Lesson 13

The purpose of this Number Talk is to elicit strategies students have for adding two numbers when one number is close to a whole number of tens. These understandings help students develop fluency in addition. Students may look for and make use of structure (MP7) in a number of ways. For example, they may add 1 to the first addend to make a full ten and subtract 1 from the second addend to find each sum. They may also notice how the addends compare to those in the previous expression and use the change to find the new sum.

In this string, students may also add the tens and ones separately to find the sum. Adding by place value is the focus of upcoming work. This Number Talk also enables the teacher to learn the strategies students currently have for addition.

• 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 observan sobre cómo se relacionan las dos primeras expresiones? ¿Qué observan sobre cómo se relacionan las dos últimas expresiones?” // “What do you notice about how the first two expressions are related? What about how the last two expressions are related?” (The second sum is 10 more than the first. The fourth sum is 10 more than the third.)

The purpose of this activity is for students to find the area of figures that are composed of rectangles but are not fully gridded with squares. Partially gridded figures help to prepare students to find the area of figures with only side length measurements. Students should be encouraged to find side lengths and multiply, rather than rely on counting, as the grids disappear. If students continue to draw in the squares, ask them if there is another way to find the area.

• Groups of 2
• Sketch or display a rotated L-shape figure as shown.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Students may notice: The figure is not a rectangle. It could be split into smaller rectangles. Students may wonder: Why are there no squares inside? How can I find out how many squares will cover that shape?)
• 1 minute: quiet think time
• Share and record responses.
• “¿Qué información les ayudaría a encontrar el área de esta figura?” // “What information would help you find the area of this figure?” (The side lengths. Being able to see the squares inside the figure.)
• 1 minute: quiet think time
• Share responses.
• Display image from the first problem.
• “¿Qué información, de la que se da en la figura, podría ayudarles a encontrar el área?” // “What information is given in this figure that could help you find the area?” (Grid lines. The side lengths. Some of the squares.)
• Share responses.

• “Ahora, encuentren el área de esta figura con su pareja” // “Now work with your partner to find the area of this figure.”
• 5 minutes: partner work time
• Monitor for strategies for finding the side lengths and decomposing into rectangles.
• “Examinemos la primera figura” // “Let's look at the first figure.”
• Have students share strategies for finding the side lengths and area of figures with a partial grid.
• “Observen la siguiente figura. Piensen en cómo podrían encontrar el área de esta figura” // “Take a look at the next figure. Think about how you could find the area of this figure.”
• 1 minute: quiet think time
• “Con su pareja, encuentren el área de esta figura” // “Work with your partner to find the area of this figure.”
• 5 minutes: partner work time
• Monitor for strategies for finding the side lengths.

• “¿Cuál fue su estrategia para encontrar el área de la segunda figura?” // “What was your strategy for finding the area of the second figure?”
• “¿Qué les ayuda a encontrar el área de figuras como estas, que no están completamente cubiertas con cuadrados?” // “What helps you find the area of figures like these where the shape is not fully covered with squares?” (Imagining where the squares would be so I can count them or find the side lengths. Finding the side lengths and multiplying to find the area of rectangles in the figure.)

The purpose of this activity is for students to find the area of a figure composed of rectangles given only their side lengths. The context of paving a patio provides students a link to their experience with squares of various sizes and should help them imagine how the diagram of the patio could be covered with squares. Students decompose the patio into rectangles and can multiply to find the area of the patio, but they should make the connection that the number of pavers needed to cover the patio is the same as the area of the patio. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).

• Groups of 2
• Display the image.
• “Este problema se trata de la construcción de un patio con adoquines. Los adoquines son rocas, ladrillos o bloques que se ponen sobre el suelo para hacer un camino o un área adoquinada. Este es un patio hecho de adoquines” // “This problem is about making a patio using pavers. Pavers are stones, bricks, or blocks that are put on the ground to make a path or paved area. This is a patio that’s made of pavers.”
• “Antes hablamos sobre distintos tipos de unidades que podemos usar para medir el área. En este caso, Noah usa adoquines de 1 pie cuadrado. Eso significa que el adoquín es de 1 pie por 1 pie. Ayúdenlo a descubrir cuántos adoquines necesita para cubrir el patio. Piensen en la situación durante un minuto” // “Earlier, we’ve talked about several different types of units that we can use to measure area. Here, Noah is using pavers that are 1 square foot. That means the paver is 1 foot by 1 foot. Help him figure out how many pavers he needs to cover the patio. Take a minute to think about the situation.”
• 1 minute: quiet think time

• “Ahora, respondan las preguntas con su pareja” // “Now work with your partner to answer the questions.”
• 5 minutes: partner work time
• Monitor for multiplication expressions students use to represent the area of the rectangles within the figure, such as:
  • \(9 \times 4 = 36\), \(3 \times 2 = 6\), \(36 + 6 = 42\)
  • \(6 \times 3 = 18\), \(6 \times 4 = 24\), \(18 + 24 = 42\)
  • \(9 \times 4 + 3 \times 2\) or \((9 \times 4) + (3 \times 2)\)
  • \(6 \times 3 + 6 \times 4\) or \((6 \times 3) + (6 \times 4)\)

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

• “Dime cómo encontraste el área de la figura” // “Tell me about how you found the area of the figure?”
• “¿Cómo influiría la superposición de los rectángulos en el número de cuadrados necesarios para cubrir la figura?” // “How would overlapping the rectangles affects the number of squares it would take to cover the figure?”

• “¿Cómo descubrieron cuántos adoquines necesitaría Noah?” // “How did you figure out how many pavers Noah would need?”
• Select students to share a variety of strategies. Display any expressions students used in their explanations.
• During each explanation, ask the class which measurements were and were not used and why.
• “¿Cómo representa cada expresión una forma de encontrar el área de la figura?” // “How does each expression represent a way of finding the area of the figure?” (The expression \((9 \times 4) + (3 \times 2)\) shows the figure decomposed into a big rectangle across the top and a small rectangle below. Each part of the expression in parentheses represents one of the smaller rectangles in the figure. The expressions \(6 \times 3 = 18\), \(6 \times 4 = 24\), and \(18 + 24 = 42\) show how the figure was decomposed into a 6-by-3 rectangle on the left and a 4-by-6 rectangle on the right, then the areas were added.)