This learning module is intended for a middle school mathematics classroom. Students will construct their understanding of proportional relationships using a Project Based Learning approach along with Desmos.com.
Project Based Learning (PBL), ratio, proportion, unit rate
This module is a sixth grade ratio and proportion Project Based Learning Task. Project Based Learning (PBL) is “a teaching method in which students learn by actively engaging in real-world and personally meaningful projects” (PBLworks, 2020). These learning tasks require that the students think critically, problem solve, communicate, collaborate, and construct their knowledge through the seven essential project design elements: Challenge Problem or Question, Sustained Inquiry, Authenticity, Student Voice and Choice, Reflection, Critique and Revision, and Public Product (PBLworks, 2020).
Through PBL, students will be encouraged to partake in a productive struggle in order to help construct their knowledge. Great teachers are often missing the PBL that allows students an opportunity to lead themselves through inquiry in order to demonstrate their understanding (Fancher & Norfar, 2019). Mathematics teachers often feel that they do not have the adequate time to partake in PBL due to the need to an emphasis to raise test scores or need to follow the district mandated (Larmer, 2019). Students are able to develop both conceptual understanding and procedural fluency through the incorporation of rich mathematical tasks (Fancher & Norfar, 2019). In order to start to implement PBL models into the classroom, teachers should consider incorporating shorter projects that are aligned to the standards and may need to begin with more of a problem-based approach (Larmer, 2019). The following chart helps to decipher the difference between a true project-based approach versus problem-based approach. Problem-based learning lends itself more to the mathematics classroom with ill-structured problems being presented with students generating that they know and need to know, possible solutions, and formulate learning (Larmer, 2013).
Both aspects of PBL framework focuses on John Dewey’s work that supports the idea of constructivism. Students are intended to work collaboratively with their peers to construct their understanding. The activities have student-led aspects that allow them to construct their own knowledge as well as opportunities to reflect on their understanding in order to make necessary changes to gaining a deeper, more rich understanding of the concepts.
Being a sixth-grade math teacher, it is important for me to continue to evolve my teaching practice. Although our curriculum incorporates rigorous and challenging mathematic tasks, I would like to incorporate a series that I call “Math in Action”. These “Math in Action” tasks are intended to follow more of the PBL framework in order for students to be able to construct more of their own understanding. The following module focuses on students gathering an in-depth understanding of proportional relationships, one of the more emphasized standards in the sixth-grade curriculum, through a PBL task. The students will be writing a proposal for a local Farmer's Market Lemonade vendor in order to help them make business decisions when purchasing ingredients for their lemonade stand. The module will incorporate a combination of material that has previously been used within my sixth-grade classroom as well as material that is newly develop for the “Math in Action”, or PBL framework, incorporation. Below is an excerpt from a previous work that stresses this importance (Sayles, 2020):
“Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well (NCTM, 2000). Students are introduced to important mathematical ideas with worthwhile tasks in order to engage and challenge the students intellectual capabilities (NCTM, 2000). Teachers know that students need to be presented with challenging tasks in order to grow; however, many teachers still revert back to a more “traditional” means to teaching math where more explicit instruction is given. There is a need to do better in our math classrooms in order to encourage our students to become logical thinkers with the means to reason mathematically.
Productive struggle is an educational phenomenon that encourages students to “muddle” through a given task in order to construct their own understanding and learning, rather than being conditioned through repetition of the same types of problems (Warshauer, 2014). Productive struggle allows students to advance their conceptual understanding and apply what they learn to novel problems (Sengupta-Irving & Agarwal, 2017). It refers to the “student’s ‘effort to make sense of mathematics, to figure something out that is not immediately apparent’ (Warshauer, 2014). Incorporating more tasks within our math curriculum in order to encourage this productive struggle in the classroom will enhance our students’ learning. “
This module is intended to provide students with rich and meaningful mathematics tasks that will enhance the learning environment as well as deepen their understanding of proportional relationships through actively engaging in authentic tasks that are intended to create a productive struggle for them to take part in. The following statistics support the PBL model and the reasoning for this modul (Merritt, Mi, Rillero, & Kinach, 2017).
KeyWords: Project Based Learning (PBL), ratio, proportion, unit rate
This learning module is intended for use in a sixth-grade middle school math course.
Students will have prior knowledge of fractions and decimals. This module will connect their understanding of fractions to ratios and proportions. Students will construct an understanding of proportional relationships through PBL and inquiry- based tasks. Students will use the proportional relationships to help to identify the better buys.
NCTM Learning Principles:
Common Core State Standards:
Understand ratio concepts and use ratio reasoning to solve problems.
CCSS.MATH.CONTENT.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
CCSS.MATH.CONTENT.6.RP.A.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.
CCSS.MATH.CONTENT.6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems
CCSS.MATH.CONTENT.6.RP.A.3.A
Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
CCSS.MATH.CONTENT.6.RP.A.3.B
Solve unit rate problems including those involving unit pricing and constant speed.
CCSS.MATH.CONTENT.6.RP.A.3.D
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
This module will require 3 weeks to complete.
Learning Objective: “I can write ratios and find equivalent ratios.”
You will be working through a Desmos Activity. The teacher will provide you with the access code.
Khan Academy Introductory videos to clarify understanding:
Khan, S. (2014, March 25) Introduction to Ratios. Retrieved from www.khanacademy.com from Khan Academy
Khan, S. (2017, October 27). Equivalent Ratios for Recipes. Retrieved from www.khanacademy.com from Khan Academy
Go to Khan Academy – Practice: Equivalent Ratios
Comment: What are equivalent ratios? Explain in your own words.
Update: Give examples of other ratios or rates that you encounter is everyday life. Create a problem with equivalent ratios for your classmates to solve.
Materials:
This introductory lesson will introduce students to the concept of ratios as well as equivalent ratios using Desmos: Balloon Float activity along with Kahn Academy.
Warm-Up:
Ask students what they notice about the fractions. Begin to connect the warm-up to the idea of part-to-part and part-whole ratios. This will link ratios to the prior knowledge of fractions. Students should start to make connections that equivalent fractions are similar to equivalent ratios. The students knowledge of multiples will help them create equivalent ratios.
Teacher: What if I asked you how many students where on stability balls to students sitting on chairs?
Begin to write the responses on the board and discuss the concept of ratios. Be sure to touch on how to write them in word, with a colon, or in fractional form.
Continue to address the other warm-up scenarios as well.
Teacher: Now that we have been introduced to ratios. Let’s explore them further.
Students will work through the Desmos learning in order to construct their own knowledge and understanding of ratios and proportions prior to working through a Khan Academy Equivalent Ratios.
Teacher Note: The teacher will need to create free accounts at teacher.desmos.com as well as a free teacher and student account at Khan Academy.
Desmos is an interactive tech tool that allows students to learn the concepts through the constructivism approach to learning. Students will construct their understanding of what a ratio is through this interactive lesson. In the Balloon Float activity, students will watch a video that will pique their curiosity about a man that is actually floating due to a large amount of balloons.
Video:
CBC News Full Video – Shorter clip is used in lesson.
Students will then work through a series of problems that involve the mass of an object, person, or animal in order to identify how many balloons would be necessary to make that object float. The students will construct an idea of equivalent ratios through the interactive lesson.
Teacher: Let’s get started.
Provide the necessary Code to access the Desmos Activity.
Be sure to facilitate conversation around the ratios and proportions that are being created. Allow students to share how they are coming to the correct conclusion for the number of balloons necessary to make the object, person, or animal float.
Teacher: What did you notice? How were you solving each problem?
Be sure to discuss that the students were starting to identify equivalent ratios.
The teacher will reinforce the concept using Khan Academy. The students may watch instructional videos about introductions to ratios as well as equivalent ratios. Students will work through Khan Academy Practice Problems for Equivalent Ratios in order to reinforce their understanding. Khan Academy’s program allows students to learn through scaffolded hints, should they struggle with the concept. The practice problems allow for a behaviorism model of learning with immediate feedback and praise when certain problems are answered correctly.
Learning Objective: “I can think critically about a situation and ask appropriate questions.”
Project Based Learning Question: How can you help your local Farmer’s Market Lemonade stand make a profit on their lemonade without wasting money?
Situation/Problem: The local Farmer’s Market Lemonade is a must buy item. The owners have hired you to help make some important business decisions. They are having trouble buying the correct number of lemons for their Sunday Farmer’s Market stand. Some weeks they waste lemons that do not go to use and other weeks they have to close the stand early because they do not have enough lemons. You have been hired as a consultant to the owners in order to help make business decisions about what lemons to buy and how many? They want to be sure that they are getting the best profit possible.
You and your consulting team will create a proposal for the owners in order to provide important information to ensure success.
Comment: As a consulting group, what questions do you have for your client? What things must you consider as the consulting group?
Update: Research any information that would be useful for this consulting task. Inform your classmates of some pertinent information that would be useful to know or consider when going about this task.
***Idea adapted and created from Project-Based Learning in the Math Classrooms (Fancher and Norfar, 2019).***
Materials:
This is the introduction into the PBL task that will allow students to learn about ratios and proportions through a constructivism approach to learning or creating meaning of the concept of proportions by doing. The students should be placed in a group of 3-4 students in order to “consult” for the local Farmer’s Market Stand. Students should have clear collaborative norms establish for working in their collaborative groups. If no norms are not yet established, take time to establish group norms prior to moving forward with the task. The authenticity of the problem presented is intended to pique student interest (Fancher & Norfar, 2019).
Teacher: Today, you and your collaborative group are consulting for our local Farmer’s Market Lemonade Stand. You are being hired in order to provide a business proposal on how to increase their profit while ensuring that they are making appropriate buying decisions when buying lemons.
Introduce the situation and problem.
Below are some possible proposal templates:
You may choose to share a template /provide guidelines, allow students to google on their own to research a proposal outline, or allow the students to create their own versions of a proposal presentation. A true PBL project would allow the students to find the answer to what a proposal may be or create their own idea of a proposal.
Teacher: You and your consulting team need to think through some of the information that is still needed in order to properly write a proposal for the owners of the lemonade stand. They have provided you with the recipe. Begin to collaborate together to ask yourself: What do you notice? What do you wonder? What information do we still need?
Set timer for 10-15 minutes in order for collaborative groups to have an opportunity to think through the information that would still be needed.
Teaching Tip: Use a collaborative technology tool such as jamboard, padlet, or Teams/Google Classroom in order for each collaborative group to share their wonderings. They may also include ways to research the information still needed.
Discuss as a class some of the information that was noticed and things that have been brought up as wondering questions.
Questions that may be considered:
As time permits, students should brainstorm ways to come to the answer for these different questions. The teacher may help facilitate contacting the local Farmer’s Market Lemonade vendor. As a follow up to today’s discussion, students should begin searching for answers.
Teacher Tip: Create collaborative group roles in order to provide jobs to each student.
Learning Objective: “I can identify the average ounces per lemons and write a ratio to represent the situation.”
Problem: How many lemons are needed to make one batch of the lemonade?
Comment: What additional questions do you have, if any? Help answer any questions for other groups as well.
Update: Make a model that represents your findings that could possibly be used as part of your group’s proposal. How can you demonstrate your findings to your client?
Materials:
This activity allows the students to construct their understanding and problem solve and authentic real-life situation. Students will actually take part in an experiment to collect their information.
Teacher: Today we are going to conduct experiments to gather information about the lemons that your client could use to make their lemonade.
Allow students time to collect data. Each group should juice one smaller lemon from the 3 lb. bag and a larger lemon sold separately. You may choose to have the students find averages from repeating the experiment over multiple trials. Students should create a rate of the number of ounces of lemon juice per lemon. (20 - 30 minutes)
Teaching Tip: Use a collaborative technology tool such as jamboard, padlet, or Teams/Google Classroom in order for each collaborative group to share their findings.
Discuss as a class each group’s findings.
Teacher: What were some similarities and differences in rate? Why might this be?
Determine a class average ounce of lemon juice per lemon.
Students will work together in order to determine how many of each type of lemon are needed to create one batch of lemonade. Remind students to clearly show their work using proportions.
Remember, students will ultimately need to decide on or be given how many people they will need to serve at the Farmer’s Market in order to determine how many batches of lemonade will need to be made. In order to do this, the students must decide or inquire about what size each serving is. This is all information that is going to be in an upcoming lesson; however, it may be addressed within this lesson or considered by students.
Learning Objective: “I can represent a ratio on double number lines and charts.”
You will be working through a Desmos Activity. The teacher will provide you with the access code.
Watch the following Khan Academy Videos for clarification:
Khan, S. (2017, November 8). Ratios and Double Number Lines. retrieved from www.khanacademy.com from Khan Academy
Khan. S. (2017, August 11). Ratio Problems Tables Examples. retrieved from www.khanacademy.com from Khan Academy
Go to Khan Academy - Practice: Create double number lines, ratios with double number lines, relate double number lines and ratio tables, ratio tables
Comment: Explain how to use a chart or double number line to represent a proportional relationship.
Update: Using the information gathered about our Farmer’s Market Lemonade, create a chart and double number line to represent the ratio of lemon juice to lemons and lemons to batches of lemonade.
Materials:
Students will work through the Desmos: Nana’s Chocolate Milk learning in order to construct their own knowledge and understanding of multiple representations of proportions through charts and double numbers lines.
Desmos is an interactive tech tool that allows students to learn the concepts through the constructivism approach to learning. Students will construct their understanding of double number lines as well as ratio table through working through the Desmos tasks. In the Nana’s Chocolate Milk, students will watch a video that will pique their curiosity:
Meyer, D. (2015, March 19). Desmo's Nana's Chocolate Milk Introductory Video retrieved from www.desmos.com
Students will then work through a series of problems that involve the trying to fix the mix up for mixing the chocolate milk incorrectly. The students will construct an understanding of double number lines and charts through while revisiting creating equivalent ratios.
Teacher: Today we are going to gain an understanding of ways to represent ratios. We will begin with a Desmos Activity.
Provide the necessary Code to access the Desmos Activity.
Be sure to facilitate conversation around the interaction with double number lines as well as ratio charts.
The teacher will reinforce the concept using Khan Academy. The students may watch instructional videos about introductions to ratios as well as equivalent ratios. Students will work through Khan Academy Practice Problems for Create double number lines, ratios with double number lines, relate double number lines and ratio tables, ratio tables in order to reinforce their understanding. Khan Academy’s program allows students to learn through scaffolded hints, should they struggle with the concept. The practice problems allow for a behaviorism model of learning with immediate feedback and praise when certain problems are answered correctly.
Learning Objective: “I can work collaboratively to figure out the better buy with the lemons and sugar.”
Problems:
Your client only likes to use Domino sugar. They buy the following bags wholesale:
Comment: What were your successes today? What was challenging today? What questions do you still have?
Update: Find a recipe and create a proportion problem for your classmates to solve. Answer your group members' updates.
(2 Days)
Materials:
The students will continue to construct their knowledge of ratios and proportional reasoning by working collaboratively to continue to gather information for their client’s proposal.
Teacher: What are some things that you need to consider about the client’s Farmer’s Market Lemonade? What answers do you still need to answer?
Teacher Tip: Use padlet to have each group share.
If students have yet to make the connection about the size of the cups used, the teacher should start to make this connection for students. (You could choose to have students find cups online or may suggest a serving size such as 12 oz.)
Also, students need to make a connection to the number of people that will buy lemonade. If students have interviewed an actual company, they should continue in that route. If not, make a suggestion such as 250-300 people. (A range is given to allow students to productively struggle with this task.)
Teacher: You and your consulting team will need to decide which type of lemons are best for your client to purchase as well as a mathematical reasoning why? If the client sells 250-300 – 12 oz. cups of lemonade (optional scenario), which type of lemon do you recommend? How many batches will you need to make for the
Students should collaborate with their teams in the task of trying to figure out which types of lemons are the best to purchase and how many. They also will calculate the cost including information about the sugar needed as well. Students will find the cost per batch as well as the cost of all the batches needed for the weekend. This part of the task is intended to produce a productive struggle in which the collaborative group has opportunities to "muddle" through the task to come to a solution.
Any additional time should be spent starting to create the group proposal and presentation.
Learning Objective: “I can use unit rates to solve proportions.”
You will be working through a Desmos Activity. The teacher will provide you with the access code.
Watch the following Khan Academy Video for Clarification:
Khan, S. (2010, August 11). Unit Rates Problems. retrieved from www.khanacademy.com from Khan Academy
Go to Khan Academy - Practice: Unit Rate Problems
Comment: What is a unit rate? How may it be useful?
Update: Create your own unit rate problems and work through the problems that your group members post.
Materials:
Students will work through the Desmos: Click Battle learning in order to construct their own knowledge and understanding of unit rate and how to use unit rates to solve problems.
Desmos is an interactive tech tool that allows students to learn the concepts through the constructivism approach to learning. Students will construct their understanding of double number lines as well as ratio table through working through the Desmos tasks. In the Click Battle, students will work through a series of tasks to figure out how to use the unit rate to problem solve.
Students will then work through a series of problems that involve unit rate.
Teacher: Today we are going to gain an understanding of unit rates. What do you think a unit rate may mean? Share with a partner. We will begin with a Desmos Activity and you should be trying to define what a unit rate is.
Provide the necessary Code to access the Desmos Activity.
Facilitate conversations about how the students are able to use the unit rate to solve the situations in Desmos.
The teacher will reinforce the concept using Khan Academy. The students may watch instructional videos about unit rates. Students will work through Khan Academy Practice Problems for Unti Rate Problems.
in order to reinforce their understanding. Khan Academy’s program allows students to learn through scaffolded hints, should they struggle with the concept. The practice problems allow for a behaviorism model of learning with immediate feedback and praise when certain problems are answered correctly.
Learning Objective: “I can work collaboratively on the 3-Act math task to come to a mathematically reasonable solution.”
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Comment: How do you determine the better buy? Give your mathematical reasoning. What were the successes of you and your group? What could you have done better?
Update: Research your own better buy options. Create your own problem for your classmates to solve. Respond to your groupmates’ problems by solving them.
Materials:
In this activity students will be expected to collaboratively, problem-solve to figure out the better buy.
The task is intended to allow the students to productively struggle with an authentic math concept in order to problem-solve and come to a solution. This is an important aspect of the mathematics classroom and supported by NCTM principles.
In Act 1, students will watch a video about candy being gone and going to the store to buy more.
Teacher: What do you notice? What do you wonder?
Allow the students to work within their groups to write down information that they notice and things that they wonder about. Record all of the students answers.
Teaching Tip: Use a collaborative technology tool such as jamboard, padlet, or Teams/Google Classroom in order for each collaborative group to share what they notice and what they wonder.
Students will investigate which sized Sour Patch Kids in the Better Buy.
In Act 2, students will be presented with the pounds or ounces of each Sour Patch bag along with the prices. Note, one bag is given in pounds and will require students to make the connection to need to convert to ounces first. Conversion may be given, or you may allow students to problem solve to figure out the conversion rate (typically students have conversions available via state testing reference sheets and/or assignment notebooks).
Allow students time to go about solving the focus question.
Students should write their mathematical explanation prior to creating a 2- minute video (or presentation) about what their solution is and how they came to that solution.
Watch the students’ videos/ listen to presentations. Be sure to encourage students to explain their mathematical reasoning. Discuss the similarities and differences between each group’s responses.
Reveal Act-3 that answers the focus question.
Allow time for students to reflect on the task.
Learning Objective: “I can create a proposal and present with reasonable explanation.”
Complete the following survey in order to reflect how well you have learned the following learning targets from this unit.
You and your consulting team will work on completing your proposal. Be prepared to give a 5-7 minute presentation (or record your presentation) with the proposal and suggestions for your client. Be sure to include visuals that help to make your case.
Prior to presenting, you will have an opportunity to share your presentation with two other groups. They will provide feedback to your group using the grading rubric and peer feedback form.
Comment: What were some similarities and differences between your presentation and other groups’ presentations? How can you make adjustments to improve your presentation?
(2-4 days)
Materials:
Allow students to have the time to work together to create their proposal presentations. Students should make proper recommendations for the Farmer’s Market Lemonade. This is the project that the students are completing a the product to show their learning. In the constructivism theory, students create meaning through the project that they create.
Students should be given an opportunity to complete their presentations with their partners. Once their work is complete, groups should share their presentation with two other groups to gather feedback prior to completing their final proposal.
Students will reflect on their proposal and the feedback from peers in order to make necessary changes prior to the final presentation.
Students will present their final proposal. This may be done live or via a video presentation.
Optional Suggestion: Students could also create a marketing component to the proposal in order to include additional creativity aspects to the PBL project. The students could share their marketing strategy with an actual Farmer's Market Lemonade vendor
Learning Objective: “I can write a CER to show my understanding of ratios and proportions.”
Problem solve the given CER problem and write your mathematical reasoning to support your answer.
Comment: Reflect on the process of writing your proposal. What were the successes of you and your group? What could you have done better? What have your learned through this process?
Materials:
Claim, Evidence, and Reasoning formats are generally used in science courses. With that being said, the connection between using the same writing format across disciplinaries allows students to deepen their understanding of providing reasoning to their answers.
Students will be given their CER to demonstrate their understanding of ratios and proportion. The CER is a mathematical reasoning released items from the Illinois Assessment of Readiness. NCTM as well as other research shows that students should provide written communication about their mathematical problem solving. This will be a formative assessment that shows what the students are able to do independently.
*Fancher, C., & Norfar, T. (2019). Project-based learning in the math classroom: Grades 6 - 10. Waco, TX: Prufrock Press.
*Khan, S. (2014, March 25) Introduction to Ratios. Retrieved from www.khanacademy.com
*Khan, S. (2017, October 27). Equivalent Ratios for Recipes. Retrieved from www.khanacademy.com
*Khan, S. (2017, November 8). Ratios and Double Number Lines. retrieved from www.khanacademy.com
*Khan. S. (2017, August 11). Ratio Problems Tables Examples. retrieved from www.khanacademy.com
*Khan, S. (2010, August 11). Unit Rates Problems. retrieved from www.khanacademy.com
*Larmer, J. (2013, November 14). Project based learning vs. problem based learning vs. XBL. Retrieved July 15, 2020, from https://www.pblworks.org/blog/project-based-learning-vs-problem-based-learning-vs-xbl
*Larmer, J. (2019, April 19). PBL in the math classroom: a new book, plus more resources. Retrieved July 15, 2020, from https://www.pblworks.org/blog/pbl-math-classroom-new-book-plus-more-resources
*Merritt, J., Mi Yeon Lee, Rillero, P., & Kinach, B. M. (2017). Problem-based learning in K-8 mathematics and science education: a literature review. Interdisciplinary Journal of Problem-Based Learning, 11(2), 1–12. https://doi-org.proxy2.library.illinois.edu/10.7771/1541-5015.1674
*Meyer, D. (2015, March 19). Desmo's Nana's Chocolate Milk Introductory Video retrieved from www.desmos.com
*PBLworks. Retrieved July 13, 2020, from https://www.pblworks.org/what-is-pbl
*Sayles, M. (2020). Productive struggle to enhance student learning