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.
Launch
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
Activity
Share and record answers and strategy.
Repeat with each statement.
Decide si cada afirmación es verdadera o falsa. Prepárate para explicar tu razonamiento.
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿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?”
Activity 1
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 multiply a whole number by a fraction greater than 1 in a way that makes sense to them.
Monitor for and select students with the following approaches to share in the Activity Synthesis:
Explain why the area of the shaded region is or by counting the number of shaded whole square units and then counting the number of shaded third of a square units.
Explain how to use the expression to find the area of the shaded region.
Explain how to use the expression to find the area of the shaded region.
The approaches are sequenced from more concrete to more abstract to support students with applying what they know about equivalent fractions to multiplying a mixed number and a whole number. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently. For examples for the approaches, look at the Student Responses.
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 Activity 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).
MLR8 Discussion Supports.Display sentence frames to support small-group discussion: “Me pregunto si . . . ” // “I wonder if . . . ”, “_____ y _____ se parecen porque . . .” // “_____ and _____ are the same because. . . .”, and “_____ y _____ son diferentes porque . . . ” // “_____ and _____ are different because . . . .” Advances: Conversing, Representing
Launch
Groups of 2
Activity
1–2 minutes: quiet think time
5 minutes: partner work time
As you monitor for the approaches listed in the Activity Narrative, consider asking:
“¿Cómo calcularon el área de la región coloreada?” // “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?”
Encuentra el área de la región coloreada, en unidades cuadradas. Explica o muestra tu razonamiento.
Selecciona todas las expresiones que representan el área de la región coloreada, en unidades cuadradas. Para cada expresión correcta, explica tu razonamiento.
Activity Synthesis
Invite previously selected students to share in the given order. Record or display their work for all to see.
Connect students’ approaches by asking:
“¿Cómo se usaron números enteros en cada manera de encontrar el área? ¿Cómo se usaron fracciones en cada manera?” // "How did each way of finding the area use whole numbers? How did each way use fractions?" (I notice that and both represent the area of the shaded region. You can count the number of shaded whole square units to get 16 and then count the number of shaded third of a square units to get . If you group the into wholes, you have for a total of , which is .)
“¿De qué manera representa el área de la región coloreada?” // “How does represent the area of the shaded region?” (There are 56 pieces shaded in and each piece has an area of of a unit square.)
Connect students’ approaches to the learning goal by showing the equivalent expressions and asking:
Display:
“¿Cómo sabemos que estas expresiones tienen el mismo valor?” // “How do we know these expressions have the same value?” (They both represent the shaded area in the diagram. I know that 3, 6, 9, 12 thirds is 4 wholes and then there are two thirds left.)
Activity 2
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 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.
Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share how they use the diagram to calculate the area of the desk or the garden and the method they used with a classmate. Supports accessibility for: Conceptual Processing, Attention
Launch
Groups of 2
Activity
5 minutes: individual work time
5 minutes: partner work time
Escribe una expresión de multiplicación que represente el área de la región coloreada, en unidades cuadradas.
¿Cuál es el área de la región coloreada?
Escribe una expresión de multiplicación que represente el área de la región coloreada, en unidades cuadradas.
¿Cuál es el área de la región coloreada?
Student Response
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Advancing Student Thinking
If students write a multiplication expression that does not represent the area of the shaded region, consider asking:
“¿Por qué escogiste esa expresión?” // “How did you choose that expression?”
Write an expression that represents the area of the shaded region. “¿De qué manera esta expresión representa el área de la región coloreada?” // “How does this expression represent the area of the shaded region?”
Activity Synthesis
Ask several students to share their responses to the second problem.
Display:
“¿De qué manera estas expresiones representan el área de la región coloreada?” // “How do these expressions represent the area of the shaded region?” (They are the width and length of the shaded region. In one expression 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 and add them together. For the second one, I can see how many fourths I have using multiplication.)
Lesson Synthesis
“Hoy aprendimos que podemos aplicar lo que ya sabemos sobre la multiplicación para encontrar el área de un rectángulo en el que una longitud de lado es una fracción mayor que 1” // “Today we learned that we can apply our understanding of multiplication to find the area of a rectangle with a side length that is a fraction greater than 1.”
Display the image:
Display the expression .
“¿De qué manera el diagrama representa la expresión?” // “How does the diagram represent the expression?” (There are small pieces and each one has an area of square unit.)
Display the expression .
“¿De qué manera el diagrama representa la expresión?” // “How does the diagram represent the expression?” (There are two rows of shaded pieces and each row has area square units.)
Display the expression .
“¿De qué manera el diagrama representa la expresión?” // “How does the diagram represent the expression?” (There's a 2 by 2 array of whole square units and then there are 2 shaded pieces each having area square unit.)
Standards Alignment
Building On
Addressing
5.NF.B.3
Interpret a fraction as division of the numerator by the denominator . Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret as the result of dividing by , noting that multiplied by equals , and that when wholes are shared equally among people each person has a share of size . If people want to share a -pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
If students only select one expression, consider asking:
“¿En qué parte del diagrama están representados los valores de la expresión de multiplicación?” // “Where are the values in the multiplication expression represented in the diagram?”
Refer to a different expression. “¿Qué representa cada cuadrado del diagrama? ¿Cómo está partido cada cuadrado? ¿Cómo puedes usar eso para encontrar de qué maneras _____ podría representar la región coloreada?” // “What does each square in the diagram represent? How is each square partitioned? How can you use that to find ways that _____ (expression) might represent the shaded region?”
