Lesson 10

The purpose of this Estimation Exploration is for students to practice the skill of estimating a reasonable answer based on experience and known information. It gives students a low-stakes opportunity to estimate the area of a rectangle when one side is not a unit fraction.

• Groups of 2
• Display the image.
• “¿Cuál puede ser el área de la región sombreada?” // “What might be the area of the shaded region?”

• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.

• “¿Está sombreada más de la mitad o menos de la mitad del rectángulo?” // “Is more than half or less than half of the rectangle shaded?” (More than half)
• “¿Cómo les puede servir esto para estimar?” // “How can you use this to help make your estimate?” (Half of 7 is \(3 \frac{1}{2}\) so it’s a little more than that.)
• “Después de esta discusión, ¿alguien quiere revisar su estimación?” // “Based on this discussion does anyone want to revise their estimate?”

The purpose of this activity is for students to find the area of rectangles with a side length that is a non-unit fraction. Students may use a variety of strategies to find the areas of the shaded region. Monitor for students who are noticing and using the structure of the rectangle and expressions to determine the area (MP7) by:

• grouping the shaded pieces with a fractional area to make whole unit squares
• multiplying the whole number of units by the numerator of the fractional side length and dividing the result by the denominator

• Display image from warm-up.
• “Si la altura de la región sombreada fuera \(\frac{5}{6}\) de una unidad cuadrada, ¿qué expresión pueden escribir para representar el área de la región sombreada?” // “If the height of the shaded region was \(\frac{5}{6}\) of a square unit, what expression could you write to represent the area of the shaded region?” (\(\frac{5}{6} \times 7\) or \(7 \times \frac{5}{6}\))
• Groups of 2

• 1–2 minutes: quiet think time
• 5–8 minutes: partner work time

• Select previously identified students to share their strategies.
• “¿En qué se parecen la segunda y la tercera región sombreada?, ¿en qué son diferentes?” // “How are the second and third shaded regions the same? How are they different?” (They are each 4 units long. There are 12 shaded pieces in each. The 12 pieces in the second example are \(\frac{1}{4}\) of a unit square. The 12 shaded pieces in the third shaded region are \(\frac{1}{5}\) of a unit square.)
• Display the expression: \(4 \times 3\)
• “¿Cómo se relaciona esta expresión con la segunda y la tercera región sombreada?” // “How does this expression relate to the second and third shaded regions?” (In both of them, the number of shaded pieces is \(4 \times 3\).)
• “¿Por qué las áreas de estas dos regiones sombreadas son distintas?” //  “Why are the areas of these two shaded regions different?” (The small pieces are not the same size. They are \(\frac{1}{4}\) of a unit square in one and \(\frac{1}{5}\) in the other.)
• Display the expressions: \(12 \times \frac{1}{4}\) and \(12 \times \frac{1}{5}\)
• “Las expresiones de las áreas de la segunda y la tercera región muestran que en ambas hay 12 partes sombreadas, pero son de tamaños diferentes” // “The expressions for the area show that there are 12 shaded pieces in both but they are different sizes.”

The purpose of this activity is for students to find the area of rectangles with a fractional side length. The side lengths are not labeled, so students will have to determine them by considering the number and size of the shaded pieces.

In the second problem, students are given expressions and they determine whether or not each expression represents the shaded area in a given diagram. The numbers in the expressions are similar, so students need to consider the structure of the expressions and the shaded regions. While they may match the expressions with the diagrams based on their value, students are encouraged to look for and clearly express how the diagrams represent the different expressions (MP6, MP7).

• Groups of 2

• 2–3 minutes: quiet think time

• 5–6 minutes: partner work time

• Monitor for students who: 

  • correctly determine the side lengths of the shaded region.

  • use the side lengths to determine a correct multiplication expression.

  • refer to the number of shaded pieces and the size of the shaded pieces when determining the area.

If students do not write the correct expression, show them the correct expression and ask, “¿De qué manera la expresión representa el área de la región sombreada en el diagrama correspondiente?” // “How does the expression represent the area of the corresponding diagram?”

• “¿Cómo encontraron las longitudes de los lados de la región sombreada en el diagrama X?” // “How did you determine the side lengths for shaded region in diagram X?” (I looked at the side lengths of the squares and the partitions, then I counted how many are shaded.)
• Display the expression: \( 3 \times 4 \times \frac{1}{5}\)
• “¿De qué manera esta expresión representa el área de la región sombreada del diagrama X?” // “How does this expression represent the shaded region in diagram X?” (The vertical rectangle has 3 rows of 4 shaded pieces, so that’s \(3 \times 4\) shaded pieces. Each one has an area of \(\frac{1}{5}\) square unit.)