This Warm-up prompts students to compare four different shaded regions in order to introduce the new type of region that will be considered in this lesson, namely regions where neither side length is a unit fraction. It gives students a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminologies students know and how they talk about regions with fractional side lengths. During the discussion, focus on Diagram A, which has a rectangular region where both sides are non-unit fractions.
Launch
Groups of 2
Display the image.
“Escojan 3 que vayan juntas. Prepárense para compartir por qué van juntas” // “Pick 3 that go together. Be ready to share why they go together.”
1 minute: quiet think time
Activity
“Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
2–3 minutes: partner discussion
Share and record responses.
Student Task Statement
¿Cuáles 3 van juntas?
A
B
C
D
Student Response
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Advancing Student Thinking
Activity Synthesis
Display Diagram A.
“¿Cuál es el área de la región coloreada? ¿Cómo lo saben?” // “What is the area of the shaded region? How do you know?” ( because there are 6 shaded pieces and there are 12 pieces in the whole square.)
Activity 1
15 mins
Muchas expresiones
Standards Alignment
Building On
Addressing
5.NF.B.4.b
Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
The purpose of this activity is for students to relate the structure in an expression to an area diagram (MP7). As students work with their partners, make sure both partners have an opportunity to verbally explain how the diagram represents each expression.
Launch
Groups of 2
Activity
2–3 minutes: independent think time
5–8 minutes: partner work time
Monitor for students who can explain how each expression is represented in the diagram.
Student Task Statement
¿De qué manera cada expresión representa el área de la región coloreada? Explica o muestra tu razonamiento.
Student Response
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Advancing Student Thinking
If students only explain how one expression represents the area of the shaded region, consider asking:
“¿Cómo describirías el área de la región coloreada?” // “How would you describe the area of the shaded region?”
Write one of the expressions. “¿Dónde ves esta expresión en la región coloreada?” // “Where do you see this expression in the shaded region?”
Activity Synthesis
Ask previously selected students to share their thinking.
“¿De qué manera representa el diagrama?” // “How does represent the diagram?” (There are 8 pieces shaded and each piece is of the square.)
“¿De qué manera la expresión representa el diagrama?” // “How does the expression represent the diagram?” (The shaded region is a 2 by 4 array and each of the pieces in the array is of of the whole square.)
Activity 2
20 mins
Más patrones
Standards Alignment
Building On
Addressing
5.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
The purpose of this activity is for students to observe and use the structure of diagrams to find areas of shaded regions with non-unit fraction side lengths. Students build on what they learned in the previous activity, solidifying their understanding of why the numerator of a product of two fractions is the product of the numerators and the denominator of a product of fractions is the product of the denominators.
Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for writing multiplication expressions to represent the area of the shaded rectangle in each figure before they begin. Students can speak quietly to themselves, or share with a partner. Supports accessibility for: Organization, Conceptual Processing, Language
Launch
Groups of 2
Activity
“Empiecen a completar la tabla de manera independiente. Después de un par de minutos, van a terminar de completar la tabla y responder el resto de las preguntas con su compañero” // “Start working on completing the table independently. After a couple minutes, you’ll work with your partner to complete the table and answer the rest of the questions.”
1–2 minutes: independent work time
5-8 minutes: partner work time
Monitor for students who:
Notice the area of the shaded regions is always twentieths.
Write the expression to represent the shaded region of the last diagram in the table.
Explain that the expression represents the shaded part of the last diagram in the table because represents the number of pieces that are shaded and represents the number of those pieces in the unit square.
Student Task Statement
Completa la tabla.
¿Qué patrones observas en la tabla?
¿De qué manera la expresión representa el último diagrama de la tabla? Explica o muestra tu razonamiento.
Student Response
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Advancing Student Thinking
Activity Synthesis
Ask previously selected students to share their reasoning.
“¿De qué manera las expresiones de la tabla representan el número de partes coloreadas y el tamaño de las partes coloreadas?” // “How do the expressions in the table represent the number of pieces shaded in and the size of the pieces shaded in?” (If we multiply the numerators, we get the number of pieces that are shaded in. If we multiply the denominators, we get the size of the pieces.)
Refer to diagrams and draw on each diagram to show how the multiplication of the numerators and denominators represents the number of shaded pieces and the size of the shaded pieces (the number of those pieces in the whole).
Lesson Synthesis
Display Diagram A from the last activity.
Display .
“Podemos multiplicar los numeradores para encontrar el numerador del producto. ¿De qué manera el diagrama representa ?” // “We can multiply the numerators to find the numerator in the product. How does the diagram represent ?” (The shaded pieces are a 2 by 3 array and there are 6 of them.)
“Podemos multiplicar los denominadores para encontrar el denominador del producto. ¿De qué manera el diagrama representa ?” // “We can multiply the denominators to find the denominator in the product. How does the diagram represent ?” (The unit square is a 4 by 5 array so there are 20 pieces in the whole unit square.)
“¿De qué manera el diagrama representa ?” // “How does the diagram represent ?” (There are 6 pieces shaded and they are each of the unit square.)
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Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.