Lesson 18

The purpose of this Number Talk is to encourage students to apply properties of operations (especially the commutative and associative properties of addition) to mentally find sums of three or more whole numbers. The reasoning elicited here will be helpful later in the lesson when students are to add three or more tenths and hundredths.

To mentally add several two- and three-digit numbers, students need to look for and make use of structure (MP7), such as finding pairs of numbers that add up to 10 or 100, or numbers that end in 0 or 5. 

• 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.

• “¿Qué estrategias les ayudaron a sumar varios números?” // “What strategies were helpful for adding multiple numbers?” (Sample responses:
  • Find two or three numbers that add up to 10 or 100.
  • Add numbers that end with 5 or 0 first. Add familiar numbers first.
  • Put the numbers into groups of two, and then add what’s in each group before adding the groups.)
• 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?”

In this activity, students solve problems involving tenths and hundredths in a context about coins. Given information about the thickness of some Mexican coins, students compare the heights of different combinations of stacked coins. To complete the task, students need to write equivalent fractions, add tenths and hundredths, and compare fractions. Some students may choose to use multiplication to reason about the problems. Though the mathematics here is not new, the context and given information may be novel to students. Students have a wide variety of approaches available for these problems with no solution approach suggested (MP1). For example, to compare the peso coins of Diego and Lin, students could reason that they each have a 5 peso and a 20 peso coin and then compare the remaining coins, a 1 peso coin and 2 peso coin on the one hand and a 20 peso coin on the other. This method would require minimal calculations. Other students may add the thicknesses of Lin's coins and Diego's coins and compare these values. 

To help students visualize stacked coins, prepare some coins of different thicknesses or include an image of stacked coins. (Access to the Mexican coins would be interesting to students but is not essential.) Some students may be curious about the equivalents of centavos or pesos in U.S. dollars. Consider checking the exchange rates before the lesson.

• Gather a few coins of different thicknesses for display.

• Display the image of Mexican coins.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• Share responses.
• Explain that pesos and centavos are units of Mexican money, just as dollars and cents are units of American money.
• “¿Cuáles son los valores de las monedas estadounidenses que se usan hoy en día?” //  “What are the values of the American coins we use today?” (1 cent, 5 cents, 10 cents, 25 cents, half a dollar, and 1 dollar.)
• Explain that Mexico uses many more types of coins than the U.S. does.
• Display coins of different thicknesses.
• “Las monedas estadounidenses (un centavo, diez centavos, cinco centavos y veinticinco centavos) tienen grosores y pesos diferentes. Lo mismo ocurre con las distintas monedas mexicanas de centavos y de pesos” //  “Just like pennies, dimes, nickels, and quarters, centavo and peso coins have different weights and thicknesses.”
• Refer to the table in the task, showing the thicknesses in tenths or hundredths of a centimeter.

• “Trabajen con su compañero en los tres primeros problemas” // “Work with your partner on the first three problems.”
• 7–8 minutes: partner work time
• “Busquen otra pareja de compañeros y compartan sus respuestas. Discutan cualquier desacuerdo que tengan” //  “Find another group of classmates and share your responses. Discuss any disagreement you might have.”
• 3–4 minutes: group discussion
• “En silencio, respondan la última pregunta durante unos minutos” // “Take a few quiet minutes to answer the last question.”
• 3 minutes: independent work time
• Consider asking each combined group to share their response to one assigned problem.

• Invite groups to share their responses and reasoning. Highlight the different approaches students took to solve the same problems.
• If no students mentioned using multiplication to find the height of Diego and Lin’s centavo stacks, ask if any of the heights could be found using multiplication.

The purpose of this activity is for students to practice finding sums of three or more fractions in tenths and hundredths and applying properties of operations to facilitate that addition. (Students are not expected to use the terms “commutative property” or “associative property,” but should recognize from the work in earlier grades that numbers can be added in different orders and in different groups.)

This activity can be done in the format of a gallery walk. Ask students to visit at least three of six posters (or as many as time permits). The last three expressions include one or more mixed numbers. In the last expression, the fractional parts add up to a sum greater than 1, which would need to be decomposed into a mixed number and a fraction before being added to the whole number. Consider assigning this as a starting expression for students who could use an extra challenge.

• Create six posters with an addition expression from the activity on each one. 

• Groups of 2

If done as a gallery walk:

• Consider assigning each group a starting poster and giving directions for rotation.
• “Van a encontrar seis pósteres con expresiones de suma en cada uno. Vayan a ver al menos tres pósteres y encuentren el valor de las expresiones” // “You’ll find six posters with addition expressions on each one. Visit at least three posters and find the value of the expressions.”

• “Trabajen con su compañero en las primeras dos expresiones e individualmente en al menos una de ellas. Muestren su razonamiento” // “Work with your partner on the first two expressions and independently on at least one of them. Show your reasoning.”
• 10 minutes: group work or gallery walk

• See lesson synthesis.