This Notice and Wonder asks students to consider 2 diagrams representing a shaded region with the same side lengths. The first diagram shows the unit square and the grid lines and the second diagram just shows the side lengths of the shaded region. This prepares students to transition from the gridded diagrams they have worked with in previous lessons to the diagrams they will work with in this lesson. In the Activity Synthesis, students discuss different equations that represent different ways of finding the area.
Launch
Groups of 2
Display the image.
“¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
1 minute: quiet think time
Activity
“Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
1 minute: partner discussion
Share and record responses.
¿Qué observas? ¿Qué te preguntas?
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Cómo sabemos que las regiones coloreadas tienen la misma área?” // “How do we know that the shaded regions have the same area?” ( and . or , so they must be the same.)
“¿De qué manera el primer diagrama representa la ecuación ?” // “How does the first diagram represent this equation: ?” (Each of the .)
Activity 1
Standards Alignment
Building On
Addressing
5.NF.B.4.a
Interpret the product as parts of a partition of into equal parts; equivalently, as the result of a sequence of operations . For example, use a visual fraction model to show , and create a story context for this equation. Do the same with . (In general, .)
The purpose of this activity is for students to find the area of rectangles with fractional side lengths where the scaffold of an area diagram is not provided. For the first product, the subdivision of each side into unit fractions is shown and then that is taken away. Without these divisions, students can either try to sketch them or use their understanding of how the numerator and denominator of the fractions relate to:
The total number of shaded pieces in an area diagram.
The total number of pieces in the unit square.
When students find products of fractions without an area diagram for support, they rely on their understanding of the meaning of the numerator and denominator and the patterns they have repeatedly observed when finding these products with area diagrams (MP8).
Launch
Groups of 2
“Van a encontrar algunos productos de fracciones. Van a poder usar un diagrama como ayuda” // “You are going to find some products of fractions. You will have a choice of using a diagram to help you.”
Activity
5–8 minutes: independent work time
2 minutes: partner discussion
Monitor for students who:
Fill in the first diagram which shows the division into smaller pieces.
Try to divide the second diagram into smaller pieces to show the product.
Understand that drawing all of the pieces will be difficult for the third product.
Encuentra el valor de cada producto. Si te ayuda, dibuja un diagrama.
¿Cómo decidiste si dibujabas o no un diagrama? ¿Cómo los diagramas te ayudaron a encontrar los productos?
Diego dibujó este diagrama para representar el producto . ¿Cómo puede el diagrama ayudar a Diego a encontrar el valor de ? Explica o muestra tu razonamiento.
Student Response
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Advancing Student Thinking
Activity Synthesis
Display expression:
Invite students to share their calculations for .
“¿Fue útil un diagrama?” // “Was a diagram helpful?” (Yes, it helped me to visualize the product and understand why is the total number of pieces and is the number of those pieces in a whole.)
Display expression:
“¿Alguien dibujó un diagrama para esta expresión?” // “Did anyone draw a diagram for this expression?” (I started to and I got the eighths. Then I gave up for elevenths as that is a hard fraction to show.)
“¿El diagrama de Diego ayuda a encontrar el valor de ? ¿Cómo ayuda?” // “Does Diego’s diagram help to find the value of ? How?” (Yes, even though it doesn’t show any of the parts, having the numbers there helps me see that there would be 9 columns and 5 rows so 45 pieces. There would be 11 columns and 8 rows in a whole so 88 pieces in a whole.)
Activity 2
Standards Alignment
Building On
Addressing
5.NF.B.4.a
Interpret the product as parts of a partition of into equal parts; equivalently, as the result of a sequence of operations . For example, use a visual fraction model to show , and create a story context for this equation. Do the same with . (In general, .)
The purpose of this activity is for students to find missing values in equations that represent products of fractions. The numbers are complex so students will rely on their understanding of products of fractions rather than on drawing a diagram.
MLR8 Discussion Supports. Synthesis: Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking
Engagement: Provide Access by Recruiting Interest. Provide choice. Invite students to decide the order to complete the problems. Supports accessibility for: Attention, Social-Emotional Functioning
Launch
Groups of 2
Activity
1–2 minutes: independent think time
5–8 minutes: partner work time
Monitor for students who:
Use what they know about whole number multiplication to determine that .
Use what they know about equivalent fractions to rewrite as 2.
En cada caso, encuentra el valor que hace que la ecuación sea verdadera. Si te ayuda, dibuja un diagrama.
Student Response
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Advancing Student Thinking
If students only find some correct missing factors, consider asking:
“¿Cómo encontraste el valor desconocido en las otras ecuaciones?” // “How did you find the value for the other equations?”
“¿Cómo puedes adaptar tu estrategia para encontrar los factores desconocidos?” // “How can you adapt your strategy to find the missing factors?”
Activity Synthesis
Display:
“¿Cómo encontraron el valor que hace que la ecuación sea verdadera?” // “How did you find the value that makes the equation true?” (I knew the numerator was 7 since and the denominator had to be 4 since .)
Display:
“¿Cómo supieron que el valor que hace que la ecuación sea verdadera es ?” // “How did you know the value that makes the equation true is ?” (I doubled .)
Display equation:
“¿En qué es diferente esta ecuación de las demás?” // “How is this equation different from the others?” (It has a whole number as a factor. The others are all fractions.)
“¿Cómo resolvieron este problema?” // “How did you solve this problem?” (I knew that the denominator of the missing number had to be 8 to get and then the numerator needed to be 3.)
Lesson Synthesis
“Hoy encontramos el área de rectángulos sin una cuadrícula y encontramos productos de fracciones sin referirnos al área” // “Today we found the area of rectangles without a grid and we found products of fractions without referring to any area.”
“¿Qué saben acerca de la multiplicación de fracciones?” // “What do you know about multiplying fractions?” (It is kind of like multiplying whole numbers, but different. We can use some of the strategies we use to multiply whole numbers. We can draw diagrams to represent equations. The numerator of the product represents the number of pieces and the denominator of the product represents the number of pieces in the whole.)
“¿De qué manera los diagramas representan productos de fracciones?” // “How do diagrams represent products of fractions?” (They show that the product of the numerators is the total number of pieces shaded and the product of the denominators is the size of the pieces that are shaded.)
Display a diagram that shows .
“Este diagrama representa “ // “This diagram represents .”
“Podemos ver que hay rectángulos coloreados, y rectángulos en total. Así que...“ // “We can see that there are shaded rectangles, and rectangles in the whole. So ...”
Display:
“Esto siempre funciona. Cuando multiplicamos fracciones, multiplicamos los numeradores y los denominadores para encontrar el producto“ // “This always works. When we multiply fractions, we multiply the numerators and the denominators to find the product.”
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.
