How big is this blob?
Broad topic area(s):
· Shape and Space: Measurement
Specific Learning Outcomes (Grade 3):
· Demonstrate an understanding of perimeter of regular and irregular shapes by:
• Estimating perimeter, using referents for cm or m
• Measuring and recording perimeter (cm, m)
• Constructing different shapes for a given perimeter (cm, m) to demonstrate that many shapes are possible for a perimeter.
[C, ME, PS, R, V]
Materials/Resources:
· String
· Graph paper
· Pencils/Pens
Description:
Give students the following word prompt:
You want to find the area of this “blob.” Your friend has a suggestion: “Lay down a string around its perimeter. Arrange the string to form the shape of a square and then figure out its area. The area of the square will be just about the same as the area of the blob.” Do you agree or disagree with your friend’s method? Or are you unsure? Explain your thinking. Use a sketch if it helps you explain your ideas.
Collect their responses. Then have students estimate the area by counting the number of square units the blob takes up on a sheet of graph paper. Compare that estimation with the calculation of using a string the same length arranged into a square. Were they similar or different?
1. Fixed Perimeters
Students work with a loop of string 24 units long. They decide what different-sized rectangles can be made with a perimeter of 24 inches. Students record their rectangles on grid paper, noting the area and perimeter of each shape.
Readiness suggestions: Students with lower achievement in multiplication can use a shorter string (12 units) to remove some of the burden of calculation. Students of higher readiness could forego the manipulative, using only grid paper instead.
2. Fixed Areas
Students work with a set of 36 color tiles to make as many different rectangles as possible with the tiles (filled-in rectangles, not just borders). Students record their rectangles on gird paper, recording the perimeter as well.
Readiness suggestions: Students with lower achievement in multiplication can use a smaller set of tiles (12) to remove some of the burden of calculation.
Interdisciplinary Extensions:
Given a “blob” of land, farmers tend to divide their field into neat rectangular polygons, but there will always be areas along lakes or the coast which do not fit into corners. To be fair amongst each other, they need a way to calculate the areas of non-rectangular polygons. Students can examine aerial photos of farmland from different countries and discuss how geography would shape the process of farming. How do people divide a flat field compared to a rugged mountainous range? Is there a relationship between the land, its shapes and the culture? Do people who grow up in rectangular fields and rectangular houses think differently from people who grow up with winding curved fields and round houses?
Have students then consider how scientists would measure the perimeter of a shoreline. What is the perimeter of Canada? What is the total area of Canada? How does this compare to the United States of America? China? Russia? Which country can boast to possess the largest area?
Four weeks later, return the papers and ask students to review their answers and explain in writing if they were more firmly convinced of their original response or if their thinking had changed and if so, how.
Assessment:
This task can be used as a diagnostic tool to keep track of student’s mathematical reasoning throughout the unit by comparing their explanations before and after.
Why this task is worthwhile:
This task puts the math into context: why people have a need to measure the perimeter and area of non-rectangular polygons. Students have an opportunity to make cross-curricular connections, communicate their understanding and reasoning.
Source(s):
Rectanus, C. 2006. So You Have to Teach Math? Sound Advice for Grades 6-8 Teachers. Sausalito, CA: Math Solutions Publications. Pgs. 18-19.
Hoppe, A. Perimeter and Area—a Mathematical Unit for Grade Four. Retrieved from: http://differentiationcentral.com/examples/G4_Area_and_Perimeter_Unit.pdf
· Shape and Space: Measurement
Specific Learning Outcomes (Grade 3):
· Demonstrate an understanding of perimeter of regular and irregular shapes by:
• Estimating perimeter, using referents for cm or m
• Measuring and recording perimeter (cm, m)
• Constructing different shapes for a given perimeter (cm, m) to demonstrate that many shapes are possible for a perimeter.
[C, ME, PS, R, V]
Materials/Resources:
· String
· Graph paper
· Pencils/Pens
Description:
Give students the following word prompt:
You want to find the area of this “blob.” Your friend has a suggestion: “Lay down a string around its perimeter. Arrange the string to form the shape of a square and then figure out its area. The area of the square will be just about the same as the area of the blob.” Do you agree or disagree with your friend’s method? Or are you unsure? Explain your thinking. Use a sketch if it helps you explain your ideas.
Collect their responses. Then have students estimate the area by counting the number of square units the blob takes up on a sheet of graph paper. Compare that estimation with the calculation of using a string the same length arranged into a square. Were they similar or different?
1. Fixed Perimeters
Students work with a loop of string 24 units long. They decide what different-sized rectangles can be made with a perimeter of 24 inches. Students record their rectangles on grid paper, noting the area and perimeter of each shape.
Readiness suggestions: Students with lower achievement in multiplication can use a shorter string (12 units) to remove some of the burden of calculation. Students of higher readiness could forego the manipulative, using only grid paper instead.
2. Fixed Areas
Students work with a set of 36 color tiles to make as many different rectangles as possible with the tiles (filled-in rectangles, not just borders). Students record their rectangles on gird paper, recording the perimeter as well.
Readiness suggestions: Students with lower achievement in multiplication can use a smaller set of tiles (12) to remove some of the burden of calculation.
Interdisciplinary Extensions:
Given a “blob” of land, farmers tend to divide their field into neat rectangular polygons, but there will always be areas along lakes or the coast which do not fit into corners. To be fair amongst each other, they need a way to calculate the areas of non-rectangular polygons. Students can examine aerial photos of farmland from different countries and discuss how geography would shape the process of farming. How do people divide a flat field compared to a rugged mountainous range? Is there a relationship between the land, its shapes and the culture? Do people who grow up in rectangular fields and rectangular houses think differently from people who grow up with winding curved fields and round houses?
Have students then consider how scientists would measure the perimeter of a shoreline. What is the perimeter of Canada? What is the total area of Canada? How does this compare to the United States of America? China? Russia? Which country can boast to possess the largest area?
Four weeks later, return the papers and ask students to review their answers and explain in writing if they were more firmly convinced of their original response or if their thinking had changed and if so, how.
Assessment:
This task can be used as a diagnostic tool to keep track of student’s mathematical reasoning throughout the unit by comparing their explanations before and after.
Why this task is worthwhile:
This task puts the math into context: why people have a need to measure the perimeter and area of non-rectangular polygons. Students have an opportunity to make cross-curricular connections, communicate their understanding and reasoning.
Source(s):
Rectanus, C. 2006. So You Have to Teach Math? Sound Advice for Grades 6-8 Teachers. Sausalito, CA: Math Solutions Publications. Pgs. 18-19.
Hoppe, A. Perimeter and Area—a Mathematical Unit for Grade Four. Retrieved from: http://differentiationcentral.com/examples/G4_Area_and_Perimeter_Unit.pdf
The Integer Dice
Broad topic area(s):
· Number Sense
· Statistics and Probability: Chance and Uncertainty
Specific Learning Outcomes (Grade 6):
· Demonstrate an understanding of integers, concretely, pictorially and symbolically. [C, CN, R, V]
· Demonstrate an understanding of probability by:
• identifying all possible outcomes of a probability experiment
• differentiating between experimental and theoretical probability
• determining the theoretical probability of outcomes in a probability experiment
• determining the experimental probability of outcomes in a probability experiment
• comparing experimental results with the theoretical probability for an experiment.
[C, ME, PS, T] [ICT: C6–2.1, C6–2.4]
Materials/Resources:
· Dice
· Sticky-tack
· Scissors
· Paper
· Pencils and/or pens
Description:
Have student work in small teams to solve the following word problem:
Travis and Nathan were trying to be clever and invent a new dice game to fool a friend. They made two dice with the following six numbers on it: -3, -2, -1, 0, 1, 2. Before they could invent a game that ensured that they would win, they did some problem solving to determine the probabilities with the dice. If you roll the two dice 100 times, what sum will occur most often? Explain your thinking and your answer completely.
Students can use tables (a matrix), charts, lists, fractions, percentages and writing to explain and clarify their thinking. Collect their responses.
Extension: Would the sum change if they rolled the dice 1000 times? Have them try to complete creating a game that would ensure they would win. Is it possible? Using pen, paper scissors and sticky-tack, student can create the new dice and test their game.
Assessment:
Students are assessed according to the clarity of their explanations, in addition to the correct answer.
Why this task is worthwhile:
Students get another opportunity to practice their communication skills and problem solving when they must explain their reasoning and collaborate in their game creation. They also experience how math is an integral part to creating games involving chance and probability, which can lead to a discussion about gambling and how the casino makes sure they stay in business.
Source(s):
Rectanus, C. 2006. So You Have to Teach Math? Sound Advice for Grades 6-8 Teachers. Sausalito, CA: Math Solutions Publications. Pgs. 86-90.
· Number Sense
· Statistics and Probability: Chance and Uncertainty
Specific Learning Outcomes (Grade 6):
· Demonstrate an understanding of integers, concretely, pictorially and symbolically. [C, CN, R, V]
· Demonstrate an understanding of probability by:
• identifying all possible outcomes of a probability experiment
• differentiating between experimental and theoretical probability
• determining the theoretical probability of outcomes in a probability experiment
• determining the experimental probability of outcomes in a probability experiment
• comparing experimental results with the theoretical probability for an experiment.
[C, ME, PS, T] [ICT: C6–2.1, C6–2.4]
Materials/Resources:
· Dice
· Sticky-tack
· Scissors
· Paper
· Pencils and/or pens
Description:
Have student work in small teams to solve the following word problem:
Travis and Nathan were trying to be clever and invent a new dice game to fool a friend. They made two dice with the following six numbers on it: -3, -2, -1, 0, 1, 2. Before they could invent a game that ensured that they would win, they did some problem solving to determine the probabilities with the dice. If you roll the two dice 100 times, what sum will occur most often? Explain your thinking and your answer completely.
Students can use tables (a matrix), charts, lists, fractions, percentages and writing to explain and clarify their thinking. Collect their responses.
Extension: Would the sum change if they rolled the dice 1000 times? Have them try to complete creating a game that would ensure they would win. Is it possible? Using pen, paper scissors and sticky-tack, student can create the new dice and test their game.
Assessment:
Students are assessed according to the clarity of their explanations, in addition to the correct answer.
Why this task is worthwhile:
Students get another opportunity to practice their communication skills and problem solving when they must explain their reasoning and collaborate in their game creation. They also experience how math is an integral part to creating games involving chance and probability, which can lead to a discussion about gambling and how the casino makes sure they stay in business.
Source(s):
Rectanus, C. 2006. So You Have to Teach Math? Sound Advice for Grades 6-8 Teachers. Sausalito, CA: Math Solutions Publications. Pgs. 86-90.
One Grain of Rice
Broad topic area(s):
· Number Sense
· Patterns and Relations
Specific Learning Outcomes (Grade 1-6):
· Demonstrate an understanding of place value, including numbers that are:
• Greater than one million
• Less than one thousandth.
[C, CN, R, T]
· Determine the pattern rule to make predictions about subsequent elements. [C, CN, PS, R, V]
· Represent and describe patterns and relationships, using graphs and tables. [C, CN, ME, PS, R, V] [ICT: C6–2.3]
· Demonstrate an understanding of the relationships within tables of values to solve problems. [C, CN, PS, R] [ICT: C6–2.3]
Materials/Resources (depending on grade level):
· One Grain of Rice: A Mathematical Folktale by Demi
· Computers or calculators
· Spreadsheet program
Description:
Read to the class the children’s book, One Grain of Rice: A Mathematical Folktale by Demi:
For Grade 1: Show what 1,073,741,823 looks in numerals and name each place value.
For Grade 3: Read the story and spent about fifteen minutes afterwards discussing the numbers. Write 1,073,741,823 on the board and review the place values. Even though they may only know up to the thousands place, most of them know or have heard of the billions place (and beyond). Next, ask them to say the number in words, which some students may have show difficulty — particularly the “seven-hundred forty one thousand” part. Then, write the number 1 and have students tell you the next numbers in the sequence. When students cease to be able to add mentally in their head, allow them to continue the sequence using a pocket calculator. Have the students take turns computing the next number with it until the calculator overflows. Tell the class that when the numbers get so big, you need a computer or several computers to figure them out. One boy asked: “what if you got 20 grains every minute for 30 days — would that be more?” I said, “Great question: I think you’d still get more by doubling though.” The calculator showed him this to be true — his method accumulated only 864,000 grains.
Extension: Following this activity, the teacher could pose the following question: Would you rather get one cent a week doubled every week starting now, or each week receive one dollar for every year of your age?
For Grade 9-10: Have students create a spreadsheet, graph and equation to calculate how much rice Rani would be given on day x, measuring by grain, bowls, bags, baskets and storehouses.
Assessment:
Students can be assessed by a checklist of whether or not they can remember the place values and can articulate them, as well as their participation in the activity.
If the grade level is higher, then students will have more problem solving to do and explain how they developed their formula to calculate the amount of rice Rani would receive for day x.
Why this task is worthwhile:
This task can introduce technology into the classroom to help develop number sense, explore and demonstrate mathematical relationships and patterns, allowing students to organize and display data. Because 1,073,741,823 is such a big number to wrap your head around, the technology can help decrease the time spent on calculations and show the exponential growth of doubling in a graph. To further reinforce the concept, students can be assigned a particular day and draw a picture of how much rice Rani was rewarded. The pictures can then be displayed in sequence to further show the effect of doubling, powers of two, geometric sequences, geometric series, and exponents.
This story can also open discussion to math’s origins around the world.
If you search the Internet for “One Grain of Rice” you’ll find other ideas on how to incorporate this story into a math lesson. For example, this worksheet uses graphs and logarithms: http://www.utdanacenter.org/highered/alg1/downloads/IV-B-CourseContent-AlgI/AlgI_3-3-2.pdf
Source(s):
NCTM. 2008. “One Grain of Rice,” in Math II—Unit 10: Exponential Functions, Modelling Exponential Growth. Retrieved from: http://www.gwinnett.k12.ga.us/PhoenixHS/math/grade10/Unit05/Exponential_Task.pdf Regan, R. 2011. 1,073,741,823 Grains of Rice. Retrieved from: http://www.exploringbinary.com/1073741823-grains-of-rice/ Willis, J. 2010. Learning to Love Math: Teaching Strategies that Change Student Attitudes and Gets Results. Alexandria, VA: ASCD. Pg. 101.
· Number Sense
· Patterns and Relations
Specific Learning Outcomes (Grade 1-6):
· Demonstrate an understanding of place value, including numbers that are:
• Greater than one million
• Less than one thousandth.
[C, CN, R, T]
· Determine the pattern rule to make predictions about subsequent elements. [C, CN, PS, R, V]
· Represent and describe patterns and relationships, using graphs and tables. [C, CN, ME, PS, R, V] [ICT: C6–2.3]
· Demonstrate an understanding of the relationships within tables of values to solve problems. [C, CN, PS, R] [ICT: C6–2.3]
Materials/Resources (depending on grade level):
· One Grain of Rice: A Mathematical Folktale by Demi
· Computers or calculators
· Spreadsheet program
Description:
Read to the class the children’s book, One Grain of Rice: A Mathematical Folktale by Demi:
For Grade 1: Show what 1,073,741,823 looks in numerals and name each place value.
For Grade 3: Read the story and spent about fifteen minutes afterwards discussing the numbers. Write 1,073,741,823 on the board and review the place values. Even though they may only know up to the thousands place, most of them know or have heard of the billions place (and beyond). Next, ask them to say the number in words, which some students may have show difficulty — particularly the “seven-hundred forty one thousand” part. Then, write the number 1 and have students tell you the next numbers in the sequence. When students cease to be able to add mentally in their head, allow them to continue the sequence using a pocket calculator. Have the students take turns computing the next number with it until the calculator overflows. Tell the class that when the numbers get so big, you need a computer or several computers to figure them out. One boy asked: “what if you got 20 grains every minute for 30 days — would that be more?” I said, “Great question: I think you’d still get more by doubling though.” The calculator showed him this to be true — his method accumulated only 864,000 grains.
Extension: Following this activity, the teacher could pose the following question: Would you rather get one cent a week doubled every week starting now, or each week receive one dollar for every year of your age?
For Grade 9-10: Have students create a spreadsheet, graph and equation to calculate how much rice Rani would be given on day x, measuring by grain, bowls, bags, baskets and storehouses.
Assessment:
Students can be assessed by a checklist of whether or not they can remember the place values and can articulate them, as well as their participation in the activity.
If the grade level is higher, then students will have more problem solving to do and explain how they developed their formula to calculate the amount of rice Rani would receive for day x.
Why this task is worthwhile:
This task can introduce technology into the classroom to help develop number sense, explore and demonstrate mathematical relationships and patterns, allowing students to organize and display data. Because 1,073,741,823 is such a big number to wrap your head around, the technology can help decrease the time spent on calculations and show the exponential growth of doubling in a graph. To further reinforce the concept, students can be assigned a particular day and draw a picture of how much rice Rani was rewarded. The pictures can then be displayed in sequence to further show the effect of doubling, powers of two, geometric sequences, geometric series, and exponents.
This story can also open discussion to math’s origins around the world.
If you search the Internet for “One Grain of Rice” you’ll find other ideas on how to incorporate this story into a math lesson. For example, this worksheet uses graphs and logarithms: http://www.utdanacenter.org/highered/alg1/downloads/IV-B-CourseContent-AlgI/AlgI_3-3-2.pdf
Source(s):
NCTM. 2008. “One Grain of Rice,” in Math II—Unit 10: Exponential Functions, Modelling Exponential Growth. Retrieved from: http://www.gwinnett.k12.ga.us/PhoenixHS/math/grade10/Unit05/Exponential_Task.pdf Regan, R. 2011. 1,073,741,823 Grains of Rice. Retrieved from: http://www.exploringbinary.com/1073741823-grains-of-rice/ Willis, J. 2010. Learning to Love Math: Teaching Strategies that Change Student Attitudes and Gets Results. Alexandria, VA: ASCD. Pg. 101.
Are we happy?
Broad topic area(s):
· Patterns and Relations: Patterns
· Statistics and Probability: Data Analysis
Specific Learning Outcomes (Grade 6):
1. Represent and describe patterns and relationships, using graphs and tables. [C, CN, ME, PS, R, V] [ICT: C6–2.3]
2. Demonstrate an understanding of the relationships within tables of values to solve problems. [C, CN, PS, R] [ICT: C6–2.3]
3. Create, label and interpret line graphs to draw conclusions. [C, CN, PS, R, V]
4. Select, justify and use appropriate methods of collecting data, including:
• Questionnaires
• Experiments
• Databases
• Electronic media.
[C, CN, PS, R, T] [ICT: C4–2.2, C6–2.2, C7–2.1, P2–2.1, P2–2.2]
5. Graph collected data, and analyze the graph to solve problems. [C, CN, PS, R, T] [ICT: C6–2.5, C7–2.1, P2–2.1, P2–2.2]
Materials/Resources:
· Paper
· Pencils/Pens
· Spreadsheet Program or graph paper
· SMARTboard or White board
Description:
Teacher initiates discussion about how feelings change from moment to moment and you can measure or estimate your disposition towards optimism or pessimism based on the mean of how you feel throughout a day, a week, a month or over the course of a year.
If students struggle with working in groups, then the teacher can initiate discussion about the importance of working together and develop guidelines (lists of agreed behaviours) with students for showing respect and what negative comments to avoid.
If the class is starting a large research or inquiry project, then the teacher can initiate discussion about how self-reflection involves examining both the cognitive thinking domain and the affective feeling domain.
Incorporate student’s opinions and feedback into construction of a survey. Include questions that encourage reflection such as:
On a scale of 1 to 5, with 1 being poor and 5 being excellent:
1. How do you rate our classroom atmosphere?
2. What would you like the class atmosphere to be?
3. Has my behaviour in our class improved the classroom climate? (1=agree; 5=disagree)
Have students complete a feeling/cooperation/self-reflection survey which scores how happy they feel at several different times of the day or different stages of the project. On a 10 or 5 point scale, students rate their feelings, which can then be put into a table and graphed over time.
The graph can be an individual graph for the student’s responses, or with the permission of the students, a class graph comparing everyone’s responses (with an option of anonymity). This could be done traditionally on paper or digitally and then printed, if the class agrees they want a permanent reminder in the classroom.
Depending on the authentic purpose the teacher uses for the activity, the class can observe where it is common to feel up or down at various stages of inquiry, or how their feelings carry over from one day to the next, or how their cooperative behaviours have increased or decreased according to each individual’s effort.
Students will interpret the data and identify areas for growth, as well as insight into social and emotional intelligence.
This activity can also be used for different purposes over the course of the year, to look at social status and bullying behaviours, providing a means for students to document and analyze both in their class and in the school (by distributing questionnaires) the wellbeing of their community. Students must take into consideration validity and reliability of their methods, and be able to justify their methods in collecting data.
Assessment:
This activity can be used as formative assessment, self-reflection and a diagnostic tool.
Why this task is worthwhile:
Students learn that the touchy-feely or qualitative descriptions are also quantifiable. This is an authentic and meaningful activity that shows students how math can be applied to solve problems into their own lives, in this case, to better the atmosphere that affects how they learn in the classroom. This is especially applicable if students come with negative attitudes towards math and the class behaviour has not been agreeable.
Source(s):
Rectanus, C. 2006. So You Have to Teach Math? Sound Advice for Grades 6-8 Teachers. Sausalito, CA: Math Solutions Publications. Pgs. 32-36.
· Patterns and Relations: Patterns
· Statistics and Probability: Data Analysis
Specific Learning Outcomes (Grade 6):
1. Represent and describe patterns and relationships, using graphs and tables. [C, CN, ME, PS, R, V] [ICT: C6–2.3]
2. Demonstrate an understanding of the relationships within tables of values to solve problems. [C, CN, PS, R] [ICT: C6–2.3]
3. Create, label and interpret line graphs to draw conclusions. [C, CN, PS, R, V]
4. Select, justify and use appropriate methods of collecting data, including:
• Questionnaires
• Experiments
• Databases
• Electronic media.
[C, CN, PS, R, T] [ICT: C4–2.2, C6–2.2, C7–2.1, P2–2.1, P2–2.2]
5. Graph collected data, and analyze the graph to solve problems. [C, CN, PS, R, T] [ICT: C6–2.5, C7–2.1, P2–2.1, P2–2.2]
Materials/Resources:
· Paper
· Pencils/Pens
· Spreadsheet Program or graph paper
· SMARTboard or White board
Description:
Teacher initiates discussion about how feelings change from moment to moment and you can measure or estimate your disposition towards optimism or pessimism based on the mean of how you feel throughout a day, a week, a month or over the course of a year.
If students struggle with working in groups, then the teacher can initiate discussion about the importance of working together and develop guidelines (lists of agreed behaviours) with students for showing respect and what negative comments to avoid.
If the class is starting a large research or inquiry project, then the teacher can initiate discussion about how self-reflection involves examining both the cognitive thinking domain and the affective feeling domain.
Incorporate student’s opinions and feedback into construction of a survey. Include questions that encourage reflection such as:
On a scale of 1 to 5, with 1 being poor and 5 being excellent:
1. How do you rate our classroom atmosphere?
2. What would you like the class atmosphere to be?
3. Has my behaviour in our class improved the classroom climate? (1=agree; 5=disagree)
Have students complete a feeling/cooperation/self-reflection survey which scores how happy they feel at several different times of the day or different stages of the project. On a 10 or 5 point scale, students rate their feelings, which can then be put into a table and graphed over time.
The graph can be an individual graph for the student’s responses, or with the permission of the students, a class graph comparing everyone’s responses (with an option of anonymity). This could be done traditionally on paper or digitally and then printed, if the class agrees they want a permanent reminder in the classroom.
Depending on the authentic purpose the teacher uses for the activity, the class can observe where it is common to feel up or down at various stages of inquiry, or how their feelings carry over from one day to the next, or how their cooperative behaviours have increased or decreased according to each individual’s effort.
Students will interpret the data and identify areas for growth, as well as insight into social and emotional intelligence.
This activity can also be used for different purposes over the course of the year, to look at social status and bullying behaviours, providing a means for students to document and analyze both in their class and in the school (by distributing questionnaires) the wellbeing of their community. Students must take into consideration validity and reliability of their methods, and be able to justify their methods in collecting data.
Assessment:
This activity can be used as formative assessment, self-reflection and a diagnostic tool.
Why this task is worthwhile:
Students learn that the touchy-feely or qualitative descriptions are also quantifiable. This is an authentic and meaningful activity that shows students how math can be applied to solve problems into their own lives, in this case, to better the atmosphere that affects how they learn in the classroom. This is especially applicable if students come with negative attitudes towards math and the class behaviour has not been agreeable.
Source(s):
Rectanus, C. 2006. So You Have to Teach Math? Sound Advice for Grades 6-8 Teachers. Sausalito, CA: Math Solutions Publications. Pgs. 32-36.