Lesson 3

The purpose of this warm-up is to elicit the idea that we can think about multiple equal parts in a diagram and use fractions to refer to them, which will be useful when students identify fractions in diagrams and shade diagrams to show a specific fraction in a later activity. While students may notice and wonder many things about these images, the fact that more than one of the equal parts of the square is shaded, there is a fraction underneath the third diagram, and how the shaded parts could be described are the important discussion points.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “¿Qué creen que representan el 3 y el 4 del número que está debajo del tercer cuadrado?” // “What do you think the 3 and the 4 stand for in the number below the third square?” (The 4 stands for the 4 equal parts in the square. The 3 stands for the number of parts that are shaded.)
• “Es posible que a veces queramos hablar sobre más de una parte. Podemos describir esas partes con un número” // “Sometimes we may want to talk about more than one part and we can describe those parts with a number.”
• “¿Cómo se llaman las partes de cada cuadrado?” // “What do we call the parts in each square?” (Fourths)
• “¿Cuántos cuartos están sombreados en el primer cuadrado?” // “How many fourths are shaded in the first square?” (1)
• “¿Cuántos cuartos están sombreados en el segundo cuadrado?, ¿en el tercer cuadrado?” // “How many fourths are shaded in the second square? Third square?” (2 and 3)
• “¿Cuántos cuartos están sombreados en la última imagen?” // “How many fourths are shaded in the last image?” (4)
• “Podemos referirnos a las partes sombreadas de cada imagen al describir el número de cuartos: un cuarto, dos cuartos, tres cuartos y cuatro cuartos” // “We can refer to the shaded parts in each image by describing the number of fourths: one fourth, two fourths, three fourths, and four fourths.”
• “En la siguiente actividad, vamos a ver cómo escribir estas cantidades” // “We’ll look at how to write these amounts in the next activity.”

The purpose of this activity is for students to make sense of non-unit fractions and the notation used to describe them. They learn that the denominator tells the number of equal parts the whole was partitioned into and the numerator tells the number of parts that are being described. Students write non-unit fractions that represent the shaded portions of area diagrams.

If needed, especially with the fractions greater than one, clarify that each rectangle represents one whole. The activity concludes with students practicing how to read non-unit fractions. The terms “numerator” and “denominator” will be introduced in a later lesson.

When students notice that the bottom part of the fraction stays the same and the top part of the fraction changes, representing the number of equal parts that are shaded, they look for and make use of structure (MP7).

• Groups of 2
• Display the table.
• “Examinemos la primera tabla” // “Let's look at the first table.”
• “Las primeras tres imágenes son los cuadrados que vimos antes. Nombrémoslos otra vez” // “The first three images are the squares we saw earlier. Let's name them again.” (One-fourth, three-fourths, four-fourths)
• “Completemos juntos la segunda fila de la tabla. Este es el cuadrado con el que trabajamos en el calentamiento y el número que representa la cantidad total que está sombreada ya está en la tabla. ¿Cuántas partes están sombreadas?” // “Let’s complete the second row of the table together. This is the square we just worked with in the warm-up and the number that represents the total amount shaded is already in the table. How many of the parts are shaded?” (Three)
• “¿Cuál es el tamaño de cada parte?” // “What is the size of each part?” (\(\frac{1}{4}\))
• “Anoten ‘tres cuartos’ para tener escrita la forma como leemos esta fracción” // “Write ‘three-fourths’ to record how we read this fraction.”

• “Completen la tabla con su compañero” // “Work with your partner to complete the table.”
• 5–7 minutes: partner work time

• Display the completed table.
• “Ahora que la tabla está completa, ¿qué observan?, ¿qué se preguntan?” // “Now that the table is complete, what do you notice? What do you wonder?” (The size of each part always has a 1 in the top part of the fraction. The bottom part of the fraction is the same for each part and the total amount shaded. The top part of the fraction is how many parts are shaded.)
• “Pensemos en cómo describir o nombrar las partes sombreadas de cada imagen” // “Let’s think about how we should describe or name the shaded parts in each image.”
• 30 seconds: quiet think time
• Invite students to name the shaded parts in each image.
• As students name the shaded part in each image, have them share how they recorded the name using fraction notation.

The purpose of this activity is for students to match fractions to shaded diagrams. Reiterate that each rectangle represents one whole. After one round of matching, students pause to create 4 new pairs of cards to add to their set. Give students the Fraction Match Part 2 cards when they create their own pairs of cards.

Students observe and use structure as they identify that the top number in the fraction represents the number of shaded pieces while the bottom number represents the number of those pieces in one whole rectangle (MP7).

• Create a set of cards from the Fraction Match Part 1 blackline master for each group of 2.
• Create a set of 8 cards from the Fraction Match Part 2 blackline master for each group of 2.

• Groups of 2
• “Vamos a jugar un juego en el que van a emparejar fracciones y diagramas. Lean las instrucciones del juego con su compañero y discutan cualquier pregunta que tengan sobre el juego” // “We’re going to play a game in which you match fractions and diagrams. Read the directions to the game with your partner and discuss any questions you have about the game.”
• Answer any questions about the game.
• Give each group one set of cards created from Fraction Match Part 1.

• “Jueguen una ronda de ‘Parejas de fracciones’ con su compañero” // “Play one round of Fraction Match with your partner.”
• 5–7 minutes: partner work time
• Give each group one set of Fraction Match Part 2.
• “Antes de jugar otra ronda, con su compañero, hagan 4 parejas nuevas de tarjetas y agréguenlas al grupo. Partan y sombreen un diagrama para que sea la pareja de cada fracción” // “Before you play another round, work with your partner to create 4 new pairs of cards to add to the set. Partition and shade a diagram to match each fraction.”
• 3–5 minutes: partner work time
• “Ahora jueguen otra ronda de ‘Parejas de fracciones’ con su compañero usando todas las tarjetas” // “Now, play another round of fraction match with your partner using all the cards.”
• 5–7 minutes: partner work time
• Monitor for students who notice that \(\frac{3}{3}\) means the whole rectangle is shaded.

• Ask 1–2 groups to display a rectangle they partitioned and shaded to match one of the fractions.
• For each rectangle students share, discuss the fraction that it represents.
• Display a diagram of \(\frac{3}{3}\) with all the parts shaded.
• “_____ se dio cuenta de que para mostrar \(\frac{3}{3}\), tenemos que sombrear todo el rectángulo. Revisen las dos actividades anteriores y busquen otras fracciones que se muestren al sombrear todo el rectángulo” // “_____ noticed that to show \(\frac{3}{3}\), we have to shade the whole rectangle. Look back over both activities for other fractions that are shown by shading the whole rectangle.” (\(\frac{2}{2}\), \(\frac{4}{4}\), and \(\frac{6}{6}\))
• Consider asking: “¿Cuántos _____ se necesitan para formar una unidad? ¿Cómo lo saben?” // “How many _____s would it take to make a whole? How do you know?”
• “Estas fracciones son equivalentes a 1 porque representan todas las partes de la unidad. A medida que sigamos aprendiendo sobre las fracciones, vamos a trabajar con otras fracciones como esta” // “These fractions are equivalent to 1 because they represent all the parts of the whole. We’ll work with other fractions like this as we learn about fractions.”