A Focus on Proportional Reasoning, Grades 4-8 February, 2015 Marian Small
Agenda What does/can proportional reasoning look like in Grades 4 8?
Agenda What have we seen Ontario students do when confronted with proportional reasoning problems?
Agenda What manipulatives are useful to evoke proportional reasoning?
Agenda Creating rich proportional reasoning problems
Proportional reasoning Proportional reasoning involves the use of multiplicative relationships to compare quantities and to predict the value of one quantity based on the values of another.
Proportional reasoning But it s not about actually seeing multiplication signs.
Proportional reasoning For example, if I ask a child what 8 cookies should cost, s/he is thinking proportionally whether adding 89 + 89 or thinking 2 x 89. 89
In the curriculum Most obvious spots Grades 4 8: sections under number entitled Proportional Relationships
But there are SO MANY more Examples Grade 4 on: any measuring activity using units If I ask you to predict how many metres long a room is if I let you see one or two metre sticks against the wall.
But there are SO MANY more How long is this room?
So what do you think? Which is your choice? Type in the chat box. 1. Measuring is always about proportional reasoning. 2. When you use a partial measurement to predict a full one, that s when you use proportional reasoning.
But there are SO MANY more Grade 4 on: exchanging coins If I ask you to show me 60 with fewer coins and you exchange 2 dimes and a nickel for a quarter, you are changing to bigger units to get fewer coins.
But there are SO MANY more Grade 4 on: multiplication/division If I ask you how many children are in a class if there are 6 tables, each with 4 children at the table. I am changing from a unit of table (6 units) to a unit of child (24 units)
But there are SO MANY more Grade 4 on: fractions Any work with fractions involves the multiplicative relationship between numerator and denominator; for example Fractions are equal to 1/2 if the denominator is twice the numerator
But there are SO MANY more Grade 4 on: fractions How far apart numerators and denominators are tells me nothing about relative size. For example, 2/3 > 3/5 (1 apart vs 2) but 1/2 < 7/9 (1 apart vs 2)
But there are SO MANY more Grade 4 on: graphs with scales Using a many-to-one correspondence on a graph (e.g. one icon represents 4 people) is proportional thinking--- thinking of a number as groups of, e.g. 4
But there are SO MANY more A circle graph or any other graph that shows fractions or percents of people in categories involves a multiplicative comparison between the part and the whole. Make a multiplicative comparison about this graph in the chat box.
But there are SO MANY more Grade 4 on: probability All probability work involves comparing, fractionally, the desired events to the total number of events e.g. the probability of rolling 1 on a die is 1 out of 6
But there are SO MANY more Grade 4 on: solving problems relating to magnitudes of 1000, etc. These problems normally involve unitizing. For example, if there are 200 sheets of paper in a pack costing $1.20, how much would 1000 sheets cost?
But there are SO MANY more Grade 4 on: area/volume formulas The area of a rectangle describes the number of squares that form equal rows of squares. 3 units of 5 squares
But there are SO MANY more Grade 4 on: area/volume formulas The volume of a prism is about the number of cubes that form equal layers of cubes. 3 layers of the area of the base
But there are SO MANY more Grade 4 on: unit conversions Determining the number of metres for 430 cm involves unit changes. There will be 1/100 as many units.
But there are SO MANY more Grade 4 on: linear patterns Determining what the 100 th term of 5, 10, 15, 20,. is involves thinking of 100 units of 5. Determining what the 100 th term of 4, 9, 14, 19, is involves thinking of 100 units of 5, less 1.
But there are SO MANY more Grade 5 on: mean of a set of data Calculating the mean is about replacing n pieces of data with n identical units; the mean is the size of that unit e.g. the mean of 3, 4, 5 is 4
But there are SO MANY more Grade 6 on: percent work Any percent is a multiplicative comparison between a number and 100. Thinking of 50% of 32 is about relating the relationship between 50 and 100 to a number and 32.
But there are SO MANY more 0 50 100 0?? 32
But there are SO MANY more Grade 6: rotation work with patterns Asking what the 50 th term of the pattern below looks like requires you to think of 50 as groups of 4.
But there are SO MANY more Grade 7 on: Two shapes are similar if the proportions relating their side lengths are maintained. 1 5 3.75 0.75
But there are SO MANY more Grade 7 on: solving linear equations by, for example, multiplying both sides by the same amount You can multiply both sides of an equation by 3 since if one item equals another, 3 of them match 3 of the other.
But there are SO MANY more Grade 7 on: linear relationships When students look at relationships between two variables and see a line, through (0,0), they are recognizing that one variable s value is always the same multiple of the other s
For example Days vs. Weeks (2,14) (1,7)
Your turn How might one of these curriculum expectations be about proportional reasoning? Gr 4: demonstrate an understandng of place value Gr 6: identify composite and prime numbers Gr 8:measure circumference and area of circles
Big Ideas of PR It is often useful to think of one amount as groups of another amount. e.g. one loonie as 4 quarters 14 days as 2 weeks 20 eggs as 1 2/3 dozen 25 as ¼ of 100
In fact Any number can be compared to any other number multiplicatively, e.g. 8 can be compared to 2 by thinking of it as 4 twos. And 2 can be compared to 8 as 2/8 (or ¼) of an 8.
In fact And 9 can be compared to 2 by thinking of it as 4 ½ twos. And 2 can be compared to 9 as 2/9 of a 9.
Comparing changes Which price changed the most? $5.99 to $2.99 $46 945.00 to $44 999.00
Related important ideas If you use a bigger unit, you need fewer of them. If units are related, you can use that relationship to predict how many of one unit if you know how many of the other.
Related important ideas How far apart numbers are additively has nothing to do with how far apart they are multiplicatively. For example, 2 and 2000 are far apart both ways. But 1000 and 2000 are only far apart additively.
Related important ideas Using a fraction, decimal or percent is a way of comparing numbers multiplicatively. For example, 2/3 tells us that 2 is only 2/3 of a 3. 0.4 is a way to compare 4 to 10 35% is a way to compare 35 to 100
What does it look like? What sorts of problems involve proportional reasoning?
Dogs 1 out of every 3 Canadian households has a dog. About how many dogs would you predict for the students in your class? How would you envision a Grade 4 solving this?
Or.. On average, Canadians consume 18% of their daily calories at breakfast. Is that true in your class?
Probability You are pulling out a counter from each bag. Which bag gives you the best chance of pulling out a red?
Speeds A car goes 280 km in 3 hours. How far, at that speed, will they go in another 1.5 hours? Why was it smart to ask about 1.5? To use 280 and not 270?
Length How long is a line of 1 000 000 pennies? 19 cm
How much faster? You normally drive 90 km/h on a certain road. How much faster would you have to go to save 15 minutes on a 400 km trip on that road?
Estimation A Fermi problem, e.g. Estimate the number of square centimetres of pizza that all of the students in Toronto eat in one week.
An EQAO video http://www.youtube.com/watch?v=lpkqvn3r8js
Let s look at the types of problems that involve proportional reasoning that students around the province have been solving.
Problems we ve tried Which sequence gets past 1000 first? 15, 25, 35, 45, 55,. 500, 502, 504, 506, 508, Why is this about proportional reasoning?
Problems we ve tried You have linking cubes to build a rectangle. The perimeter has to be three times as much as the length. What do you know about the length and width?
Problems we ve tried A yellow pattern block is worth A. Build a design worth B. Choice 1: A is 6 and B is 20 Choice 2: A is 5.1 and B is 17 Choice 3: A is ½ and B is 1 2/3
Problems we ve tried A light green Cuisenaire rod is worth 9 (or 15). What should the other rods be worth?
Problems we ve tried Make a rectangle. Figure out its perimeter. Then make a rectangle with half the area. Figure out that perimeter. P = 18 P = 12
Problems we ve tried What fraction of the big perimeter is the small one? Try more times. What fractions are possible and which are not?
Problems we ve tried You model a number with base ten blocks. There are twice as many rods as flats. There are 3 times as many unit blocks as rods.
Problems we ve tried What could the number be? Think of as many numbers as you can that are less than 1000.
Problems we ve tried
Problems we ve tried Make a design with pattern blocks that is half yellow.
Maybe
Maybe
Maybe
Problems we ve tried Make a design that is 2/3 red and 1/3 green.
Maybe
Useful manipulatives Pattern blocks The block is worth. What are the other blocks worth? 12
Useful manipulatives Pattern blocks The block is worth. Make a design worth. 12 44
Useful manipulatives Pattern blocks Make a design where ¾ of the area is yellow.
Useful manipulatives Cuisenaire rods Find rods that are ½ (or 2/3 or 5/6) as long as other rods.
Useful manipulatives Cuisenaire rods One rod is 2 ½ times as long as another. What rods could they be?
Useful manipulatives Cuisenaire rods A line of 8 of one colour rod matches a line of 5 of another colour rod. What rods could you use?
Useful manipulatives Cuisenaire rods What single colour rods can make a line as long as 4 orange rods?
Useful manipulatives Cuisenaire rods You measure something with orange rods. It takes 4 orange rods. How many yellow would it take? How many pink?
Useful manipulatives Square tiles What does 3 x 4 look like?
Useful manipulatives Square tiles Why did 8 x 3 have to be the same as 4 x 6?
Useful manipulatives Square tiles Build a rectangle with a width of 3. How does the area relate to the length? Could the perimeter be a multiple of the length?
Useful manipulatives Square tiles Build a shape with 3 times as many blue squares as yellow ones, but 2 times as many red squares as yellow ones.
Useful manipulatives Square tiles Make a design that is 2/3 red and 1/4 green.
Base Ten Blocks Show any 2-digit number with base ten blocks. Now show a number 10 times as big.
Your turn Have any of you used other manipulatives in a valuable way for proportional reasoning?
Creating good PR problems The purpose of the problem should be to draw out proportional reasoning ideas. Here are a number of examples.
You could ask: You can arrange a batch of ABOUT 50 counters into equal groups. How many groups and of what size might they be?
Follow up by asking Why did nobody have 100 groups? What was the biggest group size anyone had? Why? When did someone have a lot of groups? When did someone have a big group size? When could there be 2 groups?
You could ask: How many marbles do you think the big container could hold? Choice 1: Choice 2: 10 10
Common questions: Are there more than 10 marbles in the big container? How do you know? Do you think there are more than 20 marbles? Why or why not? Did it matter how wide the dark blue container (with 10 marbles) was? How?
Common questions: Did it matter how high the dark blue container of 10 was? How? How did you decide how many marbles? What if there had only been 5 marbles in the small can? How would your answer change?
You could ask: How many ears would I draw if I draw 8 cows? How many legs?
You could ask: How many numbers would I need to write (say) to continue this way to get to 50? 12, 14, 16, 18, 20,.
You could ask: You can show an amount of cookies exactly using groups of 6 cookies. How do you know that you can also show it exactly using groups of 3 cookies? What about using groups of 4 cookies?
You could: Regularly use multiplicative language such as: Twice as much Four times as big Half as many Two thirds as heavy
You could ask: My brother has 2.5 times as many games as I have. How many might we each have? Do you think I have 9 games?
A Colourful Spinner I spin a spinner. I am twice as likely to get red as blue. I am half as likely to get blue as green. What could the probability of green be?
Possibilities Blue Red Red Green Green Green Blue Red Green Red Green Green Blue Red Blue Red Red Green Yellow Red Green
You could ask: A sentence has 40 letters in it. What number of words do you think it probably has? Why?
You could ask: About how many ceiling tiles are there in the whole school?
You could ask: You draw a scale diagram and a m distance is represented as cm. Choose values for the blanks. Then describe how a 17 m and 3.2 m distance would be represented.
You could ask: The perimeter of one square is 1/3 as long as the perimeter of another. What do you know about the side lengths? How could you represent this?
You could ask: A jacket price is reduced by 40%. A shirt price is reduced by 20%. They end up costing the same amount (on sale). How were the original prices related?
Ministry resources http://www.edu.gov.on.ca/eng/teachers/ studentsuccess/proportionreason.pdf http://www.edugains.ca/resources/ LearningMaterials/ ContinuumConnection/ BigIdeasQuestioning_ProportionalReas oning.pdf
Ministry resources Math camppp materials on proportional reasoning http://gains-camppp.wikispaces.com/ CAMPPP+2010 http://gains-camppp.wikispaces.com/ CAMPPP+2011+Home
Message Proportional reasoning is about unitizing, grouping and counting groups, thinking of comparisons multiplicatively. Proportional reasoning comes out if you model it, talk about it, present tasks that allow for it, and encourage it.