Lesson 20

The purpose of an Estimation Exploration is to practice the skill of making a reasonable estimate based on experience and known information. When students notice that they can make a more accurate estimate when the single cubes are grouped into 10s they make use of base-ten structure (MP7).

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.
• “Veamos otra imagen de la misma colección” // “Let’s look at another image of the same collection.”
• Display the image.
• “Teniendo en cuenta la segunda imagen, ¿quieren ajustar o cambiar sus estimaciones?” // “Based on the second image, do you want to revise, or change, your estimates?”

• “¿Alguien cambió la estimación ‘razonable’ que tenía originalmente? ¿Por qué la cambió?” // “Did anyone change their original ‘about right’ estimate? Why did you change it?” (I changed it because I see there are at least 50 cubes in the 5 towers.)
• “Examinemos nuestras estimaciones ajustadas. ¿Por qué nuestras estimaciones fueron más precisas la segunda vez?” //  “Let’s look at our revised estimates. Why were our estimates more accurate the second time?” (Some of the cubes are organized.)
• “Hay 76 cubos” // “There are 76 cubes.”

The purpose of this activity is for students to represent a two-digit number in multiple ways. Students do not need to come up with every way, but they may find a method that results in them doing so. Students may choose to use connecting cubes as they work and then show their thinking with drawings, numbers, or words. If students use expressions to represent to 94 as tens and ones (for example, 90 + 4, 80 + 14, 70 + 24), ask them to explain which addend represents the value of an amount of tens and which represents a value of ones. During the activity synthesis, students discuss whether all of the different ways to represent 94 have been found and how they know. When students explain that when the number of tens decreases by 1, the number of ones increases by 10 because a ten is the same as 10 ones, they are using the base-ten structure of the numbers to express regularity in repeated reasoning (MP7, MP8).

• Groups of 2
• Give each group access to connecting cubes in towers of 10 and singles.

• “El reto de hoy es encontrar todas las maneras posibles de formar 94 usando decenas y unidades. Si les ayuda, pueden usar cubos. Cada manera en la que formen 94 debe tener un número diferente de decenas” // “Today’s challenge is to find as many ways as you can to make 94 using tens and ones. You can use cubes if they will help you. Each way you make 94 should have a different number of tens.”
• 10 minutes: independent work time
• 4 minutes: partner discussion
• Monitor for students who:
  • use connecting cubes to physically break apart a ten at a time to move between representations
  • use tens and ones notation
  • use addition expressions

If students believe that they have found all the ways to make 94 with tens and ones, consider asking:

• “¿Cómo sabes que encontraste todas las maneras posibles?” // “How do you know you've found all the different ways?”
• “¿Cómo podrías enumerar las diferentes maneras en las que formaste 94 con decenas y unidades para demostrar que encontraste todas las maneras?” // “How could you list the different ways you made 94 with tens and ones to prove you found all the ways?”

• Invite previously identified students to share.
• “¿Piensan que encontramos todas las maneras? ¿Por qué sí o por qué no?” // “Do you think we found all the ways? Why or why not?”
• If needed, ask “¿Qué tienen en común todas estas maneras? ¿Qué patrones observan?” // “What do all of these have in common? What patterns do you notice?” (Every time I break apart a ten into ones, the number of ones increases by 10.)
• “¿Con cuál representación de 94 preferirían trabajar? ¿Con cuál menos les gustaría trabajar? ¿Por qué?” // “Which representation of 94 would you like to work with the most? Which would you like to work with the least? Why?” (I would like to work with 9 tens and 4 ones because it is the easiest. You can easily count 9 tens and you can just see there are 4 ones. I would like to work with 94 ones the least. It is really hard to know how many you have when there are so many.)

The purpose of this activity is for students to identify two-digit numbers or a part of a number represented in different ways, with different amounts of tens and ones. Students determine how many connecting cubes are in each bag, given clues about how many tens and ones are in the bag. Students also determine how many tens or ones are in a bag, given the total number of cubes and either the number of tens or ones. Students may use connecting cubes to make sense of the problems, and show their thinking using drawings, numbers, or words. 

• Groups of 4
• Give students access to connecting cubes in towers of 10 and singles.

• “Van a resolver problemas sobre cubos encajables dentro de bolsas misteriosas. Si les ayuda, pueden usar cubos encajables. Muestren cómo pensaron. Usen dibujos, números o palabras” // “You are going to solve problems about connecting cubes in mystery bags. You can use connecting cubes if they will help you. Show your thinking using drawings, numbers, or words.”
• “Primero, van a trabajar individualmente. Después, van a compartir con un compañero de su mesa cómo pensaron” // “You will begin by working on your own. Then you will share your thinking with a partner at your table.”
• 6 minutes: independent work time
• “Compartan con un compañero de su mesa lo que pensaron sobre el problema 1” // “Share your thinking for problem 1 with a partner at your table.”
• 1 minute: partner discussion
• “Compartan con otro compañero de su mesa lo que pensaron sobre el problema 2” // “Share your thinking for problem 2 with a different partner at your table.”
• 1 minute: partner discussion
• Repeat for problems 3 and 4.
• Monitor for students who use connecting cubes in these ways to solve for mystery bag C:
  • Shows 49 as 4 tens 9 ones and moves 2 tens over to the ones cubes to have 29, shows 2 tens left.
  • Shows 29 ones, adds towers of 10 to get to 49.

If students show they are adding the total number of cubes and the known tens or ones for Bag C or Bag D, consider asking:

• “¿Cómo podrías actuar este problema?” // “How could you act out this problem?”
• “¿Qué sabes? ¿Qué no sabes?” // “What do you know? What don’t you know?”
• “Hemos aprendido a formar números con diferentes cantidades de decenas y unidades, ¿cómo podrías usar esto para resolver el problema?” // “How could you use what we’ve learned about making numbers with different amounts of tens and ones to solve the problem?”

• Invite previously identified students to share.
• “¿En qué se parecen estas formas de encontrar el número desconocido de decenas? ¿En qué son diferentes?” // “How are these ways for finding the mystery number of tens the same? How are they different?” (They both used tens and ones. One person started with 29 ones and added the tens, the other person started with 4 tens 9 ones and broke apart tens until there were 29 ones.)