Lesson 21

The purpose of this estimation exploration is for students to apply what they know about area and multiplication of decimals to a situation where the side length of the rectangle are decimals. Students will approximate the length and width to obtain a product of decimals which they can find mentally. This prepares them for work in the lesson where they will find more complex products of decimals since making an estimate is a good way to check work. 

• Groups of 2
• Display the image.
• “¿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

• “Discutan con su pareja cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• Invite students to share their estimates.
• “¿Cómo saben que el área es mayor que 2 kilómetros cuadrados?” // “How do you know the area is greater than 2 square kilometers?” (I know that \(3 \times 0.7\) is 21 tenths or 2.1 and it’s more than that.)
• “¿Cómo saben que el área es menor que 3.2 kilómetros cuadrados?” // “How do you know the area is less than 3.2 square kilometers?” (I know 3.85 is less than 4 and 0.79 is less than 0.8. Then \(4 \times 0.8\) is 32 tenths or 3.2.)

In previous lessons, students have found products of a whole number and a decimal and products of two decimals. They used diagrams, place value reasoning, and expressions to explain their reasoning. The purpose of this activity is for students to find both kinds of products with larger numbers. For each product, students show that an expression using whole number products is equal to the given decimal product. Then they calculate the decimal product. Students may use a strategy, other than the given equivalent expressions, to make the calculations. For example, they might decompose the numbers by place value and use the distributive property (partial products). All of these methods focus on the place value of each digit in the products (MP7).

• Groups of 2

• 1–2 minutes: quiet think time
• 8–10 minutes: partner work time

• Invite students to explain why \(7.2 \times 5.3\) is equivalent to \(72 \times 53 \times 0.01\).
• Display equations: \(7.2 = 72 \times 0.1\) and \(5.3 = 53 \times 0.1\)
• “¿Cómo saben que las ecuaciones son verdaderas?” // “How do you know the equations are true?” (Multiplying 72 or 53 by 0.1 changes the tens to ones and the ones to tenths.)
• Display equation: \(7.2 \times 5.3 = (72 \times 53) \times 0.01\)
• “¿Cómo saben que la ecuación es verdadera?” // “How do you know the equation is true?” (I use the equations for 7.2 and 5.3 and multiply 0.1 and 0.1 to get a hundredth or 0.01.)
• “¿Cómo pueden usar la ecuación para encontrar el valor de \(7.2 \times 5.3\)?” // “How can you use the equation to find the value of \(7.2 \times 5.3\)?” (I can multiply 72 and 53 and then multiply that by 0.01.)

The purpose of this activity is for students to find products of decimals and whole numbers with no scaffold. As in the previous activity, the products are either a whole number and a decimal to the hundredths or two decimals to the tenths. Students may use any strategy including partial products or using products of whole numbers and place value understanding. The final problem, the product of a three-digit decimal number and a two-digit decimal number, is new but all of the strategies students have used to multiply two-digit decimals apply here as well.

• Groups of 2

• 5-8 minutes: independent work time
• 2-5 minutes: partner discussion

• Invite students to share their responses for the first two products.
• “¿Cómo usaron su comprensión del valor posicional para encontrar los productos?” // “How did you use your understanding of place value to find the products?” (I used whole number products and then remembered that I have that many hundredths so I had to multiply that product by 0.01.)
• “¿En qué se parece el último producto a los otros productos que han encontrado? ¿En qué se diferencia?” // “How is the last product the same as the other products you have found? How is it different?” (It is also a whole number and a decimal and the decimal has tenths and hundredths. The whole number is bigger. I can use the same methods but the product is more complicated since one factor has three digits.)