Lesson 13 Slides | Fishtank Learning

Anchor Problems - Problem 1

Melissa orders business cards to promote her new website. She previews the cards online before placing her order. The scale image she sees online and the actual size of the cards are shown below.

Online scale image and actual size of Melissa

What scale factor was used to create the online image from the actual size of the business card?

Purpose

This task engages students in defining scale factor and calculating scale factor using corresponding side lengths, connecting to what students already know of the constant of proportionality. This problem builds conceptual understanding of scale factor as a multiplicative value applied to all dimensions and prepares students to draw scale images using a given scale factor in the next Anchor Problem.

Facilitation Guidance

Launch the problem by connecting the context to students' experiences with images and representations on screens CRMT Culturally Responsive Math Teaching . You might ask:
- Think about a time when you saw an image online or on a phone that was smaller or larger than a real-life object. What objects are shown in different sizes?
- Why might screen previews shrink or enlarge an item before purchase?
Invite students to notice and describe the differences between the online image and the actual business card, focusing on relative sizes rather than numbers initially. Be sure that students highlight that all dimensions seem to be affected proportionally when creating a scaled image.
Have students work in pairs to calculate the scale factor. If students need help getting started, encourage them to identify corresponding lengths first, and consider whether the scale factor will be greater than or less than 1. As you circulate, listen for students’ strategies to highlight in the debrief, and ask guiding questions, such as:
- Which lengths in the actual and online images correspond?
- What operation would allow you to go from the actual length to the online length?
- How does this number relate to the concept of a constant of proportionality?
Debrief the problem by inviting students to share their scale factors and the methods that they used to find them. Center students’ expertise throughout, prompting them to explain reasoning, verify each other’s ideas, and appreciate multiple correct approaches CRMT Culturally Responsive Math Teaching . Share student approaches that include:
- Writing an equation and solving for the unknown.
- Using understanding of the constant of proportionality and familiar methods such as proportions or tape diagrams.
- Articulating reasoning about why the scale factor is less than 1 when shrinking.
- Using a table or other organizational tool to record actual vs. online lengths and the scale factor, emphasizing the patterns across multiple measurements.
Stamp student understanding by defining scale factor and writing an agreed-upon definition on the board or an anchor chart for future reference. This statement should include the connection between scale factor and constant of proportionality, and show that the same factor multiplies all corresponding dimensions to create the scale image. You might extend the discussion by asking:
- How would the scale factor change if we calculated from the online image to the actual card?
- How could we use the scale factor to draw a new scale image of the business card?
- What percent of the actual length does the scaled image represent, and how does that connect to the scale factor?

Notes

Watch a video analysis of this Anchor Problem in the Unit Launch, Big Idea 2, Step 2 starting at 0:57.

Anchor Problems - Problem 2

Dzifa and her group are designing a game room for teenagers at the local library.

a. Dzifa wants to include a small square table for chess or checkers games. The table will be 3 ft by 3 ft. If the rest of the drawing has a scale factor of $$ \frac{1}{18 }$$, how big should she draw the table? Draw the table to scale in the space below, and label the lengths of the sides.

b. Part of the group’s design is a large rug in the center of the space in the shape of a hexagon. When Dzifa and her group redraw their design onto poster paper, the size of their model changes by a factor of 150%. Use the grid below to redraw the scale image of the rug.

Fishtank Learning curriculum image

Purpose

This problem asks students to draw scale images in two contexts: shrinking a real-world square table in a floor-plan sketch and enlarging a polygonal rug on a grid. Students apply scale factor as a single multiplicative value that changes all dimensions, and reason about what the factors $$ \frac{1}{18 }$$ and 150% do to lengths and overall size. The work reinforces connections between scale drawings and proportional relationships and prepares students to generate accurate, labeled sketches for their own design tasks in Anchor Problem 3 for the culminating project.

Facilitation Guidance

Launch the problem by orienting students to the design context and the two different scale factors. Briefly ask the class to predict, without computing yet, what each factor does to size:
- What do you notice about the two situations? What do you know about the numbers $$ \frac{1}{18 }$$ and 150%?
- What do you already know about the scale factors here?
- How do you expect the images to change based on the factors provided?
Invite one or two quick explanations about why one represents a shrink and the other an enlargement.
During work time, have students begin Part (a) independently, then check with a partner before moving to Part (b). As you circulate, listen for how students coordinate scale and units for the table. Many will convert 3 ft to 36 inches first, then apply the factor $$ \frac{1}{18 }$$ to get a 2-inch side; others may scale first $$\left ( 3\,\text{ft} × \frac{1}{18}=16\,\text{ft} \right ) $$ and then convert (16 ft = 2 inches).
Transition pairs to Part (b) and prompt them to agree on a practical method to enlarge the hexagon by 150% on the grid. Circulate and look for two common pitfalls to discuss during the debrief: stretching the figure in only one direction (changing width or height but not both) or changing the lengths and height but forgetting to keep the angles the same (or not knowing how to do that). Possible questions to ask:
- How did the horizontal and vertical distances change in your new drawing?
- Where can you see the idea of the constant of proportionality in your sketch?
- How can you check your work?
Collect student sketches (or prepare a few yourself) that show how students are applying the idea of scale, including some with misconceptions. Look for the following to display during the debrief:
- Hexagons stretched only horizontally.
- Hexagons stretched disproportionately in either direction.
- Hexagons with horizontal edges that are scaled appropriately, and a height that is scaled appropriately, but end points that didn’t move, resulting in changed angles.
- Hexagons scaled accurately by 150%.
Debrief the problem by first comparing solutions for Part (a). If some students left their units in feet, compare their thinking to students who converted to inches, and consider if the order of operations mattered (e.g. converting units before or after scaling). Then discuss Part (b) by showing student work and contrasting the different versions that they see. Ask:
- What's the same? What’s different?
- Which ones are true scale drawings? How can you tell?
- What happens if you only scale one dimension?
Allow students time to discuss, and press for specific evidence, such as pointing out how a segment that was 8 units is now 12, or how a single point moved compared to the center point of the image.
- This practice and discussion prepares students to use scale factor throughout the rest of the unit and also supports prerequisite understanding for dilations in 8th grade.
Close by articulating the throughline across both parts: a single scale factor multiplies every corresponding length in a scale image to preserve shape and angles while changing size. MP.7 Look for and make use of structure.
Extensions:
- Ask students to calculate the area of the table in real life and the area of the scale image and then compare the values.
- Provide other objects or furniture for students to scale by $$ \frac{1}{18 }$$ or images to scale by 150% (or another factor).

Notes

Students may benefit from rulers, grid paper, or tracing paper to mark reference lengths and ensure all sides are scaled consistently.
Students may misinterpret a percentage scale factor in one of two ways: (1) using the percentage as a whole number multiplier or (2) thinking that “scaled by 150%” means increasing the original length by 150% (thus resulting in a length equivalent to 250% of the original. Encourage students to convert percentages to decimals and reason about the resulting side lengths. Prompt students to reason about 150% and connect to the idea of “about 1 and a half times larger” if needed.
This Anchor Problem connects to the work students have starting on their culminating projects and the concepts they will apply in Anchor Problem 3 to their own design. If students do not make this connection naturally, it may help them to explicitly name the connection or to change the context slightly to match a group’s project from class.

Anchor Problems - Problem 3

When architects or designers create a scale drawing, they have to choose a scale factor that makes sense for the size of the project. A room, a playground, and a large park would all need different scales so the drawing can fit on paper but still show enough detail.

a. For your project, what will be the real-life outer dimensions for the community space? If helpful, explain or draw a sketch to show the perimeter.

b. What scale factor could make sense? Write one possible scale for your design. Explain why this scale is reasonable for your project.

c. Draw the outer edges of your community space to scale. As you start drawing, you may decide to adjust your scale factor. Show your calculations and explain your thinking.

Purpose

This problem is part of a connected sequence leading to the unit’s culminating project, where students create a scale drawing for a community space. In this Anchor Problem, students determine what scale factor to use for their scale model, reasoning about the need for precision, usefulness, and feasibility. Students must justify why a chosen scale is reasonable, which encourages proportional reasoning and precision in communication as they convert back and forth between inches, feet, and scale factors (without units). MP.2 Reason abstractly and quantitatively.

Setup

Optional Prepare and print a handout for students to be able to use to approximate values for the real-life objects in their projects. This handout could include benchmark values or research with approximate or average dimensions for some objects or features of spaces that students plan to design. See the Notes section for additional details.

Facilitation Guidance

Launch the problem by inviting students to imagine being architects or game designers who need to shrink something large, like a playground or even a video game world, so that it fits on paper or on a screen. Ask them to consider what would happen if the scale was too big or too small. CRMT Culturally Responsive Math Teaching
First, have students talk with their group about the process that they could use to determine a possible scale factor. Be sure to discuss the following:
- Students must estimate and decide on the anticipated real-life dimensions before they can calculate a scale factor.
  - It’s important that the real-life values are realistic. We don’t want to plan a playground that is 6 feet by 6 feet, if a single slide is larger than that.
  - Have students consider how to estimate these values. Students may suggest using benchmark measurements that they are familiar with or conducting research. If offering prepared benchmarks, hand them out or project them at this point.
- Students should name that the drawing must fit on a sheet of paper (or large poster paper).
- Students should practice writing the scale in different formats, and converting between units. This might look like: $$1\,\text{inch} = 10 \,\text{feet} → 1\,\text{inch} = 120 \,\text{inches} → \frac{1}{120 }$$ is the scale factor.
Once groups have a plan for what they need to know, provide work time for them to estimate their dimensions and test out possible scale factors. Circulate and look for groups who try out multiple scales and recognize when it leads to scale images that are either too large or small. Check that students are understanding the relationships between inches, feet, and the scale factors, and refer them back to the work in Anchor Problem 2 as needed. Possible questions include:
- How will you know if a scale factor is too big?
- How will you know if a scale factor is too small?
- What can you do to check if your scale factor is appropriate?
Supporting Access for Students

If students are struggling to manage the various steps, consider any of the following scaffolds:
- Provide a chart with several common scales, such as 1 in = 2 ft or 1 in = 5 ft, etc. Show how to calculate the scale factor based on the scales, and then prompt students to test different scales and compare.
- Give students graph paper with a pre-drawn rectangle that is a certain, given length. Have students reason about whether a rectangle at this scale would work and then determine what the scale factor is based on their estimated dimensions.
- Offer actual rulers or measuring tapes so that students can physically see what 1 inch, 5 inches, etc. will look like on their papers, and so they can measure the sizes of their papers.
Debrief the problem by highlighting different scale choices and the reasoning behind them. Ask groups to share initial scales that they tried that did not work and encourage students to ask questions of one another to promote shared understanding CRMT Culturally Responsive Math Teaching . Different groups may select different but equally reasonable scales, which creates a natural opening for comparison and discussion of efficiency, precision, and trade-offs. Anchor the conversation in the idea that scale choice is not arbitrary - mathematicians make decisions based on the purpose and the constraints. Possible debrief questions include:
- How did your group balance “fits on the page” with “still shows detail”?
- Which scales were not practical, and why?
- What if your design needed to fit on a poster board instead of regular paper? How would your scale change?
- What would you adjust if you wanted to include even more detail (like small plants, benches, or signs)?
- How might scale choice differ if you were drawing a room, a whole playground, or an entire city block?

Notes

This task builds directly from Anchor Problem 4 of Lesson 12, where students selected a community design project and sketched a possible design. In today’s lesson, students apply the skills from Anchor Problems 1 and 2 by selecting a scale that makes sense for their design. Their initial decision lays the groundwork for representing their designs accurately, and it creates a throughline across lessons as they see how scale functions both as a flexible tool and as a constraint.
Different groups may reasonably choose different scales, which opens opportunities to highlight mathematical practices such as modeling with mathematics MP.4 Model with mathematics. , reasoning quantitatively MP.2 Reason abstractly and quantitatively. , and comparing the efficiency of different representations. You may emphasize that choosing and adjusting a scale is an iterative process, and that students’ designs – and their mathematical reasoning – will continue to strengthen as they revisit and fine-tune their choices.
For students who need support estimating the real-life outer dimensions for their space, allow them to conduct their own research online, or give them benchmark values to use as they plan.
- If you choose to provide benchmarks, consider what community spaces students are designing and research the approximate or average dimensions for some objects or features of those spaces. For example, if students are designing a playground, provide the length of a swingset, slide, bicycle, or other relevant objects; for a community library space, provide the sizes for tables, armchairs, bookshelves, etc.
- Model how students might think through these estimations, or ask groups to share their explanations out loud. Possible reasoning might sound like: “I knew that I wanted to have a climbing structure with multiple slides, big enough for multiple kids to play at the same time. If a single slide without the climbing structure is 8 feet long, and I want at least two slides, with several feet of climbing and stairs in between, then I might want the playground area to be 25 or 30 feet long.”
Have students add their work from this Anchor Problem to their project log to use and refine in future lessons.
This Anchor Problem and the entire culminating project connects to the themes of Fishtank ELA Grade 7, where students explore identity, belonging, and community. You may choose to highlight these themes by surfacing how space, design, and representation can influence whether people feel welcome or excluded. This supports students in making cross-disciplinary connections and positions mathematics as a tool for thinking about real issues of identity and belonging. ELA connection

Fishtank Learning

7th Grade Math
Unit 5: Percent and Scaling
Lesson 13

Objective

Warm Up

Anchor Problems - Problem 1

Anchor Problems - Problem 2

Anchor Problems - Problem 3

Problem Set

Target Task

Fishtank Learning

7th Grade MathUnit 5: Percent and ScalingLesson 13

Objective

Warm Up

Anchor Problems - Problem 1

Anchor Problems - Problem 2

Anchor Problems - Problem 3

Problem Set

Target Task

7th Grade Math
Unit 5: Percent and Scaling
Lesson 13