Lesson 10

This Number Talk encourages students to use what they learned about products of a whole number and a fraction, the relationship between each pair of factors, and the structure in the expressions to mentally solve problems. 

Students may write all the products as fractions, including the ones greater than 1. If everyone expresses the last three products only as \(\frac{18}{12}\), \(\frac{36}{12}\), and \(\frac{360}{12}\), during the synthesis, ask if students could write whole-number or mixed-number equivalents for these fractions. The reasoning elicited here will be helpful later in the lesson when students decompose whole numbers in order to subtract fractional amounts.

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “¿En cuáles expresiones puede ser útil usar 1 unidad —o 12 doceavos— para encontrar el producto? ¿Cómo?” // “In which expressions might it be helpful to use 1 whole—or 12 twelfths—to find the product? How?” (The first three. We know 4 groups of \(\frac{3}{12}\) make 1 whole, so:
  • 2 groups of \(\frac{3}{12}\) make \(\frac{1}{2}\)
  • 6 groups of \(\frac{3}{12}\) make  \(1\frac{1}{2}\)
  • 12 groups of \(\frac{3}{12}\) make 3
• “¿Por qué puede ser un poco más difícil pensar en la última expresión en términos de 12 doceavos?” // “Why might it be a little harder to think of the last expression in terms of 12 twelfths?” (The 30 in \(\frac{30}{12}\) is not a factor or a multiple of 12.)
• Consider asking:
  • “¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____'s reasoning in a different way?”
  • “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
  • “¿Alguien pensó en la expresión de otra forma?” // “Did anyone approach the expression in a different way?”
  • “¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”

This is a 5 Practices activity. Students use any strategy that makes sense to them to reason about subtraction of a fraction from a whole number. They begin by using an image to support their reasoning. Later, when no image is given, students may use a variety of ways to find differences. In the synthesis students share, explain, and relate different strategies for solving the problem (MP3).

Monitor for the following strategies, listed from concrete to abstract, and select students to share during synthesis:

• Draw a diagram showing whole numbers and unit fractions and remove parts of the diagram.
• Create a number line that is partitioned into thirds and draw jumps to the left.
• Reason in terms of addition (for instance, how many thirds to add to \(\frac{5}{3}\) to reach 4?).
• Subtract in multiple rounds (for example, to subtract \(2\frac{1}{3}\) from 4, first subtract 2 wholes, and then subtract 1 third).
• Decompose or rewrite a whole number or a fraction before subtracting.
  • To subtract \(\frac{5}{3}\) from 4, first decompose \(\frac{5}{3}\) into \(\frac{3}{3} + \frac{2}{3}\), or \(1 + \frac{2}{3}\), then subtract.
  • To subtract \(2\frac{1}{3}\) from 4, first decompose 4 into \(3 + \frac{3}{3}\) and decompose \(2\frac{1}{3}\) into \(2 + \frac{1}{3}\), and then subtract. Or, rewrite 4 and \(2\frac{1}{3}\) as \(\frac{12}{3}\) and \(\frac{7}{3}\) and then subtract.

At this point, students are not expected to decompose or rewrite numbers using expressions and equations. They may perform the reasoning intuitively and informally. Later, they will formalize the different ways of recording the decomposition of numbers for subtraction.

• Groups of 2
• Display the image of the graduated pitcher.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• “Ahora resolvamos problemas en los que se sirve una bebida de una jarra” // “Let’s now solve some problems about pouring a drink out of a pitcher.”

• 7–10 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for the different strategies used, as noted in the Activity Narrative.

To support students in visualizing the context, consider revisiting it in several rounds and with a limited scope each time. Consider asking:

• “¿Qué sucede en esta situación?” // “What is happening in this situation?”
• “¿Qué queremos averiguar?” // “What are we trying to find out?” and “¿Qué información tenemos?” // “What information do we have?”
• “¿Qué operación matemática nos puede ayudar a responder esta pregunta?” // “What mathematical operation can help us answer this question?”
• “¿Qué expresión usaríamos para representar la cantidad de líquido?” // “What expression would we use to represent the amount of liquid?”

• “¿Cómo le restaron cada fracción a 4, sin tener que servir en la vida real \(\frac {1}{3}\), \(\frac{5}{3}\) o \(2\frac{2}{3}\) tazas de agua de la jarra que contiene 4 tazas?” // “How did you find the result of subtracting each fraction from 4, without physically pouring out \(\frac {1}{3}\), \(\frac{5}{3}\), or \(2\frac{2}{3}\) cups of water out of a 4-cup pitcher?”
• Select students who used different strategies in the second problem to share their responses, in the order listed in the Activity Narrative.
• Record all the ways students decomposed or rewrote the numbers in the problem to facilitate subtraction (whether or not they wrote expressions or equations).

This activity makes explicit what students may have noticed in earlier activities, namely, that we can subtract a fraction from a whole number by:

• writing an equivalent fraction for the whole number
• decomposing the whole number into a sum of smaller numbers, which could be fractions, or a whole number and a fraction

Students sort a set of cards (with a number or an expression) into two groups and justify the categories. Each card has a value equivalent to either \(1 – \frac{5}{12}\) or \(2 – \frac{5}{12}\). Though students may sort the cards in a few valid ways, the discussion should focus on the value on the cards. Students then apply their insights from the sorting work to subtract a fraction from whole numbers. This sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Here is a list of the numbers or expressions on the blackline master, for reference:

This activity uses MLR2 Collect and Display. Advances: conversing, reading, writing

• Create a set of cards for each group of 2.

• Groups of 2 or 4
• Give each group one set of cards from the blackline master.

• “Con sus compañeros, clasifiquen estas tarjetas en dos grupos. Prepárense para explicar por qué las tarjetas de cada grupo van juntas” // “Work with your group to sort these cards into two groups. Be prepared to explain why the cards in each group belong together.”
• 5 minutes: group work time on the first problem
• Monitor for the ways students sort the cards. Identify those who sort the cards by their value or by their equivalence to \(1 – \frac{5}{12}\) and \(2 – \frac{5}{12}\).

MLR2 Collect and Display

• Circulate, listen for, and collect the language students use to describe the features of the expressions on the cards or the connections between expressions. Listen for terms such as “equivalent,” or “equivalent fractions,” “equal,” “sum,” “difference,” and “decompose.”
• Record students’ words and phrases on a visual display and update it throughout the lesson.
• Pause before students proceed to the next problem. Select a few groups to share their sorting decisions, ending with a group that sorted the cards by value.
• “Veamos esa última forma de clasificar. Las tarjetas de cada grupo tienen el mismo valor y ese valor es el resultado de restarle una fracción a un número entero” // “Let’s look at that last way of sorting. The cards in each group have the same value, and that value is a result of subtracting a whole number by a fraction.”
• Display the cards as shown:

  

  

• “Hablen con un compañero sobre cómo se relaciona cada tarjeta con la que está debajo. Hagan esto con todos los grupos de tarjetas” // “Turn to a partner. Talk about how each card is related to the one below it. Do this for each group of cards.”
• 2 minutes: partner discussion
• Select 2–3 students to share the connections between the expressions in each group. Highlight that:
  • In the first group, the 1 can be written as an equivalent fraction, \(\frac{12}{12}\), which is helpful for subtracting \(\frac{5}{12}\).
  • In the second group, the 2 can be rewritten as an equivalent fraction, \(\frac{24}{12}\), or decomposed into a sum of \(1+1\), which is equivalent to \(1 + \frac{12}{12}\). Both strategies help us to subtract \(\frac{5}{12}\).
• 5 minutes: independent work on the second problem

• See lesson synthesis.