Adding and subtracting (age 9+)

Activities to try

1. 1

   Plan-then-compute with a multi-bar model

   Why it works: A multi-step problem is not finished until the final question is answered; the structure of the situation, modelled before computing, determines which operations and in what order, while the numbers determine the best method per step.

   1. Before calculating, lay out the whole problem as a bar model: for the bottle problem, draw Week 1 (12,450) and Week 2 (8,372) as combining bars, then set that combined bar against the 25,000 target bar with the shortfall shown as an unknown gap.
   2. Label every step and name each intermediate result explicitly (e.g. write '20,822 = two-week total, NOT the answer').
   3. State an up-front estimate of the final answer (e.g. 'about 25,000 - 21,000 = roughly 4,000 more').
   4. Choose a method for each step by looking at the numbers: count up for 25,000 - 20,822; use a column method for an awkward sum like 12,450 + 8,372.
   5. Answer the actual question asked (how many MORE are needed = the gap, 4,178) and check it against your estimate and the story.
2. 2

   Walk me through your plan: justify each step and method

   Why it works: Deciding which operations and methods to use AND WHY is the explicit objective; articulating the plan and justifying each method choice surfaces the reasoning that number-grabbing hides.

   1. Take a multi-step scenario (e.g. the two-step problem 'A shop had £41,000 in the bank, spent £18,650, then took £9,200 in sales. How much is in the bank now?', or a longer one with three or more steps) and, out loud or in writing, say which step comes first and why.
   2. For each step, state the operation and whether you will use a mental or written method, and justify that choice from the numbers.
   3. Say what each intermediate number means in the story before moving on (e.g. '22,350 is the balance after spending, not the final answer').
   4. Answer the probing follow-ups: 'What is the question actually asking for?', 'Roughly how much should it be before you calculate exactly?', 'For this subtraction, would you take away or count up — which is quicker here?', 'Why did you subtract first to follow the story, and would adding the £9,200 first give the same final balance?'
3. 3

   Sequence the steps and choose the method for each

   Why it works: Recognising the right step order, the better method, and the trap of stopping at an intermediate value reinforces that structure beats surface keywords.

   1. Given a multi-step problem (two or more steps), put the steps in the correct order rather than accepting the order the words appear.
   2. For each step, select the operation and the better method (mental counting-up vs written column), and give a one-line reason.
   3. Spot and reject a keyword-trap option (an operation suggested only by a word like 'more' or 'total') and an intermediate-as-answer distractor (e.g. choosing 20,822 instead of 4,178).
   4. Estimate the final answer before committing and use it to sanity-check the chosen sequence.