Creating Mathematical Conversations using Open Questions

Marian SmallSydney

August, 2015

#LLCAus#LoveLearning@LLConference

Let’s warm up

• The answer to a question is 100.• What might the question have been?

Maybe

• What is 10 x 10?• What comes after 99?• How old are you when you are REALLY old?• What is a number in the pattern 10, 20, 30,…?• What is a number you can represent with

one base ten block?

Agenda

• Why open-ended?• We will look at lots of examples• We will look at strategies to create them.• We will create them.

Why open-ended?

• To be accessible• To challenge• To evoke rich conversations

Balance

• You put yellow cubes on one side of a pan balance.

• You put blue cubes on the other side.• What would make the balance look like this?

We might discuss

Which might work?• 1 and 2?• 5 and 3?• 9 and 1?• What sort of relationship are we looking for?

Patterns

• The 10th shape in a pattern is .• What could the pattern be?

We might discuss

• Is it repeating?• If it is, how long could the core be?• If the core is 3, where would the triangles in

the core be?• Could it be a growing pattern?

Representing numbers

• A two-digit number is represented with twice as many one blocks as ten blocks.

• What number could it be?

We might discuss

• Are you allowed to have 5 ten rods and 10 ones?

• What do you notice about the possible numbers: 12, 24, 36, 48, 60,….

Adding mentally

• Two numbers are really easy to add in your head. What might they be?

We might discuss

• Why is it easy to add 0?• 1?• 10?• 100?• What else is easy?

Words

• A sentence has 40 letters in it.• What number of words do you think it has?

Why?

We might discuss

• What assumptions we make• How valid those are• The fact that we can’t know

Fill in the blanks

• 5 of ______ is less than 2 of ______.• What amounts could go in the blanks?

We might discuss

• Is the first number more or less than the second? Why?

• Is it more than half or less than half? Why?

Addition puzzle

• You add two numbers.• The answer is twice as much as if you

subtract them.• What might the numbers be?

We might discuss

• Why it works for so many numbers.• What visual might show us why it is true• What algebra might show us why it is true

Visual

10 10 10

10

Visual

5 5 5

5

Visual

8 8 8

8

Algebra

• x + y = 2y – 2x, so• 3x = y

• x + 3x = 4x• 3x – x = 2x

Square tiles

• Use the squares.• Make a design that is ALMOST half red.

We might discuss

• How to start with half and half and add either one more of another colour or subtract one red.

Multiplication

• You multiply two numbers and the product is ALMOST 400. What could the numbers have been?

We might discuss

• Why we can’t be sure what almost means• How if you start with, e.g. 2 x 200, you are

more likely to change the 200 than the 2• How you change• Whether you go up or down

Rectangles

• A length is four times the width of a rectangle.

• What do you notice about the relationship between perimeter and width?

We might discuss

• Why the perimeter is always 2 ½ of the length.

I can see that

I can see that

I might ask….

• You add 2 fractions.• The sum is []/15.• What fractions might you have added?

Someone might say

• 3/15 + 6/15 OR• 1/3 + 2/5 OR• 2/3 + 1/15 OR• 3/5 + 2/15 OR• 2/4 + 16/32 (if the answer is 15/15)

We would discuss

• That we usually use a common multiple as the denominator when we add two fractions, but not necessarily.

A problem I tried in Grade 4

We would discuss

• Some properties were easier to use than others (e.g. a very small angle OR angle bigger than a right angle OR symmetry) to make the problem accessible.

• Could talk about what combos didn’t work• Could talk about what automatically

happened when certain combos were chosen

Or

• Certain properties don’t mix (e.g. some, but not all, equal side lengths AND four equal side lengths).

• Some properties automatically come together (e.g. 4 equal side lengths and some parallel sides)

Adding fractions

• You add two fractions less than 1.• The answer is a little less than 5/4.• What might the fractions have been?

We could discuss

• Why 9/10 + 1/4 (or something like that works)

• Why at least one fraction has to be more than ½

• Why both fractions can be more than half, but not if one is too close to 1

Consider this problem

• A sweater was on sale, 40% off.• A pair of pants was on sale, 20% off.• The sale prices were the same.• How did the original prices compare?

Could be solved numerically

• If the original sweater price was $100, the sale price is $60.

• If $60 is 80% of the pants price, 20% is $15, so 100% is $75.

Could be solved numerically

• If the original sweater price was $50, the sale price is $30.

• If $30 is 80% of the pants price, 40% is $15, so 100% is $15 + 15 + 7.50 = $37.50.

• The pants price is 3/4 of the sweater price.

Could be solved algebraically

• 6/10 s = 8/10 p• 6s = 8p• 3s = 4p• So p/s = 3/4 , so the pants cost 3/4 of the

sweater.

Could be solved visually

We could discuss

• When you know a relationship between percents, you can’t know the absolute values, only relative values.

Right triangles

• One side of a right triangle is 10 cm long. What might be the lengths of the other two sides?

We might discuss

• Whether or not the triangle could be isosceles

• The idea that the 10 could be a leg or a hypotenuse

Multiplying integers

• You multiply two integers. The result is about 50 less than one of them.

• What might they be?

We could discuss

• Why it is likely one is positive and one is negative, but

• Why it could be 0 and 50 or 0 and 49, etc.• Why there can’t be a really big positive

integer

Algebra

• Create a story that you would represent with the equation 3x + 5y = 60.

Maybe

• I build triangles and pentagons with toothpicks. I used 60 toothpicks. How many of each?

• Some kids were in groups of 3 and some in groups of 5. There were 60 kids. How many of each size group?

• I bought a bunch of notebooks that each cost $3 and a bunch of packs of paper that each cost $5. I spent $60. How many notebooks and how many packs of paper?

Maybe

• I bought 3 identical pairs of shorts and 5 identical t-shirts and spent $60. How much did each cost?

• There were 3 identical groups of girls and 5 identical groups of boys. There were 60 kids. How many girls were in a group? How many boys in a group?

We could discuss

• that 3x can either mean a lot of 3s (x of them) or 3 of the same thing.

Volume

• A cone and cylinder have the same volume.• The cylinder is taller.• Create possible dimensions.

We could discuss

• The relationship forced by the formulas between the radii if the heights are the same

• The relationship forced by the formulas between the heights if the radii are the same

Powers

• Write 88 as the sum of powers in different ways.

We could discuss

• How using the exponent 1 can be very helpful

Linear Relations

• A system of equations is graphed. The solution is a point in Quadrant II.

• What might the equations be?

We might discuss

• Why if you know the solution is, e.g. (–3,1), it is easy to come up with equations numerically.

• e.g. since –3 = 3 x (–1), one equation is x = 3y.• Since 2 x (–3) + 4 x (1) = –2, another equation

is 2x + 4y = –2, etc.

Mathletics tasks

Task

Video and interactive

• Let’s check them out

Opening up Questions

• Start with the answer. Ask for the question.• The answer is “a square”.• The answer is “2/3”.• The answer is √10.• The answer is 4x – 2.

Opening up Questions

Similarities and differences• How are 7 and 10 alike and different?• 350 and 550?• 3x and 2x• Adding and subtracting

Opening up Questions

Choose your own values for the blanks.• Add 5[] + []9• [][] is a little less than 4[]. What can go in the

blanks?• [] is about 25 less than []. What values can go

in the blanks?

Opening up Questions

Use “soft” words.• The square root of a number is ABOUT 30.

What could it be?• The quotient of two numbers is a LITTLE LESS

than 20. What could they be?• A line is VERY steep. It goes through (4,2).

What could the equation be?

Your turn

• Choose three curriculum topics.• Create different sorts of rich open questions

you could use.

• We will share!

Download

• You can download this atwww.onetwoinfinity.caSydney2