Lesson 1 Slides | Fishtank Learning

Anchor Problems - Problem 1

You have the following collection of shapes:

Group of shapes: 10 circles, 4 squares. Ratio of circles to squares: 10:4.

These shapes can be sorted into different groups. For example, you can make a group of squares and a group of circles. We can use ratios to describe how these groups are related; for example, the ratio of squares to circles is 4 to 10, or 4:10.

Sort the shapes into as many different groups as you can think of. Write statements that describe the association between different groups using a ratio.

Purpose

This Anchor Problem invites students to make sense of and define the idea of a ratio by comparing quantities in a visual, low-floor context. Students explore multiple ways to group the same set of shapes and describe relationships between those groups using emerging ratio language. This problem establishes norms for mathematical discussion, sense-making, and precise communication that will be built on throughout the year.

Setup

Make enough copies of the Ratio Shapes Handout so that each group can have a half-sheet. Cut out shapes or provide students with scissors to cut the shapes out themselves at the start of the activity.
- Alternatively, you may use a variety of small physical objects, such as colored paperclips or geometric shapes. To support whole class discussion, ensure all groups receive the same numbers in each group and subgroup to be able to compare ratios.

Facilitation Guidance

Launch the problem by displaying the collection of shapes and prompting students to notice and wonder. Ask:
- What’s similar amongst some of the shapes? What’s different?
- What are some different groups that you could create?
- How might you compare two of those groups?
Allow students to consider and discuss some of what they already know about comparing different sized groups, and notice if multiplicative relationships show up. For example, students may say, “There are twice as many shaded circles as there are shaded squares.”
- Have you heard of the term “ratio” before? What do you think it might mean?
Introduce the task and provide the following sentence frames for students:
- The ratio of _____ to _____ is _____ to _____.
- The ratio of _____ to _____ is _____:_____.
- For every _____ _____, there are _____ _____.
- There are _____ _____ for every _____ _____.
Have students work in pairs or small groups with physical shapes (or a visual representation) to create and record as many different groupings and comparisons as they can.
- Consider pausing briefly to model what productive partner talk can sound like, supporting students in building strong habits at the start of the year. For example: “I noticed…,” “Can you explain why…?,” or “I agree/disagree because…”.
- Emphasize that there is not just one correct way to solve the problem — this is about exploring possibilities and making sense of relationships. This is an important early opportunity to establish that mathematical thinking includes exploring, revising, and explaining ideas — not just getting answers quickly. Encourage students to ask questions of one another and build on each other’s observations.
As students work, circulate and listen for informal language describing the comparisons as well as early attempts to use the sentence frames. Support students in connecting their ideas to the provided sentence frames:
- How could you say that using “the ratio of _____ to _____”?
- What does your 2 represent? What does your 6 represent?
- If you switch the order, how does the statement change?
- Ask your partner for a specific ratio and then have them respond in two or more ways.
Debrief the problem by bringing the class together to share different groupings and ratio statements. Center student voice by validating multiple ways of grouping and describing relationships, reinforcing that there is not just one “right” way to see the math. CRMT Culturally Responsive Math Teaching
- As students share, reinforce that different perspectives can help the class notice new mathematical relationships. Position students as contributors whose thinking helps shape the class understanding of ratio.
- Highlight examples where students used the same groups but described them differently (e.g., 2:6 vs. 6:2) to emphasize the importance of order and meaning.
- As a class, co-construct a working definition of ratio based on student thinking (e.g., “A ratio compares two quantities and tells how much of one there is compared to another”).

Notes

This first Anchor Problem is as much about establishing classroom norms as it is about content. Prioritize routines like think time, partner talk, and sharing and listening to reasoning as a class. Keep the focus on exploration and language development, rather than correctness or efficiency. Students will revisit and refine the definition and use of ratios throughout the unit.
Watch a video analysis of this Anchor Problem in the Unit Launch, Big Idea 1, Step 2 starting at 0:49.

Anchor Problems - Problem 2

Some teachers and students are playing games during recess.

16 students play basketball
6 teachers play basketball
8 students jump rope
2 teachers jump rope

After recess, Mr. Hill shares this information with his class. He asks his students to compare the students and teachers playing different games.

Mika said: “Eight more students played basketball than jumped rope.”

Chaska said: “For every student who jumped rope, two students played basketball.”

Mr. Hill said, “Mika compared the students by looking at the difference, and Chaska compared the students using a ratio."

a. Compare the number of teachers who played basketball and jumped rope using the difference. Write your answer as a sentence as Mika did.

b. Compare the number of teachers who played basketball and jumped rope using a ratio. Write your answer as a sentence as Chaska did.

c. Compare the number of students who played basketball to the number of teachers who played basketball using a ratio. Write your answer in two different ways.

Purpose

In this problem, students compare quantities using both differences and ratios, building on prior understanding of additive and multiplicative thinking. Students learn to distinguish between these two types of comparisons and express each using precise language.

Facilitation Guidance

Launch the problem by presenting the scenario and the two student statements. Ask:
- What do you notice about how Mika and Chaska are comparing the students?
- What feels similar or different about their thinking?
Invite a brief partner discussion, then surface the idea that Mika is describing a difference while Chaska is describing a multiplicative relationship between quantities. Let students explain what they think each comparison means in their own words.
- Avoid correcting language too quickly during initial discussion. Instead, use students’ informal phrasing as a resource for clarifying meaning together.
Have students work independently or with a partner to complete Parts (a), (b), and (c). Encourage them to refer back to Mika’s and Chaska’s statements as models for how to structure their own comparisons.
As students work, circulate and support sense-making with prompts like:
- How did you determine the difference in Part (a)?
- What quantities are you comparing in Part (b)? How do you know?
- How can you say that ratio in another way?
- How does the order of the quantities affect your statement?
Supporting Access for Students

If helpful, encourage students to represent the situation visually (e.g., quick sketches or grouping) or even physically model the groups to make the relationships more concrete.
Debrief the problem by focusing on the distinction between additive and multiplicative comparisons.
- Start by revisiting Mika’s and Chaska’s statements and asking: How are these two types of comparisons different? When might each be useful?
- Then invite students to share their responses:
  - Who wrote a difference statement for Part (a)? What does it tell us?
  - Who wrote a ratio statement for Part (b) in a different way?
  - What are the two different ways you described the ratio in Part (c)?
- Encourage students to compare reasoning rather than evaluate whose response is “best.” Early in the year, emphasize that mathematical discussions are about making ideas clearer together.
Highlight that differences describe “how much more or less”, while ratios describe “how many times as much” or “for every” relationships. Emphasize precise language and the corresponding order of quantities, especially when comparing students to teachers versus teachers to students.

Notes

Students may need support connecting Chaska’s statement to the information in the bullets since he does not use the numbers “8” and “16.” A diagram may be helpful to make the connection between the two ways this ratio is represented. Diagrams will be explored more in the next lesson. It may also be possible to have students physically model the ratios in class, standing in front of the room.
Watch a video analysis of this Anchor Problem in the Unit Launch, Big Idea 1, Step 2 starting at 2:10.

Anchor Problems - Problem 3

Yu-Mi and her family are preparing cheonggukjang-jjigae, a Korean stew made with a fermented soybean paste (cheonggukjang). She mixes 2 cups of rice water with 6 tablespoons (T) of paste.

Two cups of white paint to six tablespoons of blue paint.

Write as many ratio statements as you can to describe this portion of Ahn Yu-Mi’s recipe.

Purpose

In this problem, students describe a real-world situation using ratio language, attending to units and the order of quantities. Students build precision by clearly communicating what each number in a ratio represents and what the relationship means in context. MP.6 Attend to precision.

Facilitation Guidance

Launch the problem by inviting students to think about cooking and recipes as a familiar context.
- Have you ever helped cook something or seen someone follow a recipe? What kinds of measurements do people use?
- Why do the amounts matter? What happens if you change them?
Invite students to share examples of recipes from their own lives or communities, and briefly collect a few ingredients and potential measurements on the board. CRMT Culturally Responsive Math Teaching
- Create space for students to pronounce ingredients, dishes, or cooking terms from their own communities in ways that feel comfortable and respectful. This can help establish the classroom as a place where students’ experiences and languages are valued resources for learning.
- If students are unfamiliar with the recipe or pronunciation, frame curiosity and respectful questions as part of learning rather than something to avoid.
Have students begin by independently annotating the situation to identify the quantities, units, and how they are related:
- What do you notice about the amounts?
- What are the units of each? Does that matter with ratios?
- How are these two quantities connected?
Then have students work independently for a few minutes to brainstorm some initial drafts of ratio statements. Encourage use of sentence frames as needed, while also allowing students to experiment with their own phrasing. As students work, circulate and look for:
- Attention to units (cups vs. tablespoons)
- Maintaining or switching the order within a ratio statement.
- Informal vs. increasingly precise language
Supporting Access for Students

Consider having students act out or visualize the quantities (e.g., with measuring tools or additional discrete drawings) to ground the ratios in concrete meaning.
Move students into partners or small groups to discuss and share their ideas. Support students in refining their thinking, as needed:
- What does each number represent in your ratio?
- How do the units help make your meaning clear?
- What would your statement mean if you reversed the order?
Some students may notice that the relationship can be described in different but consistent ways (e.g., per 1 cup). This line of reasoning previews equivalent ratios, which students will spend a significant amount of time on throughout the rest of the unit.
In the debrief, discuss a variety of examples and nonexamples of ratio statements that show the comparison from the diagram and recipe. Potential nonexamples to introduce if they do not arise naturally:
- “The ratio is 2:6.”
- “There are 2 cups for every 6.”
- “The ratio of rice water to paste is 6:2.”
Invite students to critique:
- What is unclear or missing?
- How could we revise this to make the meaning precise?
- Does this match the situation? How do you know?
Reinforce that critiquing a statement means examining the mathematical clarity of the idea, not criticizing the person who shared it.
Extension: Provide additional recipes to students, or additional information about the recipe for cheonggukjang-jjigae. Ask students to sketch a diagram to represent two measurements and brainstorm additional ratio statements to match them.

Notes

You may wish to share a short pronunciation clip with students to support accurate reading and discussion of cheonggukjang-jjigae. Here is one pronunciation. Hearing the term spoken aloud can help students build confidence when engaging with unfamiliar terms. CRMT Culturally Responsive Math Teaching
This problem is an early opportunity to establish expectations around precision in mathematical language, especially naming quantities and including units when describing ratios. Emphasize meaning over notation — students should understand what a ratio is describing before focusing on symbolic forms.
Encourage students to generate multiple statements to reinforce flexibility, but avoid requiring equivalent ratios at this stage. If students naturally notice patterns such as “for every 1 cup,” you may briefly highlight this as an emerging structure, without formalizing it.

Fishtank Learning

6th Grade Math
Unit 1: Understanding and Representing Ratios
Lesson 1

Objective

Warm Up

Anchor Problems - Problem 1

Anchor Problems - Problem 2

Anchor Problems - Problem 3

Problem Set

Target Task

Fishtank Learning

6th Grade MathUnit 1: Understanding and Representing RatiosLesson 1

Objective

Warm Up

Anchor Problems - Problem 1

Anchor Problems - Problem 2

Anchor Problems - Problem 3

Problem Set

Target Task

6th Grade Math
Unit 1: Understanding and Representing Ratios
Lesson 1