The purpose of this activity is for students to use properties of operations to develop strategies for multiplying decimals. They sort expressions into groups of expressions that 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 sorting task gives students opportunities to analyze expressions closely and make connections (MP7).

This activity uses MLR2 Collect and Display. Advances: conversing, reading, writing.

• Invite previously selected students to share 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 .
• “¿Por qué  es una expresión útil para encontrar el valor de ?” // “Why is  a useful expression for finding the value of ?” (I can find those products in my head.)
• “¿Por qué  es una expresión útil para encontrar el valor de ?” // “Why is a useful expression for finding the value of ?” (I know how to find products of whole numbers like and then multiplying by 0.1 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.

If students find values that are partially correct, consider asking:

• “¿Cómo resolviste el problema?” // “How did you solve the problem?”
• “¿Entre cuáles 2 números enteros estará el valor?” // “Which 2 whole numbers will the value be between?”

• Invite selected students to share their reasoning for the value of .
• Display expression:
• “¿Por qué esta expresión nos puede ayudar a encontrar el valor de ?” // “Why is this expression helpful to find the value of ?” (I know is 56. I can do that with whole number multiplication. Then it’s that many tenths so 5.6.)
• Display expression:
• “¿Por qué esta expresión nos puede ayudar a encontrar el valor de ?” // “Why is this expression helpful to find the value of ?” (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 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.

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