Lesson 11

The purpose of this True or False is for students to demonstrate strategies they have for relating division of two whole numbers to multiplication of a fraction by a whole number. The reasoning students use here helps to deepen their understanding of the relationship between multiplication and division. It will also be helpful later when students find the area of rectangles with mixed number side lengths.

• Display one equation.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

• Share and record answers and strategy.
• Repeat with each statement.

• “¿Cómo pueden explicar su última respuesta sin encontrar el valor de ambos lados?” // “How can you explain your answer to the last statement without finding the value of both sides?”

The purpose of this activity is for students to multiply a whole number by a fraction greater than 1 in a way that makes sense to them. Monitor for students who:

• can explain why the area of the shaded region is \(18 \frac{2}{3}\) or \(16 \frac{8}{3}\) by counting the number of shaded whole square units and then counting the number of shaded third of a square units
• can explain how to use the expression \(4 \frac{2}{3} \times 4\) to find the area of the shaded region
• can explain how to use the expression \(\frac{14}{3} \times 4\) to find the area of the shaded region

As students work with fractions greater than 1, they may choose to write or rewrite them as mixed numbers. Students may also relate the expressions to the diagrams in different ways. Encourage them to interpret the diagram and find the area of the shaded region in whatever way makes sense to them. During the synthesis, show the relationships between the different ways of finding and representing the area.

Identifying which expressions represent the area of the rectangle requires careful analysis of the expressions and the figure and the correspondences between them (MP7).

• Groups of 2

• 1–2 minutes: quiet think time
• 5 minutes: partner work time
• As students work, consider asking:
  • “¿Cómo calcularon el área de la región sombreada?” // “How did you calculate the area of the shaded region?”
  • “¿Cómo saben que su respuesta tiene sentido?” // “How do you know your answer makes sense?”

If students do not interpret the factors as side lengths of rectangles, encourage them to listen to a partner explain where they see the multiplication expression and then describe what their partner says in their own words.

• Ask selected students to share in the given order.
• “¿De qué manera \(\frac{56}{3}\) representa el área de la región sombreada?” // “How does \(\frac{56}{3}\) represent the area of the shaded region?” (There are 56 pieces shaded in and each piece has an area of \(\frac{1}{3}\) of a unit square.)
• Display: \(4 \frac{2}{3} \times 4= \frac{14}{3} \times 4\)
• “¿Cómo sabemos que estas expresiones son iguales?” // “How do we know these expressions are equal?” (They both represent the shaded area in the diagram or I know that 3, 6, 9, 12 thirds is 4 wholes and then there are two thirds left.)

The purpose of this activity is for students to find areas of rectangles where one side is a whole number and the other side is a fraction that is greater than 1.  Students should solve the problems in a way that makes sense to them. Ask students to explain how the diagrams show the multiplication expressions.

• Groups of 2

• 5 minutes: individual work time
• 5 minutes: partner work time

If students do not write multiplication expressions to represent the area of the shaded region, prompt them to explain how they found the area of the shaded region. Then, write expressions to represent the student’s strategy and ask, “¿De qué manera estas expresiones representan tu estrategia?” // “How do these expressions represent your strategy?”

• Ask several students to share their responses to the second problem.
• Display:
  • \(3 \times 3\frac{3}{4}\)
  • \(3 \times \frac{15}{4}\)
• “¿De qué manera estas expresiones representan el área de la región sombreada?” // “How do these expressions represent the area of the shaded region?” (They are the width and length of the shaded region. In one, the length is a whole number and some fourths, and in the other, it is all fourths.)
• “¿En qué se diferencian las maneras de encontrar el valor de estas dos expresiones?” // “How is finding the value of these two expressions different?” (For the first one, I can multiply 3 by 3 and then 3 by \(\frac{3}{4}\) and add them together. For the second one, I can see how many fourths I have using multiplication.)