Having fun with the vertical algorithm

Prior knowledge

Background

While calculators can be expected to be used for more complicated additions and subtractions, learning the "good old vertical algorithm" still has its place - once students are able to use the place value system properly so have a chance of understanding how it works. These exercises, however, while giving some practice with this method, also show students a nice strategy for avoiding "all that borrowing", and introduces a "complementary numbers" strategy that can be very handy for decimals and percentages.

These activities are designed as an investigation that students undertake with little teacher supervision. After each stage/exercise, it is envisaged that the teacher will facilitate a discussion with and between students about what they have learned and discovered. In some cases, where students have struggled to make progress, individually or collectively, the teacher may need to 'help' students to make a discovery by showing a problem, or a range of problems. Highlighting what the students need to look at, as in the example below and asking leading questions like "what do you notice about these numbers?" is a useful approach.

When using these activities, note that the discussions around not only how to use the strategy, but when to use it, are central to the learning being undertaken by students. As this is student, and not teacher, directed the discussions also act as a reporting back process on the work students have undertaken. It may pay to highlight the expectation that all can contribute to a discussion about what they have tried and have learned before each section of the work is set to ensure the students pay attention to more than simply 'getting the answers'.

The final challenge is a good research project for a few - but some may twig immediately that the answer plus the number subtracted should give the number started with. If that number has a lot of zeros in it, adding 1 will roll all the other digits to zero. 999,999 + 1 = 1,000,000, so were are making sure that the answer and the subtracted number add to 999,999 + 1.

Comments on the Exercises

Exercise 1
Asks students to subtract numbers from 1000 using a "complementary numbers" strategy.

Exercise 2
Asks students to subtract numbers from multiples of a 1000 using a "complementary numbers" strategy.

Exercise 3
Asks students to subtract 1000s numbers from 1000s numbers using a "complementary numbers" strategy and check with a calculator. 

Exercise 4
Asks students to decide if the "complementary numbers" strategy is a good strategy to use on a selection of subtraction problems. 

Exercise 5
Asks students to explore the "complementary numbers" strategy with decimal problems.

Exercise 6
Asks students to solve decimal subtraction problems in the context of percentages using complementary numbers.

Exercise 7
Asks students to make a poster to explain how the "complementary numbers" strategy works.