1 of 13

The Arizona STEM Acceleration Project

Using Mathematical Models in Engineering and Science – Position/Time Graphs using Cams

2 of 13

Using Mathematical Models in Engineering and Science – Position/Time Graphs using Cams

A Middle School STEM Lesson

Kathy Hartley

12/10/2023

3 of 13

Notes for teachers

This lesson looks at mathematical models – graphing pe. The use of cams to change rotational motion to linear motion.
Timing - 2 to 3 periods
Students will work individually, with elbow partners and participate in group discussions.
Students will be introduced to graphing mechanical motion and to cams and their uses.
Terminology: cam, follower, axle, oscillating, rotary, and linear

List of Materials

Karakuri: How to make mechanical paper models that move by Keisuke Saka
A composition notebook
YouTube Video: What are Cams?
Rulers, graph paper, card stock, skewers with tips removed, glue, tape
Slow Reveal Graph: ASAP - Slow Reveal Graph - Position vs Time Pendulums.docx

Optional Resources:

Video: Cams are simple to design but oh so versatile - Guide with examples
Website: Karakuri Workshop

Notes and Materials:

These lessons were used in an Engineering class. The class consisted of 30 8^th grade students. Periods were 53 minutes, which was not optimum, as with all lab classes this would have worked well with block schedules.

Materials are all easily obtained. I used the book Karakuri: How to make mechanical paper models, and copied the parts for Cam A and the cam from Cam B. 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 cam designs.

This was not a terminology lesson but some very basic vocabulary was introduced. I also did go over the parts of a graph with the students.

Videos: Cams are simple to design but oh so versatile - Guide with examples – This video was a good source of info for the teacher (13 min).

4 of 13

Arizona State Science Standards

Systems and System Models: Models can be used for understanding and predicting the behavior of systems.

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

Math State Standards

8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph.

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

3.3 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)

�

5 of 13

Objective(s):

Today I will be able to use graphing to analyze the periodic motion of a cam by measuring the change in position of the follower as the cam rotates.
Today I will be able to discuss how a cam is used to convert rotational motion into linear motion by building and experimenting with different cams.
Today I will be able to predict the shape of a cam by analyzing the position graph that is produce by the cam.

6 of 13

Pacing (2-3 periods)

Opening Activity: Slow Reveal Graph (5-10 min)

Introduction to new material:

Video/Discussion: What are Cams? (10 min)

Guided Practice:

Activity: Karikuri paper cams (75 min)
Graphing: graphing cam motion (20 min)

Independent Practice:

Graphing: graphing/identifying graphs of different cams (20 min)

Closing: Discussion (15 min)

Teacher Tips:

Opening Activity - Slow Reveal Graphs - Advanced Teacher Preparation

Print single-sided copies of the blank graph for them to fill in as it is revealed. Print double-sided, stapled copies of the notice and wonder pages for students to complete as you go through the frames of the slow reveal graph. These documents are included at the end of this file.

Guided Practice – Advanced Teacher Preparation

Karakuri: paper cams - Copy Cam A and Cam B – use heavy card stock. Make an example Cam A using the pear shape cam from Cam B. It helps to make an example fold board that show how the different fold look – mountain vs. valley
Graph paper

Independent Practice – Advanced Teacher Preparation

Pre-cut several different cam shapes – snail, heart, off-centered, inline double lobed (if you have the time students can cut out their assigned cams).

7 of 13

Opening Activity: Slow Reveal Graph

This lesson was designed to introduce position/time graphs for harmonic motion.

Students will view a series of slides. Each slide will reveal more of the original graph and diagram.

For each Frame there are 2 slides. The first slide allows the student to make observation/wondering on their own. The second slide allows students to work with an elbow partner to talk through their ideas.
For each Frame have students add to their copy of the graph any new information that is revealed.
For each Frame have students record their noticings and wonderings – they should also add any information that is uncovered through discussion.

At this time, it is not important for students to have the right answer - Use this activity to identify student’s prior knowledge

Materials needed:

Blank graph for students to fill in
Hard copy of Student’s Notice and Wonder worksheets

Engagement Suggestions:

The whole point of a slow reveal graph is for students to not know what the graph is about until it is revealed so try not to give a hint or accidentally reveal what the graph is about until you get to that point in the lesson.
Bring up the Slow Reveal Graph and go through the frames one at a time in slideshow view. Allow time for students to think on their own during the Alone Zone time so they can notice, wonder, and make guesses about the graph topic without distraction chatting with others. Then let elbow partners share out ideas. Keep repeating this process as indicated on the slow reveal graph frames.

8 of 13

Introduction to New Materials

Video: What are Cams?

Discussion Question:

How does a cam change the motion of the follower?

Look at some of the Karakuri mechanical toys.

9 of 13

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 cam
Step 5 – mark the handle disk in 15° increments
Step 6 – record the height of the follower every 15° in a table in their notebooks
Step 7 – graph their data

Hands-On Activity:

The objective of this activity is to demonstrate:

that a cam is a mechanical device used to convert rotational motion into linear motion.
The position of an object attached to a cam changes over time, resulting in a periodic motion.
Graphs can be used to represent the relationship between position and time in periodic motion.

Teaching Tips-

Provide students with karakuri mechanical paper models of cams and guide them through the process of building a working cam mechanism.
Instruct students to rotate the cam and observe the motion of the attached object.
Ask questions to help students make connections between the rotational motion of the cam and the linear motion of the object.
Guide students in graphing the position-time relationship of the cam motion using a provided graph template.
Scaffold the questioning from simple to more complex, such as asking students to identify the maximum and minimum positions, time period, and amplitude of the motion.

10 of 13

Independent Practice:

Hands-on Activity Instructions

Students will work in pairs.
Students will be given one of the following cam types: snail, heart, off-centered, inline double lobed.
Students will change out the original cam for their new cam in their gear box.
Student will collect the new data created by the movement of the follower and the new cam.
Students will graph the new data.

11 of 13

Homework (optional)

Assign students to research and find real-life applications of cams in engineering, robotics, or other fields.

Require students to write a short paragraph explaining how the cam mechanisms they researched are used to achieve specific motions.

12 of 13

Assessment

Students will be given an offset double lobe cam. They will be asked to do the following:

Predict the effect of the cam on the follower’s action.
Collect the data needed to graph the motion of this cam.
Graph the collected data.
Brainstorm how this cam might be used.

13 of 13

Differentiation

Provide example of the karakuri designs that are being built.

Give students graph paper that already has a coordinate plane.

Pair students together based on skill levels.

Remediation

Extension/Enrichment

For students who finish early, provide them with additional cam shapes and challenges to graph.

Students can also design and build their own cam mechanisms using various materials.

Students can use tinkercad to design a cam.