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The Arizona STEM Acceleration Project

Lesson #6: Using Mathematical Models in Engineering and Science –Using Gears to Learn About Ratios and Proportions

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Lesson 6: Using Mathematical Models in Engineering and Science – Using Gears to Learn About Ratios and Proportions

A Middle School STEM Lesson

Kathy Hartley

12/10/2023

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Notes for Teachers

In this lesson, students will explore the concepts of gear ratios and proportions. They will construct a working gear system and observe how the ratio between gears affects their motion. They will use proportional relationships to find the ratios between different gears.
Timing - 3 to 5 periods
Students will work individually, with elbow partners and participate in group discussions.
Terminology: diver, follower, gear, gear ratio, rotational speed, work and torque

List of Materials

“Karakuri: How to make mechanical paper models that move” by Keisuke Saka
A composition notebook
YouTube Videos:

What is a Ratio?

How to Simplify a Ratio

Equivalent Ratios Explained

Determining if Two Ratios are Proportional

Rulers, graph paper, card stock, skewers with tips removed, glue, tape, toy pull-back cars

Optional resources:

Materials are all easily obtained. I used the book “Karakuri: How to make mechanical paper models that move”, and copied the parts for Gear A using the heavy card stock. Both 64 lbs and 110 lbs work, 64 lbs is not quite as stiff but is easier to work with, but I found that the 110 lbs card stock makes better gears. The website: Karakuri Workshop has a lot of information that I found helpful as a teacher. We do not have a 3D printer, but I did use Tinkercad as an extension activity to look at other gears designs. I also use Gear Generator as a simulator, but there is a bit of a learning curve. I used it to demonstrate gear ratio and speed. It could be used for extension/enrichment.

Pull-back cars can be purchased on Amazon for about $20 for 20 cars. You can sometimes find them cheaper at Oriental Trading Company online.

This was not a terminology lesson, but some very basic vocabulary was introduced.

Videos:

What are Ratios?

How to Simplify a Ratio

Equivalent Ratios Explained

Determining if Two Ratios are Proportional

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Standards

Physical Science

7.P3U1.4 – Use non-algebraic mathematics and computational thinking to explain Newton’s laws of motion.

Systems and System Models: Models can be used to represent systems and their interactions—such as inputs, processes and outputs

Math State Standards

6.RP.A.3 - Use ratio and rate reasoning to solve mathematical problems and problems in real-world context.
8.EE.B - Understand the connections between proportional relationships, lines, and linear equations.

ELA State Standards

8.SL.1 - Engage effectively in a range of collaborative discussions with diverse partners on grade 8 topics, texts, and issues, building on others’ ideas and expressing their own clearly.

Arizona State CTE Standards

STANDARD 3.0 Apply mathematical laws and principles relevant to engineering technology

3.2 - Use data collection and analysis to display data and verify its accuracy
3.3 - Display data graphically using diagrams and working drawings

NGSS

Systems and System Models

Models can be used to represent systems and their interactions—such as inputs, processes and outputs—and energy and matter flows within systems. (MS-PS2-1),(MS-PS2-4)

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National Standards

Next Generation Science Standards (NGSS)

MS-ETS1-4: Develop a model to generate data for iterative testing and modification of a proposed object, tool, or process such that an optimal design can be achieved.

Connection: Students design and build a gear system using cardboard and skewers, testing it to achieve specific speed or torque goals.

MS-PS2-1: Apply Newton’s Third Law to design a solution to a problem involving the motion of two colliding objects.

Connection: While focused on rotation, the interaction of gear teeth involves forces that transfer motion, connecting to physical principles of force and motion.

Common Core Mathematics (CCSS.MATH)

6.RP.A.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

Connection: Students explore the relationship between the number of teeth on driver and follower gears to understand the concept of a gear ratio.

6.RP.A.3: Use ratio and rate reasoning to solve real-world and mathematical problems.

Connection: Students calculate gear ratios and rotational speeds, determining how input speed translates to output speed in their designed systems. * 7.RP.A.2: Recognize and represent proportional relationships between quantities.
Connection: Students observe and graph the proportional relationship between the turns of the driver gear and the turns of the follower gear.

Common Core English Language Arts (CCSS.ELA)

RST.6-8.3: Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks.

Connection: Students follow instructions (or video tutorials) to construct complex mechanical gear models from simple materials like cardboard and straws.

SL.6-8.4: Present claims and findings, emphasizing salient points in a focused, coherent manner with relevant evidence, sound valid reasoning, and well-chosen details.

Connection: The assessment requires students to present their gear systems to the class, explaining their design choices and the ratios they achieved.

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Objective(s):

Today I will be able to:

Review and apply the concept of ratios.
Build a working gear model to use to understand gear ratios.
Explore how gear ratios impact the speed of rotation.
Practice the use of proportional relationships to find gear ratios.

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Pacing (3-5 periods)

Opening Activity: reviewing ratios (5-10 min)

Introduction to new material:

Pull-back cars (10 min)
Video/Discussion: Simple Machines – Gears (10 min)

Guided Practice:

Activity: Karakuri paper gears (75 min)
Introduce the gear ratio formula (20 min)
Discussion of proportional relationships and gear ratios (20 min)

Independent Practice:

Gear ratios and Speed (20 min)
Challenge: Gear systems (60 min)

Closing: Group Presentation of Gear Systems (50 min)

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Opening Activity: Reviewing Ratios

Begin the lesson by asking students to recall and review the concept of ratios.
Watch the video: What is a Ratio?
Write a few ratio problems on the board and ask students to solve them individually. (Math-aids: Ratios)
After a few minutes, facilitate a class discussion to review and solve the ratio problems together.
Emphasize the importance of understanding ratios for today's lesson on gear ratios.

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Introduction to Gears

Pull-back car activity

Give pairs of students a pull-back car to observe.
Have them predict how the car operates (write their predictions down in their notebooks).
Next have them take apart the cars and draw what they find on the inside (particularly the gears).

Video: Simple Machines – Gears

Discussion Question:

What other machines use gears? Collect a list.

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Guided Practice: Hands- on Activity Instructions

Students will work in pairs

Step 1 – they will read through the directions – review the different folds
Step 2 – they will label all the parts (must be done before cutting parts out)
Step 3 – cut out all parts – may need to review use of exacto knife.
Step 4 – assemble gear box
Step 5 – record the # of teeth in the driver and in the follower
Step 6 –observe the gear's motion, how the gear ratio affects the speed of rotation, and the direction each gear turns.

Hands-On Activity:

The objective of this activity is to demonstrate:

that a gear is a mechanical device used to change the speed and direction rotational motion.
The number of teeth in each gear can be used to determine the gear ratio

Teaching Tips-

Introduce the concept of gear ratios and explain that they are used to determine the relationship between the number of teeth on different gears. Show visual examples of different gear ratios and explain how they affect the speed of rotation. The Gear Generator works well to introduce gear ratios.
Divide the students into small groups and provide students with karakuri mechanical paper models of gears.
Instruct the groups to follow the instructions provided with the kit to assemble a working gear.
Encourage them to observe the gear's motion and discuss how the gear ratio affects the speed of rotation.

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Explain/Practice:

Facilitate a class discussion to share and compare the observations made by the small groups. Ask guiding questions such as:

How does changing the gear ratio affect the speed of rotation?
What do you notice about the proportions between the number of teeth on different gears?

Introduce the gear ratio formula: Gear Ratio = Number of Teeth on Driven Gear / Number of Teeth on Driving Gear.
Discuss how proportional relationships can be used to predict gear ratios by setting up equivalent ratios between the numbers of teeth on the driving and driven gears.
Worksheet: Gear Ratio Worksheet

Videos:

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Independent Practice: Gear System Challenge

Students will work in small groups
Using materials such as cardboard, straws, bottle caps, and tape students will design and build their own gear system (video: How to Make Simple Gears from Cardboard)
Students will brainstorm and sketch out their design ideas, considering the gear ratios they want to achieve.
Once the students have a plan, they will build a gear system.
After the gear systems are built, students will test and compare the speed of rotation turning one gear and observing the motion of the other gears in the system.

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Assessment

Students will present their gear systems to the class, explaining their design choices and the gear ratios they achieved.

Assessment:

Observation: Observe students as they build their gear systems and engage in discussions about gear ratios. Note: if they are able to apply their understanding of gear ratios to their designs and if they can explain how different gear ratios affect the speed of rotation.
Group Presentations: Have each group present their gear systems to the class. During their presentations, ask them to explain the gear ratios they achieved and how they determined them. Evaluate their ability to effectively communicate their understanding of gear ratios.

Alternative assessments:

Written Reflections: Ask students to write a reflection on their gear system design process. Have them explain the gear ratios they chose and why, as well as any challenges they encountered. Assess their understanding of gear ratios based on the clarity and accuracy of their explanations.
Performance Tasks: Create a performance task where students are given a specific gear system and asked to calculate the gear ratio. Provide them with the number of teeth on each gear and ask them to use the gear ratio formula to determine the ratio. This will assess their ability to apply the gear ratio formula correctly.

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Differentiation

Provide example of the Karakuri designs that are being built.
Allow students to use basic calculators.
Pair/group students together based on skill levels.

Remediation

Extension/Enrichment

For students who finish early, have them work on finding the gear ratio when there is an idle gear.
Students can use Tinkercad to design their gears or explore Gear Generator.
Students can research real-world applications of gear ratios and present their findings to the class. This could include industries such as automotive, robotics, or manufacturing.