Lesson 11 Setters and solvers

# Episode 1

Reasoning Resources: Worksheet 1, Calculators
Whole class preparation
Hiding a digit here makes routine number work impossible and focuses the attention on the mathematical ideas involved. This is even more so if two or more digits are hidden.

The pupil must combine the value of the individual digit with the value of its position to keep a sense of overall magnitude. A basic step in numeracy is ease in comparing two numbers so that 4899 is quickly seen as smaller than 6110, even though most of the digits are larger.

Rehearse with the class: what a digit is; the alternative words for ‘digit’ such as ‘figure’ and ‘numeral’; How is a digit different from a number? and How many digits are there? Tell a story of historians finding sheets of calculations in a castle with some digits damaged or erased, and having to construct them, and making mistakes.

Start with very simple addition calculations with one or two missing digits, such as 32+ ?=35 and 72 + ?? = 97. Let pupils talk about their methods, which could simply be from ‘adding on’ to'l just know it’ referring to knowledge of number bonds.

Introduce the word “inverse” as a mathematical word for the natural language of doing the opposite or reverse operation. Clarify where they have only used the units position. Pupils should also notice that they can find individual missing digits separately in the units and tens columns.Then work with the whole class on the first three questions A,B and C on Worksheet 1. The questions ‘across; A, B and C, are of different types. So are the questions ‘down; A, D and G. One question can be solved through steps of reasoning, one is impossible to solve, and the other has more than one possible correct answer.
Pair work
Coordinating place value while carrying out an operation or its inverse may involve breaking up or combining numbers either side of tens, variously labelled bridging, trading, or decomposition. Give out Worksheet 1 to pairs to work on the remaining questions, They should see which can be solved and then sort them into the three types - one solution, many solutions, cannot be solved. Pairs who finish early could check with other pairs and then give names or labels to each of the three types of problem.

Allow pupils to use calculators but suggest their best use is for checking answers occasionally. Calculators will be more useful for Episode 3, which deals with multiplication and division. During the lesson highlight instances where thinking through questions is much easier and faster than trial and error on the calculator.
To include impossible sums and sums that allow more than one answer fosters a critical outlook, based on belief in consistency in mathematics itself. Whole class sharing and discussion
Subtraction examples: 3?-?2 =45 and ?7 - 83 =2? are impossible. 28 — 3?=09 and 6? —?8 = 23 have unique answers 12? -56 = 6? and ?1 —?2 = 69 have more than one answer each. Agree as a class which sums have unique, multiple or impossible solutions. They share alternative methods for finding solutions and how these combine using place value with inverse operations.

Encourage pupils to explain in words the conditions on the ‘units position’ and ‘tens position; and the least and most that the number can be (the maximum and minimum possible total of addition of 2 two-digit numbers). The agenda here is for them to realise they are using a chain of reasoning, each step of which is fully proved to be true. That is a form of proof.
Elaboration and extension
• Write some subtraction questions on the board in the same vertical format, leading to the sum recognition that the top number must be the bigger one.
• Write three-digit addition and subtraction questions in a horizontal format, allowing the pupils either to organise them themselves or to solve them without such organisation.