Big Ideas & Better Questions

Marian SmallApril, 2009

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Agenda

• Linking pattern and algebra• Measurement• Fraction computation• Proportional reasoning

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Linking Pattern and Algebra

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What is algebra?

• If you had to define algebra, what would you say?

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Some might say…

• Algebra is a way to represent and explain mathematical relationships and to describe and analyze change.

• It is invariably about “generalizing”.

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For example…

• If I say… What comes next in the pattern 4, 7, 10, 13,…., that is not algebra.

• If I say… What is the 100th number in the pattern 4, 7, 10,…. and you figure it out without doing all 100 terms, that is algebra.

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Is there more than one way?• You bet!• Someone might say: - I notice that the number endings go: 4, 7,

0, 3, 6, 9, 2, 5, 8, 1, and then repeat. So the 100th one must end in 1. Since you add about 300 to 4 and it has to end in 1, it’s probably 301.

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Is there more than one way?• Or….• You add 99 x 3 to 4, so it must be 297 + 4

= 301.

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Is there more than one way?• Or….• You say the numbers in this pattern are

each 1 greater than the numbers in the pattern 3, 6, 9, 12,….. The 100th number there is 300, so it must be 301.

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Is there more than one way?• Or….

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If….

• The pattern had been 3, 7, 11, 15,…

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You might ask a student

• Why might you think differently to figure out what the 88th number is than the 6th number?

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The Handshake Problem

• Create a table of values to show the number of handshakes for 2, 3, 4, 5, … people if each shakes hands once with each other person.

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How can you predict?

• How can you now predict the number of handshakes if there are 100 people?

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Table of Values

2 3 4 5 6 7 8 9 10

1 3 6 10 15 21 28 36 45

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P

H

Look at the Doubles

2 3 4 5 6 7 8 9 10

1 3 6 10 15 21 28 36 45

2 6 12 20 30 …

1x2 2x3 3x4 4x5 5x6

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Or Visually

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And again…

• How did what we did show that we used algebra?

• How was using algebra useful?

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Some questions to ask• How does the greatest number of

handshakes for one person in a group relate to the number of people in the group? Why?

• One number in the pattern for the total number of handshakes is 20 more than another one. How many more people were shaking hands that time?

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Some questions to ask

• Why doesn’t the number of handshakes go up by the same amount each time?

• Jeff said that if there are 10 people, they each shake hands with 9 other people. That’s 10 groups of 9, or 90 handshakes. Do you agree? Explain.

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A problem

• Look for patterns in a 100 chart. Describe four different patterns that you see.

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Possible Follow-up• Look for patterns in a 100 chart. Describe four

different patterns that you see.

• Would the pattern continue if you went past 100?

• Would the pattern be the same if there had been 5 columns? 6 columns?

• Why did that pattern work?

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Possible Follow-up• The sum of two consecutive numbers is 967.

Show two ways to determine the numbers. Why could the sum not be 968?

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Possible Follow-up• The sum of two consecutive numbers is 967.

Show two ways to determine the numbers. Why could the sum not be 968?

• You add two consecutive numbers and get a sum. You double the two consecutive numbers. What happens to the sum? Why?

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Possible Follow-up• The sum of two consecutive numbers is 967.

Show two ways to determine the numbers. Why could the sum not be 968?

• You add three consecutive numbers. The sum is 303. What are they?

• You add three consecutive numbers. What is true each time?

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Possible Follow-up• The sum of two consecutive numbers is 967.

Show two ways to determine the numbers. Why could the sum not be 968?

• What if you had used algebra to describe the situation; should you call the first number n and the next one n+1 or should you use n-1 and n? Does it matter?

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Possible Follow-up• The sum of two consecutive numbers is 967.

Show two ways to determine the numbers. Why could the sum not be 968?

• What kinds of numbers do you get if you add consecutive odd numbers? Consecutive even numbers?

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Or this…

• A pattern is built by adding in some way to get from one term to the next term. There is a 10 somewhere between the fourth term and the tenth term. What could the pattern be? Think of as many possibilities as you can.

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Or this..

• Draw pictures that might help someone predict the next four terms of 1, 4, 9, 16,….

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More Open Examples

• You colour squares to form a capital letter on a 100-chart. If you add the values the letter covers, the sum is between 100 and 120. What and where could the letter be on the chart?

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1 2 3 4 5 6 7 8 9 10

11 13 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

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More Open Examples

• An expression involving the variable k has the value 20 when k = 6. What could the expression be?

• There is an input/output machine. When you put in a 6, out comes a 20. What could the expression be?

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More Open Examples

• If [] = 4 and * = 5, these statements are all true:

• 3([] + *) = 27• 2[] + 3* = 23• 2x* – 2x[] = 2.

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When is each true?

• How many solutions to:- a number + 5 = 10- a number + 5 = 5 + the number- a number squared = 25- a number is 2 more than another numberDo these all show number relationships?

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Other “versions” of algebra• P = 2l + 2w• a/c + b/c = (a+b)/c

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Measurement

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What measurement ideas do we consider in

7-10?

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Include….

• Metric conversions• Areas of trapezoids/composite shapes• Circumference and area of circles• Angle measurement “stuff”• Pythagorean theorem• Volumes/surface areas

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Include….

• “Optimal” rectangles• Similarity and trig stuff

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What do you think?

• What do you think are THE MOST important ideas about these topics that students learn?

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A possible point of view

• The unit chosen affects the measurement; you can be more precise using a smaller unit.

• Measurement formulas allow us to use simpler measurements to access more complicated ones.

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A possible point of view

• Various measures of a shape can be independent.

• Knowing one measurement can sometimes help you figure out another one.

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Metric conversions

• The area of a piece of land is 4 000 000 cm2. Is it big enough for a shopping mall?

• I want students to know that the number of square metres must be smaller than square centimetres. How do we help them see it is 1/10 000 as many?

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Or….

• You want a measurement that isn’t really that big to sound big. What might you do (without lying)?

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Areas of trapezoids…

• Melissa said that as long as you know the formula for areas of triangles, you can figure out the area of any polygon.

• Do you agree? Explain.

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Is there more than one?

• We all know the formulas for the areas of rectangles, parallelograms, trapezoids, triangles.

• Are there other area formulas for 2-D shapes?

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Build a shape

• On your grid paper, build shapes where the vertices are ONLY at lattice points.

• Build shapes with these characteristics and I’ll predict the areas without even seeing your shape.

• 1: 4 points on border and 1 inside• 2: 6 points on border and 2 inside• 3: 10 points on border and 5 inside

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Pic’s formula

• A = B/2 + I - 1

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Circle stuff

• Jeff said that the circumference of a circle can be close to 10, but not exactly 10. Do you agree? Explain.

• Erica said that the number of centimetres in the circumference of a circle is usually less than the number of square centimetres in the area. Do you agree? Explain.

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More circle stuff

• Lianne said that a square and circle can never have exactly the same area. Do you agree? Explain.

• Ryan said that if a circle is just as wide as a square, the area will be less. Do you agree? Explain.

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Angle measure stuff

• When you cut up a pentagon into triangles, you can see that the sum of the angles is 540°. Can you be sure it will be the same if you cut it up a different way? Explain.

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How can you be sure ..?

• How can you be sure that these lines are parallel?

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Pythagorus

• One side of a right triangle is 5 cm long. How long might the other two sides be? How do you know?

• How do a2 + b2 and c2 compare if the triangle is not a right triangle?

• How do you know that there is only one square but many rectangles with a diagonal that is 10 cm long?

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Volumes/surface areas

• These two shapes have the same volume. Which has more surface area? How do you know?

• How do you know that there have to be more than two prisms with the same volume?

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Volumes/surface areas

• Why do you need 3 numbers to calculate the volume of a rectangular prism, but only 2 to calculate the volume of a cylinder?

• Can a cylinder have exactly the same volume as a cube? How?

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Volumes/surface areas

• Why is it faster to calculate the surface area of a cube than the surface area of a cylinder?

• Why does the volume of a cube have a fraction in the formula?

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Similarity/trig

• When I tell you ALL the angle sizes in a triangle, why can you still not be sure exactly what triangle to draw?

• Why can you tell me everything about a right triangle if I tell you either two side lengths or one angle and one side length OR can you?

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Fraction computation

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Big Ideas

We want students to realize that:• The operations mean the same thing they

always did before.• That renaming fractions is often, but not

always, key to calculating with them.

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Fraction computation

• That there are many ways to perform a calculation.

• That you can often estimate the size of the answer prior to calculation.

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Addition and subtraction

• Let’s look at these models: fraction strips, number line, grid for :

• ¾ + 1/8• 2/3 + 1/4• ¾ - 1/8• 2/3 – 1/4

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Mixed numbers

• Why might it be better to leave mixed numbers in that form, rather than as improper fractions, to add them or subtract them?

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Questions to ask

• The sum of two fractions is 8/5. Could one or both be greater than 1?

• The sum of two fractions is 8/5. What could the fractions be?

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More questions to ask

• Two fractions have a sum of 11/12 and a difference of 5/12. What could they be?

• The sum of two fractions is 5/12 . Could their denominators have been 2 and 24?

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Multiplying fractions

• Why does it make sense that 2/3 x 3/5 = 2/5 (without doing any work)?

• Why does it make sense that 1/3 x 4/5 is less than 4/5?

• Why does it make sense that the square root of a fraction between 0 and 1 is greater than the fraction?

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How can you model?

• How can you model each:2/3 x 1/62/3 x 7/61 ½ x 2 ½

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Division of fractions

• What does each mean (not what is the answer)?:

• 3/5 ÷ 1/5• 3/5 ÷ 2/5• 1 ÷ 2/5

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Questions

• How do you know that 3/5 ÷ 2/8 must be greater than 1 without doing any calculation?

• How do you know that 4/[] ÷ 2/[] = 2 no matter what the denominators are?

• How do you know that a/b ÷ c/b = a/c?

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How would you solve..

• You can finish 2/5 of your chores in ¾ hour? How long would it take you to finish all of your chores?

• Why is this a division of fractions question?

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Questions…

• Create a pair of fractions where the product is less than the quotient.

• Create a pair of fractions where the product is greater than the quotient.

• What division of fractions questions are super easy to do?

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Proportional reasoning

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What is proportional reasoning?

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You might ask…

• Choose a price for 12 cookies.• How much should you charge for 20

cookies?

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You might ask…

• 60% of the student body voted for a student council candidate. He received 387 votes. How many students are there?

• Think of 3 different ways to solve the problem.

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Let’s consider

• Diagrams• Ratio tables• Double number lines

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Diagram

387

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Ratio table

6055 30 300 90 390 3 387

100 50 500 150 650 5 645

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Double number line

0 20 40 60 80 100

387

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Use the models

• Use these models to calculate 85% of 455.

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Some questions

• You know what 40% of a number is.• How can you calculate 20%?• 35%?

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Some questions

Notice that 6 is:• 6% of 100• 12% of 50• 24% of 25• What else do you know?• Why can you halve/double?

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Some questions

• Why is it easy to calculate 32% of 50 using mental math?

• Is there another reason?• Is 30% of a number halfway between 20%

and 40%? Why or why not?

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Some questions

• Why might you not choose to calculate a unit rate to determine the price of 36 boxes of something if you know the price of 9 boxes?

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To summarize

• To ask better questions, you need to focus on what part of an expectation (or topic) really matters.

• You need to call on problem solving and reasoning and not just recall.

• You need to think of being inclusive.• You need to build connections.

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Would you try…..

• I’m hoping that you might try asking your students some of the questions we worked on today if the topic is one you are working on. Then you can let us know how it went.

• Otherwise… think about one of the big ideas you think describes the topic you are teaching and create a question to try and report on.

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Download

• You can download this presentation for about a week or so at:

• www.onetwoinfinity.ca • Quick Links/Renfrew

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