Lesson 16

The purpose of this number talk is to find the value of subtraction expressions with decimal numbers which encourage adding on as a strategy. While the first two expressions can be found readily by taking away, finding the difference or adding on to the smaller number is an effective strategy. The second pair of expressions also encourage adding on or compensation strategies. Students will have an opportunity to use these strategies in the lesson as they continue to add and subtract decimals.

• 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 estrategia usaron para encontrar el valor de \(2.57 - 0.99\)?” // “What strategy did you use to find the value of \(2.57 - 0.99\)?”
  • I subtracted 1.57 to get 1 and then one more hundredth to get 0.99.
  • I added 0.01 to 0.99 to get 1 and then added 1.57 more to get 2.57.

The purpose of this activity is for students to find the value of various decimal differences. Most of the numbers do not have the same number of decimal digits so students need to subtract carefully if they make vertical calculations, making sure to align place values correctly (MP6). Students may use the standard algorithm or they may choose a different technique. Some of the differences are designed to bring out other techniques such as adding on or compensation.

• Groups of 2

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who
  • use the standard algorithm correctly for each calculation
  • use other techniques such as adding on or subtracting in different ways by place value

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

• Display expression: \(100.8 - 6.03\)
• “¿Cómo encontraron el valor de la diferencia?” // “How did you find the value of the difference?”
  • I used the standard algorithm.
  • I first took the 6 from 100 and then the 0.03 from 0.8. 
• Display expression: \(465 - 463.14\)
• “¿Cómo encontraron el valor de esta diferencia?” // “How did you find the value of this difference?”
  • I used the standard algorithm.
  • I added on to 463.14 since they are so close.
• “¿En qué se parecen las distintas estrategias que usaron para restar? ¿En qué son diferentes?” // “How are the different strategies you used to subtract the same? How are they different?” (They all make sure to use the right place value for each digit. The standard algorithm starts from the smallest place value and then breaks up larger place values when needed. The other strategies subtract an amount that you can do mentally or use addition.)

The purpose of this activity is to find sums and differences of decimals using any method. The variety of problems encourages different strategies including

• adding or subtracting by place value
• using the standard algorithm
• adding on in order to calculate a difference

The goal of the synthesis is to discuss how students decided which strategies to use to find the values of the different expressions. As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).
 

• Groups of 2

• Invite students to share how they found the value of the expression \(36.51 - 4.3\).
• “¿Por qué restar usando el valor posicional es una buena estrategia para esta expresión?” // “Why is subtracting by place value a good strategy for this expression?” (I can take 4 from 6 and 3 tenths from 5 tenths and that gives me 32.21.)
• “¿También funciona el algoritmo estándar?” // “Does the standard algorithm also work?” (Yes, you just need to make sure to subtract ones from ones and tenths from tenths.)
• Display image of calculation of \(36.51 - 4.3\) with the standard algorithm from student solution.
• Invite students to share how they found the value of the expression \(100 - 31.05\).
• “¿Alguien usó el algoritmo estándar?” // “Did anyone use the standard algorithm?” (I tried but there was a lot of borrowing so I decided to use a different strategy.)
• Display image of calculation of \(100 - 31.05\) with the standard algorithm from student solution.
• “El algoritmo estándar siempre funciona para encontrar sumas y diferencias. Pero, dependiendo de los números, otros métodos pueden ser más eficientes” // “The standard algorithm always works to find sums and differences but sometimes the numbers make other methods more efficient.”

The purpose of this activity is for students to find the value of subtraction expressions using a strategy of their choice. The standard algorithm which students learned in a previous activity will always work to successfully to calculate a difference but some of the problems are deliberately chosen to encourage other techniques such as adding on or compensation.  If students choose to use the standard algorithm to calculate the differences, they will need to pay close attention to place value as several of the differences have one decimal with only tenths while the other has hundredths (MP6).

• Groups of 2

• 10 minutes: independent work time
• 5 minutes: partner discussion

• Display expression \(43.14 - 18.6\)
• “¿Qué estimación harían del valor de esta expresión?” // “What estimate would you make for the value of this expression?” (about 20, about 25)
• Invite students to share their strategies and solutions.
• “¿Por qué usaron el algoritmo estándar?” // “Why did you use the standard algorithm?” (The numbers are complicated so I can't see what the difference is.)
• “Tendiendo en cuenta la estimación que hicieron, ¿su solución tiene sentido?” // “Did your solution make sense, based on your estimate?” (Yes, it is between 24 and 25.)
• Display expression: \(261.25 - 260.7\)
• Invite students to share their strategies and solutions.
• “¿Pueden encontrar mentalmente esta diferencia? ¿Cómo?” // “Can you find this difference mentally? How?” (Yes, I can add 0.3 to 260.7 and that gives me 261 and then I need 0.25 more. That's 5 tenths and 5 hundredths so 0.55.)