Asking Better Questions: Fractions & Integers · Asking Better Questions: Fractions & Integers...

Asking Better Questions:Fractions & Integers

©Marian Small 2019

Today

• What is the difference between knowledge and understanding in math Gr 7 - 9 • We will see lots of examples.• We will work together to create understanding questions.

Today

• What is the difference between applicaton and thinking in Gr 7 – 9 math?• We will see lots of examples.• We will work together to create understanding questions.

How are these different?

• Knowledge: Write 5/9 as a decimal or percent.• Understanding: WITHOUT GETTING AN EXACT ANSWER, tell how you

know that 5/9 cannot be the decimal 0.42.

To build understanding

• Instead of focusing on exact values, I am focusing on meaning and estimates. • If students know 5/9 means 5 parts out of 9, and so is more than one

half, they can answer this question.


• Knowledge: Write 3/5 as a decimal and percent.• Understanding: What are possible percents equivalent to 0.3[]?


• By using a form like 0.3[] instead of being exact, allowing for a range, students still have to have a sense of what 0.3[] means in terms of part of a whole, but do not go immediately to a procedure.

What is the difference?

• Knowledge: What is 0.75 as a fraction?• Understanding: In what specific situation might it be useful to think of

0.75 as ¾?


• I focus on why we do what we do.


• Knowledge: What is 5/8 + 3/4?• Understanding: What’s wrong with this picture to show 5/8 + 3/4?


• I see if students realize what addition means, as well as that the wholes have to be the same.


• Knowledge: What is 5/8 + 3/4?• Understanding: WITHOUT ADDING, tell what a good estimate for 5/8

+ 3/4 would be and why.


• I ask for estimates and not exact answers.• For example, 4/7+ 3/4 is less than ¾ + ¾ , 1 ½, so maybe 1 1/3.• Or 4/7 + 3/4 is more than 8/7, so maybe 1 1/3.• Or ½ + ¾ is close and that is 1 ¼. • Notice I used denominators that did not make it overly simple to

get the exact answer.


• Knowledge: Find an equivalent fraction for 2/3 with a denominator of 18.• Understanding: Draw a picture to show why 2/3 can be []/18, but not

[]/17, if [] is a whole number.


• I focus, again, on how a visual shows the idea and move away from just rules about multiples.


• Knowledge: What is 3/5 + 2/3?• Understanding: Draw a picture to show what 3/5 + 2/3 is and why.

Then when I teach…

• I don’t just talk about getting equivalent fractions with the same denominator and add.• I make sense of it.

For example….

• To add 3/5 and 2/3, I need to think about something that is easy to show fifths of and thirds of.• It could be a picture or a number.

If it’s a picture.

x x x x x x x x

x x x x x x x x

x x x

If it’s a picture.x x x x x

x x x x x

x x x x x

x x

x x

Or a number

• It is easy to take fifths and thirds of 30.• 1/5 of 30 is 6.• 1/3 of 30 is 10.• 3/5 of 30 is 18.• 2/3 of 30 is 20.• So 3/5 + 2/3 is 38 compared to 30, or 38/30 (or 19/15).

How are they different?

• Knowledge: What is []/[] – []/[]?• Understanding: You subtracted two fractions and the answer was

7/15. What could the denominators have been?


• I help kids see why it’s easy to subtract when the units are the same without seeing the items, but not easy when the units are different.• Compare , e.g. figuring out how much more 12 nickels is than 8

nickels vs how much more 12 dimes are than 8 nickels.


• Knowledge: What is 1 1/3 – 4/5?• Understanding: Draw a picture to show why 1 1/3 – 4/5 = 2/3 + 4/5.

When I teach for understanding

• I help kids see that when I subtract from a mixed number, I can subtract the fraction part from the whole and then add on the rest of the mixed number.• This makes subtraction much easier.• I also help them see you can subtract by adding up.

What’s the difference?

• Knowledge: What is 4 x 2/3?• Understanding: Why does knowing how to add fractions help you

figure out 4 x 2/3?


• We focus on meaning


• Knowledge: Which is greater: -4/5 or -1/3?• Understanding: -4/[] > -3/∆. What could [] and ∆ be?


• -4/[] > -3/∆. What could [] and ∆ be?• The student realizes that this means that 4/[] < 3∆.• The student realizes that if 3 parts are bigger than 4 parts, the parts

must be bigger, so [] < ∆.• For example, 3/5 > 4/8, so -4/8 > -3/5


• Notice that I didn’t give specific values, This is a “strategy” for understanding questions.


• Knowledge: What is 2/3 x 5/6?• Understanding: WITHOUT SAYING THE ANSWER, why is 2/3 x 5/6 less

than 5/6?


• This is another example of focusing on meaning and not just answers.


• Knowledge: What is 5/8 x 7/6?• Understanding: I want to multiply two fractions and the answer has to

be just a LITTLE BIT less than each of them. What could I multiply?

So when I teach

• My focus is on how multiplication by particular fractions changes numbers, e.g. what happens if I multiply by ½ vs 1/10 vs 9/10, etc.- a variation of the last question you saw


• Knowledge: What is 2 1/3 x 5 4/7?• Understanding: WITHOUT GETTING AN ANSWER, what’s a good

estimate for 2 1/3 x 5 4/7?• OR


• Knowledge: What is 2 1/3 x 5 4/7?• Understanding: Draw a picture that would help you figure out what 2

1/3 x 5 4/7 is?


• Knowledge: What is 4 ÷ 1/3?• Understanding: How many 1/3s are in 4? How do you know? Why

might you write that as 4 ÷ 1/3?

So when I teach

• My focus is on the meaning of the operation.


• Knowledge: What is 1/2 ÷ 1/3?• Understanding: Draw a picture that would help you see the answer to

½ ÷ 1/3.


• Knowledge: What is 3/4 ÷ 5/8?• Understanding: WITHOUT GETTING AN ANSWER, what is a good

estimate for ¾ ÷ 5/8?


• Knowledge: What is 3/4 ÷ 5/8?• Understanding: Suppose you know that a/b ÷ c/d is less than 1. What

do you know about c/d ÷ a/b? Why?


• This leads to a generalization.• Since 2/3 ÷ 1/5 asks how many 1/5s are in 2/3, it is more than 1 only

if 1/5 < 2/3.• That means 1/5 ÷ 2/3 asks how many big things are in a little thing, so

it’s less than 1.


• Knowledge: What is 8 ÷ ½?• Understanding: How do you figure out how far you go in 1 hour if you

know how far in 2 hours? How do you figure out how far you go in 8 hours if you know how far in ½ hour?


• Knowledge: What is 3 ÷ 2/3?• Understanding: How is figuring out how many 1/3s are in 3 help you

figure out how many 2/3 are in 3? What additions, subtractions, multiplications and/or divisions did you just do?


• This helps students relate division of fractions to unit rate to help explain the invert and multiply rule.

• E.g. 5 ÷ 1/3 indicates that if you move 5 km in 1/3 of an hour, you would go 5 x 3 in 1 hour.• 5 ÷ 2/3 indicates that if you move 5 km in 2/3 of an hour, you would

go 5/2 km in 1/3 of an hour and so 5 x 3/2 km in 1 hour.


• Knowledge: Show -2 on a number line.• Understanding: Why does it make sense to call -2 the opposite of 2?• OR


• Knowledge: Show -2 on a number line.• Understanding: Why does it make sense that -2 is 0 – 2?


• This is going to lead to an understanding related, later, to multiplication with negatives.


• Knowledge: List the integers between -3 and +4.• Understanding: Between two integers, there are both negative and

positive integers, but there are more negative ones than positive ones. What could the integers be?


• This leads to a generalization that it could be any –a and +b as long as b is closer to 0 than a.


• Knowledge: What is -5 + 7?• Understanding: You add two integers and the answer is negative.

When can that happen?

To build on understanding

• There is an element of generalization.


• Knowledge: What is –4 – (–9)?• Understanding: When you subtract two negative integers, the result

is –10. What could they be?


• My focus is on what -4 – (-9) means.

-9 -4 0


• I would also focus on the fact that lots of questions result in the same answer.

-9 -4 0 1 5


• Knowledge: What is (-4) x (-3)?• Understanding: Why does it make sense that (-4) x (-3) = 0 - 4 x (-3)?• OR


• Knowledge: What is (-4) x (-3)?• Understanding: How could you use counters or a number line to

EXPLAIN why (-4) x (-3) = 12?


• Knowledge: Put these in order: -3/5, 5/7, -5/3, 2/8, -5/17• Understanding: -3/[] is less than -5/8. What could [] be?


• Knowledge: Show each of these on a number line: - 2/3, 4/(-5), (-2)/8• Understanding: Why does it make sense that 4/(-5) is the same as the

opposite of 4/5?


• Knowledge: What is the solution: 4/x = 3/7.6?• Understanding: WITHOUT GETTING A SOLUTION, estimate the answer

to 4/x = 3/7.6.


• 4/x = 3/7.6• I know that 7.6 is about 2 ½ groups of 3, so I need 2 ½ groups of 4;

that is 10.• OR• I know that 3 is ¾ of 4, so I need to think of what 7.6 is ¾ of. I know

that 7.5 is ¾ of 10.

Some strategies you might use

• Show… without getting the answer (or solving)• Focus on meaning, sometimes by drawing a picture to show why• Build toward a generalization• Build a sense of why we do something

You try

• Turn one of these Knowledge questions and 2 of your own related to fractions or integers into Understanding ones.

Maybe

• What is 2 1/3 + 3 4/5?• What is 3/5 x (4/3 – 3/8)?• What is 5 – 7 (2 - (-8))?• Solve 8/x = 27/103.

Now

• Now let’s look at the differences between application and thinking.


• Application: A recipe uses 1/3c of flour for one part and 1/8 c of flour for another part. How much flour is used?• Thinking: Create a problem that requires the addition and subtraction

of fractions and where the result is 2/3.


• Application: A recipe uses 1/3c of flour for one part and 1/8 c of flour for another part. How much flour is used altogether? How much more is used in the first part than the second part?• Thinking: The sum of two fractions is 2/3 more than the difference

between them. What could they be?


• Application: Some kids ate 1 ¼ pepperoni pizzas, 2 cheese pizzas and 2/3 of a vegetarian pizza. How many pizzas did they eat?• Thinking: Use different values in each box to make this true: []/[] +

[]/[] + []/[] = 4/3.


• Application: In a particular school, 1/9 of the students are in Grade 7 and 1/6 are in Grade 8. What fraction of the students are in other grades?• Thinking: 2/9 of the students in Linda’s school is the same number as

1/3 of the kids in Dan’s school. What are possible and what are some impossible sizes for the two schools?


• Application: You watched the same movie 3 times in a row (with almost no time between) for 4 ¾ hours. How long was each movie?• Thinking: You added 3 mixed numbers for a total of 4 ¾. The second

was double the first. The third was triple the second. How long was each movie?


• Application: The high temperatures this week were -5, -3, +4, -15, -8, 2, 1. What was the average high temperature.• Thinking: The average of 7 high temperatures was -2. The highest was

+2. What could the average of the 2 lowest temperatures be?


• Application: Jessica is awake 2/3 of the day. She spends 5/8 of this time at home. What fraction of the day is she awake at home?• Thinking: The product of two fractions is 4/9 of the quotient. What

could the fractions be? OR• ¾ x 2/3 = ½• ¾ ÷ 2/3 = 9/8


• Application: A printer prints 20 pages in 1 1/3 minutes. How many pages does it print per minute?• Thinking: 20 is divided by a mixed number and the result is a whole

number. What could the mixed number be?


• Application: A large bottle holds 1 ¾ times the amount of liquid as a small bottle. How many large bottles would 10 ½ small bottles’ worth of liquid fill?• Thinking: The amount of liquid in 5 ½ small bottles is the same as in 2

½ large bottles. If the difference in size between the two bottles is 132 mL, how much does each hold? x = 110 y = 242


• Application: A shirt costs $18.95. You get a 20% discount. What do you pay?• Thinking: You pay the same for a shirt that is 20% off as for a jacket

that is 40% off. How were the original prices related?

Just plain thinking

• You have two rational numbers.• You add them, subtract them, multiply them and divide them.• Which ordering of ASMD is possible for the results?• (e.g. for 1/3 and ½ (in that order), S M D A is the order.)

You try

• Start with an application question and change it to thinking.

Asking Better Questions: Fractions & Integers · Asking Better Questions: Fractions & Integers...

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