Prior knowledge
Use the count on strategy
Background
The hundreds board is an ideal piece of equipment to introduce the development of patterning and part-whole concepts. The particular hundreds board used in these exercises is the reverse to most. This is because in some exercises we are developing symbols and movement conventions. On this hundreds board, to add 1 we go to the right, and to add 10 we go up (so numbers to the right and higher up are larger). This is consistent with the conventions we use with graphing.
Comments on the Exercises
Exercise 1
This activity develops the concept of multiples and gets students to look for patterns in these multiples (visual patterns that can be seen on the hundreds board as well as patterns that can be seen in the numbers). Stage 4 students need practice developing their skip counts, and continuing these skip counts beyond the first few terms. This activity also gives practice in these.
Exercise 2
This exercise introduces the sieve of Eratosthenes to find the prime numbers. Initially students are using the same procedure as in exercise 1, but may find it harder to “count on” in twenty nines, hence the suggestion that a calculator be used. This could also be useful for reinforcing that counting on 29 is the same as adding 29.
Exercises 3 and 4
These exercises are designed to reinforce a teaching session that develops the concept of place value. The questions that are being addressed are “how do numbers change when you add ten?”, and “how do numbers change when you subtract ten?” Part of the learning here relies on the fact that the students know about numbers in columns (and the term digit for a number in a column), so if they do not, they may not be able to manage the learning step that “you can add ten by making the number in the tens column one bigger”. Exercise 4 builds on the questions in exercise 3, using numbers rather than the hundreds board, though some may still want to use the board to check answers. Later problems look at adding numbers bigger than ten.
Exercise 5
This exercise builds on the learning in exercise 3, with adding and subtracting 10. It is still predominantly counting based, working with forward and backward number word sequences, but is also developing place value through adding and subtracting tens in this manner.
The exercise is designed to follow on from a teaching session in which students participate in the creation of a set of symbols, and meanings for those symbols. The shared meaning they develop mirrors how mathematicians operate when using symbols. That is, understanding is communicated because of the shared meanings held by the community. This exercise is also part of developing an understanding that we can use symbols in certain situations to mean certain things. In other situations they may mean something different.
During teaching, initially students should use hundreds boards and a counter as they develop the use of:
Develop a notational form for them to follow, thus 54→→↓ means 54, +1 +1 -10, giving 46. (Examples of what to do when at the start of a row and you get ← should also be discussed during this lesson). Move students onto imaging and checking and working with the numbers alone if possible before setting them to the exercise.
As an extension to successful initial work, an additional symbol can be developed
to mean ‘add one hundred’.
The challenge looks at what happens when starting in the bottom row and going down. Here the pattern of -10 is maintained, but this is not visibly subtracting 10 in the way the place value system works with positive numbers. For example, 3 – 10 = -7. Note that students are directed to discuss their inventions and answers for the challenge and the final question with you, their teacher.
Exercise 6
This exercise is aimed at moving students on from counting in ones to add 9, to adding 9 “by adding 10 and subtracting 1”. In other words using a part-whole strategy. This is done through investigative or discovery, though the exercise could be used as a follow-up to an initial teaching session. Students may notice that for adding nine, you can make the number in the tens column one bigger, and the number in the ones column one smaller (56 becomes 65). However, this does not work for tidy numbers (90 becomes 99). A similar issue exists for subtracting 9, so this exercise needs to be reviewed after students have attempted it as many students who are still counters are not confident with numbers. Students may indeed discover the exception for themselves, but almost certainly need to discuss the fact that all useful strategies work for some problems, and not others. Practice using the “add ten then subtract” should then be regularly given with numbers where it does work – as the move to part-whole understanding is a critical development that needs careful nurturing.
Challenge: A rule can also be found for adding 11 – but does not work when choosing numbers at the end of the row.
Extending the exercise to adding and subtracting 8 can be worthwhile – provided numbers ending in 0 and 1 are avoided in the examples you give early on.
Exercise 7
This exercise is a development of the symbol work done in exercise 4, but with letters. In this context, the letters are used as abbreviations of words, which is consistent to their usage when we develop formulae.
Here a teaching session needs to be used to introduce the concept of using words for directions.
Up for moving up one row on the hundreds board (adding 10)
Right for moving right on the hundreds board (adding 1)
Left for moving left on the hundreds board (subtracting 1)
Down for moving down one row on the hundreds board (subtracting 10)
Some oral instructions can be given for students to follow, before abbreviations (33 UDLR) are introduced. This work then mirrors what was done in exercise 4.
Exercise 8
This exercise provides an additional development of the work in exercise 6, by considering repeated instructions. When wanting to go down twice (down 20 not down 10) the notation 2D needs to be introduced (as in algebra DD means something different, but this does not need to be explained). An addition sign is also introduced to mean “and”. Thus 2D + R means to subtract 20 and add 1. Students may want to revert to the hundreds board for working with this exercise.