Two Equations for Each Relationship
5 min
Teacher Prep
Setup
Narrative
This Warm-up prompts students to compare four geometric patterns. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three images that go together and can explain why. Next, tell students to share their response with their group, and then together find as many sets of three as they can.
Student Task
Which three go together? Why do they go together?
Sample Response
Sample responses:
A, B, and C go together because:
- The ratio of the number of blue tiles to the number of yellow tiles is 2:3.
- They have an even number of blue tiles and an even number of yellow tiles.
- You can use vertical cuts to make groups of 5 tiles (with 2 blue and 3 yellow in each group).
A, B, and D go together because:
- They have both colors on the top and on the bottom.
- They have some blue tiles on the bottom.
- They have some yellow tiles on the top.
A, C, and D go together because:
- The pattern ends (on the right) with blue on top and yellow on bottom.
- They have fewer than 15 total tiles.
B, C, and D go together because:
- The pattern starts (on the left) with blue on top and yellow on bottom.
- Each image has 6 yellow tiles on the bottom.
Activity Synthesis (Teacher Notes)
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “row,” “group,” “partition,” “even,” “odd,” “horizontal,” “vertical,” “ratio,” or “area,” and to clarify their reasoning as needed. Consider asking:
- “How do you know . . . ?”
- “What do you mean by . . . ?”
- “Can you say that in another way?”
If time allows, invite 2–3 students to briefly share what they notice all of the figures have in common. For example:
- They are all rectangles composed of smaller rectangles.
- Half their area is blue and half is yellow.
- They all have an even number of total tiles.
The purpose of this concluding share out is to reinforce the importance of using precise terminology. For example, saying “the ratio of blue to yellow” is not specific enough. The ratio of blue area to yellow area is 1:1 for all of the figures, while the ratio of blue pieces to yellow pieces is either 2:3 or 1:3.
Standards
- 6.RP.1·Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. <em>For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."</em>
- 6.RP.A.1·Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. <span>For example, “The ratio of wings to beaks in the bird house at the zoo was <span class="math">\(2:1\)</span>, because for every <span class="math">\(2\)</span> wings there was <span class="math">\(1\)</span> beak.” “For every vote candidate A received, candidate C received nearly three votes.” </span>
15 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Two Equations for Each Relationship
5 min
Teacher Prep
Setup
Narrative
This Warm-up prompts students to compare four geometric patterns. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three images that go together and can explain why. Next, tell students to share their response with their group, and then together find as many sets of three as they can.
Student Task
Which three go together? Why do they go together?
Sample Response
Sample responses:
A, B, and C go together because:
- The ratio of the number of blue tiles to the number of yellow tiles is 2:3.
- They have an even number of blue tiles and an even number of yellow tiles.
- You can use vertical cuts to make groups of 5 tiles (with 2 blue and 3 yellow in each group).
A, B, and D go together because:
- They have both colors on the top and on the bottom.
- They have some blue tiles on the bottom.
- They have some yellow tiles on the top.
A, C, and D go together because:
- The pattern ends (on the right) with blue on top and yellow on bottom.
- They have fewer than 15 total tiles.
B, C, and D go together because:
- The pattern starts (on the left) with blue on top and yellow on bottom.
- Each image has 6 yellow tiles on the bottom.
Activity Synthesis (Teacher Notes)
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “row,” “group,” “partition,” “even,” “odd,” “horizontal,” “vertical,” “ratio,” or “area,” and to clarify their reasoning as needed. Consider asking:
- “How do you know . . . ?”
- “What do you mean by . . . ?”
- “Can you say that in another way?”
If time allows, invite 2–3 students to briefly share what they notice all of the figures have in common. For example:
- They are all rectangles composed of smaller rectangles.
- Half their area is blue and half is yellow.
- They all have an even number of total tiles.
The purpose of this concluding share out is to reinforce the importance of using precise terminology. For example, saying “the ratio of blue to yellow” is not specific enough. The ratio of blue area to yellow area is 1:1 for all of the figures, while the ratio of blue pieces to yellow pieces is either 2:3 or 1:3.
Standards
- 6.RP.1·Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. <em>For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."</em>
- 6.RP.A.1·Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. <span>For example, “The ratio of wings to beaks in the bird house at the zoo was <span class="math">\(2:1\)</span>, because for every <span class="math">\(2\)</span> wings there was <span class="math">\(1\)</span> beak.” “For every vote candidate A received, candidate C received nearly three votes.” </span>