Solving by substitution

Solving by substitution

Pencil and paper
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This task is about working out the value of an algebraic expression by substitution.

Show how to find out what each expression equals by substituting the given value into it.

a) If  p = 7, show how to find the value of 4(p + 3)

 
 
 
 
 

Answer: __________  

b)
If m = ⁻3, show how to find the value of 4 + 2m

 
 
 
 
 

Answer: __________  

c)
If r = 3, show how to find the value of 4r2

 
 
 
 
 

Answer: __________  

d)
If s = ⁻3, show how to find the value of (s + 6)(s + 3)

 
 
 
 
 

Answer: __________  
Task administration: 
This task is completed with pencil and paper only.
Level:
5
Curriculum info: 
Maths, Number and Algebra, Equations and expressions
Keywords: 
algebraic expressions, substitution
Description of task: 
Students show how to solve a series of algebraic expressions by substituting given numbers.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Using symbols and expressions to think mathematically, sets 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y10 (09/2008)
a) 40
Working that shows any 1 of:

  • 4(7 + 3) = 4 × 10  Substitution into the equation
  • 7 + 3 = 10; 4 × 10  Evaluation of sub-steps of the equation
  • 4 × 7 + 4 × 3 = 28 + 12  Expanding the brackets
  • (7 + 3) × 4  Reversing the order of the equation
  • Other correct methods 

NOTE: Do not accept 7 + 3 = 10 × 4 = 40 or other untrue statements

 easy
easy
b) ⁻2
Working that shows any 1 of:

  • 4 + (2 × ⁻3) = 4 + ⁻6  Substitution into the equation
  • 2 × ⁻3 = ⁻6; 4 + ⁻6 = ⁻2  Evaluation of sub-steps of the equation
  • 2 × ⁻3  + 4  Reversing the order of the equation
  • Other correct methods 

NOTE: Do not accept 2 × ⁻3 = ⁻6 + 4 = ⁻2 or other untrue statements

 easy
easy
c) 36
Working that shows any 1 of:

  • 4 × 32 = 4 × 9  Substitution into the equation
  • 32 = 9;  4 × 9  Evaluation of sub-steps of the equation
  • Other correct methods 

NOTE: Do not accept 32 = 9  × 4 =36  or other untrue statements

 moderate
moderate
d) 0
Working that shows any 1 of:

  • (⁻3 + 6)( ⁻3 + 3) = 3 × 0  Substitution into the equation
  • ⁻3 + 6 = 3; ⁻3 + 3 = 3 × 0  Evaluation of sub-steps of the equation
  • s2 +9s + 18; 9 - 27 + 18

or 9 - 18 - 9 + 18  Expanding the brackets

  • Other correct methods 

NOTE: Do not accept ⁻3 + 6 = 3 × 0 = 0 or other untrue statements

 moderate
easy

Based on a representative sample of 161 students.

NOTE: Accept method even if computational details of it are incorrect.

Diagnostic and formative information: 
  Common errors Likely calculation Likely misconception
a)
b)
c)
d)		 7+3=10×4=40
2×–3=–6+4=–2
32=9×4=36 
–3+6=3×0=0		   1. Incorrect notation
Does correct substitution and computation, but writes a string of untrue statements connected by equals signs.
a)
b)
c)		 31
–18
144		 4 × 7 + 3
(4 + 2) × –3
(4 × 3)2		 2. Incorrect order of operation
Ignores the brackets
Adds before multiplying
Multiplies before squaring
c) 24 4 × 3 × 2 3. Confuses r2 with r  × 2
b)
c)

c)

 3
13

or 49

or 10

or 9


1849 or
86

 4 + 2 + –3
4 + 32
or (4 + 3)2

or 4 + 3 × 2

or 4 + 3 + 2


432 or
43 × 2

 4. Does not know the algebraic convention for multiplication
Interprets ab as a + b rather than a × b
Interprets ab as a + b rather than a × b
Interprets ab as a + b rather than a × b and adds before squaring
Interprets ab as a + b rather than a × b and confuses r2 with r × 2
Interprets ab as a + b rather than a × b and confuses r2 with r + 2

Interprets 4r as 43, then squares the number
Interprets 4r as 43, and confuses r2 with r × 2
a)
d)		 14
3		 4 + (7 + 3)
(–3 + 6) + (–3 + 3)		 5. Does not know the algebraic convention for brackets
Interprets a(b + c) as a + b + c rather than a × (b + c)
Interprets (a + b)(c + d) as (a + b) + (c + d) rather than (a + b) × (c + d)

 

Next steps: 
1. Incorrect notation
Encourage the student to expand their incorrect expressions into a series of correct equivalent statements. Start each new statement on a new line may help with the logical flow of the argument, and lead to fewer errors.
Example: For a),  instead of  7 + 3 = 10 × 4 = 40
7 + 3 = 10
so 4(7 + 3) = 4 × 10
so 4(7 + 3) = 40

2. Incorrect order of operations
Firstly, ensure the student knows the rule for the order of computations (BEDMAS rule) for arithmetic expressions.

Click on the link or use the keyword order of operations for further resources that test this.
Then get them applying the same rules to algebraic expressions.

3. Confuses r2 with r × 2
Explore the convention for exponents

4. Concatenation rather that substitution
Explain that mathematicians write 4 × b as 4.b  or 4b
and that more generally they write a × b as a.b  or ab
The CAS calculator needs the ".", so a.b  = a × b, but ab is a variable with the name "ab"

5. Does not know the algebraic convention for brackets
Explain that mathematicians write 4 × (a + b) in the shortened form of 4(a + b)

Strategies
There were four different correct strategies that students used (see Answers/responses). They were as follows:

  1. Substitution into the equation  4(p + 3) = 4(7 + 3) = 4 × 10 = 40
  2. Evaluation of sub-steps of the equation  4(p + 3) = 7 + 3 = 10; 4 × 10 = 40
  3. Expanding the brackets  4(p + 3) = 4 × 7 + 4 × 3 = 28 + 12
  4. Reversing the order of the equation  4(p + 3) = (7 + 3) × 4

Each of these strategies had high success rates for obtaining the correct answer, typically in the range 85% – 100%. The exception was in part d) where expanding the bracket only had a success rate of 58%. This was because errors were made in the expansion, and errors were also made when multiplying by the –3.

The more mathematically able students tended to use direct substitution into the equation (Method 1 above).
More average ability students tended to use sub-steps, or expanding the brackets (Methods 2 or 3).
Students who reversed the order of the equation (Method 4) tended to be mathematically less able than those using the other strategies.