Lesson 13

The purpose of this Estimation Exploration is for students to estimate a sum of two decimals. The numbers are complex to encourage students to make an estimate which means identifying that the leading digits, rather than the decimal places, are the most important for making a good estimate. 

In this lesson, students start to work with sums of larger numbers. Making a mental estimate before calculating is a valuable skill to help confirm the reasonableness of a solution.

• Groups of 2
• Display the expression.
• “¿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 compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “¿Cuáles dígitos de los números fueron los más importantes para hacer su estimación?” // “Which digits in the numbers were most important for making your estimate?” (The 1 and first 9 from from the first number and the 6 and 5 from the second number.)

The purpose of this activity is for students to analyze a common error when using the standard algorithm to add decimals. The standard addition algorithm requires students to add digits with the same place value. In the given example the two numbers are “right aligned” as when adding whole numbers but this leads to the error of adding tenths from one number to hundredths of the other number as if they had the same place value. When students analyze the incorrect calculation, explain why it is incorrect, and correct it they critique the reasoning of others (MP3).

This activity uses MLR3 Clarify, Critique, and Correct. Advances: Reading, Writing, Representing.

• Groups of 2
• “Resuelvan individualmente el primer problema” // “Solve the first problem on your own.”
• 2–3 minutes: independent work time
• “Ahora trabajen unos minutos individualmente en el segundo problema y luego discútanlo con su compañero” // “Now, work on the second problem on your own for a few minutes, and then talk to your partner about it.”

• 1–2 minutes: independent work time
• 6–8 minutes: partner work time
• Monitor for students who:
  • use estimation to reason (For example, they notice that adding almost 70 to 620 must be a lot greater than 628, as shown in Elena’s calculation.)
  • notice the misalignment of the decimal places being added in Elena’s calculation

If students do not have a strategy to solve the first problem, ask “¿Qué estrategia usarías para encontrar el valor de 621 + 72?” // “What strategy would you use to find the value of 621 + 72?”

• Display Elena’s partially correct answer and explanation.
• Read the explanation aloud.
• “¿Qué creen que quiere decir Elena? ¿Hay algo que no sea claro?” // “What do you think Elena means? Is anything unclear?” (She says the answer will be more than 621, but she doesn’t say how much more. The answer will be a lot more than 628.68.)
• “Escriban con su compañero una solución y una explicación ajustadas” // “With your partner, work together to write a revised solution and explanation.”
• (Optional) Display and review the following criteria:
  • Explanation of mistakes
  • Specific words and phrases
    • Decimal point
    • Place value
  • Labeled diagram
  • Correct solution
• 3–5 minutes: partner work time
• Select 1–2 groups to share their revised explanation with the class.
• “¿En qué se parecen y en qué son diferentes las soluciones y explicaciones ajustadas de estos grupos?” // “What is the same and different about the revised solutions and explanations?”
• Invite previously selected students to share.
• Display Andre’s solution.
• “¿Cómo sabe Andre que necesita sumarle un poco más de 3 a 690?” // “How does Andre know that he needs to add a little more than 3 to 690?” (He hasn’t added the ones, tenths, or hundredths places yet and \(1.45 + 2.30\) is a little more than 3.)
• “¿Por qué creen que Andre usa 72.30 en vez de 72.3?” // “Why do you think Andre uses 72.30 instead of 72.3?” (It helps him keep the place values and decimals in line so he doesn’t mix up the places when he is adding. 72.30 is equivalent to 72.3.)

The purpose of this activity is for students to find the value of various decimal sums with no scaffold. Most of the numbers do not have the same number of decimal digits so students need to add carefully if they make vertical calculations, making sure to align place values correctly (MP6).

• Groups of 2

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who
  • use the standard algorithm
  • use place value understanding but do not organize their calculations vertically

If students do not find the correct value of a sum, ask, “¿Entre cuáles 2 números enteros estará el valor de la suma?” // “Which 2 whole numbers will the value of the sum be between?”

• Invite previously selected students to share.
• Display: \(463 +3.14\)
• “¿Cómo se encuentra esta suma usando el algoritmo estándar?” // “How do you add these using the standard algorithm?” (I made sure that the 3 in 463 lines up with the 3 in 3.14 so I am adding numbers with the same place value. I added 0’;s in the tenths and hundredths places of 463 to help add the numbers.)
• “¿Alguien usó otra estrategia para sumar estos números?” // “Did anyone use a different strategy to add these numbers?” (I knew that 463 and 3 is 466 and then I also have 14 hundredths so it's 466.14.)