Lesson 19

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for place value relationships and the properties of operations as they find the value of different products (MP7). The products all have the same value, 6, and also all have a decimal factor of 0.1 or 0.01. The whole number factors are organized differently and this encourages students to think flexibly about how to find products of a whole number and a decimal.

• 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 qué se diferencia la última expresión de las demás expresiones?” // “How is the last expression different from the others?” (It has a hundredth in the product instead of a tenth.)
• “¿Cómo encontraron el valor de esta expresión?” // “How did you find the value of this expression?” (I knew \(40 \times 20 = 800\) and then 800 hundredths is 8.)

The purpose of this activity is for students to use properties of operations to develop strategies for multiplying decimals (MP7). They first sort expressions into groups which can be used to find the value of a given decimal product. Then they choose one of the expressions to find the value. Many of the expressions use whole number products and the associative property which students have seen in previous lessons. Some of the expressions use subtraction and a compensation strategy. This strategy is new for decimals but will be familiar to students for whole number products.

This activity uses MLR2 Collect and Display. Advances: Conversing, Reading, Writing.

• Groups of 2
• Distribute one set of pre-cut cards to each group of students.

• “En esta actividad, van a clasificar algunas expresiones en categorías que ustedes mismos inventen con su compañero” // “In this activity, you will sort some expressions into categories of your choosing. When you sort the expressions, work with your partner to come up with categories.”
• 3 minutes: partner work time
• “Cada expresión corresponde a una de las expresiones que tienen las letras A, B o C. Con su compañero, agrupen las tarjetas. Justifiquen sus elecciones” // “Each expression matches one of the expressions marked A, B, or C.  Work with your partner to match the cards. Justify your choices.”
• 8 minutes: partner work time

• Circulate, listen for and collect the language students use to explain how they know expressions are equal. Listen for words such as groups of, the same as, and break apart. For example, students may say:
  • 4 groups of 39 tenths is the same as 4 groups of 3 and 9 tenths.
  • 4 times 3 tenths plus 4 times 5 hundredths is the same as 4 times 35 hundredths.
  • 2 groups of 2 groups of three and 5 tenths is the same as 4 groups of 3 and 5 tenths.
• Record students’ words and phrases on a visual display and update it throughout the lesson.

• Invite students to share the matches they made and explain how they know those cards go together.
• Refer to the language on the display as students describe their justification for a match, giving them opportunities to describe the relationship more precisely.
• “¿Qué otras palabras o frases importantes deberíamos incluir en nuestra presentación?” // “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• Invite selected students to share their strategies for calculating \(4 \times 3.9\).
• “¿Por qué \((4 \times 4) - (4 \times 0.1)\) es una expresión útil para encontrar el valor de \(4 \times 3.9\)?” // “Why is \((4 \times 4) - (4 \times 0.1)\) a useful expression for finding the value of \(4 \times 3.9\)?” (I can find those products in my head.)
• “¿Por qué \((4 \times 39) \times 0.1\) es una expresión útil para encontrar el valor de \(4 \times 3.9\)?” // “Why is \((4 \times 39) \times 0.1\) a useful expression for finding the value of \(4 \times 3.9\)?” (I know how to find products of whole numbers like \(4 \times 39\) and then multiplying by 0.1 just changes place values.)

The purpose of this activity is for students to find products of a whole number and a decimal where the decimal has more than one place value, either a whole number and some tenths or some tenths and some hundredths. Monitor for these strategies which students saw in the previous activity

• multiplying whole numbers and then multiplying the result by 0.1 or 0.01
• using the distributive property and multiplying by place value
• using the distributive property and compensation 

• Groups of 2

• 1–2 minutes: quiet think time
• 6–8 minutes: partner work time
• Monitor for students who find the products using
  • whole number products and place value understanding
  • the distributive property

If students need support with the placement of the decimal in the product, ask them to use the language on the display from activity 1 to describe each of the expressions.

• Invite selected students to share their reasoning for the value of \(4 \times 1.4\).
• Display expression: \((4 \times 14) \times 0.1\)
• “¿Por qué esta expresión nos puede ayudar a encontrar el valor de \(4 \times 1.4\)?” // “Why is this expression helpful to find the value of \(4 \times 1.4\)?” (I know \(4 \times 14\) is 56. I can do that with whole number multiplication. Then it’s that many tenths so 5.6.)
• Display expression: \((4 \times 1)  + (4 \times 0.4)\)
• “¿Por qué esta expresión nos puede ayudar a encontrar el valor de \(4 \times 1.4\)?” // “Why is this expression helpful to find the value of \(4 \times 1.4\)?” (It breaks it up by place value. I first find 4 ones and then 4 times 4 tenths. That’s 4 and 16 tenths or 5 and 6 tenths so it’s 5.6.)
• “¿Cómo escogieron una estrategia para resolver cada problema?” // “How did you choose a strategy for each problem?” (I like to multiply whole numbers so I always thought of products of whole numbers and then took that many tenths or hundredths. I noticed 3.9 is really close to 4 and I know \(5 \times 4\) so I started there and figured out what I needed to subtract.)

The purpose of this optional activity is to find more complex products of a whole number and a decimal using any strategy. For the more complex numbers, the strategies that students have seen all apply but the most reliable one is to find a product of whole numbers and then identify the number of tenths or hundredths that is. The distributive property is still an effective tool but a product of 2 two-digit numbers gives 4 single digit products. The problems are scaffolded so that students can use their answers for the first two problems to find the answer to the third.

• 5 minutes: independent work time
• Monitor for students who use the first two calculations for the third and who use place value understanding for the last calculation.

• Invite students to share their calculations.
• “¿Cómo usaron el valor de \(35 \times 0.08\) y el valor de \(35 \times 0.7\) para encontrar el valor de \(35 \times 0.78\)?” // “How did you use the values of \(35 \times 0.08\) and \(35 \times 0.7\) to find the value of \(35 \times 0.78\)?” (I added them.)
• Display equation: \(35 \times 0.78 = (35 \times 0.7) + (35 \times 0.08)\)
• “¿Cómo saben que esta ecuación es verdadera?” // “How do you know this equation is true?” (It’s the distributive property.)
• “¿Cómo calcularon \(42 \times 0.66\)?” // “How did you calculate \(42 \times 0.66\)?” (I found \(42 \times 66\) and then knew I had that many hundredths.)