Bancroft - One, Two.. Infinity
Transcription
Bancroft - One, Two.. Infinity
Leading effective math instruction Marian Small April 2015 Agenda • Planning a classroom problem • Going into the classroom • Debriefing what happened • Issues you raised: 2 Agenda • Value of manipulatives • Open-ended questions vs practice; how many • Where do they fit? Before or after direct instruction. 3 Agenda • Value of math talk/documenting/ translating to report cards and grades • Keeping proportional reasoning going • Consolidation 4 Possible problems • Jason has a box of books. • Trisha has 2 boxes. • Alyson has 3 boxes. • All the boxes hold the same number of books. • Decide how many books you want in a box. • Tell how many books there are altogether. Possible problems • Repeat trying different numbers of books in each box. • What do you notice? Consolidation • Who had the most books: Jason, Trisha or Alyson? Does it depend on how many books are in the box? • Did Alyson have twice as many as Trisha or not? • Were your total numbers of books even or odd? • Why do you think that happened? Consolidation • How did the total number compare to how many books Trisha had? • Why do you think that happened? • Could the total have ended up with 60 books? How about 50? Minds on • Andrew picked a number. • Zayden picked one twice as big. (We’ll make sure they understand twice as big.) • If you add the two numbers, what do you notice? Another problem • A yellow pattern block is worth 10. • Build a design worth more than 30 but less than 40. • It should be closer to 40 than 30. Consolidate • • • • • How do you know it’s worth more than 30? How do you know it’s worth less than 40? Why is it not all yellow? How many colours could it be? How did you know it’s closer to 40? Consolidate • How do you know your design is worth more than 30? • How do you know your design is worth less than 40? • Why is it not all yellow? • How many colours could it be? Consolidate • How did you decide whether it was closer to 30 or to 40? • How would you change your design to be worth between 60 and 80? Minds On • If a yellow block were worth 12, what would a red be worth? A blue? A green? An alternate problem • Use linking cubes to make rectangles. • The perimeter of the rectangle has to be three times as much as the length. • What do you notice about how the length and width compare? Consolidating • Was your perimeter an even or odd number? • Did it have to be? • Was your length an even or odd number? • Did it have to be? • What did you notice about how the width compared to the length? • Why does that make sense? Minds on • One number is three times as much as another number. • What could those numbers be? Another choice • A number is represented using base ten blocks. • There are twice as many rods as flats. • There are twice as many unit blocks as rods. • What could the number be if it’s less than 1000? • Think of lots of possibilities. Consolidating • Was your number of rods even or odd? Why? • What’s the most number of flats you could use? Why? • What’s the least number of units blocks you could have used? Why? • Were your numbers even or odd? Why? • If a number works, does its double? Minds On • A number is shown with 6 ones and a lot of rods. • What could it be? My last choice • Three events took place between 10:30 am and 11:30, one right after the other, with 5 minutes between them. • The second event was twice as long as the first one. • The third event was twice as long as the second one. • When could each have started and ended? Consolidation • Could the shortest event have been 20 minutes? Why or why not? • Could the longest event have been 20 minutes? Why or why not? • Could the shortest event have been 10 minutes? Why or why not? Consolidation • Is there a shortest amount of time for each event? • Is there a longest? • When did your events start and end? Minds On • How much time is your walk if you started at 10:45 and ended at 11:30? Manipulatives to support PR • How can we effectively use a variety of manipulatives to support proportional reasoning? Using linking cubes • For perimeter/area problems For example • Use a train of linking cubes to build the width of a rectangle. • Then use a train to build a length that is four times as long as the width. • Create the rectangle with that length and width. • How does the perimeter relate to the length? the width? • What if the length were five times as long as the width? For example • Build multiple “trains” of five cubes. • Use different numbers of those trains to build polygons. • How are the perimeters changing? • Why does that make sense? • How are the areas changing? • Why does that make sense? • Measure the angles in the corners. • What do you notice? Using linking cubes • Relating to mean and median and mode For example • Make trains of 8, 6 and 13 cubes, each a different colour. • How can you use your cubes to find the mean of those three numbers? Using linking cubes • To show ratios For example • Build a linking cube train to meet each condition. • It shows the ratio 4:5, but also the ratio 4:9. • It shows the ratio 4:5, but also the ratio 3:4. • It shows at least six ratios. Using linking cubes • To show factors and multiples For example • Use linking cubes to make it easy to see that 4 x 5 = 20. • Use linking cubes to make it easy to see that 4 is a factor of 28. • Use linking cubes to make it easy to see that 24 is a common multiple of 3 and 8. • Use linking cubes to make it easy to see that 3 is a common multiple of 21 and 15. Cuisenaire rods • One rod is twice as long as another. • What could the rods be? Cuisenaire rods • How could you used rods to show that 6 is a factor of 12? • That 24 is a multiple of 4? • That 30 ÷ 6 = 5? Cuisenaire rods • It takes 8 of one rod to make the same length as 3 of another. • What could the rods be? • What other rods (of one colour) could make the same length? Cuisenaire rods • What is the probability that a random three Cuisenaire rods could make a triangle? Square tiles • How could you show 3 x 4 using the tiles? • How could you see that 3 x 4 = 6 x 2 using the tiles? • How could you show the fraction 2/5 using the tiles? How else? • How could you make a design more than ½ red and more than ¼ blue using tiles? How can you tell you are right? Open-ended questions • vs practice; how many • Where do they fit? Before or after direct instruction. Tracking math talk • What are you listening for? • How much do you have to document? • Can you grade it? Where is PR in other strands? • Have a look at my presentation called PVNCC Consolidation • What should be involved? • You saw many examples in my sample problems. • For every expectation, you could decide what is important and focus the expectation on those ideas. Grade 8 • solve problems involving percent that arise from real-life contexts What ideas do you want to come out? • I want students to realize that solving percent problems ALWAYS involves coming up with an equivalent ratio. • I want students to use reasonableness to check their answers. Grade 7 • model real-life relationships involving constant rates (e.g., speed, heart rate, billing rate), using algebraic equations with variables to represent the changing quantities in the relationship What ideas do you want to bring out? • I want students to know WHY graphs of constant rates go through the origin; why the rate is the y-coordinate when x = 1; and why and how the size of the rate affects the steepness of the line and the numbers in the table Grade 6 • multiply and divide decimal numbers to tenths by whole numbers, using concrete materials, estimation, algorithms, and calculators What ideas do you want to bring out? • I want students to understand why dividing or multiplying with decimals is the SAME process as with whole numbers; it’s about a unit change. Grade 5 • demonstrate and explain the concept of equivalent fractions, using concrete materials What ideas do you want to bring out? • I want students to realize why and that the multiplicative relationship between numerator and denominator of equivalent fractions are identical. Grade 4 • describe, through investigation, the relationship between various units of length What ideas do you want to bring out? • I want students to realize that knowing how many of unit A makes unit B allows the same process to be used to change units no matter how many of the units there are. An expectation Grade 1: • Demonstrate an understanding of the use of non-standard units of the same size for measuring Possible learning goal • When you measure an item with a nonstandard unit, the measure depends both on the item and the size of the unit. So…. • I might ask kids, How many baby steps make a giant step? Consolidation • • • • Who got a big answer? Why do you think your answer was big? Who got a small answer? Why do you think your answer was small? Consolidation • What else could you measure the length of to get a big answer? How would you measure it? • What else could you measure the length of to get a small answer? How would you measure it? Grade 2 • Add and subtract money amounts to 100¢ using a variety of tools and strategies Possible learning goal • That subtraction might help us solve different kinds of money problems. My problem • I subtracted 35¢ from 75¢ to answer a problem about money. • Create several problems that sound really different that I might have been solving. To consolidate • Let’s hear a problem. • What is it about your problem that led you to subtract? • Who has one that is really different? What makes it different? To consolidate • Did anyone have a problem that really is a subtraction but sounds like an addition? • Did anyone have a problem that involved two people in it? And then I might ask… • I had 37¢. • I bought something that cost 10¢. • Do I find out what’s left by adding or subtracting? Grade 3 • represent, compare, and order whole numbers to 1000, using a variety of tools I might ask • Choose a 2-digit or 3-digit number. • Represent it as many ways as you can. • Tell one thing each representation of the number helps you know about that number. For example • If I showed 7 in a ten-frame, I am helping you see that 7 is 3 less than 10 and also that it is 2 more than 5. But • If I showed 7 this way, I am helping you see that it is odd and that it is 4 + 3. To consolidate • My entire focus is on what each representation does or does not show. Or you might ask • A number is shown with base ten blocks. • There are a lot more ones than tens. There are a few more tens than hundreds. • What might the number be? My consolidation • How many digits could the number have? • Do there have to be hundreds? • Are there more tens or more hundreds? Why? My consolidation • Could the number 71 work? Why or why not? • Could the number 140 work? • How would you have to represent it to make it work? My consolidation • Give me another number that can’t work. • Give me another number that can. Download at www.onetwoinfinity.ca Bancroft