Lesson 20

The purpose of this What Do You Know About ____? is for students to think about \(0.1 \times 0.1\) before working with this kind of expression more formally in the lesson. Students know that 0.1 is the same as \(\frac{1}{10}\) and they know how to find products of fractions. The goal of the synthesis is to highlight this before students find products of decimals in the lesson. 

• Display the expressions.
• “¿Qué saben sobre \(1 \times 0.1\) y \(0.1 \times 0.1\)?” // “What do you know about \(1 \times 0.1\) and \(0.1 \times 0.1\)?”
• 1 minute: quiet think time

• Record responses.
• “¿Cómo podríamos representar estas expresiones?” // “How could we represent these expressions?” (I could use a hundredths grid or area diagram.)

• “¿Pueden encontrar el valor de \(0.1 \times 0.1\)?” // “Can you find the value of \(0.1 \times 0.1\)?” (Yes, 0.1 is \(\frac{1}{10}\) so that’s \(\frac{1}{10} \times \frac{1}{10}\) and I know that’s \(\frac{1}{100}\).)

The purpose of this activity is for students to find products of a number of tenths and a number of tenths written as decimals. Students can think of find these products in many ways including

• using a diagram
• using whole number arithmetic and place value reasoning or properties of operations (MP7)

• Groups of 2

• 1–2 minutes: quiet think time
• 6–8 minutes: partner work time
• Monitor for students who:
  • use grids
  • use whole number facts and place value reasoning

• Invite students to share their reasoning for \(0.2 \times 0.4\).
• Display a student generated diagram of \(0.2 \times 0.4\) or the diagram from the student solution.
• “¿Cómo se muestra \(0.2 \times 0.4\) en el diagrama?” // “How does the diagram show \(0.2 \times 0.4\)?” (There is 2 tenths of 4 tenths of the rectangle shaded.)
• “¿Cómo supieron que la región sombreada tiene un área de 0.08 unidades cuadradas?” // “How did you know that the shaded region has area 0.08 square units?” (There are \(2 \times 4\) shaded pieces and each one is \(\frac{1}{100}\) of the full square.)
• Display equation \(0.2 \times 0.4 = 2 \times 4 \times (0.1 \times 0.1)\).
• “¿Cómo se muestra esta ecuación en el diagrama?” // “How does the diagram show this equation?” (The shaded part is 2 tenths of 4 tenths of the rectangle so that's \(0.2 \times 0.4\). It’s \(2 \times 4 \times (0.1 \times 0.1)\) because there are \(2 \times 4\) pieces and each one has area \(0.1 \times 0.1\) or one hundredth of a square unit.)

The purpose of this activity is for students to multiply decimals by decimals, building on the strategies they saw in the previous activity. Monitor for these strategies:

• using a diagram
• using whole number products and place value understanding
• using expressions to show their thinking

• Groups of 2
• Make copies of hundredths grid blackline master available.

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who:
  • use the grids
  • multiply two whole numbers and then multiply their product by \(0.01\)

• Invite students to share their responses and reasoning for the product \(1.8 \times 0.4\).
• Display student generated diagram or diagram in student solution.
• “¿Cómo se muestra \(1.8 \times 0.4\) en el diagrama?” // “How does the diagram show \(1.8 \times 0.4\) ?” (There is a full group of 0.4 and then there is 8 tenths of another group of 0.4.)
• “¿Cómo se muestra \(18 \times 4 \times 0.01\) en el diagrama?” // “How does the diagram show \(18 \times 4 \times 0.01\)?” (There is an 18 by 4 array of pieces and each piece is a hundredth of the whole.)
• Display: \(1.8 \times 0.4 = (18 \times 4) \times 0.01\)
• Invite students to share their responses about the products \(74 \times 6\), \(7.4 \times 6\) and \(7.4 \times 0.6\).
• “¿Cómo pueden usar un producto de números enteros para encontrar un producto de números decimales?” // “How can you use the whole number product to find decimal products?” (I just think about how many tenths or hundredths I have.)