Roger Hatlen, MEd
Secondary Mathematics Consultant
District and School Transformation Division
North Carolina Department of Public Instruction
roger.hatlen@dpi.nc.gov
How is it possible to actually engage students in 21st century
mathematics??
FEED THEM!!!
Abstract: Many people incorporate the idea of food or other real-life
applications into math, but often it turns into another handout or worksheet. I
have never been big on algebra tiles as manipulatives. They provide opportunities
for tactile learning, but they don’t have any inherent meaning. On the other hand,
EVERYONE understands food. My plan is to use food itself as a manipulative. We
will walk through a multitude of activities over a broad range of mathematics,
providing useful, fun activities for Middle School math through Calculus, using
food that is fun but inexpensive for the teacher.
IMPORTANT SAFETY TIP: *Always use paper towels for food activities to keep food off of desks.
Teacher Notes
1)
Food: Smarties®
Target Audience:
Middle School Math – Math I
Rationale:
Many of our students do not come into “algebra” level courses with the
necessary pre-requisite skills. These three activities form a good foundation for factoring,
working with radicals, rational/irrational numbers and quadratics as well as revisiting
multiplication skills, divisibility rules, patterns and iterations. Knowledge of prime numbers,
perfect squares and the Pythagorean Theorem are all foundation topics built upon later in
mathematics, but introduced in middle school or before.
Part 1:
The Sieve of Eratosthenes (Finding prime numbers)
Pre-requisite skills:
Multiplication facts, divisibility rules, identification of
patterns, basic math vocabulary
Provide students with a 10 x 10 grid of numbers, 1 – 100. Have student groups (2 or 3) place a
smartie on every square. The goal is to remove all smarties from numbers that are not prime.
Remaining smarties at the end of the procedure should be all of the primes from 1 to 100.
Initial questions -whole group
Is 1 a prime number? (Think definition of “prime”)
Is 2 a prime number ? If so, what do you know about all other even numbers?
If no other even number except 2 can be prime, how can we eliminate the rest?
Instructions
Using patterns is the key point for students to understand. Repeat the same procedure for the
number 3, eliminating it’s multiples. *Note: This is a useful exercise to introduce students
to iterations. They should continue on their own to move to the next number and repeat the
practice until they are confident they have removed all the non- primes (composite numbers)
Closure
Check final results- see if everyone agrees. * Also a good use of handheld technology- on smart
phone (Google) prime numbers 1 -100 to see results
Discuss strategies-Include vocabulary like divisibility rules, multiples, prime, composite
-How do you prove a number is composite?
SIEVE OF ERATOSTHENES
Name____________________
Place a smartie® on each square
Remove all numbers that are not prime
When finished, check answers, place an X on every prime number from 1 - 100
Part 2:
Perfect Squares
Pre-requisite skills:
Definition of a square
Initial questions -whole group
The goal is for students to discover/remind themselves about what a “perfect square” is, and identify
which numbers from 1 – 100 qualify as perfect squares. This knowledge is particularly useful when
students are introduced to or utilizing square roots, discovering irrational numbers, utilizing the
Pythagorean Theorem or Quadratic Formula.
Instructions
Students will make squares using smarties. Start with 1 smarty –then 2 per side, etc. Have student
groups record the total number of smarties they used to make each square on their 1 – 100 chart.
*Cheez-its work well for this as well.
Etc.
Total # smarties:
1
4
Closure
Discuss what the only number of smarties make squares.
Is it possible to make a square using 20 smarties? Why/why not?
How many X’s are on each student’s paper? What do those numbers mean?
Can a perfect square be prime? Explain.
Knowing these numbers helps quantify values of square roots. Great introduction to irrational numbersimportant to know prior to the Pythagorean Theorem.
Perfect Squares
Goal:
Name___________________________
Make squares out of smarties®. Do this on your paper towel/napkin
Count how many total smarties are in each square-Put an X over that number
Use the chart below to record your results.
Example:
A square with 1 smartie per side has a total number of 1 smartie. (Put an X on 1)
*Repeat by adding 1 more smartie per side until it takes over one hundred to fill in the square.
How many X’s did you end up with?
Activity 3:
Pythagorean Triples
Pre-requisite skills:
Knowledge of the Pythagorean Theorem
Initial questions -whole group
Introduce the Pythagorean Theorem: a2 + b2 = c2. Which side must be represented by “c”? Use 3,4
and 5 as a simple example of how the theorem works. Demonstrate how this can be found using
smarties.
Remove to test hypotenuse
*There should be no extra space
Instructions
Students will use trial and error-keep adding smarties to bottom and left and see if smarties fit perfectly
along the diagonal/hypotenuse (example: try 4 and 4). *Keep only smarties on the end points on grid
when testing the hypotenuse, otherwise the shape of the candy gets in the way!!! Use a ruler or other
clean straight edge to keep smarties straight on the diagonal.
*Note: Excellent way to demonstrate perseverance in problem solving!
Use 103% zoom Graph Paper: 1/2 inch tkawas@mathwire.com
Closure
Hopefully, some students will have found 6,8 and 10 work in the formula. Have students compare that
set with 3,4,5. Show how multiples of the original set also work in the formula. Students can be assigned
homework to research the web to find other sets of Pythagorean triples that they didn’t find in the
activity. *Important to note that any other lengths for the two legs will result in an irrational number for
the length of c.
Name____________________________
Finding Pythagorean Triples Using Smarties®
Pythagorean Triples
Check
_3_, _4_, _5_
_3_2 + _4_2 = _5_2
___, ___, ___
___2 + ___2 = ___2
___, ___, ___
___2 + ___2 = ___2
___, ___, ___
___2 + ___2 = ___2
___, ___, ___
___2 + ___2 = ___2
___, ___, ___
___2 + ___2 = ___2
___, ___, ___
___2 + ___2 = ___2
___, ___, ___
___2 + ___2 = ___2
___, ___, ___
___2 + ___2 = ___2
___, ___, ___
___2 + ___2 = ___2
___, ___, ___
___2 + ___2 = ___2
2)
Food: Dum-dums/tootsie pops (any spherical suckers)
Activity 1:
Proportional Reasoning
Rationale:
*Discovers /reinforces 1 dimensional measurements (lengths,perimeter,circumference)
get multiplied by k, 2 dimensional measurements (areas/surface areas) get multiplied
by k2, and 3 dimensional measurements (volume) get multiplied by k3.
Pre-requisite skills:
Can solve a proportion for an unkown, solving formulas for a single variable,
calculating circumference, surface area of a sphere and volume of a sphere.
Initial questions-whole group
Does a sucker stay “similar” to its original shape as you suck on it? How could you tell? Will it be the
same for everyone?*
*Avoid the “rate that you suck” type discussions!
Instructions
Students will need to measure the circumference of their original sucker. Wrap dental floss around the
widest part, then lay the floss straight alongside a ruler.
Have students suck on the suckers for 5 minutes then re-measure and record in data table.
Repeat for another 5 minutes, record data in the row titled After 10 min.
Once they have the data, depending on the level of your class, walk them through the calculations in the
table. Calculate radius from C = 2πr. You may need to work through an example depending on the
algebraic level of your classroom.
Divide the current radius of the sucker by the original radius to find the constant of proportionality ∆
R(scale factor). Repeat with the circumference, surface area, volume
Closure
Did radius and circumference each increase by the same scale factor? How about surface area? Volume?
Was the rate of change the same from 5 – 10 minutes as from the original to 5 minutes? Should it be
consistent?
Do suckers shrink proportionally?
Name_________________
Items you need:
Paper towel
Spherical sucker
Dental Floss
Ruler *Use metric!
Procedure
1.
Unwrap sucker-place dental floss around the outside of the sucker at its widest spot.
Lay the floss straight alongside the ruler (without touching it) to measure the circumference.
Use centimeters! Record in the table in the Circumference column and the row titled Original..
2.
Suck on the sucker for 5 minutes. At the end of the 5 minutes, wrap the floss around the sucker
again, and measure the circumference along the ruler. Record in the table in the Circumference
column and the row titled After 5 min.
3.
Suck on the sucker for another 5 minutes. At the end of the 5 minutes, wrap the floss around
the sucker again, and measure the circumference along the ruler. Record in the table in the
Circumference column and the row titled After 10 min.
Radius ∆ R Circumference ∆ C Surface Area ∆ SA Volume ∆ V
Original
After 5 min.
After 10 min.
Calculations
4.
Calculate the radius of each using C = 2πr. Record in the table.
5.
Calculate the surface area of the sucker for each different radius (SA = 4πr2). Record in the table.
6.
Calculate the volume of the sucker for each different radius (V =
7.
Divide the radius found after 5 minutes by the original radius. Record under ∆ R (top half).
Repeat this procedure with the new circumference divided by the original circumference (in ∆ C)
and the same with surface area and volume.
8.
Do the same calculations with your radius after 10 minutes divided by the radius after 5
minutes, then the circumference after 10 divided by circumference after 5, etc.
. Record in the table.
Questions
1.
Does your first ∆R from the first 5 minutes equal ∆R from the second 5 minutes? Should it?
2.
Does ∆R = ∆C? Does ∆R = ∆SA or ∆V? Should any of them be the same?
Activity 2:
Related rates
Rationale:
Great way to demonstrate how the rate of one variable changing effects how other
variables change. I recommend using this just after introducing the idea of related rates.
Pre-requisite skills:
Basic knowledge of related rates
Initial questions
When you suck on a sucker and it shrinks, is the rate that the radius changes related to how the volume
changes?
Instructions
Repeat the same procedure as previous activity, but take measurements repeatedly at 1 minute
intervals for 8 to 10 iterations. Fill in chart:
Circumference|
Radius
|
Volume
|
dr/dt
|
dV/dt
Students can calculate empirical values for dr/dt and dV/dt and use related rates with their
dr/dt values to solve for dV/dt. Then you’ll know how long it takes to get to the center of a
tootsie pop!
Closure
Check data to see if any students got a reasonably consistent dr/dt value-use theirs as an example.
Show how calculus determines what the rate of change of the volume should be at any time t.
Compare to their empirical data-discuss why they might be different.
TOOTSIE POP PROJECT
Name____________________
Items you need:
Paper towel
Spherical sucker
Dental Floss
Ruler *Use metric!
Goal:
To measure the rate of change of the circumference of a tootsie pop, and use it to calculate the
rate of change of the radius and volume of the sucker.
Procedure
1.
Unwrap sucker-place dental floss around the outside of the sucker at its widest spot.
Lay the floss straight alongside the ruler (without touching it) to measure the circumference.
Use centimeters! Record in the table in the Circumference column for t = 0.
2.
Suck on the sucker for 1 minute. At the end of the minute, wrap the floss around the sucker
again, and measure the circumference along the ruler. Record in the table under t = 1.
3.
Repeat this procedure 7 times-fill in the table for the circumference as you go.
t Circumference radius
0
1
2
3
4
5
6
7
8
Experimental
Theoretical
XXXXXXXXX
XXXXXXXXX
Volume
XXXXXXXX
Calculations
4.
Use your circumference measurements to calculate the radius and volume by formula.
5.
Find dr/dt by dividing each radius by the previous radius.
6.
Find dV/dt (Experimental) by dividing each volume by the previous volume.
7.
Find dV/dt (Theoretical) by plugging your dr/dt value into the related rates formula for dV/dt.
Questions
1.
Was the rate of change of the radius fairly consistent over time? If not, explain?
2.
Did your experimental and theoretical values for dV/dt agree? If not, what could be some
reasons for the discrepancy?
3.
Let us assume the center of the tootsie pop is also a sphere with a radius of .5cm. Using your
formula for dV/dt, how long would it take to reach the center of a tootsie pop? Explain.
3)
Food: M&Ms (or skittles or miniature candy bars in some instances))
Activity 1:
Proportional Reasoning (m&ms or miniature candy bars)
Can students determine if mini-M&Ms (or Hershey’s miniatures, etc.) are truly shrunken
versions of the original. Give students one of each size and provide a multitude of measuring
tools. They need to “prove” (yes or no) that mini m&ms are the same proportion as regular.
Students need to document their findings and support their conclusion mathematically.
*This works well with miniature candy bars and original size, or jumbo size.
Student Handout-could be a half sheet or instructions on board/smartboard
Similarity: Are “mini” versions of candy proportional to the originals?
Task:
Prove mathematically whether or not the 2 different sized candies are similar by definition.
Requirements: Student groups will turn in one report (1 page maximum) and must disclose all
measurements, calculations, and how their final conclusion was determined.
Materials:
Assorted candy and measuring devices.
Activity 2:
Exponential Functions (m&ms or skittles)
Rationale:
Students get to discover growth factors using data. This not only helps reinforce what
the parts of an exponential formula stand for, but displays asymptotes as a very real
visible boundary.
Pre-requisite skills:
Knowledge of y = abx and what “a” and “b” each stand for.
Instructions
Most teachers should know of this activity. However, it works well with some additional twists: Not only
can you demonstrate growth or decay by how you construct the rules, you can insert blank
candies
x
to alter the “a” in y = ab for a more comprehensive view of the function. Place each group’s candies in a
Dixie cup.
For growth, pick a number to start with, like 4 m&ms (that is the data point for x = 0). Roll out
randomly on paper towels and count how many m&ms have the “m” showing. Add that many
m&ms to your total, record as x = 1 and graph, keep repeating until they run out of m&ms.
For decay, start with all m&ms (record as x = 0) and repeat procedure, this time removing all
m&ms with “m”s showing. Have them stop when they get to 1 m&m
*Both activities work even better if you add a certain number of similar candies without
the “m” on them! It obviously changes the horizontal asymptote and makes the
growth factor less obvious, particularly for higher level courses.
Closure
Explain why they didn’t start with zero m&ms (growth) or end with zero m&ms (decay).
What is the meaning of the y-intercept?
What should the growth factor have been?
M & M Lab
Name_________________
Items you need:
Cup of m&ms
Paper towel
1.
Shake 4 m&ms out onto your paper towel. This will be your initial value. Record in your table as
trial #0 and on the graph as (0,4). Count how many of the m&ms have their “m” showing. Add
that number of m&ms to your paper towel. Shake them up.
2.
Record this new total under trial #1. Count how many now have “m” showing. Keep repeating
this procedure and recording your data in the table and on the graph.
3.
Repeat until you run out of m&ms.
Trial # of
#
m&ms
0
4
1
2
3
4
5
6
7
8
# of m&ms
Trial #
4.
Find an equation for your model.
5.
Assuming that a typical m&m would have a 50% chance of having it’s “m” showing, what should
your equation be? Why does your model differ?
M & M Lab
Name__________________
Items you need:
Cup of m&ms
Paper towel
1.
Shake out all of your m&ms onto the paper towel. Count all of your m&ms. Record as t = 0.
2.
Count how many have their “m” showing. Remove that number of m&ms, record the total of
the remaining m&ms as t = 1. Replace in the cup and shake out again.
3.
Count how many have their “m” showing. Remove that number of m&ms, record the total of
the remaining m&ms as t = 2. Keep repeating this procedure and recording your data in the
table and on the graph. Repeat until you run out of m&ms.
Trial # of
#
m&ms
0
1
2
3
4
5
6
7
8
# of m&ms
Trial #
4.
Find an equation for your model.
5.
Assuming that a typical m&m would have a 50% chance of having it’s “m” showing, what should
your equation be? Why does your model differ?
Activity 3:
Statistics (m&ms or skittles)
Binomial probability:
Collect data from snack pack or random sample of a large bag in
a dixie cup
Always draw from full compliment of m&ms! What is the probability of drawing
consecutive Green m&ms? 3 out of 5? Etc.
Geometric probability: Similar, except draw until you get 1st Green; record how many
m&ms were chosen to get 1st success. Have them repeat many trials. Graph and
calculate probabilities for success on 1st trial, success on 2nd trial, etc.
Chi-Squared:
Compare sample to company’s claim of % for each individual color.
BINOMIAL DISTRIBUTION
1.
Shake cup of m&m’s to mix them up.
2.
Shake out 10 m&ms from your cup. Record how many ORANGE m&ms you get.
3.
Repeat 20 times. Use tally marks to keep track of how many ORANGE ones you got.
4.
Make a histogram of your results.
# Orange Frequency
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4 5 6 7 8
# of ORANGE m&ms
9
10
5.
What seems to be the center of the distribution (mean)?
6.
Does the distribution appear symmetric?
7.
Does this activity meet the following criteria:
Fixed # of trials (same # of m&ms chosen each time)
Same probability of success each time
Each trial was independent
Only 2 possible outcomes
???
8.
If the MARS corporation claims that 20% of their m&ms are ORANGE, do we have any reason to
suspect that our sample is different?
9.
Assuming the probability of drawing an ORANGE m&m is .2, what is the probability of drawing 5
m&ms and getting exactly 2 ORANGE? *Keep in mind that there are multiple combinations of
ways to get 2 successes out of 5.
GEOMETRIC DISTRIBUTION
Procedure: Each group needs 1 snack-pack of m&ms.
1.
Shake bag of m&m’s to mix them up.
2.
Open the package by making a small tear across a corner, just big enough to pull out a single
m&m.
3.
Remove one m&m at a time until you get a green one.
4.
Record the number of trials it took to get the green m&m.
5.
Replace all the m&m’s into the bag and repeat steps 1-4 above. Do this 10 times.
___
___
___
___
___
___
___
___
___
___
Class Results:
6.
Include your results on the Class Dotplot on the Smartboard.
7.
Create a probability distribution for all outcomes on the class dotplot.
(Let x = the number of m&m’s drawn before you got a green one, ex. How often did you get
green on the 1st m&m, etc.)
8.
Draw a histogram that approximates the class dotplot.
Develop the Concept:
9.
What value do you think is the mean of the distribution? How did you decide?
10.
Is this a binomial distribution? Why or why not?
Proportion of colors in m&m packages
Name________________________
According to the MARS® company, each bag of m&ms produced (any size) will contain
approximately 13% Brown, 13% Red, 14% Yellow, 16% Green, 20% Orange and 24% Blue.
Instructions
1.
Take your sample/bag of m&ms and count how many of every color you have.
Record in the table in the Observed row.
2.
Based on the total of the m&ms you received, calculate the Expected counts by
multiplying the total number of m&ms you have by the given percents by color. Record
these Expected counts in your table.
3.
For each color, perform this calculation:
(
(
Record these values in the table.
Brown Red Yellow Green Orange Blue TOTAL
Observed, O
Expected, E
(
4.
Add the values in the bottom row. This total =
Questions
2
or your Chi-Squared Statistic.
1.
Compare your observed counts and the expected counts for each color. Is the
company’s claim accurate, in your opinion?
2.
Were there any disparities between what you got and what you were supposed to get?
If so, which color?
COMBINE ALL STUDENT TOTALS AND REPEAT
Brown
Red Yellow Green Orange Blue TOTAL
Observed, O
Expected, E
(
3.
Calculate the 2 statistic for your whole class. Look up the p-value for 5 degrees of
freedom in the table of 2 values given. How rare is your classes’ sample?
4)
Jello
Activity 1:
Surface area and volume. Have students calculate surface area and volume by
measuring each dimension necessary for prisms/cylinders/pyramids/cones/spheres.
*Can give non-traditional shapes (like a bell) and see how they can approximate
surface area and/or volume. This can be done with any odd shaped candyespecially around holidays.
Activity 2:
Proportional reasoning. Give 2 shapes, ask students to measure and decide if
they are similar.- Same instructions as candy given earlier
Activity 3:
Volumes of solids/Riemann sums. Students can slice figures horizontally and
develop approximations for volumes and visualize what these slices look like, leading to
the development of formulas for integration to find volume.
JELLO LAB-VOLUMES OF SOLIDS
Geometric formulas do not apply to all shapes. We need to find a way to calculate volumes of 3dimensional solids. In order to do this, we will first approximate them using Riemann sums. If possible,
we will then compare your answer to the geometric formula’s volume.
Procedure: Slice the figure so that the face of each slice is the same shape. Use a constant width
(usually 1 centimeter). Find the area of the flat surface of your slice starting with top for the Left hand
sum, then the bottom for the Right hand sum.
Shape
1.
LHS
RHS
Actual volume
2.
3.
4.
Questions:
1)
Which shapes should have the exact volume as a Reimann sum? Why?
2)
How can the volumes be more closely approximated?
3)
Is it true that you could approximate any volume by slicing, then adding each slice’s area x
width to get the volume?
5)
Bundt Cake/Jello Mold
Activity 1:
Demonstration: Volumes of revolution using the cylindrical shell method
Use a function like f(x) =
revolved around the y-axis. This would create a bowl
shape. Use cookie cutter concentric slices to show how to develop a formula for volume
unrolling a single slice.
↓Slice↓
∆x
 f(x)
|
Pies? Definition of pi, area of a sector, arc length
circumference
|