Quick Questions for Uncovering Student Thinking

Some of my favorite

quick questions for Uncovering Student Thinking

in Middle School Mathematics

and beyond…

Bob earns $5000 a month. What is his annual income?

a) $5000

b) $10,000

c) $50,000

d) $60,000

What is the MEDIAN of the following numbers?

5, 10, 8

Quiz Question 1 Week Later

What is the MEDIAN of the following data set?

2, 4, 4, 4, 4, 5, 6, 6, 8, 6, 7, 9, 9, 10, 10, 11, 15

8 + 7 = + 12

3 + 4 x 5 = 35

What is the name of the shape pictured below?

a) Square

b) Rectangle

c) Parallelogram

d) Trapezoid

e) Rhombus

f) Quadrilateral

What’s wrong with the following?

Which is greater?

0.8 or 0.79

Use a calculator

10)

Graph (-3, 4)

List 3 things students typically do wrong when graphing coordinate points.

11)

-1 - -3 - -4 – (-6) – 8 =

12)

4x – x = 4

13)

57 + + =

14)

Which is greater?

23 or 32

15)

Each Calculus student spent 1 hour and 45 minutes on their homework. How long did the 23 students work altogether?

16)

2.5 million ÷ 5000 =

17)

How many square inches are in the rectangle pictured below?

12”

10” 10”

12”

a) 22 inches2

b) 44 inches2

c) 120 inches2

d) 144 inches2

18)

What are the odds of pulling a white cube out of the bag pictured below?

19)

What is 8% of 50?

Answer: 40

What did the student do wrong when answering the question above? What is the correct solution?

20)

Which is larger?

9.348x102 OR 2x109

21)

Use a calculator. 1 piece of candy costs 8¢. What is the cost of 425 pieces of candy?

22)

Find the mean of the following numbers.

47, 58, 56

23)

-52=

(-5)2=

24)

True or False?

0.9 + 0.1 = 0.10

25)

True or False?

5÷30=6

26)

True or False?

8-12=4

27)

True or False?

3x + 5x = 8x2

28)

True or False?

3x + 5y = 8xy

29)

True or False?

= 0.17

30)

True or False?

2.1

x0.4

8.4

31)

Simplify

5x – 2y – 3(x+7)

32)

Respond to this statement:

Multiplication always increases a number.

33)

What is the same and what is different about how the letter m has been used in the following three statements?

Alissa jumped 3m.

3 x m = m x 3, for all numbers m.

If 3m+2 =14, m=4.

34)

Solve the following pair of simultaneous equations:

3a = 24

a + b = 16

35)

For homework last night, Jorge had to simplify four expressions. He wrote his answers without copying the original expressions. What might they have been?

a. 10x + 3

d. -3a-6

Follow-up question:

Note: This question focuses on combining like terms. Look for the common misconceptions in students' answers.

What did each student do incorrectly?

a. 10x + 3 = 12 – 2x + 3

b. = + x

c. = -

d. -3a – 6 = -3(a – 2)

36)

Simplify (if possible):

2a + 3b

37)

Ms. Rhee’s math class was studying statistics. She brought in three bags containing red and blue marbles. The three bags were labeled as shown below:

75 red 40 red 100 red

25 blue 20 blue 25 blue

Bag X Bag Y Bag Z

Total = 100 marbles Total = 60 marbles Total=125 marbles

Ms. Rhee shook each bag. She asked the class, “If you close your eyes, reach into a bag, and remove 1 marble, which bag would give you the best chance of picking a blue marble?”

Which bag would you choose? Explain why this bag gives you the best chance of picking a blue marble. You may use a diagram in your explanation.

38)

Item 1 (88% success rate on TIMSS)

Which fraction is the smallest?

a) b) c) d)

Item 2 (46% success rate on TIMSS)

What fraction is the largest?

a) b) c) d)

39)

Three fractions add together to give one fifth.

If all the question marks represent the same number, what is that number?

What if all the question marks represent different numbers?

40)

Use your calculator to find which whole number divided by another whole number gives the answer:

1.36363636...

41)

True or False?

= 0.67

42)

Follow-up Question:

Use a Calculator. True or False?

2 ÷ 3 = .6666666667

43)

Is this a right triangle? Why or why not?

48 mm

20mm

52mm

44)

True or False?

FOIL always works as a method for multiplying polynomials?

For example: (x+4)(x+5)

45)

Give a number between 0.3 and 0.4

46)

Describe the following shape:

4 6

47)

Is –x a negative number?

48) Classic 6th grade questions to diagnose conceptual understanding of fractions:

What fraction is represented by this section of the circle? Why?

49) Potential Algebra Errors

50)

"The most powerful learning experiences often result from making mistakes".
I usually address my students with the above phrase after handing out marked papers, tests and exams. I then provide time for my students to carefully analyze their errors. I also ask them to keep a running record/journal of the patterns of their errors. Understanding how and where you go wrong will lead to enhanced learning and improved grades - a habit often developed by strong math students. It's not unlike me to develop my next test based on a variety of student errors!

1. A number with three digits is always bigger than one with two
Some children will swear blind that 3.24 is bigger than 4.6 because it's got more digits. Why? Because for the first few years of learning, they only came across whole numbers, where the 'digits' rule does work.

2. When you multiply two numbers together, the answer is always bigger than both the original numbers
Another seductive 'rule' that works for whole numbers, but falls to pieces when one or both of the numbers is less than one. Remember that, instead of the word 'times' we can always substitute the word 'of.' So, 1/2 times 1/4 is the same as a half of a quarter. That immediately demolishes the expectation that the product is going to be bigger than both original numbers.

3. Which fraction is bigger: 1/3 or 1/6?
How many pupils will say 1/6 because they know that 6 is bigger than 3? This reveals a gap in knowledge about what the bottom number, the denominator, of a fraction does. It divides the top number, the numerator, of course. Practical work, such as cutting pre-divided circles into thirds and sixths, and comparing the shapes, helps cement understanding of fractions.

4. Common regular shapes aren't recognised for what they are unless they're upright
Teachers can, inadvertently, feed this misconception if they always draw a square, right-angled or isosceles triangle in the 'usual' position. Why not draw them occasionally upside down, facing a different direction, or just tilted over, to force pupils to look at the essential properties? And, by the way, in maths, there's no such thing as a diamond! It's either a square or a rhombus.

5. The diagonal of a square is the same length as the side?
Not true, but tempting for many young minds. So, how about challenging the class to investigate this by drawing and measuring. Once the top table have mastered this, why not ask them to estimate the dimensions of a square whose diagonal is exactly 5cm. Then draw it and see how close their guess was.

6. To multiply by 10, just add a zero
Not always! What about 23.7 x 10, 0.35 x 10, or 2/3 x 10? Try to spot, and unpick, the 'just add zero' rule wherever it rears its head.

7. Proportion: three red sweets and two blue
Asked what proportion of the sweets is blue, how many kids will say 2/3 rather than 2/5? Why? Because they're comparing blue to red, not blue to all the sweets. Always stress that proportion is 'part to whole'.

8. Perimeter and area confuse many kids
A common mistake, when measuring the perimeter of a rectangle, is to count the squares surrounding the shape, in the same way as counting those inside for area. Now you can see why some would give the perimeter of a two-by-three rectangle as 14 units rather than 10.

9. Misreading scales
Still identified as a weakness in Key Stage test papers. The most common misunderstanding is that any interval on a scale must correspond to one unit. (Think of 30 to 40 split into five intervals.) Frequent handling of different scales, divided up into twos, fives, 10s, tenths etc. will help to banish this idea.

Probing Questions:

· Give me two equivalent fractions. How do you know they are equivalent?

· Are 16 and 25 equivalent? Explain.

20 20

· How do you know that 4 is not equivalent to 5?

5 6

· If I cancel a fraction by 2 and then 3 is this the same as cancelling by 6. Justify your answer.

· Give me a fraction that cannot be simplified. How do you know?

· 10% = 1 ; 20% = 1 Are these statements true or

10 20 false? Why?

· Is 17% > 1/5?

· Give me a percentage > 1/5.

· If 6a + 4 = ___ what is the meaning of a?

· If 5 + __ = 18 what else do you know?

· Is n + 2 the same as 2n?

· Is n +2 ever the same as 2n?

· Is x squared the same as 2x?

· a+b= 10. What is a and what is b?

· Is 2n always different from n squared?

· a + b is 10. What is a? What is b?

· Why does my calculator give an incorrect answer for 2 + 4 x 6? Why do I get different answers from different calculators?

· Why do some calculators give us a different answer to this calculation? 4 + 3 x 7

· How might the inclusion of brackets affect the answer to this calculation 2 + 3 x 4

· Is (2x) squared = 2x squared?

· Give me the three angles of a triangle
a) one must be acute
b) one must be obtuse
c) two obtuse

· Which is bigger 200cm or 20000mm? Explain how you worked it out.

· Give me 5 numbers where the mean/median/mode is 6 and the range 8. How did you do it?

· Does addition always produce a bigger answer?

· Is -4 bigger than -3? OR is -4 ‘more’ than -3?

· Explains what happens with 0 in the 4 rules

· If I increase a number by 10% and then increase the answer by 10%, is this the same as a 20% increase?

· 3 (x+2) = 3x+2 explore!

· 2a + 3b= 5ab isn’t it? Discuss

· 3(a+b) = 3ab *or vice versa

· or 3(2+6) = 6+6 do you agree?

· X+y = 7 find values of x and y that fit this equation.

· If p = 2a +3b, give me some values of a and b, if p = 60.

· The area of a triangle is 60cm squared, using the formula A = ½ bh What values of b and h are possible?

· How can you recognize parallel lines by their equations?

· Y= 3x +4. What can you tell me about the graph?

· How do you know from the equation that the line passes through the origin?

· Give me the equations of some graphs // to y = 2x +3

· Will y = 3x-5 give a straight line?How can you tell? What about y= 3x?

· Do the angles of a triangle always add up to 180º? Why? (leads to different stages of proof) (same for quadrilateral)

· How does knowing the angles of a triangle total 180° help you work out the angles of a quadrilateral?

· 3/5 +1/2 = 4/7 is this true?

· ½ + ¼ = 2/6 is this true?

· Is it true that ½+1/2 = 2/4?

· Is 1/2x1/3 smaller than both, bigger than both or in the middle?

· The area of this rectangle is 10cm². What is its width? What do we need to find?

· 6 = 2p – 8. Give me other equations with the same solution? Why? How do you know?

· 1, 3. What could the next 2 terms be? Why?

· How would you convince somebody that the exterior angles of a polygon add up to 360°

· What is pi?

· Can you have a square measurement exact area of 32cm2

· Can you give me a question with the answer xº?

· Can you give me the two factors of 28?

· Which has the greatest value (23)4 or (24)3 ? Why?

· Explain why xº = 1

· Five people weighing 80kg each get in a lift. Is it safe if the maximum weight is 401kg?