Lesson 24

The purpose of this estimation exploration is for students to consider quotients of a decimal to the hundredth by a whole number. Students can draw or visualize a diagram or they may reason about place value in order to make a meaningful estimate. 

• 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
• 1 minute: partner discussion
• Record responses.

• “¿Cómo saben que \(0.42 \div 5\) es más que 8 centésimas?” // “How do you know \(0.42 \div 5\) is more than 8 hundredths?” (I know 5 groups of 8 hundredths is 40 hundredths and that’s 0.4 which is 2 hundredths less than 0.42.)
• “¿Por qué 8 centésimas es una buena estimación del valor de \(0.42 \div 5\)?” // “Why is 8 hundredths a good estimate for the value of \(0.42 \div 5\)?” (Because 40 hundredths is really close to 42 hundredths.)

The purpose of this activity is for students to find quotients of a decimal by a whole number. Students may choose to use diagrams to show their reasoning. Since the diagram is a hundredths grid, this will naturally encourage students to think of decimals like 0.8 and 0.5 as 0.80 and 0.50, that is, think about them in terms of hundredths. This is an important step since finding a quotient like \(0.5 \div 2\) will require thinking about hundredths. Students may also apply their understanding of dividing fractions by whole numbers (MP7).  Monitor for students who

• use the grid to identify the amount being divided and think geometrically about finding equal shares
• think about the number of pieces that they need to partition into equal groups and use whole number division to find those shares

• Groups of 2
• Give students access to blackline master of grids.

• 2 minutes: quiet think time
• 10 minutes: partner work time
• Monitor for students who:
  • use equivalence to think about tenths as hundredths
  • decompose the dividend into a sum of dividends that are divisible by the divisor

• Ask previously identified students to share their thinking for the last two problems.
• Display expression: \(0.5 \div 2\)
• “¿Cómo les ayuda pensar en centésimas a encontrar el valor de esta expresión?” // “How does thinking about hundredths help find the value of this expression?” (Since 0.5 is 50 hundredths I can find half of that and that's 25 hundredths or 0.25.)
• Display expression: \(0.75 \div 5\)
• “¿Cómo les ayuda pensar en partes iguales a encontrar el valor de esta expresión?” // “How does thinking about equal shares help find the value of this expression?” (If I have 75 things I can make 5 groups of 10 with 25 left so that’ s 5 more groups of 5. So 75 is 5 groups of 15 and 75 hundredths is 5 groups of 15 hundredths.)

The purpose of this activity is for students to find quotients of a decimal by a whole number. In the previous activity students saw that whole number quotients can be used to find the quotient of a decimal by a whole number. In this activity, they begin with a whole number quotient which they can use to find decimal quotients and then they find the value of decimal quotients without a scaffold.

• Groups of 2

• 5 minutes: quiet work time
• 3 minutes: partner discussion
• Monitor for students who use whole number facts to find the decimal quotients.

• Invite students to share responses for the second question.
• “¿Cómo les ayudó pensar en 0.7 y en 0.45 como centésimas a encontrar el valor de estas expresiones?” // “How did thinking of 0.7 and 0.45 as hundredths help to find the value of these expressions?” (I can think about whole number division. I have 70 hundredths and splitting that in half would give me 35 hundredths. Or I have 45 hundredths and that’s 5 equal groups of 9 hundredths.)
• “¿Pueden usar un diagrama para encontrar estos cocientes?” // “Could you use a diagram to find these quotients?” (I can shade 7 tenths in a hundredths grid and then cut that in half. I think seeing 5 groups of 9 hundredths in 45 hundredths might be challenging.)