Lesson 6

This warm-up elicits the idea that squares of different sizes can be used to measure area, which will be useful when students encounter different square units later. While students may notice and wonder many things, focus the discussion on how different-size squares could be used to tile the rectangle.

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “Hemos estado pensando en cómo podemos recubrir rectángulos con cuadrados de diferentes tamaños. Tengan esto en mente cuando pasemos a la siguiente actividad” // “We spent some time thinking about how rectangles could be tiled with different-size squares. Keep this in mind as we move on to the next activity.”

The purpose of this activity is for students to see that there are different types of square units that can be used to measure area and that an area with the same number of square units can be larger or smaller depending on the unit that is used. To facilitate comparison, one partner works on inch grid paper and one works on centimeter grid paper. In the synthesis, students are introduced to square inches and square centimeters.

• Groups of 2
• Give each group 1 copy of the blackline master.
• “Con su compañero, creen un rectángulo para cada expresión. El compañero 1 usará la cuadrícula 1 y el compañero 2 usará la cuadrícula 2. Piensen en qué podrían ser diferentes los rectángulos” // “Work with your partner to create a rectangle for each expression. Partner 1 will use grid 1 and partner 2 will use grid 2. Think about how the rectangles might be different.”
• 30 seconds: quiet think time

• 3–5 minutes: partner work time
• “Discutan con su compañero en qué se parecen y en qué son diferentes los rectángulos” // “Discuss how the rectangles are the same and how they are different with your partner.”
• 2–3 minutes: partner discussion

If students say that rectangles with the same number of squares have the same area, even if the squares are different sized, consider asking:

• “¿Qué estamos midiendo cuando medimos el área?” // “What are we measuring when we measure area?”
• “Si hay la misma cantidad de cuadrados, ¿cómo influye cambiar el tamaño de los cuadrados en el área que ellos cubren?” // “If there’s the same amount of squares, how does changing the size of the squares affect the area that the squares cover?”

• “¿En qué se parecen y en qué son diferentes los rectángulos?” // “How are the rectangles the same and different?” (The rectangles have the same number of squares, but they do not have the same area because they do not cover the same amount of space.)
• Give students rulers.
• “Con su pareja, investiguen los cuadrados de las dos cuadrículas. Prepárense para compartir lo que encuentren” // “Work with your partner to investigate the squares on both grids. Be prepared to share what you find.” (On the grid with the large squares, each side is 1 inch. On the grid with the small squares, each side is 1 centimeter.)
• “Estos son 2 tipos de cuadrados que podemos usar para medir el área. La pulgada cuadrada tiene una longitud de lado de una pulgada. El centímetro cuadrado tiene una longitud de lado de 1 centímetro” // “These are 2 types of squares we can use to measure area. The square inch has a side length of 1 inch. The square centimeter has a side length of 1 centimeter.”
• “La cuadrícula 1 es papel cuadriculado de pulgadas. Los cuadrados del papel cuadriculado de pulgadas tienen una longitud de lado de 1 pulgada. La cuadrícula 2 es papel cuadriculado de centímetros. Los cuadrados del papel cuadriculado de centímetros tienen una longitud de lado de 1 centímetro.” // “Grid 1 is inch grid paper. Inch grid paper has squares with side lengths of 1 inch. Grid 2 is centimeter grid paper. Centimeter grid paper has squares with side lengths of 1 centimeter.”

The purpose of this activity is for students to estimate and then find the area of a square using different standard units of area (square inches and square centimeters). Students discuss why using square inches and square centimeters gives different measurements for the same area.

To measure the area of the square, students trace the square or cut it out, place the traced or cut square on the grids from the previous activity, and determine the number of squares covered on each grid. The measurement in square centimeters is not exactly 25 square centimeters because the square is 2 inches by 2 inches. The main idea is that it takes more of a smaller unit than a larger unit to cover the same area. When students reference units in their area measurements they are attending to precision in language (MP6).

This activity uses MLR2 Collect and Display. Advances: conversing, reading, writing

• Prepare additional copies of the grids from Same Rectangles, Different Units so students can have a fresh copy to measure the area of the square.
• Have patty paper available, in case requested.

• Groups of 2
• “Estimen cuántas pulgadas cuadradas y cuántos centímetros cuadrados necesitarán para recubrir este cuadrado. Anoten sus estimaciones” // “Estimate how many square inches and how many square centimeters you think it will take to tile this square. Record your estimates.”
• 1 minute: independent work time
• “Compartan su estimación con su pareja. Pueden revisar y ajustar su estimación si cambia su forma de pensar” // “Share your estimate with your partner. You may revise your estimate if your thinking changes.”
• 1–2 minutes: partner discussion
• Give students access to scissors.

• “Ahora van a medir el área del cuadrado usando el papel cuadriculado de pulgadas y el papel cuadriculado de centímetros” // “Now you’re going to measure the area of the square using your inch grid paper and your centimeter grid paper.”
• “Si quieren, pueden trazar el cuadrado o recortarlo” // “You can trace the square or cut it out if you want to.”
• “Después, pueden poner el cuadrado sobre las cuadrículas de la actividad 1 para encontrar el área, en pulgadas cuadradas y en centímetros cuadrados” // “Then, you can place the square over the grids from Activity 1 to find the area in square inches and square centimeters.”
• “Por último, escriban una expresión de multiplicación que corresponda al área que encontraron” // “Finally, write a multiplication expression to match the area you found.”
• 5 minutes: partner work time

MLR2 Collect and Display

• Circulate, listen for, and collect the language students use to discuss square inches and square centimeters. Listen for: square inch, square centimeter, 1 inch on each side, 1 centimeter on each side, takes up less space than square inches, takes up more space than square centimeters, takes more square centimeters than square inches to cover the same space, or takes fewer square inches than square centimeters to cover the same space.
• Record students’ words and phrases on a visual display and update it throughout the lesson.

• “¿Qué otras palabras o frases deberíamos incluir en nuestra presentación?” // “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• “¿Cómo influyó medir con diferentes unidades cuadradas en la medición del área del cuadrado?” // “How did measuring with different square units affect the measurement of the area of the square?” (It took more of the smaller squares to tile the same area. It took fewer of the larger squares to tile the same area.)