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Activity 1

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Coins Index cards Paper Two-color counters

Activity Narrative

The purpose of this activity is for students to visualize and make sense of the context of problems they will solve in the next activity. They will also consider representations that can be used to model the quantities and actions in the situation (MP4) and try creating them. Students will not have time to make a complete representation, so the focus is on making sense of the situation and persevering in finding effective representations (MP1).

Launch

Groups of 3 or 4
Display the image of the lockers.
“In your small group, discuss what you notice and wonder about the picture.”
1 minute: quiet think time
1 minute: group discussion
Share and record responses. If students are not familiar with lockers, explain that lockers are used to hold students’ coats, books, and backpacks in some schools.

Activity

Read the problem aloud to students.
“Take some time on your own to make sense of the problem. You may want to try to visualize it or reread the problem.”
2 minutes: quiet think time
Give students access to two-color counters, index cards, coins, and other materials that may be helpful to them.
“Based on what you know, create a representation of the problem.”
5 minutes: independent work time
Monitor for and select 3 different representations that:
- Represent lockers in different ways.
- Show instances of contact with lockers (or of opening and closing).
- Show correspondence between each student and the lockers touched.

The picture shows lockers in a school hallway.

The 20 students in Tyler’s fourth-grade class play a game in a hallway that has 20 lockers in a row. The lockers are numbered from 1 to 20.

20 rectangles labeled 1 to 20 to represent 20 lockers.

The 1st student starts with the 1st locker, and while going down the hallway, opens all the lockers.
The 2nd student starts with the 2nd locker, and while going down the hallway, shuts every other locker.
The 3rd student stops at every 3rd locker and opens the locker if it is closed or shuts the locker if it is open.
The 4th student stops at every 4th locker and opens the locker if it is closed or shuts the locker if it is open.
This process continues through the 20th student, so that all 20 students in the class touch the lockers.

Create a representation to show what you understand about this problem. Consider:

How does your representation show lockers?
How does your representation keep track of students who touch lockers?
How does your representation show which lockers are open or closed?

Student Response

to access Student Response.

Advancing Student Thinking

Activity Synthesis

While discussing each representation, ask students:
- “Where are the lockers in this representation?” (Numbers or pictures used to indicate lockers)
- “How can we tell if a locker is open or closed in this representation?” (Symbols or colors were used to indicate open or closed.)
- “How does this representation show which student is touching the lockers?” (Students and their actions were listed in a table or a T-chart.)
“In the next activity, you will answer questions about the game. You may use your representation or one shared by a classmate to help you think through the problems.”

Activity 2

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

The purpose of this activity is for students to solve problems about a locker game by using the ideas of factors and multiples and by observing patterns in the numbers. The situation offers many possible explorations, but the questions around which lockers are touched and by how many people are designed to elicit understandings about multiplication.

Discussion about which lockers will be open or closed or by whom will likely move students away from thinking about factors and multiples and should not be the focus. The final question challenges students who are ready for more to look for additional patterns and generalize their observations. It is not expected that all students complete the final problem. Students who are ready may simulate opening and closing the lockers in different ways, such as:

Using actual lockers, if available.
Using a printed set of squares labeled 1 through 20.
Using two color counters and 10-frames, if available.
Using the head and tail sides of coins

MLR7 Compare and Connect. Synthesis: Invite groups to prepare a visual display that shows the strategy they used to determine which lockers will be open. Encourage students to include details that will help others interpret their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams or drawings. Give students time to investigate each other’s work. During the whole-class discussion, ask students, “Which students open or close locker 10? What about locker 20? What patterns do you notice about the number of factors for a locker number and whether it is open or closed?”
Advances: Representing, Conversing

Representation: Internalize Comprehension. Support students in transferring their knowledge to a new context. Begin by asking, “Does this problem remind anyone of the concepts we have been studying?” Invite students to reference the vocabulary display, and continue to refer them to it throughout the lesson.
Supports accessibility for: Conceptual Processing, Language, Attention

Launch

Groups of 3–4
“Let’s solve some problems about a game you read about earlier, where students take turns opening and closing lockers.”
“Silently read and think about each question.”
1 minute: quiet think time
“Work in your group to answer each question. Consider the representations you created or saw earlier to help you think about the problems.”

Activity

15–20 minutes: group work time
Monitor for students who:
- Represent each contact with the lockers in a systematic way.
- Represent the open and closed lockers in various ways.
- Reason about factors and multiples and use the vocabulary in their explanations.
Encourage students who use a pattern to answer the final question to share their explanation. Invite them to consider a situation with more lockers such as 50 or 100.

Tyler’s class plays the same locker game again.

Your goal this time is to find out which lockers are touched as each of the 20 students takes their turns.

Which locker numbers does the 3rd student touch?
Which locker numbers does the 5th student touch?
How many students touch locker 17? Explain or show how you know.
Which lockers are touched by only 2 students? Explain or show how you know.
Which lockers are touched by only 3 students? Explain or show how you know.
Which lockers are touched the most? Explain or show how you know.

If you have time: Which lockers are still open at the end of the game? Explain or show how you know.

Activity Synthesis

“Take a few minutes to share your answers with students in a different group.”
Invite students to share how they kept track of the number of touches for each locker.

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Lesson 6

The Locker Problem

Warm-up

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Advancing Student Thinking

Activity Synthesis

Activity 1

Standards Alignment

Building On

Addressing

4.OA.B.4

Building Toward

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Student Response

Advancing Student Thinking

Activity Synthesis

Activity 2

Standards Alignment

Building On

Addressing

4.OA.B.4

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Lesson Synthesis

Standards Alignment

Building On

Addressing

4.OA.B

Building Toward

Student Response

Advancing Student Thinking