Lesson 5

The purpose of this How Many Do You See is for students to use grouping strategies to describe the images they see. Students’ descriptions are recorded using equations and expressions to support the goal of creating equivalent expressions. The synthesis encourages students to think about why two expressions can represent the same amount.

• Groups of 2
• “¿Cuántos tercios ven? ¿Cómo lo saben?, ¿qué ven?” // “How many thirds do you see? How do you see them?”

• Display the image.
• 1 minute: quiet think time

• For each way that students see thirds, ask: “¿Qué expresión deberíamos usar para representar los grupos de tercios que vio _____?” // “What expression should we use to represent the groups of thirds that _____ saw?”
• If no students suggest “4 groups of \(\frac{2}{3}\)”, ask them how that might be visible in the diagram. (By combining 2 of the thirds from each strip, we can make 4 groups of \(\frac{2}{3}\).)
• Write \(8 \times \frac{1}{3} =4 \times \frac{2}{3}\), and ask students if they agree or disagree with the statement.

The purpose of this activity is for students to think of different ways of using multiplication expressions to represent a non-unit fraction. Students informally use the associative property as they work towards generalizing that \(n \times \frac{a}{b} = \frac{n \times a}{b} = (n \times a) \frac{1}{b}\).

• Groups of 2

• “Completen el primer problema con su compañero. Hablen sobre cómo saben qué números hacen que las ecuaciones sean verdaderas” // “Work with your partner to complete the first problem. Talk about how you know what numbers make the equations true.”
• 3 minutes: partner work time
• Monitor for students who use the factors of 12 to complete the equations.
• “Ahora, durante unos minutos, completen el resto de problemas individualmente. Después, compartan sus respuestas con su compañero” // “Now take a few minutes to complete the rest of the problems independently. Afterwards, share your responses with your partner.”
• 7 minutes: independent work time
• 3 minutes: partner discussion
• “¿Escogieron los mismos números que su compañero? Si no, ¿ambas ecuaciones son correctas?” // “Did you choose the same numbers as your partner? If not, are both equations correct?”

• “¿Cómo supieron qué fracciones usar para completar cada ecuación del primer problema?” // “How did you know what fractions to use to complete each equation in the first problem?” (We looked for the numbers to multiply to get \(\frac{12}{5}\). The denominator stays the same, so we find out what is going to be the whole number and numerator by knowing the factor pairs of 12.)
• Select 2–3 partners to share the following equations from the second problem:
  • \(\frac{6}{7} = 6 \times \frac{1}{7}\)
  • \(\frac{6}{7} = 3 \times \frac{2}{7}\)
  • \(\frac{6}{7} = 2 \times \frac{3}{7}\)
  • \(\frac{6}{7} = 1 \times \frac{6}{7}\)
• “¿Por qué son verdaderas todas estas ecuaciones?” // “Why are all these equations true?” (To get \(\frac{6}{7}\), we can multiply the whole number by the numerator and then keep the denominator the same. In all of these, multiplying the whole number and the numerator gives 6. There are different ways to multiply to get 6 as a numerator: \(6\times 1\), \(3\times 2\), \(2 \times 3\) and \(1\times 6\).)

In this activity, students analyze multiplication expressions, match each to one of a given set of fractions, and explain how they know that certain expressions represent the same fraction (MP7). 

• Groups of 2

• “Asocien cada expresión con una fracción que muestre su valor. Prepárense para explicar cómo saben cuáles expresiones se asocian con cuál fracción” // “Match the expressions to a fraction that shows its value. Be prepared to explain how you know which expressions match which fraction.”
• “Es posible que no todas las fracciones tengan el mismo número de expresiones asociadas” // “Each fraction may not have the same number of matching expressions.”
• 5–7 minutes: partner work time
• Pause for a discussion before students continue to the second half of the activity.
• Select students to share their matches and their explanations. 
• “¿Hubo expresiones que no pudieron asociar? ¿Cómo lo saben?” // “Are there any expressions without a match? How do you know?” (Yes, expressions G and I. Their values are \(\frac{16}{9}\) and \(\frac{4}{12}\).)
• “Se han dado cuenta de que varias expresiones pueden representar la misma fracción. Algunas de las expresiones tienen dos factores y algunas tienen tres. En todas hay fracciones unitarias” // “You've seen that multiple expressions can represent the same fraction. Some of the expressions have two factors, some have three. All of them show unit fractions.”
• “Ahora completen cada ecuación del segundo problema con dos factores que hagan que la ecuación sea verdadera. Traten de usar factores que no sean fracciones unitarias” // “Now complete each equation in the second problem with two factors that would make the equation true. See if you can use factors that are not unit fractions.”
• 5 minutes: independent work time

• Ask selected students to share expressions for each fraction.

• Display the possible expressions for the final equation:

  \(8 \times \frac{1}{9}\)
  \(2 \times 4 \times \frac{1}{9}\)
  \(2 \times \frac{4}{9}\)
  \(4 \times 2\times \frac{1}{9}\)
  \(4 \times \frac{2}{9}\)

• “¿Cómo podemos explicar por qué el valor de estas expresiones es \(\frac{8}{9}\)?” // “How can we explain why these expressions have the value of \(\frac{8}{9}\)?”