Lesson 13

This Number Talk encourages students to think about equivalent expressions and to rely on the properties of operations to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students match diagrams to expressions.

• 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.

The purpose of this activity is for students to analyze area diagrams and use the properties of operations to interpret expressions. The diagrams are decomposed in different ways and the expressions all have a fractional part but sometimes it is written as a mixed number and sometimes the whole number and fraction are separated using the distributive property. The numbers in the diagrams, both the whole number part and the fractional part, are deliberately chosen to resemble one another so students need to analyze the expressions carefully to make matches. 

• Create a set of cards from the blackline master for each group of 2.

• Groups of 2
• Give each group a set of cards from the blackline master.

• 1–2 minutes: independent work time
• 8–10 minutes: partner work

• Display Diagram C and Expressions J and O.

\(3 \times 5 \frac{2}{5}\)

\((3 \times 5) + (3 \times \frac {2}{5})\)

• “¿De qué manera cada expresión representa el área de la región sombreada?” // “How does each expression represent the area of the shaded region?” (There are 3 rows and each row has an area of \(5\frac{2}{5}\) square units.)
• Display Diagram B and Expression L.

\((5 \times 3) -\left(5 \times \frac{2}{5}\right)\)

• “¿De qué manera la expresión representa el área de la región sombreada?” // “How does the expression represent the area of the shaded region?” (\(5 \times 3\) is the area of the full rectangle and then I take away the unshaded part, \(5 \times \frac{2}{5}\).)

The purpose of this activity is for students to write expressions that represent the area of shaded regions. Monitor for students who are writing a variety of expressions that represent the distributive property.

• Groups of 2

• 1–2 minutes: quiet think time
• 5–7 minutes: partner work time

If students only write one expression for a diagram, partner them with a student who found a different expression and ask:

• “¿De qué manera esta expresión representa la región sombreada?” // “How does this expression represent the shaded region?”
• “¿Cómo pueden dos expresiones diferentes representar la misma región sombreada?” // “How can two different expressions represent the same shaded region?”
• “¿Qué otras expresiones podemos escribir para representar la región sombreada?” // “What other expressions can we write to represent the shaded region?”

• Select 2–3 students to share their expressions for the first problem.
• If not mentioned by students, display:

  \(4 \times 5\frac{1}{3}\)

  \(4 \times \frac{16}{3}\)

• “¿De qué manera cada expresión representa el primer problema?” // “How does each expression represent the first problem?” (There are 4 rows and each row has \(5 \frac{1}{3}\) square units or \(\frac{16}{3}\) square units.)
• “¿Su estrategia cambió cuando escribieron expresiones para el último diagrama? Si sí, ¿cómo cambió?” // “Did your approach change when writing expressions for the last diagram? If yes, how?” (It was easier to see the square units and apply the distributive property.)
• If not mentioned by students, display these expressions for the area of the last diagram in square units:

  \((7 \times 3) +\left(7 \times \frac{3}{4}\right)\)

  \((7 \times 4) - \left(7 \times \frac{1}{4}\right)\)