Lesson 17

This warm-up prompts students to carefully analyze and compare features of base-ten diagrams, looking not only at the number and types of shapes in each diagram, but also the value each diagram represents. The activity also enables students to recall what they know about representations of numbers in base-ten and enables the teacher to hear how they talk about these representations.

The analysis here prepares students for the activities in the lesson, in which they use base-ten diagrams to find whole-number quotients.

• Groups of 2
• Display image.
• “Escojan uno que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Record responses.

• “¿Por qué ambos diagramas, A y C, muestran 111?” // “How is it that A and C both show 111?” (If a small square represents 1, then a rectangle is 10 and a large square is 100. In A: \(100 + 10 + 1 = 111\). In C: \(100 + 11 = 111\).)
• “¿Cómo sabemos que el diagrama D también muestra 111?” // “How do we know that D also shows 111?” (Each group in D represents \( (3 \times 10)\) + 7 or 37. Three groups of 37 makes 111.)
• “Supongamos que no sabemos qué valor representa un cuadrado pequeño. Solo sabemos que representa el mismo valor en todos los diagramas. ¿Podemos saber si los diagramas C y D representan el mismo valor? ¿Cómo?” // “Suppose we don’t know what a small square represents except that it represents the same value in all diagrams. Can we tell if C and D represent the same value? How?” (Yes. We know that 10 small squares make 1 rectangle and 10 rectangles make 1 large square. In D, we’d have 21 small squares and 9 rectangles. Trading 10 small squares for a rectangle gives 10 rectangles and 11 small squares, which is equal to 1 large square and 11 small squares.)

In this activity, students use base-ten diagrams to find quotients of two-digit dividends and single-digit divisors. They think about distributing the pieces in the diagram into equal-size groups, decomposing a higher-value piece with 10 of the lower-value pieces as needed to divide.

Some students may benefit from manipulating physical blocks. Provide each group of students with access to a set of base-ten blocks.

• Groups of 4
• Give students access to base-ten blocks.
• Display the first diagram. Make sure students can explain why it represents 64.

• 5 minutes: quiet think time
• 2 minutes: group discussion
• Monitor for students who see that a larger piece can be decomposed into 10 of the next smaller piece to help with distribution.

• Invite students to share their responses and reasoning.
• Make sure students see that:
  • We can think of \(64 \div 4\) as putting 6 tens and 4 ones into 4 equal groups, and \(117 \div 3\) as putting 1 hundred 1 ten and 7 ones into 3 equal groups.
  • To divide the base-ten pieces, we can decompose a piece representing a larger place value with 10 of the next smaller place value.

In this activity, students continue to use base-ten representations and to reason about equal-size groups to find whole-number quotients. The work reinforces the idea of decomposing a hundred into 10 tens as needed to perform division. 

Students explicitly use place value understanding to decompose hundreds and tens (MP7) while making sense of a students' reason to help him complete the division problem (MP3).

• Groups of 2
• Give students access to base-ten blocks.
• Display the first diagram.
• “¿Cómo está representado 235 en el diagrama?” // “How does the diagram represent 235?” (A large square represents 100. A rectangle represents 10. A small square represents 1.)

• 4–5 minutes: independent work time on the first question
• Monitor for students who see the 2 hundreds as 20 tens and those who see them as 200 ones.
• Pause after the first question. Make sure students see that the 2 hundreds can be decomposed into 20 tens (or 200 ones) and split into 5 equal groups, and the 3 tens can be decomposed into 30 ones and split into 5 groups. Complete the second diagram to illustrate this reasoning.
• 5 minutes: partner work time on the last question

• Select students to share their responses and reasoning for the last question. Display them for all to see.
• Highlight the idea that each unit can be decomposed into 10 units of a lower place value to make it possible to create equal-sized groups.