The purpose of this Number Talk is to elicit strategies and understandings students have for dividing within 100. These understandings help students develop fluency and are helpful as students use division to solve problems involving perimeter.
Launch
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
Activity
Record answers and strategy.
Keep expressions and work displayed.
Repeat with each expression.
Encuentra mentalmente el valor de cada expresión.
Student Response
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Advancing Student Thinking
Activity Synthesis
“Al encontrar el valor de las otras expresiones, ¿cómo puede ayudarlos saber cuánto es ?” // “How could knowing help you find the value of the other expressions?” (Once I knew , I was able to take away a group of 9 to find . I was able to find by splitting the value of because 45 is half of 90.)
Activity 1
Standards Alignment
Building On
Addressing
3.MD.D.8
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
The purpose of this activity is for students to practice finding the perimeter of shapes that have labeled side lengths. The Activity Synthesis focuses on approaches students have for looking for and making use of the structure of shapes as a way to find their perimeter (MP7). As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).
Monitor for and select students with the following approaches for finding the perimeter of the hexagon to share in the Activity Synthesis:
Add the individual side lengths around the shape.
Add the two 8–inch side lengths together, add the four 4–inch side lengths together, and then add those sums together.
Multiply like side lengths, then add, such as for the long sides and for the short sides, and then add those products together.
Use symmetry to split the shape in half horizontally, add for the top half of the shape, and then double that for the sides on the bottom half of the shape.
The approaches are sequenced from more concrete to more abstract to encourage students to make sense of approaches that connect the properties of the shapes (for example, same side lengths) to operations that help them find the perimeter in fewer steps (for example, multiplication). Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven't shared recently.
MLR8 Discussion Supports. Synthesis: Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking
Launch
Groups of 2
Display the image.
“¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (One rectangle has numbers on the sides. One rectangle has tick marks on the sides. The rectangles are the same size. Why are the sides of the rectangles marked differently? Could we find the distance around the rectangle with the numbers on the sides?)
1 minute: quiet think time
1 minute: partner discussion
Share and record responses.
Activity
“Encuentren el perímetro de cada figura con su compañero” // “Work with your partner to find the perimeter of each shape.”
5–7 minutes: partner work time
As you monitor for the approaches listed in the activity narrative, consider asking:
“¿Qué observaron acerca de esta figura?” // “What did you notice about this shape?”
“¿Cómo usaron lo que observaron como ayuda para encontrar el perímetro?” // “How did you use what you noticed to help you find the perimeter?”
“¿Qué otra operación u operaciones pueden usar para encontrar el perímetro de esta figura?” // “What other operation(s) could you use to find the perimeter of this shape?”
¿Qué observas? ¿Qué te preguntas?
Encuentra el perímetro de cada figura. Explica o muestra tu razonamiento.
Activity Synthesis
Invite previously selected students to share in the given order. Record or display their work for all to see.
Give students a chance to ask questions about each approach as it is shared.
Connect students’ approaches by asking:
”¿En qué se parecen estas estrategias para encontrar el perímetro? ¿En qué son diferentes?” // “How are these ways of finding the perimeter the same? How are they different?”
“¿Por qué funciona la estrategia de _____ para encontrar el perímetro de esta figura? ¿Funcionará para todas las figuras?” // “Why does ______'s way work for finding the perimeter of this shape? Would it work for all shapes?”
Connect students’ approaches to the learning goal by asking:
“¿Fue más fácil encontrar el perímetro de unas figuras que de otras en esta actividad? ¿Por qué?” // “Was it easier to find the perimeter of some shapes in this activity than others? Why?” (Yes, some of the shapes had several sides that are the same length, so we could multiply. In a rectangle, we can add two sides and then double the result to find the whole perimeter.)
“¿Cómo nos ayuda tener lados de la misma longitud a encontrar el perímetro?” // “How does having sides of the same length help us find the perimeter?” (You can look for ways to use multiplication instead of having to add many different lengths.)
Activity 2
Standards Alignment
Building On
Addressing
3.MD.D.8
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
The purpose of this activity is for students to find the perimeter of shapes when some of the side lengths are not given. Students use their knowledge of shapes to reason about the length of the missing sides before they find the perimeter of the shape (MP7).
This activity uses MLR7 Compare and Connect. Advances: representing, conversing
Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most useful to solve the problem. Display the sentence frame: “La próxima vez que encuentre el perímetro de una figura en la que algunas de las longitudes de los lados no estén dadas, prestaré atención a . . .” // “The next time I find the perimeter of a shape where some side lengths are not given, I will pay attention to . . . .“ Supports accessibility for: Memory, Visual-Spatial Processing
Launch
Groups of 2
Display the rectangle from the first problem.
“Encuentren el perímetro de este rectángulo” // “Find the perimeter of this rectangle.“
1–2 minutes: independent work time
“Discutan con su compañero cómo encontraron el perímetro de este rectángulo a pesar de que algunas de las longitudes de sus lados no estaban marcadas” // “Discuss with your partner how you found the perimeter of this rectangle even though some of the side lengths were not labeled.” (Since the shape is a rectangle, we know opposite sides of a rectangle are the same length.)
1 minute: partner discussion
Share and record responses.
Give each group tools for creating a visual display.
Activity
“Encuentren el perímetro de las otras dos figuras con su compañero. Asegúrense de escribir su razonamiento para compartirlo con toda la clase” // “Work with your partner to find the perimeter of the other two shapes. Be sure to record your reasoning to share with the class.”
6–8 minutes: partner work time
Consider asking: “¿Cómo supieron cuál era la longitud de ese lado?” // “How did you know the length of that side?”
MLR7 Compare and Connect
“Creen una presentación visual que muestre cómo pensaron en el segundo problema. Incluyan detalles, como notas, diagramas, dibujos, etc., para ayudar a los demás a entender sus ideas” // “Create a visual display that shows your thinking about the second problem. You may want to include details such as notes, diagrams, drawings, and so on, to help others understand your thinking.”
3–5 minutes: partner work time
5 minutes: gallery walk
Encuentra el perímetro de este rectángulo. Explica o muestra tu razonamiento.
Todos los lados cortos de esta figura tienen la misma longitud y todos los ángulos son ángulos rectos. Encuentra el perímetro de la figura. Explica o muestra tu razonamiento.
Todos los lados de este octágono tienen la misma longitud. Encuentra el perímetro del octágono. Explica o muestra tu razonamiento.
Student Response
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Advancing Student Thinking
Activity Synthesis
“Tuvimos que encontrar muchas longitudes desconocidas de los lados de esta figura antes de poder encontrar el perímetro” // “We had to find a lot of missing side lengths in this shape before we could find the perimeter.”
“Cuando fueron a ver las presentaciones de los demás, ¿qué observaron acerca de cómo encontraron ellos las longitudes desconocidas de los lados?” // “As you visited the displays, what did you notice about how others found the missing side lengths?” (I noticed some groups counted the number of short sides and multiplied by 40. I noticed some put the short side lengths into smaller groups before finding their combined lengths.)
Consider asking:
“¿Alguien encontró las longitudes de lado desconocidas de una manera diferente a como ustedes y su compañero lo hicieron?” // “Did anyone find the missing side lengths in a different way than you and your partner?”
“¿Alguien encontró el perímetro de una manera diferente a como ustedes y su compañero lo hicieron?” // “Did anyone find the perimeter in a different way than you and your partner?”
Lesson Synthesis
“Cuando encuentran el perímetro de una figura, siempre se pueden sumar las longitudes de los lados de una en una. ¿Qué otros métodos conocen para encontrar el perímetro de las figuras?” // “When you are finding the perimeter of a shape, you can always add the lengths of the sides one at a time. What other methods do you have for finding the perimeter of shapes?” (We can look for side lengths that are the same and group them together. In a square, we can multiply one side length by 4 since they are all the same length. In a rectangle, we can add a long side to a short side and then double that for the whole perimeter.)
Display a rhombus with side lengths that are the same length, but only one side labeled 7 inches, such as:
“¿Cómo podemos encontrar el perímetro de este rombo si solo está marcado un lado?” // “How can we find the perimeter of this rhombus if only one side is labeled?” (We know that a rhombus has four equal sides, so we can find , which is 28.)
Standards Alignment
Building On
Addressing
3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that , one knows ) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
If students add the side lengths for the hexagon, but do not find a perimeter of 32 feet, consider asking:
“¿Cómo encontraste el perímetro de esta figura?” // “How did you find the perimeter of this shape?”
“¿Qué observas acerca de las longitudes de los lados? ¿Cómo puedes usar la multiplicación para encontrar el perímetro de esta figura?” // “What do you notice about the side lengths? How could you use multiplication to find the perimeter of this shape?”
