Years 2 - 8
Zeros are used to hold the important empty places and the decimal point hasn't moved. It is still in the Ones Column as a marker separating wholes from parts. Suitable for threading.
There is no such thing as 'moving the decimal point'. The decimal point never moves. It is always in the Ones column as a marker to separate wholes from parts. No, it is not in a column of its own either. If it was there would be no symmetry in our numeral system.
Turn on a calculator. In 98% of cases the number you see is 0. not just zero. Programmers of these machines know that the decimal point is connected to, in the same column as, the Ones digit. Calculator Slido is a game designed to encourage children to see what really happens.
When you multiply or divide by powers of 10 the digits, as a block, slide left or right the necessary number of columns.
- Multiply by 10 and the digits slide one column to the left.
- Divide by 10 and the digits slide one column to the right.
- Multiply by 100 and the digits slide two columns to the left.
- Divide by 100 and the digits slide two columns to the right.
- Large digit cards 1 - 9
- Paper plates (about 10)
- One calculator per group
- One pack of Operation Cards per group
- One pack of Digit Cards per group. Playing cards work well as an alternative.
If you want children to use numbers with repeated digits (622, 343 ...) you will need multiple packs of Digit Cards per group.
- Two pieces of scrap A4 paper and a marker per group
This introduction can be threaded over a number of days and can include using two and three digit numbers. See slides of the way the activity was used in Nichola Brandon's Year 4 class in Chapter 6 of the story Fractions in Action.
Introduce the activity to the whole class through a physical demonstration. Draw columns on the whiteboard that are wide enough for a student to stand in front of. Label them (say) from thousands to thousandths with a very clear decimal point in the ones column heading. Give one student a large digit card, say 7, and hand paper plates to others. The paper plates make good representative zeros when the children hold them in front of their chest.
Two other special children are required: one to draw the cards from the operation cards above and call out the instruction; and one to use a calculator.
- Ask the digit person to stand in the correct column to show their digit.
- The calculator person enters this digit into the calculator and confirms that the digit person is in the correct place in relation to the decimal point. Develop any discussion that may be necessary.
- The card person draws the top card (cards are face down) and calls out the instruction.
- The digit person moves to where they think they should be. The class can assist with this.
- The calculator person carries out the operation on their calculator to check. They should notice that zeros are necessary to hold certain spaces between the digit and the decimal point. For example if the digit has been instructed to become 10 times bigger, they have to stand in the Tens column and a zero must hold the space in (NOT BE ADDED TO) the Ones column.
NB: If the digit is instructed to become 10 times smaller (from the Ones column), then two zeros are shown on the calculator - why?
- Paper plate people are moved in as necessary.
- Continue with the card draws until children get a feel for what happens.
- decimal calculations
- decimal interpretation
- exploring large numbers
- mathematical conversation
- numeral recognition
- operations - whole number
- pattern interpretation
- pattern recognition
- place value
- properties of number
- properties of zero
- recording - calculator
- recording - written
- writing numerals
When you perceive that children understand the effect on the digits of multiplying and dividing by powers of ten, the idea can be reintroduced as a game for two. It is useful to model this game with a 'group in a fishbowl' before asking children to do it for themselves.
- Students fold the two pieces of paper into four, mark column headings and use the Digit Cards to place any hundreds number as shown. (The back of playing cards can represent zero.) This is also the starting number written on the calculator.
Sometimes discuss with students the effect that multiplying and dividing by 10, 100, 1000... seems to be having on the digit. Sometimes ask them to write an explanation of what happens.
- Shuffle the Operation Cards and place them face down.
- Player A picks up an Operation Card and makes the digit slide the appropriate amount on both the calculator and the playing board by doing what the card says.
- Check that empty places on the playing board are held correctly with zeros.
- Player A says the new number and if correct gains one point. If they say it incorrectly, they lose a point.
- Players B, C, ... continue the game in the same way.
- Play continues until one player slides the digit into the Millions column and gains two points for doing so. Start the game over when this happens.
- If an Operation Card would cause the digit to move beyond the Millions or beyond the Ones that player misses a turn.
- Play to either reach a total of points agreed in advance, or for a set time. In either case the player with the highest score wins.
- Start with any type of hundreds number, eg: 234, 240, 204
- Extend the playing board to the left and right to include higher numbers and/or decimals
Calculating Changes ... is a division of ... Mathematics Centre