Gear Ratios  

Middle School Ratios 

Lesson Study by: EMITS Teachers and Team

SUMMARY 

Looking for a way to add some fun to

your ratios unit? This lesson is your

answer! Students use gears to discover

the inverse proportional relationship

based on gear size, and then apply that

knowledge to real-life engineering

problems. Give your students the

chance to discover their own solutions

to engineering problems.

QUICK LINKS 

Lesson This lesson is designed to help students

discover the inverse relationship gears

of different sizes have, and applying

that discovery to an engineering

problem.

Teacher Notes

Materials

Engineering Context Video

STANDARDS 

Standards of Practice (math):  

CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.

CCSS.MATH.PRACTICE.MP4 Model with mathematics.

CCSS.MATH.PRACTICE.MP8 Look for and express regularity in

repeated reasoning.

Content Standards (math):

CCSS.MATH.CONTENT.6.RP.A 1-3

CCSS.MATH.CONTENT.7.RP.A 1-3

CCSS.MATH.CONTENT.8.EE.B.5

CCSS.MATH.CONTENT.HS.NQ.2

Content Standards (science/cte/etc):

CTE Manufacturing and product development: Pathway - D4.0

Materials List

It is recommended that you provide one student pack per every two students. Each pack includes the

following:

(4) Cross axle connectors (length can vary -

approximately 4 cm min.)

(1) Beam (approximately 4 – 6 inches in length

Gears

● (1) 8 tooth (Gear Wheel (8T)

● (1) 16 tooth

● (1) 24 tooth (Lego Gear Wheel Z24)

● (1) 40 tooth (Gear Wheel 40T)

Order the gears as follows: 16 tooth, 8 tooth, 24 tooth, 40 tooth

Students will assemble gears as follows:

Parts Parts Assembled

The parts and gears used came from Lego Mindstorm™ kits. Lego gear sets can be purchased on eBay

(100 gears / $25).

Set-up for student use:

Mark one tooth on each gear to provide visual as students spin driver gear and count revolutions.

Set-up for 8 tooth gear and 24 tooth gear

Set-up for 8 tooth gear and 40 tooth gear

Teacher Notes for Gear Ratios Lesson

● Scaffolding Modifications: This lesson should be adjusted to fit the grade level or content goals. The first few tasks are effective for 6th or 7th grade students who are still building proficiency in ratios. Higher grade levels may not need these tasks, and can jump to the non-unit ratios.

● Materials: This lesson can be modified based on what gears are used.

● The final section of the lesson can be used as an Exit Ticket. Lesson Summary (Crouton Trail)

1. Pose windmill problem and let students explore gears. Establish vocabulary.

2. *Find the 2:1 ratio of rotations with 6 and 12 tooth gears. 3. *Find the 3:1 ratio of rotations with 6 and 18 tooth gears. 4. Find the 5:3 ratio of rotations with 36 and 60 tooth gears. Try finding

the ratio of rotations without having the gears. 5. Return to windmill problem. Students present designs with

explanations. 6. †Follow up problems: Bicycle gears or three gears.

* These lessons can be very fast or optional, depending on the students prior knowledge. † These problems can be exit tickets or homework problems, as needed.

EMITS GEAR RAtion LESSON STUDY Date: January 26, 2016 Fall/Spring Year 2 Revised: May, 2017

Title of Lesson Gear Ratios

Team Members Heidi Bremen, Steve Camacho, John Coder, Brenda Jensen, Nicole Miller, Aly Stiles, Tim Strauch, Bruce Vieira, Debra Crane, Doug Ward, Jason Lopez, Eugen Deaconu, Julie Lewis, Greg Davis, Joey Van Steyn, Sean Duncan Deb Stetson & Rick West – Ca Math Project Facilitators Chrissy Poulsen – SCOE Facilitator

Date January, 2016 September, 2015

Grade level(s) and course(s) lesson was taught in: 6th Grade math Integrated 1, Algebra 1

SMP Goal(s) ☒ 2. Reason abstractly and quantitatively. ☒ 4. Model with mathematics.

☒ 8. Look for and express regularity in repeated reasoning.

Content Goal When students finish the lesson, they will understand the following because… (mathematical reason). Predict the number of time one gear will need to spin given the number of time the other gear spins for a variety of sizes (teeth, diameter, circumference). Apply gear ratios in an engineering context.

Common Core Content Standards

Grade, Cluster, Number 6.RP.1, 6.RP.2, 6.RP.3, HSN.1.QA.2

Previous HAVE’s What are the previous concepts/skills students need before this lesson? Students can find unit ratios, equivalent ratios. Student can create ratio tables and tape diagrams.

Mathematical Idea #1: Anticipatory set

Purpose: The purpose of this section is to pose the engineering task and for students to become familiar with the gear device. Hopefully, they would be curious. The driving question is posed.

Teacher Does or Asks

Anticipated Student Responses

Possible Back-Pocket Questions or Prompts

Where to note progress toward

SMPG or CG Intended Student

Responses

Possible Misconceptions

or Errors Pose the problem by showing the windmill video and pose this problem: This windmill needs to generate electricity. The windmill is making _20_ revolutions per minute. If we need to go 1600__ revolutions per minute to generate electricity, how can we use gears to make this happen?

The goal of the question is to find gears that will increase the revolutions per minute of the windmill.

How can we make the windmill spin faster?

Send students to groups to generate ideas and make predictions about what the gears are going to look like and what they know about gears in general.

Questions relevant to the windmill problem.

Teacher groups students and distributes preconstructed gear device.

Students play and get familiar with the gear devices.

                 

What do you notice? What are you curious about?

I notice on this gear the other turns the same, more, less.

     

      SMP#4 Students begin to make observations on the

Hopefully response: If I know how many times one gear turns, can I find how many times the other turns. The circumference or number of teeth determine the gear size.

device and use mathematics to describe what is happening.

Teacher takes guesses from students regarding windmill problem.

Students guess       Response are listed.      

Student play.                        

Mathematical Idea #2: Understand 2:1 Ratios means if A (8 teeth) turns 2 times B (16 teeth) turns 1 time.

Purpose: Students find the 2:1 ratio.

Teacher Does or Asks

Anticipated Student Responses

Possible Back-Pocket Questions or Prompts

Where to note progress

toward SMPG or CG Intended Student

Responses

Possible Misconceptions

or Errors After students play If I turn this n number of

times, how many times does this turn? If I turn this 2 time then this turns 8 times.

      What happens if I turn B (12 teeth) 3 times? How many times does A (6 teeth) turn?

     

Data collection make a t-chart A B #spins #spins Make sure a data point is : 12:6

Students chart # of turns of B to the # turns of A.

Not enough data points What happens if I turn B (12 teeth) 4 times? How many times does A (6 teeth) turn?

SMP2: Making connections between the math and the device.

How many times would the bigger one need to turn to make the smaller one turn 240 times? After 3 data points: How many times would the smaller one need to turn to make the bigger one turn 240 times?

120 times 480 times

      What happens if I turn B (12 teeth) 2 times? How many times does A (6 teeth) turn?

SMP8: Using the repeated reasoning and applying it with other numbers.

How many people have an idea of how we could have known 2:1 before we did this?

Raise hands to survey and gather data.

            SMP2: Students are being asked to contextualize.

Mathematical Idea #3: Understanding the 3:1 (8 teeth: 24 teeth)

Purpose: Students find the 3:1 ratio.

Teacher Does or Asks

Anticipated Student Responses

Possible Back-Pocket Questions or Prompts

Where to note progress toward SMPG or

CG Intended Student

Responses

Possible Misconceptions

or Errors In the previous problem (2:1), if B turned 5 times how many times did A turn? What if we turn the C (18 teeth) five times, how many times does the A (6 teeth) turn? You cannot turn the gears.

10 times Without turning the gears, students discuss and try to answer the question. Students guess.

      How many people think the small one turns more then 10, less then 10, or equal to 10? Why? - if they do not agree: TTYN, explain why? If they agree, why?

SMP2:Decontextualization without being able to turn the gears. SMP:8 Using the reapeated reasoning from above

Teacher: Go ahead and turn the gears. Data collection on the chart A C #spins #spins Make sure a data point is : 18:6

15 turns Students chart # of turns of C (18 teeth) to the # turns of A (6 teeth).

      What happens if I turn C (18 teeth) 4 times? How many times does A (6 teeth) turn?

SMP2: Making connections between the math and the device. SMP8: Using the repeated reasoning and applying it with other numbers.

How many people have an idea of how we could have known 3:1 before we did this?

Count the number of teeth.

      If there a number we can assign to the measurement of each gear? What is the same and different about these gears? Is there anything we can count to compare their size?

SMP2: Students are being asked to contextualize.

Display and go over new chart, adding data that’s already been collected: Teeth Revolutio

ns

6 12 2 1

6 18 3 1

     

                       

Mathematical Idea #4: Using non-unit fraction ratios and generalize when it is a whole number. (24 teeth: 40 teeth)

Purpose: Students to make the generalization without physical gears.

Teacher Does or Asks

Anticipated Student Responses

Possible Back-Pocket Questions or Prompts

Where to note progress toward

SMPG or CG Intended Student

Responses

Possible Misconceptions

or Errors Teacher explains the next pair has 36 teeth and 60 teeth. If I turn the 60 tooth gear 15 times, how many times does the 36 tooth gear turn?

25 times

9 times

Get 60: 36 as a data point. What is an obvious data point? Which gear turns more times? Whatever the students do: teacher

     

Can someone explain their process in context to the gears?

If a student did 15 times 60 is 900. What is the 900 in context… 900 teeth.

            SMP8: Students explain their reasoning. 

What is the ratio of revolutions for these gears?

5:3 25:15 Is there a reduced ratio that will work for these gears? How can you get from 25:15 to 5:3?

Add new data to chart.

When I turn two gears, what is the same?

tooth for tooth number of teeth times turns = number of teeth times turns

            SMP2: Students use previous examples to generalize understanding.

What happens when you pick a small gear to drive a larger gear?

The number revolutions decreases.

The number of revolutions increases.

Model with groups or whole class.

If I turn a 1000 tooth gear 12 times, how many times would a 100 tooth gear turn? Add this to the chart.

120 times             SMP8: Using the repeated reasoning and applying it with other numbers.

If the first gear revolves “x” times, what would it take for the second gear to go “y” times as fast?

Answers will vary depending on student’s previous understanding. Students should explain a relationship between the ratio of gear sized inversely affecting rotations.

Not showing an inverse relationship.

Model with student example or previous example, and allow student to modify explanation.

SMP2: Reasoning about general cases based on specific experiments.

Mathematical Idea #5: Apply learning to engineering task.

Purpose: Use mathematical learning in engineering context.

Teacher Does or Asks

Anticipated Student Responses

Possible Back-Pocket Questions or Prompts

Where to note progress toward

SMPG or CG Intended Student

Responses

Possible Misconceptions

or Errors Review problem from the start of the lesson: This windmill needs to generate electricity. The windmill is making 20 revolutions per minute. If we need to go 1600 revolutions per minute to generate electricity, how can we use gears to make this happen? Send students to groups to design a gear setup, and be prepared to share their explanations.

Students design a gear set up with a ratio of 80:1 and show how it makes the windmill go from 20 rpm to 1600 rpm.

Students create an inverse ratio, 1:80. Students not using two gears.

Have students draw a model, where they see this causes the rpms to reduce, not increase. Refer back to the chart, and ask “When we have a 12-tooth gear, what gear do we need to get twice the revolutions per minute?” Work towards 6-tooth gear, try with other examples as needed.

SMP4: Students create a mathematical model to solve an engineering problem.

Students present their designs with explanation.

Mathematical Idea #6 Extension- Cyclist Question and Three Gears Question

Purpose: Students apply their understanding of gears to a new problem.

Teacher Does or Asks

Anticipated Student Responses

Possible Back-Pocket Questions or Prompts

Where to note progress toward

SMPG or CG Intended Student

Responses

Possible Misconceptions

or Errors If a bicyclist is observed pedaling 60 revolutions per minute, the front sprocket has 51 teeth, the rear sprocket has 17 teeth, and the radius of the rear tire is 13 inches, how fast is the cyclist traveling? (Modify for grade levels that haven’t used circumference formula previously).

See that the ratio of the sprockets is 3:1, so the front tire spins 1 time for every three pedals. At 60 pedals, there are 30 tire revolutions, so 30x(26pi)=2449.2 inches/min, or 204.1 ft/min.

Trying to just use 60 rpm with the tire size. Incorrect use of circumference formula.

Draw a model showing where the pedal is in relationship to the gears. Review formula.

SMP4: Students create a mathematical model to solve an engineering problem.

If you add a third 24 tooth gear between the 8 tooth and 40 tooth gears. How many revolutions would the 8 tooth gear have to make to turn the 40 tooth gear 1 revolution? Predict. Use manipulative to confirm prediction.

The middle gear doesn’t change the revolutions of the first, so the 8 tooth gear would spin 5 times still. For reference: the 24 tooth gear would spin 1 ⅔ times.

3 turns (ratio of 3:1 for the 8 and 24 tooth gears).

If the 8 tooth gear turns 3 times, how many times will the 24 tooth gear spin? the 40 tooth gear?

SMP4: Students create a mathematical model to solve an engineering problem.