A Focus on Proportional Reasoning, Grades 4 - 8 February, 2015 - PowerPoint PPT Presentation

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. Dr. Marian Small, University of New February 2015 Brunswick

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 Dr. Marian Small, University of New February 2015 Brunswick

What does it look like? • What sorts of problems involve proportional reasoning? Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

Or.. • On average, Canadians consume 18% of their daily calories at breakfast. • Is that true in your class? Dr. Marian Small, University of New February 2015 Brunswick

Probability • You are pulling out a counter from each bag. • Which bag gives you the best chance of pulling out a red? Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

Length • How long is a line of 1 000 000 pennies? 19 cm Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

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. Dr. Marian Small, University of New February 2015 Brunswick

An EQAO video • http://www.youtube.com/watch?v=LPkQvN3r8js Dr. Marian Small, University of New February 2015 Brunswick

Let’s look at the types of problems that involve proportional reasoning that students around the province have been solving. Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

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 Dr. Marian Small, University of New February 2015 Brunswick

Problems we’ve tried • A light green Cuisenaire rod is worth 9 (or 15). • What should the other rods be worth? Dr. Marian Small, University of New February 2015 Brunswick

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 Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

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. Dr. Marian Small, University of New February 2015 Brunswick

Problems we’ve tried • What could the number be? • Think of as many numbers as you can that are less than 1000. Dr. Marian Small, University of New February 2015 Brunswick

Problems we’ve tried Dr. Marian Small, University of New February 2015 Brunswick

Problems we’ve tried • Make a design with pattern blocks that is half yellow. Dr. Marian Small, University of New February 2015 Brunswick

Maybe Dr. Marian Small, University of New February 2015 Brunswick

Maybe Dr. Marian Small, University of New February 2015 Brunswick

Maybe Dr. Marian Small, University of New February 2015 Brunswick

Problems we’ve tried • Make a design that is 2/3 red and 1/3 green. Dr. Marian Small, University of New February 2015 Brunswick

Maybe Dr. Marian Small, University of New February 2015 Brunswick

Useful manipulatives Pattern blocks • The ___ block is worth ___. What are the other blocks worth? 12 Dr. Marian Small, University of New February 2015 Brunswick

Useful manipulatives Pattern blocks • The ___ block is worth ___. Make a design worth ____. 12 44 Dr. Marian Small, University of New February 2015 Brunswick

Useful manipulatives Pattern blocks • Make a design where ¾ of the area is yellow. Dr. Marian Small, University of New February 2015 Brunswick

Useful manipulatives Cuisenaire rods • Find rods that are ½ (or 2/3 or 5/6) as long as other rods. Dr. Marian Small, University of New February 2015 Brunswick

Useful manipulatives Cuisenaire rods • One rod is 2 ½ times as long as another. What rods could they be? Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

Useful manipulatives Cuisenaire rods • What single colour rods can make a line as long as 4 orange rods? Dr. Marian Small, University of New February 2015 Brunswick

Useful manipulatives Cuisenaire rods • You measure something with orange rods. It takes 4 orange rods. How many yellow would it take? How many pink? Dr. Marian Small, University of New February 2015 Brunswick

Useful manipulatives Square tiles • What does 3 x 4 look like? Dr. Marian Small, University of New February 2015 Brunswick

Useful manipulatives Square tiles • Why did 8 x 3 have to be the same as 4 x 6? Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

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. Dr. Marian Small, University of New February 2015 Brunswick

Useful manipulatives Square tiles • Make a design that is 2/3 red and 1/4 green. Dr. Marian Small, University of New February 2015 Brunswick

Base Ten Blocks • Show any 2-digit number with base ten blocks. • Now show a number 10 times as big. Dr. Marian Small, University of New February 2015 Brunswick

Your turn • Have any of you used other manipulatives in a valuable way for proportional reasoning ? Dr. Marian Small, University of New February 2015 Brunswick

Creating good PR problems • The purpose of the problem should be to draw out proportional reasoning ideas. • Here are a number of examples. Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

You could ask: How many marbles do you think the big container could hold? Choice 2: Choice 1: 10 10 Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

You could ask: • How many ears would I draw if I draw 8 cows? • How many legs? Dr. Marian Small, University of New February 2015 Brunswick

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,…. Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

You could: Regularly use multiplicative language such as: • Twice as much • Four times as big • Half as many • Two thirds as heavy Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

Red Red Possibilities Green Blue Green Yellow Blue Green Red Green Red Red Green Green Blue Red Blue Red Green Red Green Dr. Marian Small, University of New February 2015 Brunswick

You could ask: • A sentence has 40 letters in it. What number of words do you think it probably has? Why? Dr. Marian Small, University of New February 2015 Brunswick

You could ask: • About how many ceiling tiles are there in the whole school? Dr. Marian Small, University of New February 2015 Brunswick

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. Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

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? Dr. Marian Small, University of New February 2015 Brunswick

Ministry resources • http://www.edu.gov.on.ca/eng/teachers/ studentsuccess/ProportionReason.pdf • http://www.edugains.ca/resources/ LearningMaterials/ ContinuumConnection/ BigIdeasQuestioning_ProportionalReas oning.pdf Dr. Marian Small, University of New February 2015 Brunswick