses to create the expression:

  \(\displaystyle (4 \times 5) + (5 \times 10)\)

• “Los paréntesis son símbolos de agrupación que se pueden usar en expresiones o ecuaciones. Para mostrar cómo se ve la figura, podemos mostrar el primer rectángulo con \(4 \times 5\) y el segundo rectángulo con \(5 \times 10\). Los paréntesis nos permiten poner ambos rectángulos en la misma expresión como \((4 \times 5) + (5 \times 10)\) y nos dejan ver cuál parte de la expresión representa cada rectángulo” // “Parentheses are grouping symbols that can be used in expressions or equations. To show how you saw the figure we can show the first rectangle with \(4 \times 5\) and the second rectangle with \(5 \times 10\). Parentheses let us put both rectangles in the same expression like \((4 \times 5) + (5 \times 10)\) and see which part of the expression represents each rectangle.”
• Reinforce the meaning of parentheses in a similar way with other ways students decomposed the figure.