Lesson 13

The purpose of an Estimation Exploration is for students to practice the skill of estimating a reasonable answer based on experience and known information. In this lesson they will be finding the perimeter and area of rectangles and their thinking about the size of the windows in this image prepares them for this work. 

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discutan con su pareja cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “¿Qué cosas de la imagen les podrían ayudar a estimar el área de la ventana?” // “What could you use in the image to help estimate the area of the windows?” (There are the people cleaning the windows. I used the people to estimate the height and width of the windows and then multiplied to find the area.)

The purpose of this activity is for students to plot points that represent the length and width of a rectangle with a given perimeter. Since the perimeter is twice the length plus twice the width, decreasing the length by a certain amount will mean that the width has to increase by the same amount for the perimeter to stay the same. Students have an opportunity to observe this relationship in multiple ways (MP7, MP8):

• think geometrically about the perimeter of the rectangle
• look at the table of values for length and width depending on the values they used
• look at the length and width pairs plotted in the coordinate grid

• Groups of 2

• 2 minutes: independent think time
• 5 minutes: partner work time

If students are not sure how to determine the width of Jada’s rectangle, prompt the student to draw a rectangle and ask, “¿Cómo te puede ayudar tu dibujo a completar la tabla?” // “How can you use your drawing to help you fill in the table?”

• “¿Cómo encontraron el ancho del rectángulo de Jada que medía 3.25 cm de largo?” // “How did you find the width of Jada's rectangle if it was 3.25 cm long?” (I knew that the length and width together are half the perimeter which is 6 cm. So I subtracted 3.25 from 6 and that was 2.75.)
• “¿Qué le pasa al ancho cuando el largo aumenta 1? ¿Por qué?” // “What happens to the width when the length increases by 1? Why?” (The width decreases by one. This makes sense because the sum needs to say the same or else the perimeter changes.)
• “¿Cómo se ve esto en la gráfica?” // “How does the graph show this?” (For each point I plotted, I can go right one and down one and find another possible length and width.)

The purpose of this activity is to investigate the possible lengths and widths of a rectangle with given area. Since the area is the product of length and width, this means that the main operation being used here is multiplication or division, contrasting with the previous activity where students investigated the perimeter which is the sum of the side lengths of a rectangle. This means that the calculations are more complex and some of the coordinates of the points that students plot will either be decimals or fractions depending how students express them. There are some important common characteristics between the lengths and widths for a given area and for a given perimeter which will be examined in the activity synthesis (MP7, MP8):

• when the length increases, the width decreases
• the length and width can be switched to get another possible length and width pair

• Groups of 2

• 2 minutes: independent think time
• 5 minutes: partner work time

• Invite students to share their responses for the width of a rectangle that is 5 cm long.
• “¿Cómo calcularon el valor?” // “How did you calculate the value?” (I knew that 5 times the width was 16 so the width is \(16 \div 5\) or \(\frac{16}{5}\) cm.)
• “¿Cómo supieron dónde ubicar esa pareja de largo y ancho?” // “How did you know where to plot that length and width pair?” (I looked for 5 on the horizontal axis and then I had to estimate where \(3\frac{1}{3}\) was on the vertical axis. I put it a little above 3 but closer to 3 than to 4.)
• “¿En qué fue parecido encontrar largos y anchos posibles para un área dada y encontrar largos y anchos posibles para un perímetro dado?” // “How was determining the possible lengths and widths for a given area the same as determining the possible lengths and widths for a given perimeter?” (When the length increases the width decreases. When the width decreases the length increases. I can flip the order of the length and width and get another rectangle.)
• “¿En qué son diferentes las parejas de largo y ancho de los rectángulos que tienen área de 16 y las parejas de largo y ancho de los rectángulos que tienen perímetro de 12?” // “How are the length and width pairs for rectangles with area 16 different from the length and width pairs for rectangles with perimeter 12?” (I was looking for a total of 16 instead of a total of 12. I have to multiply the side lengths rather than add them. When the length decreases by 1 for the perimeter, the width increases by 1. For area, when the length decreases the width increases but the relationship is more complex.)
• Consider drawing some rectangles with an of area 16 on the coordinate grid with the lower left corner of each rectangle at \((0,0)\). Ask students what the notice about the coordinates of the upper right corners of each rectangle. (They represent the length and width of the corresponding rectangle.)