Engineering Through Middle School Mathematics

Longwood University WebinarsWith Dr. Virginia Lewis and Mrs. Diane Leighty

Quote from Cathy Seeley: “The active engagement of students in their own learning is perhaps our most important tool in our battle for equity.”

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George Polya’s Problem Solving Process

Understand the Problem

Devising a Plan

Carry out the plan

Look Back

Engineering Skills For ALL!• Communication• Creativity/Inventiveness• Critical Thinking Skills/Problem Solving• Applied math & science• Research skills• Collaboration – team work!

NCTM Process Standards

Reasoning & Proof

Multiple Representation

s

CommunicationConnections

Problem Solving

Standards Of Learning

• Measurements, unit conversions

• Size, shape, material characteristics

• Data collection, graphical representations, algebraic statements

• Scale models; proportions

• Valid conclusions from analyzing data

• Experimental results in written form

Robo Wheel – 15 minutes for construction

2 paper bowls2 rubber bands3 feet of stringPushpin or thumbtackTape (duct or masking)Sharp pencil

Find the Center• Nest the bowls together. Wrap a rubber band around

the bowls.• Slide it right and left until it divides the bottom circle

of the bowl into two equal halves.• Make an "X" with the other rubber band. The four

quarters make four "pizza slices."• Move the rubber bands until all four "pizza slices"

are of equal size.• The center of the circle is where the rubber bands

cross.

Mark the Center – Where the holes will go

• Mark two dots on each side of the center. Make them equal distance from the center and about a half inch apart. (About the width of your index finger)

Make the Holes• With your dots as a guide, use the pushpin to

poke holes in the bowls.

Enlarge the holes Slightly

• Remove the rubber bands and separate the bowls. • Gently poke a sharp pencil into a pushpin

hole. Twist it and push gently. Stop when the hole is just a bit bigger than the string. That's usually around where the wood part of the pencil starts, just after the black lead.

Thread the String….• Begin by threading the string through the

bottom of Bowl 1, starting from the outside.• Next, thread the string through one of Bowl

2's holes, coming from the inside.

• Then thread the string through the bottom of Bowl 2, coming from the outside.• Finally, thread the string through the open

hole in Bowl 1, coming from the inside.

• Tie the ends of the string together with a knot.

Tear off four 2-inch squares of tape. For now, stick them where they will be easy to grab.Line up the bowls so the holes are even with each other. Stick the tape so the pieces are across from each other.

• Slide your wheel to the middle of the string loop and hold both ends.• To twist the string, push the wheel across

the table or floor or ask a friend to help you spin the wheel to wind up the string.

Spin the wheel – 5 to 10 minutes for this part• Pull outward on the string. The wheel will spin as the string

untwists. Pull hard.• Stop pulling just before all the twists unwind.

• Bring your hands together so the string is loose and the wheel sags down a bit. The Robo Wheel will keep spinning and will twist the string in the other direction. • When the wheel stops spinning, pull

out again, hard.

• Now that you've practiced spinning the wheel, try releasing it. • Hold the string with your thumbs in the loop. Hold the wheel just above

where you want to launch it. • Spin the wheel forwards and backwards a few times to get it revved up.• Wait until the wheel is spinning away from you to begin your release. • Let the string unwind until it is almost completely untwisted. (At this

point, there will be lots of room for your thumbs to release the string.)• Drop the string, and watch your wheel go!

What is the circumference of your wheel?

How far does your wheel travel?

How many rotations did your wheel make?

What mathematics can be learned or practiced from this activity?

Extensions….10 minutes, including video & discussion• Discussion of what mathematics can be learned or practiced using this

activity.

• Video from Zoom PBS website…http://pbskids.org/designsquad/build/robo-wheel/

Pop-Up Card – 5-10 minutes to create• How to make: Watch video at • www.pbskids.org/designsquad/build/triangle-pop-card/

• Try making your own pop-up card

Pop-Up Card (continued) – 5 minutes discussion• What mathematics can be explored or reinforced with this activity?

• How else can this be used in the classroom?

Study Mathematics Through Data – 5 min.Bones Data• Measure the length of your arm from your elbow to the tip of your

middle finger in inches. This is called a cubit.

• Compare this measurement with your height in inches.

Cubit Height Ratio =A2/B2

Cubit Height Ratio

Cubit Height Ratio

Analyze the data – 10 min. • Scatter plot of the data using Excel. Is there a line of best fit? Will this

relationship change for students or younger children? Will it change for older adults?

• Calculate the ratio of the cubit to height. In what ways can you analyze this singular data?

Concrete to Abstract:

Data • (2,3), (3,7), (4.5, 11.5)Table

Graph Equation/ Expression

x y2 33 74.5 11.5

1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

Graph of Data and Regression Line

x-axis

y-ax

is

Y=3.2421x - 3.2

Forensic Science Application• Picture of bones from forearm and hand bones…..How can we use

what we have learned to estimate the height of this person?

12.5 inches 7.75 inches

10 minutes to wrap up first session• Statistics with the data…normal distribution, z-scores, probability• Box-and-whiskers plot : National Library of Virtual Manipulatives link here

• Or INTERACTIVATE site. • Advanced Data Grapher – Illuminations site

• What is needed for next session• Data collection – upper arm length to height ratio• Research other body measurement ratios that would be interesting to explore in the classroom.

Be ready to share your research and ideas with the group. • Materials needed: Cereal box or other box (NOT a cube) that can be cut with scissors; masking

tape or transparent tape; scissors; metric ruler; • Calculate the surface area and volume of your box using cm.

Start of 2nd Session – 10 minutes• Discuss data collected on upper arm vs height

String Puppet – Create an “arm” in the same proportions as your own arm.

•3 straight straws (2 narrow, 1 wider)• Fishing line (or thin string or

thread)• Scissors or single-hole punch• Tape 

Create Notches in the Straw• Use scissors or a single-hole punch to cut

notches in the side of a narrow straw. These are the straw’s “joints.”• Experiment! Where you cut the notches will

change the way the straw moves. Cut notches on the same side of one straw. Then with the second narrow straw, notch on different sides.• TIP: If you want your “arms” to bend in different

directions… Alternate the notches on either side of the straw.

• Cut a piece of fishing line three times the length of a straw. Thread it through one of the narrow straws.

• When the string pokes through the end of the straw, bend it over the tip and tape it. Leave the other end loose.• Cut another piece of fishing line and

thread it through the second narrow straw. Tape it as you did with the first.

Make It Move• Pull the loose strings. Watch your straws

bend. What ideas does this give you? What type of puppet will you make?

• Pull the loose strings of both narrow straws into and through the wider straw.

• Wedge the ends of both of the narrow straws into the wider straw, far enough down that they are secure. Both strings should now hang out of the bottom of the wider straw.

• TIP: If you wedge the narrow straws too tightly into the wider straw… Back them out a bit. The string needs to be able to slide easily without catching or rubbing.

• Tape the ends• To pull both strings easily, tape them together

and make a tab that lets you pull the two ends of string together.

• Pull the tab and watch both “arms” (or maybe they are “legs”) of your puppet move.

• Decorate your puppets! Add a head, body, and eyestalks, arms, and legs.

• http://pbskids.org/designsquad/build/string-puppet/

Make a sketch of a person whose body parts are in the correct proportions for an average adult.

Share research on body proportions• Type their comments here……

Creating New Boxes From Old OnesQuestion How can you convert a cereal box into a new, cubical box having the same volume as the original?

Grade/Subject 6-8 MathArea, volume, surface area, measurement

Background •Students should know how to determine both the surface area and volume of a rectangular prism. •Students should be able to measure lengths accurately to the nearest millimeter.

Materials needed: • Rectangular boxes, such as cereal, crackers, or pasta; and boxes containing

macaroni and cheese, pudding or cake mixes; or boxes containing hot chocolate, tea, or instant oatmeal packets. Computer diskette boxes and boxes holding contact lens solutions and toothpaste also work well. • Masking tape, transparent tape• Sturdy scissors, one per pair if possible • Rulers (metric), one per pair if possible • Large envelopes or zipper-style plastic bags (a mix of quart and gallon

size), one per pair • Large paper clips, one per pair

Design Challenge – 10 - 15 minutes• Take your box and cut it up to make a new box

that is cube-shaped. It should have the same volume as the original box.

Do The Activity: 1. Measure each dimension (length (L), width (W), and height (H)) of your box to the nearest millimeter. L = ___________ W =___________ H =_____________ 2. Calculate the surface area (SA) of your box. SA = __________

3. Calculate the volume (V) of your box. V = ___________ Open the glued edges of your box. Cut off any parts of flaps that were hidden from view when the box was still intact. The hidden parts are usually easy to spot because they generally don’t have any color on them and/or they do have dried glue on them.

4. Using the volume you determined in step 3, calculate what the length of any side of your new cube-shaped box should be. Length of any side = 5. On the inside of your opened-out box, draw the six identical squares you will need to make your cube-shaped box. Remember that their sides must all equal the length you calculated in step 5, and the sides must meet at 90° angles. You may find that you can will have to take some of the remaining scraps and will have to take some of the remaining scraps and tape them together, rather like a jigsaw puzzle, to make the last one or two sides of your cube. 6. After you have figured out how to obtain all six sides of your cube, cut them out. Important: save any remaining scraps! Put them in an envelope or zipper-type plastic bag. (It’s okay to fold them if you need to.)

7. As neatly as you can, tape the six squares together to form your cube-shaped box. It will be sturdy and look good if you use masking tape on the inside of the cube to attach adjacent squares and then use clear tape only on the outside for additional strength. 8. Calculate the surface area of your cube-shaped box. SA of cube = 9. Find the area of each of the scraps. Since some of them may be oddly-shaped, you may want to divide them into squares and rectangles that will be easier to measure and calculate areas for. After you have determined all of their areas, add them up to get one total area of the scraps. Total area of scraps =

Analysis: Compare the new surface area of the cube to the surface area of the original box. Are they the same? If not, by how much do they differ? Difference in surface areas = How does this difference in surface areas compare with the surface area of the scraps? What is the ratio of the new surface area to the old surface area? Do you think this ratio will be the same for other boxes? Why or why not? Which shape box is the most efficient? Explain.

What’s Next?• Lessons in the classroom• Start with one per 9-week grading period or per semester• Integrate into regular curriculum – NOT an add-on!• Keep it SIMPLE!• Work together with other teachers. • Ask your PTO/parents to contribute materials.

Where can we get the lessons?• Children’s Engineering websites• http://www.childrensengineering.com/• http://www.doe.virginia.gov/VDOE/Instruction/CTE/te/K-5

/Engineering/• http://

www.childrensengineering.com/everydaydesignbriefs.htm

www.mthmtcs.net for the PowerPoint and LessonsLocation Not Found Lesson: https://sites.google.com/site/mathematicscapstonecourseunits/home/tasks

PBS Websites• Building Big – PBS website

• Zoom – PBS website

Engineering for the 21st

Century Website: • Virtual Field Trips

The New State Capitol

UNIT PLAN• Engage – Introduce the problem – something to spark interest

• Explore – Helpful resources to get students started. • One or two foundational activities (somewhat teacher directed/guided)

• Explain – Main task• What questions does the problem raise?• What are the expectations for “solving” the problem? • What will students be assessed/graded on? Rubrics/Checklists

• Elaborate – Students working in pairs/groups to develop a workable solution (or partial solution) to the problem.

• Evaluate – Provide rubrics for any written work, presentation, product produced by the group. Also peer and self-evaluations.