## Reducing Squares

*Reducing Squares*belongs to a category of puzzles called “matchstick puzzles” which were very popular in America during the last century. Most adults in those days carried small boxes of matches with them to light the many candles or lamps in their homes. Many of these same people had a favorite repertoire of matchstick puzzles to share with friends. Matchstick puzzles were just one of the indicators of a strong general interest in puzzles of all kinds during the 1800s newspapers in those days included a much greater variety of puzzles than today’s ubiquitous crossword puzzles and word jumbles.

In

*Reducing Squares*toothpicks are substituted for matchsticks for safety reasons. I have found that flat toothpicks work the best because they do not roll. Each student will need 24 toothpicks for this particular puzzle. You may wish to have students make small paper envelopes in which to keep their toothpicks when they are not working on the puzzle.

Arrange 24 toothpicks as shown below. Each challenge asks to you reduce the number of squares in the arrangement by taking away different numbers of toothpicks. See if you can solve each challenge, and then come up with a challenge of your own. Make a record of your solutions.

Challenge #1: Reduce the number of squares to two by removing eight toothpicks.

Challenge #2: Reduce the number of squares to five by removing four toothpicks.

Challenge #3: Reduce the number of squares to five by removing eight toothpicks.

Challenge #4: Reduce the number of squares to nine by removing four toothpicks.

Challenge #5: Create you own challenge and make a record of it.

Seeds have a life span. Their shelf life depends on the variety as well as storage conditions. Seed companies use germination tests to insure the quality of seed farmers and gardeners buy. Gardeners who often buy more seed than they can use in one season also conduct germination tests on stockpiles of old seed to determine its viability. Test results predict the percentage of seeds that will germinate in the garden.

Conducting a germination test requires both simple and sophisticated mathematical mental maneuvers. Teachers involve students in making seed germination tests to help them visualize and apply math concepts related to set, operations of addition and subtraction, fractions, ratio and percent.

A simple seed germination test uses old seeds, paper towels, water, and a ziplock bag or petri dish. Pumpkin seeds are excellent choices because they are easy and fast to germinate, familiar to students, and large. Good alternates include old bean, pea, sunflower, squash, cucumber, and radish seeds.

The following germination test instructions may be adapted for individual students or groups of students:

Each student or group started with a set of 10 pumpkin seeds. The germination test created two subsets, sprouted and unsprouted seeds. Where does instruction go from here?

For early elementary students the seed germination test is an introduction to sorting and sets. The initial ten pumpkin seed sample can be counted and sorted into sprouted and unsprouted seed sets. Children can arrange the sprouted vs. unsprouted seeds in vertical column bar graphs creating a visual of the relationship between the two sets.

Additionally, the seeds can be used to introduce fractions by modeling a denominator with ten pumpkin seeds representing the original set and a numerator with sprouted or unsprouted seed. Separate the numerator and denominator seeds with bright yarn.

In primary grades the germination test provides practice of simple addition and subtraction facts. Students can create addition and subtraction problems represented by their sprouted and unsprouted seed data for peers to solve.

For older elementary students the germination test retrieves past mathematical experience to move to more advanced concepts. Students now must compare the number of sprouted seeds with the number in the total sample by writing a ratio. For example, if 7 seeds of the 10 sprouted, the ratio would be written 7:10 or the fraction 7/10.

Then move students to write the fraction 7/10 as a decimal fraction or .7. Since percent is expressed in hundredths, have students express .7 in hundredths or .70. It may be necessary to include the following step: 7/10 x 10/10 = 70/100. Then, the student calculates 70/100 as 70%.

Repeated experience with decimal fractions will allow the student to automatically take an expression like .70 and multiply it by 100 to yield 70% and state that the germination rate of his sample is 70%.

Next, have students determine the percent of seed that did not germinate using two methods. They may follow the same procedure used to arrive at germination rate to determine percent of nonviable seed. Secondly, they may subtract 70% from 100% to determine 30% of the seed that did not germinate.

Pumpkin seeds are science manipulatives too. In spring, a germination test could double as a presprouting preparation for planting the schoolyard pumpkin patch.

Anytime of year sprouted seeds can give children concrete experiences to extend vocabulary and enhance contextual meanings of plant concepts. They view the embryonic root or radical, the first part of a seeding to emerge. As seedlings emerge from the seed coat, the protective function of the tough coat is observed. Some germination tests reveal the cotyledon or embryonic seed leaves.

Students have conducted germination tests on many varieties of seed sent on NASA space shuttle missions. Your students can explore their findings.

Pumpkin seed germination tests will produce a vine of ideas for teaching the growing minds of children.

Conducting a germination test requires both simple and sophisticated mathematical mental maneuvers. Teachers involve students in making seed germination tests to help them visualize and apply math concepts related to set, operations of addition and subtraction, fractions, ratio and percent.

**Seed Germination Test**A simple seed germination test uses old seeds, paper towels, water, and a ziplock bag or petri dish. Pumpkin seeds are excellent choices because they are easy and fast to germinate, familiar to students, and large. Good alternates include old bean, pea, sunflower, squash, cucumber, and radish seeds.

The following germination test instructions may be adapted for individual students or groups of students:

- Count out and line up ten pumpkin seeds on top of a damp paper towel.
- Roll up the paper towel with seeds inside and place all in a plastic ziplock bag. Zip the bag closed.
- Label the bag with date, seed variety, and student or group’s name.
- Leave the bag at room temperature out of direct sunlight.
- Make a germination peek every 3-4 days to see what is happening and if the towel remains moist. Close bag up well each time.
- Open the bag after 7-10 days from the starting date and count the number of sprouted and unsprouted seeds.

Mathematical SeedsMathematical Seeds

Each student or group started with a set of 10 pumpkin seeds. The germination test created two subsets, sprouted and unsprouted seeds. Where does instruction go from here?

For early elementary students the seed germination test is an introduction to sorting and sets. The initial ten pumpkin seed sample can be counted and sorted into sprouted and unsprouted seed sets. Children can arrange the sprouted vs. unsprouted seeds in vertical column bar graphs creating a visual of the relationship between the two sets.

Additionally, the seeds can be used to introduce fractions by modeling a denominator with ten pumpkin seeds representing the original set and a numerator with sprouted or unsprouted seed. Separate the numerator and denominator seeds with bright yarn.

In primary grades the germination test provides practice of simple addition and subtraction facts. Students can create addition and subtraction problems represented by their sprouted and unsprouted seed data for peers to solve.

For older elementary students the germination test retrieves past mathematical experience to move to more advanced concepts. Students now must compare the number of sprouted seeds with the number in the total sample by writing a ratio. For example, if 7 seeds of the 10 sprouted, the ratio would be written 7:10 or the fraction 7/10.

Then move students to write the fraction 7/10 as a decimal fraction or .7. Since percent is expressed in hundredths, have students express .7 in hundredths or .70. It may be necessary to include the following step: 7/10 x 10/10 = 70/100. Then, the student calculates 70/100 as 70%.

Repeated experience with decimal fractions will allow the student to automatically take an expression like .70 and multiply it by 100 to yield 70% and state that the germination rate of his sample is 70%.

Next, have students determine the percent of seed that did not germinate using two methods. They may follow the same procedure used to arrive at germination rate to determine percent of nonviable seed. Secondly, they may subtract 70% from 100% to determine 30% of the seed that did not germinate.

**Extension Activities with Sprouted Pumpkin Seeds**Pumpkin seeds are science manipulatives too. In spring, a germination test could double as a presprouting preparation for planting the schoolyard pumpkin patch.

Anytime of year sprouted seeds can give children concrete experiences to extend vocabulary and enhance contextual meanings of plant concepts. They view the embryonic root or radical, the first part of a seeding to emerge. As seedlings emerge from the seed coat, the protective function of the tough coat is observed. Some germination tests reveal the cotyledon or embryonic seed leaves.

Students have conducted germination tests on many varieties of seed sent on NASA space shuttle missions. Your students can explore their findings.

Pumpkin seed germination tests will produce a vine of ideas for teaching the growing minds of children.

**Sources**Math Projects

Math projects provide a method for connecting a number of math concepts within one math problem. This is critical step for students developing a better understanding of math; connecting several math concepts together. In addition to making connections in math, projects provide teachers with the opportunity to use of real-world applications.

The advantage of using math project teaching strategies is that it allows students to decide and apply math problem solving strategies and skills. This teaching strategy helps students develop a sense of mastery of math concepts. As students master math concepts they are more likely to retain the knowledge gained for a longer period of time, avoiding the short term gains of rote memorization.

Student Learning

Teaching strategies that include project based learning activities offer the greatest opportunity for student learning and must be hands-on, minds-on; i.e., students are actively engaged physically and mentally. Six examples of math projects include:

Tessellations Project – students create geometric shape using tessellations by hand. This provides students with an opportunity to use dividers, protractors, and other geometric tools. Students develop a sense of regular and repeating patterns in the world around them.

Math Cartoon Project – students must create a cartoon comic strip using a minimum of six panels. The key to success in this project is that students are able to communicate a math concept, math rule, or math technique using humor. Since all students are not artistically inclined, students have the option of using a computer drawing program.

Creative Math Book Project – students create a book using cartoons, illustrations, and pictures to explain everyday real-world applications of math. This project supports student learning in math as they make connections with geometry, algebra, number sense, and more concepts.

Stock Market Project - students pick a stock they want to follow. They make a brochure for prospective investors using a publishing computer program. Students then graph the sales of the stock over a several week period to display how well the stock fared during this time period. They make several math connections along with reading, using newspapers in the classroom, watching stock reports on television, comparing and contrasting, and more.

Geometry Map Project – students are required to design a map that includes lines, angles, and triangles. The map can be of a neighborhood town, city, or state. The map must include the following as a minimum:

Math projects provide a method for connecting a number of math concepts within one math problem. This is critical step for students developing a better understanding of math; connecting several math concepts together. In addition to making connections in math, projects provide teachers with the opportunity to use of real-world applications.

The advantage of using math project teaching strategies is that it allows students to decide and apply math problem solving strategies and skills. This teaching strategy helps students develop a sense of mastery of math concepts. As students master math concepts they are more likely to retain the knowledge gained for a longer period of time, avoiding the short term gains of rote memorization.

Student Learning

Teaching strategies that include project based learning activities offer the greatest opportunity for student learning and must be hands-on, minds-on; i.e., students are actively engaged physically and mentally. Six examples of math projects include:

Tessellations Project – students create geometric shape using tessellations by hand. This provides students with an opportunity to use dividers, protractors, and other geometric tools. Students develop a sense of regular and repeating patterns in the world around them.

Math Cartoon Project – students must create a cartoon comic strip using a minimum of six panels. The key to success in this project is that students are able to communicate a math concept, math rule, or math technique using humor. Since all students are not artistically inclined, students have the option of using a computer drawing program.

Creative Math Book Project – students create a book using cartoons, illustrations, and pictures to explain everyday real-world applications of math. This project supports student learning in math as they make connections with geometry, algebra, number sense, and more concepts.

Stock Market Project - students pick a stock they want to follow. They make a brochure for prospective investors using a publishing computer program. Students then graph the sales of the stock over a several week period to display how well the stock fared during this time period. They make several math connections along with reading, using newspapers in the classroom, watching stock reports on television, comparing and contrasting, and more.

Geometry Map Project – students are required to design a map that includes lines, angles, and triangles. The map can be of a neighborhood town, city, or state. The map must include the following as a minimum:

- Two sets of streets that are parallel
- Two sets of streets that are perpendicular
- One street that intersects another street to form an obtuse angle
- One street intersects another to form an acute angle
- One street that is a line segment
- One street that is a line
- One street that is a ray
- One building in the shape of an equilateral triangle
- One building that is in the shape of a scalene triangle
- One building that combines three different geometric shapes in its design

## Best Pizza deal

What’s the best pizza deal in town? How do you shop comparatively when different pizza stores have different size pans?

Are prices for some large pizzas for a particular store proportional to the amount of pizza for each size? Does any combination of two pan sizes give a better buy than a larger pan size? Is pizza available in shapes besides circles? If so, what are

the advantages and disadvantages of these shapes from a marketing viewpoint? From a consumer viewpoint?

To answer these and other questions, your team will collect statistics on pan sizes, prices, topping options, and other

information affecting a decision to purchase pizza. As a class, before you begin your research, rate the pizza available in

your area based on quality or taste and on your initial impressions of which pizza store has the best deal in town.

To begin your research, compare cost per square inch of pizza for each pan size offered by a restaurant. Then compare

cost per square inch of pizza with customer-chosen toppings versus store-established combinations of toppings. Compare

cost per square inch of pizza for small, medium, and large pizzas among various restaurants.

After the information is collected, discuss your findings among team members to determine how it will be organized for

reporting team results. Charts, tables, graphs, lists, and narrative are some choices you may want to consider for your presentation.

Are prices for some large pizzas for a particular store proportional to the amount of pizza for each size? Does any combination of two pan sizes give a better buy than a larger pan size? Is pizza available in shapes besides circles? If so, what are

the advantages and disadvantages of these shapes from a marketing viewpoint? From a consumer viewpoint?

To answer these and other questions, your team will collect statistics on pan sizes, prices, topping options, and other

information affecting a decision to purchase pizza. As a class, before you begin your research, rate the pizza available in

your area based on quality or taste and on your initial impressions of which pizza store has the best deal in town.

To begin your research, compare cost per square inch of pizza for each pan size offered by a restaurant. Then compare

cost per square inch of pizza with customer-chosen toppings versus store-established combinations of toppings. Compare

cost per square inch of pizza for small, medium, and large pizzas among various restaurants.

After the information is collected, discuss your findings among team members to determine how it will be organized for

reporting team results. Charts, tables, graphs, lists, and narrative are some choices you may want to consider for your presentation.

## How many hamburgers?

*What we know:*Charlie sells around 12 cases of hamburgers weekly.

Each case contains 80 hamburgers.

Each hamburger patty costs $.60.

New vendor offer:

Week 1 order: 50 cases at $.30 per patty

Then 15 cases for the next 12 weeks at $.45 per patty

*What we are looking for:*The price per patty quoted by the new vendor is attractive, but is this a good deal for the restaurant?

Beyond the cost savings, what other factors need to be considered before buying a lot more hamburgers than you have sold in the past?

How can Joe make this deal more attractive to Charlie?

*Solution Plan:*1. How many hamburgers does Charlie normally sell during the period of the contract?

2. What is the total cost of the hamburger inventory for the period?

3. How many hamburgers will Charlie need to purchase under the new vendor contract?

4. What is the total cost of this inventory?

5. What is the average cost of a hamburger under this deal? (Round to the nearest cent.)

6. What is the percent savings per hamburger under the new deal?

7. What is the cost savings in total, if Charlie accepts the new vendor deal rather than buying the same amount of hamburgers at his current vendor price?

8. If hamburger sales remain stable at 12 cases per week during this period, how many hamburgers will Charlie have remaining in inventory at the end of 13 weeks?

9. If hamburger sales remain stable at 12 cases per week into the future, how many weeks will it take to sell the remaining hamburgers? (Round to the nearest week.)

10. If Joe can figure out a way to sell 14 cases a week, how many weeks will it take to sell all of the new vendor inventory? (Round to the nearest week.)

11. Beyond the price per patty savings, what factors should Joe consider when advising Charlie whether or not to take the new deal?

## The Knight

**The Problem**

A strategy game which takes advantage of the movements of a specific chess piece – the Knight.

Knights can only move 2 spaces forward and then 1 space to the right or left or 1 space forward and then 2 spaces right or left.

**The Challenge**

How can a knight in the lower left hand corner of a 5×5 grid visit each square exactly once?

The knight may not revisit a square.

**The Procedures**

Students can select anywhere to start on the 5 x 5 grid.

Place a 1 in the starting square.

Then place a 2 in the next square and so forth as the strategy game continues

One excellent characteristic of this game is that there are multitude of solutions. Here is one solution:

**Extensions**

Use different size grids, such as 8 x 8 or 6 x 6.

**Seven Bridges Problem**

In Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts.

Seven bridges were built so that the people of the city could get from one part to another.

The people wondered whether or not someone could walk around the city in a way that would involve crossing each bridge exactly once.

Sketch the above map of the city on a sheet of paper and try to ‘plan your journey’ with a pencil in such a way that you trace over each bridge once and only once and you complete the ‘plan’ with one continuous pencil stroke.

The problem now becomes one of drawing this picture without retracing any line and without picking your pencil up off the paper.

Seven bridges were built so that the people of the city could get from one part to another.

The people wondered whether or not someone could walk around the city in a way that would involve crossing each bridge exactly once.

Sketch the above map of the city on a sheet of paper and try to ‘plan your journey’ with a pencil in such a way that you trace over each bridge once and only once and you complete the ‘plan’ with one continuous pencil stroke.

The problem now becomes one of drawing this picture without retracing any line and without picking your pencil up off the paper.

*Geometry Map Project*

students are required to design a map that includes lines, angles, and triangles. The map can be of a neighborhood town, city, or state. The map must include the following as a minimum:

- Two sets of streets that are parallel
- Two sets of streets that are perpendicular
- One street that intersects another street to form an obtuse angle
- One street intersects another to form an acute angle
- One street that is a line segment
- One street that is a line
- One street that is a ray
- One building in the shape of an equilateral triangle
- One building that is in the shape of a scalene triangle
- One building that combines three different geometric shapes in its design

**Repainting Tennis Courts**

Determine the total cost of supplies.

The number of gallons of paint to cover all 8 courts if they apply two coats of green paint on each court, along with two coats of white paint on the lines of each court.

The cost of all the paint combined.

The grand total spent on paint and supplies.

Contextual information needed by students include:

Dimensions of a tennis court

Total number of lines, along with line dimensions on a tennis court.

How many square feet does a gallon of exterior paint cover.

Cost of a gallon of exterior paint.

Cost of a combo pack of roller frame, roller cover, and paint tray.

Cost of an appropriately sized paint brush.

Cost of any other materials they feel they need.

The number of gallons of paint to cover all 8 courts if they apply two coats of green paint on each court, along with two coats of white paint on the lines of each court.

The cost of all the paint combined.

The grand total spent on paint and supplies.

Contextual information needed by students include:

Dimensions of a tennis court

Total number of lines, along with line dimensions on a tennis court.

How many square feet does a gallon of exterior paint cover.

Cost of a gallon of exterior paint.

Cost of a combo pack of roller frame, roller cover, and paint tray.

Cost of an appropriately sized paint brush.

Cost of any other materials they feel they need.