Transcript x + 1 - NCETM

PROBING QUESTIONS TO ADDRESS
ALGEBRAIC MISCONCEPTIONS
Slides by Teresa Robinson
[email protected]
ALGEBRA
MISCONCEPTIONS

How do the following expressions differ?
2n
and
n+
3(c + 5)
and
3c +
n²
2n²
(2n)²
and
and
2n
2
5
ALGEBRA
PROBING QUESTIONS


Can you write an expression that would simplify
to 6m – 3n?
Repeat for 8(3x + 6).
Are there others?
Can you give me an expression that is equivalent
to 4p + 3q – 2?
Are there others?
ALGEBRA
PROBING QUESTIONS



What do you look for when you have an
expression to simplify?
What are the important stages?
What hints and tips would you give to someone
about simplifying expressions?
ALGEBRA
PROBING QUESTIONS

1.
2.
3.
4.
5.
6.

Multiply out the following brackets:3(x + 1)
2(x – 3)
4(2x + 5)
-2(x + 2)
-3(2x – 4)
x(x + 1)
What hints and tips would you give to someone
about removing a bracket from an expression?
ALGEBRA
PROBING QUESTIONS

Are these expressions correct?
4(b + 2) = 4b + 2
3(p – 4) = 3p – 7
–2 (5 – b) = –10 – 2b
12 – (n – 3) = 9 – n

If not, what is the error and how can they be
corrected?
ALGEBRA
PROBING QUESTIONS


What do you get when you substitute x = -1 into
the formula y = 5x – 2?
Make up some more formulae that also give y = –
7 when x = –1?
What do you get when you substitute a = 2 and b
= 7 into the formula t = ab + 2a?
Make up some more formulae that also give t =
18 when a = 2 and b = 7?
ALGEBRA
PROBING QUESTIONS
p = 4a² and p = (4a)²
Are these two formulae the same or different?
When are they the same? Why?
Write down two formula which are written
differently, but are in fact the same.
ALGEBRA
TRIAL AND IMPROVEMENT

Use a systematic trial and improvement method
to find approximate solutions to the equation
x³ + x = 20.
x
x³
x³ + x
TRIAL AND IMPROVEMENT - X³ + X = 20
PROBING QUESTIONS
How do you go about choosing a value (of x) to
start?
 How do you use the previous outcomes to decide
what to try next?
 How do you know when to stop?
 How would you improve the accuracy of your
solution?
 Is your solution exact?
 Can this equation be solved using any other
methods? Justify your answer?

ALGEBRA
LINEAR EQUATIONS
6 = 2p – 8
How many solutions does this equation have?
Give other equations with the same solution?
Why do they have the same solution? How do you
know?
ALGEBRA
LINEAR EQUATIONS

Which of these linear equations are easy to solve?
Which are difficult and why?
What strategies are important with the difficult
ones?
3c – 7 = –13
1.7m² = 10.625
4(z + 5) = 8
4(b – 1) – 5(b + 1) = 0
12
= 21
(x + 1)
(x + 4)
ALGEBRA
GRAPH WORK




If I wanted to plot the graph y = 2x how should I
start?
Why is the point (3,6) not on the line y = x + 2?
Write down the equations of some graphs that
pass through (0,1)?
Write down the equations of some graphs that
pass through (0,0)?
ALGEBRA
GRAPH WORK

Use graphing software or graphic calculators to
plot the following graphs. What do you notice?
y=x
y=x+1
y=x+2
y=x+3
y = x + 4...
y=x
y = 2x
y = 3x
y = 4x
y = 5x...
y = -x
y=x-1
y = x²
ALGEBRA
GRAPH WORK

Write down some equations of graphs:




parallel to the x-axis
with a gradient of 2
with a gradient of 1/3
with a gradient of -1
with a gradient of -1/2
Write down the equation of a graph parallel to
the graph of y = 4x + 1. Repeat for 2y = x + 1
 Write down the equation of a graph
perpendicular to the graph of y = ½ x – 3. Repeat
for 3y + x = 1

Are these lines parallel or not?
ALGEBRA
GRAPH WORK

Describe the graphs
y = 2x + 1
 y = 4x – 2
 2y + x = 5


How can you tell if two lines are parallel?

How can you tell if two lines are perpendicular?
GRAPH WORK – DRAWING GRAPHS
PROBING QUESTIONS



How do you go about finding a set of coordinates
for the straight line graph y = 2x + 4?
How do you decide on the range of numbers to
put on the x and y axes?
How do you decide on the scale you are going to
use?
GRAPH WORK
PROBING QUESTIONS


If you increase/decrease the value of m, what
effect does this have on the graph? What about
changes to c?
What have you noticed about the graphs of
functions of the form y = mx + c? What are the
similarities and differences?
GRAPH WORK
PROBING QUESTIONS
How do you go about finding the gradient for a
straight-line graph:
• that has been drawn on a set of axes?
• from the equation given in the form y = mx + c?
• from a table of coordinates?
 What happens when m changes?
(Increases, decreases, is negative?)
 What happens as c changes?

ALGEBRA
GRAPH WORK



How would you go about identifying the graph of
y = 3x – 5 on a set of axes?
How can you draw the graph of the equation y =
½ x + 3? What different methods can you use?
Given a straight line graph, how can you find it’s
equation? What different methods can you use?
GRAPH WORK
PROBING QUESTIONS

Without drawing the graphs, compare and
contrast features of
graphs such as:
y = 3x
y = 3x + 4
y=x+4
y=x–2
y = 3x – 2
y = –3x + 4
x+y=6
EXPLAIN WHY (N+1)(N+20) IS AN EVEN
NUMBER
If n is an even number
- n+1 is
- n+20 is
- (n+1)(n+20) is odd x even = even
 If n is an odd number
- n+1 is
- n+20 is
- (n+1)(n+20) is even x odd = even
 n can only be odd or even and BOTH
CASES GIVES AN EVEN ANSWER.

PROVE THAT THE DIFFERENCE
BETWEEN THE SQUARES OF ANY
TWO CONSECUTIVE NUMBERS IS
ODD
Let the consecutive numbers be n and n+1
 (n)² = n²
 (n+1)² = (n+1)(n+1) = n²+2n+1
 Difference between the squares
= n²+2n+1-n²
= 2n+1
2n+1 is odd for all n


PROVE THAT THE DIFFERENCE
BETWEEN THE SQUARES OF ANY TWO
CONSECUTIVE ODD NUMBERS IS A
MULTIPLE OF
8
Let the odd numbers be 2n+1 and 2n+3
 (2n+1)² = (2n+1)(2n+1) = 4n²+4n+1
 (2n+3)² = (2n+3)(2n+3) = 4n²+12n+9
 Difference between the squares = 4n²+12n+9(4n²+4n+1)
= 4n²+12n+9-4n²-4n-1
Careful with the
= 8n+8
negative signs – use
brackets!!
= 8(n+1)


Always end
up with an
expression
to factorise
OOH – I do
like those
8’s!!!