# Research Ideas

Mathematics (funded by SSHRC + Fields)

1. Infinity in my hand (gr. 3: fractions, infinity, limit)
2. Making sums of 10 (gr. 3-4: patterning, linear functions)
3. How to fence a pen (gr. 2-4: area, perimeter, optimization)
4. Where parallel lines meet (gr. 2: geometry of a sphere)
5. Probability race (gr. 2-4: probability with dice)
6. Odds and evens (gr. 2, 7: growth patterns, odds, evens)
7. Low floor, high ceiling (big ideas for young mathematicians)
8. I don't like math anymore (I love it now! - math-for-teachers)
9. Math performance (what did you do in math today?)
10. Research performance (arts-informed research dissemination)
Mathematics (funded by KNAER)
1. How much is a billion? (gr. 3: Fermi questions, social justice)
2. Math pattern trains (gr. 1-4: growth patterns, slope, linear functions)
3. Math waves around us (gr. 3-4: patterning, trigonometry)
Science
1. Cough, cough (gr. 4: pollution, social justice)
2. Eating plastic (gr. 3-4: the great plastic dump, social justice)
3. Refraction action (gr. 2-3: refraction)
4. Will it float, will it sink? (gr. 2-3: density, buoyancy)
5. Gravity's pull (gr. 2-3: gravity, density)
Peter Jaffe on violence & abuse
1. Never met a happy bully (bullying and breaking the silence)
2. The lizard in your brain (violence in the media)

"The story used increases awareness of social issues among the kids."

"Ths is an interesting way of teaching two important topics (area and perimeter) in a very creative way."

"From this math-at-home activity I learned about the benefits of using concrete materials to solve problems."

# How to Fence a Pen

Grades 2 and 4 students explore area and perimeter relationships in the context of optimization: (1) If you put 16 tables in a rectangular array, which arrangement fits the fewest chairs all around?, and (2) What are the dimensions of the biggest rectangular pen that can be created with 20 metres of fencing?

In grade 2 (Part 1-5, below) students first investigate rectangular arrays for numbers. Then they use this knowledge to construct rectangular arrays for 16 patio stones (area), and determine which arrangement would need the most or least fencing (perimeter). They also explore the different ways that 16 tables (area) could be arranged to form a rectangle, and the number of chairs (perimeter) that would fit all around. Which arrangement would be best for a big party? Two of the songs written based on student thinking (from different classrooms) can be seen performed in Part 4 and Part 5 below.

In grade 4 (Part 6-13, below) students also construct rectangular arrays that can be enclosed by a given length of fence (perimeter), and determine which dimensions result in the biggest area. A song written based on student thinking can be seen performed in Part 13 below (download the mp3 version of this song).

The work of art shown on the right was created to celebrate the mathematics and student learning in this activity. See details and a larger image in Part 14 below. You can download and print a poster of this work of art.

#### Resources to support this activity

 Lesson plans The Shortest Perimeter (pdf) - what is the shortest perimeter for a given rectangular area? The Largest Area (pdf) - what is the largest rectangular area for a given perimeter? Materials linking cubes or colour tiles (for modeling area) string (for modeling fence or perimeter) grid paper (for bar graphs) markers or bingo dabbers Children's literature The Doorbell Rang (number arrays, division) - see Part 1 below Stay in Line (number arrays, division) - reinforces concepts encountered in "The Doorbell Rang" Wolf Gets Hurt (number arrays, area perimeter relationships) - used in Part 2 (grade 2) and Part 7 (grade 4) below A New Home for Scruffy (number arrays, area/perimeter relationships) - used in Part 11 below

#### Number arrays and area/perimeter relationships in Grade 2

1 - "The Doorbell Rang" - number arrays in children's literature

 Grade 2 students read the story The Doorbell Rang and investigate arrays for the number 12. They also use their imagination to create artistic representations of their arrays.

2 - "Wolf gets Hurt" - constant area and changing perimeter in children's literature

 Grade 2 students read the story Wolf gets Hurt and investigate ways of arranging 16 patio stones in a rectangular array, so as to need the least amount of fence to enclose it. They also encounter a retelling of the Three Little Pigs story, where the Wolf is the victim of the Pigs' environmental negligence.

3 - 16 tables in a rectangular array ... how many chairs?

 Grade 2 students explore the problem of arranging 16 tables in a rectangular array and determining the number of chairs that would fit all around. They make math connections between this problem and the fence problem in Part 2 above.

4 - Students sharing and celebrating their collective learning - song #1

5 - Students sharing and celebrating their collective learning - song #2

 Number arrays - song 2 We drew a picture of, an ambulance 11 people taking care Of one patient We also drew a dragon And a mouse And a caterpillar We made 12 in a story, 2 by 6 We made 12 in a story, 6 by 2 We drew a picture of, a rocket ship With 12 people inside They barely fit We also drew a robot And an airplane And 12 pretty flowers We made 12 in a story, 3 by 4 We made 12 in a story, 4 by 3 We drew a ladybug, and a bus We drew a Nintendo Wii Don’t cause a fuss 2 groups of 6 make 12 6 groups of 2 12 makes a rectangle We made 12 in a story, 1 by 12 We made 12 in a story, 12 by 1 La, la, la, la …………………. Grade 2 students perform their song "Number arrays." Their math performance was also shared at the Math Performance Festival.

#### Area/perimeter relationships in Grade 4

6 - 16 tables in a rectangular array ... how many chairs?

 Grade 4 students explore the problem of arranging 16 tables in a rectangular array and determining the number of chairs that would fit all around. They are surprised to discover that the number of chairs varies depending on the shape of the arrangement.

7 - "Wolf gets Hurt" - constant area and changing perimeter in children's literature

 Grade 4 students read the story Wolf gets Hurt and investigate ways of arranging 16 patio stones in a rectangular array, so as to need the least amount of fence to enclose it. They make math connections between this problem and the tables and chairs problem in Part 6 above. They also encounter a retelling of the Three Little Pigs story, where the Wolf is the victim of the Pigs' environmental negligence.

8 - Recording data in a table

 Grade 4 students use a table to record and organize their data.

9 - Plotting data on a bar graph

 Grade 4 students plot their data on a graph (perimeter vs length). They learn that the shape of the graph is called a hyperbola.

10 - What did you do in math today?

 Grade 4 students practice how they might share their learning with family and friends.

11 - "A New Home for Scruffy" - constant perimeter and changing area in children's literature

 Grade 4 students read the story A New Home for Scruffy and explore how to create the biggest rectangular pen with 20 metres of fencing. Their organize and represent their data using tables and graphs. They also use their learning to write an advertisement for a fencing company.
 Grade 4 students look at the shapes of their bar graphs and learn about hyperbolas and parabolas.

12 - Using a Bansho

 Grade 4 students use a Bansho to organize their solutions in terms of efficiency.

13 - Students sharing and celebrating their collective learning - song #3

 Need to fence a pen? need a 20 metre fence to make a sturdy pen keep the dog in the yard or catch bunnies off guard? call us now, call us first we're the professionals we use math to optimize no one beats our price 1 by 9, 2 by 8 3 by 7, 4 by 6 rectangular designs with four straight lines but if times are hard and funds are short then order the 5 by 5 square save money don't despair rectangles are pretty rectangles are nice but to save you money squares is what we advise squares is what we advise buy 20 metres of fence get more than all the others that's right get more than all the others visit our website check our tables and graphs don't get design fooled ours is mathematically proved rectangles are pretty rectangles are nice but to save you money squares is what we advise squares is what we advise our design gives you the maximum you get 25 square metres of area rectangles are pretty rectangles are nice but to save you money squares is what we advise squares is what we advise A professional recording of Grade 4 students' song "Need to fence a pen?" Their own performance was shared at the Math Performance Festival.

#### Seeing "How to fence a pen" through art

14 - Let's paint a math story!

Below is an artistic representation of "How to fence a pen ." Artistically rendered by Ann Langeman (Faculty of Education, UWO). Designed by George Gadanidis. Download the poster for printing. You can see this painting at St Monica Catholic School, DCDSB, Pickering, Ontario.