Lesson 1

The purpose of this How Many Do You See is to elicit ideas about equal groups of fractional amounts and to prepare students reason about multiplication of a whole number and a fraction. Students may describe the oranges with a whole number without units or without specifying “halves” (for instance, they may say “5”). If this happens, consider asking them to clarify whether they mean “5 oranges” or another amount.

• Groups of 2
• “¿Cuántas cosas ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you see them?”
• Display the image.
• 1 minute: quiet think time

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

• “¿Cómo podrían describirle esta imagen a un amigo?” // “How might you describe this image to a friend?” (There are 5 plates with \(\frac{1}{2}\) orange on each plate.)
• “¿Cuántos grupos ven?” // “How many groups do you see?” (I see 5 plates or 5 groups.)
• “Además de describir la imagen con palabras, ¿de qué otra manera pueden representar la cantidad que hay en la imagen?” // “Besides describing the image in words, how else might you represent the quantity in this image?” (I might write \(\frac{1}{2}\) five times, or write an expression with five \(\frac{1}{2}\)s being added together. I might write “5 times \(\frac{1}{2}\).”)
• “En esta lección, vamos a examinar otras situaciones de grupos y cantidades fraccionarias” // “We'll look at some other situations involving groups and fractional amounts in this lesson.”

The purpose of this activity is for students to interpret situations involving equal groups of a fractional amount and to connect such situations to multiplication of a whole number by a fraction (MP2).

Students write expressions to represent the number of groups and the size of each group. They reason about the quantity in each situation in any way that makes sense to them. Although images of the food items are given, students may choose to create other diagrams, such as equal-group diagrams used in grade 3, when they learned to multiply whole numbers. This activity enables the teacher to see the representations toward which students gravitate. 

Focus the discussions on connecting equal groups with fractions and those with whole numbers. 

• Groups of 2
• “¿Cuáles son algunas de sus meriendas favoritas?” // “What are some of your favorite snacks?”
• Share responses.
• “¿Qué meriendas podemos partir en pedazos más pequeños en vez de comerlas enteras?” // “What are some snacks that you might break into smaller pieces rather than eating them whole?”
• 1 minutes: partner discussion
• “Exploremos ejemplos de alimentos que pueden comerse enteros o que pueden partirse en pedazos más pequeños” // “Let's look at some food items that we might eat whole or cut or break up into smaller pieces.”

• “Durante unos minutos, piensen en silencio en el primer grupo de problemas sobre las galletas. Después, discutan sus ideas con su pareja” // “Take a few quiet minutes to think about the first set of problems about crackers. Then, discuss your thinking with your partner.”
• 4 minutes: independent work time
• 2 minutes: partner discussion
• Pause for a whole-class discussion. Invite students to share their responses.
• If no students mention that there are equal groups, ask them to make some observations about the size of the groups in each image.
• Discuss the expressions students wrote:
  • “¿Qué expresión escribieron para representar las galletas de la imagen A? ¿Por qué?” // “What expression did you write to represent the crackers in Image A? Why?” (\(6 \times 4\), because there are 6 groups of 4 full crackers.)
  • “¿Y para las galletas de la imagen B? ¿Por qué?” // “What about the crackers in Image B? Why?” (\(6 \times \frac{1}{4}\), because there are 6 groups of \(\frac{1}{4}\) of a cracker.)
• Ask students to complete the remaining problems.
• 5 minutes: independent or partner work time
• Monitor for students who reason about the quantities in terms of “_____ grupos de _____” // “_____ groups of _____” to help them write expressions.

If students are unsure how to name the quantity in the image, consider asking: “¿Cómo describirías la cantidad del pedazo de tarta que hay en un plato?, ¿cómo describirías la cantidad de dos de estos pedazos?, ¿y de tres de estos pedazos?” // ”How would you describe the amount of the slice of pie on one plate? How would you describe two of the same slices? Three of the same slices?”

• Select previously identified students to share their expressions and how they reasoned about the amount of food in each image. Record their expressions and supporting diagrams, if any, for all to see.
• If students write addition expressions to represent the quantities, ask if there are other expressions that could be used to describe the equal groups.
• “¿En qué se diferencian la cantidad de la situación de Clare y las cantidades de las otras situaciones?” // “How is the quantity in Clare's situation different than those in other situations?” (It involves whole numbers of items. Others involve fractional amounts.) 
• “¿En qué se diferencian la expresión que escribieron para los huevos y las otras expresiones?” // “How is the expression you wrote for the eggs different than other expressions?” (It shows two whole numbers being multiplied. The others show a whole number and a fraction.)

In this activity, students start with given multiplication expressions and consider situations or diagrams that they could represent. Situating the expressions in context encourages students to think of the whole number in the expression as the number of groups and the fractional amount as the size of each group, which helps them reason about the value of the expression. When students make explicit connections between multiplication situations, expressions, and drawings they reason abstractly and quantitatively (MP2).

Allow students to use fraction strips, fraction blocks, or other manipulatives that show fractional amounts to support their reasoning.

• Groups of 2
• “Lean el enunciado. Luego, hablen con su pareja sobre lo que deben hacer en esta actividad” // “Read the task statement. Then, talk to your partner about what you are asked to do in this activity.”
• 1 minute: partner discussion

• “Escojan una expresión con la cual les gustaría comenzar” // “Choose one expression you’d like to start with.”
• “Piensen en una historia que se pueda representar con esa expresión. Después, hagan un dibujo o un diagrama y encuentren el valor de la expresión” // “Think of a story that can be represented by the expression. Then, create a drawing or diagram, and find the value of the expression.”
• “Si les queda tiempo, pueden trabajar en ambos problemas” // “If you have extra time you can work on both problems.”
• 7–8 minutes: independent work time
• “Asegúrense de decir lo que el valor de la expresión significa en su historia” // “Be sure to say what the value of the expression means in your story.”

• Invite students to share their responses. Display their drawings or visual representations for all to see.
• “¿Cómo decidieron lo que representan en su historia los números de cada expresión?” // “How did you decide what the numbers in each expression represent in your story?” (It made sense for the whole numbers to represent how many groups there are and the fractions to represent what is in each group.)
• “¿Cómo mostraron el número entero y la fracción en su dibujo?” // “How did you show the whole number and the fraction in your drawing?” (I drew as many circles as the whole number to show the groups. I drew parts of objects or wrote numbers in each circle to show the fraction.)