Lesson 11 Setters and solvers

Episode 3

Reasoning Resources: Calculators
Formal written school methods for carrying out and checking an addition or a multiplication calculation start and rely on checking the digit combinations, usually through knowledge of number bonds, and pencil and paper procedures. In contrast, informal methods rely on a sense of the overall likely size of the answer, which effectively involves approximating the outcome, or at least subconsciously estimating lower and upper limits of the answer.

This episode covers the same reasoning steps as the previous episodes, but at somewhat higher thinking levels due to the number of features that need to be coordinated. It allows further explorations and links through the context of multiplication and combinations.

For classes who are confident with number operations some or all the questions below can be set as a homework exercise, to be followed by a discussion on methods and discoveries the pupils make, expressed in their own or in formal ways.
1. Find two solutions for this sum and explain your steps: ? X 6 = 28
2. Find the missing digits and explain your steps:

  • 8?X 76 = 2158
  • ?7 X 74= 11881.
  • Encouraging informal methods through talking and guessing provides the opportunity for the class to discuss estimation and approximation. These involve coordination between the place value (mentally rounding to the most significant value) and the meaning of the operation. These are routine techniques for approximations but they are best acquired by pupils through understanding rather than memorising. 3. Find the missing digits and explain how you found them: 572 +17 =28
    How is your method different from your method in question 2?
    4. Try to solve this question: 980+7=2?
    What do you notice?
    5. In this question each ? is a different digit: ??X??

  • Find the largest possible product and explain why it is the largest.
  • Find the smallest possible product and explain why it is the smallest.
  • 6. Make up calculations using the five digits 1,2, 3,4 and 5 only and addition or subtraction.

    For example:453 + 21=474 or 532-21=511

  • Hide two digits so that another person can find them with certainty.
  • Hide two digits so that the problem has more than one solution.
  • Can you hide more than two digits so that another person can still find them?
  • Explain the rules for making up questions.
  • 7. Use the five digits 1, 2, 3,4 and 5 and select one operation from +, -, X, +. Now using all the digits, each only once, and the chosen operation, find:

  • the largest possible answer.
  • the smallest possible answer.
  • Some pupils may need reminding that division to multiplication inversion and multiplication tables are useful, and how calculators need to be used together with estimation and approximation in order to ensure answers are sensible.
    The questions show how steps of reasoning are important. For question 4, for example, you may extract firstly that the largest possible missing digit on the right is 9, giving 29. If you divide 980 by 29 you will have an answer of more than one digit. Therefore this question is impossible. If you have less than 9 on the right that will also be impossible. Mention that this kind of chain of explanation is called ‘proof’.

    Question 5 (on the largest product of two two-digit numbers, all of which are different) has its subtleties. Not only must the largest digits be in the tens’ place but the larger of the digits of the unit places must not be in the same number containing the largest tens’ digit. That is different from a similar question on addition of 2 two-digit numbers where the order does not matter.


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