Prior knowledge.

• Write a whole number in expanded form
• Explain the role of the decimal point as separator of the wholes and the parts of a whole
• Explain what tenths are, and write them as both a fraction and a decimal (1/10 = 0.1)
• Model an improper fraction and a mixed number with materials

Background

It is important that students develop a good sense of understanding of decimal palce value.

Comments on the Exercises

Exercise 1
Asks students to write a number in expanded form using fractions.  For example, 34.5 = 30 + 4 + 5/10

Exercise 2
Asks students to write the expanded number as a one decimal place number.

Exercise 3
Asks students to colour in the decimal tenths on a diagram. For example, 0.7.

Exercise 4
Asks students to write mixed number fractions with a 10 as a denominator as a decimal.  For example, 23/10 = 2.3.

Exercise 5
Asks students to underline the digit in the tenths place.

Exercise 6
Asks students to circle around the digit in the tens place.

Exercise 7
Asks students to add one tenth to decimal numbers.

Exercise 8
Asks students to subtract one tenth from decimal numbers.

Exercise 9
Asks students to identify the digit in the tens place and in the tenths place.

Exercise 10
Asks students to identify what number an arrow is pointing to on a decimal number line.

Prior knowledge.

Write a whole number in expanded form
Explain the role of the decimal point as separator of the wholes and the parts of a whole
Write tenths and hundredths in decimal and fraction form

Background

It is important that students develop a good sense of understanding of decimal palce value.

Comments on the Exercises

Exercise 1
Asks students to write a number in expanded form.  

Exercise 2
Asks students to write the expanded number as a one decimal place number.

Exercise 3
Asks students to colour in the decimal (tenths decimals and hundredths decimals) on a diagram.

Exercise 4
Asks students to write mixed number fractions with a 100 as a denominator as a decimal.  For example, 15/100 = 0.15.

Exercise 5
Asks students to underline the digit in the hundreds place.

Exercise 6
Asks students to circle around the digit in the hundredths place.

Exercise 7
Asks students to add one hundredth (1/100) to numbers.

Exercise 8
Asks students to subtract one hundredth (1/100)  from numbers.

Exercise 9
Asks students to add one hundredth (0.01) to decimal numbers. 

Exercise 10
Asks students to subtract one hundredth (0.01) to decimal numbers.

Exercise 11
Asks students to identify the digit in the tens, hundreds, tenths, hundredths column in numbers.

Exercise 12
Asks students to identify what number an arrow is pointing to a decimal number line. This exercise taps into the measure construct of decimals, so may be harder for students to work with than the numbers alone. This is because number lines use a different set of conventions to those used in counting. For example, in this exercise it is important for students to start off by looking at the numbers at each end of the interval, then working out what each major mark, and each minor mark stands for. It is surprising how many students assume that each (major) mark stands for one, with each of the little marks being a tenth, and do not realise that they need to start out by identifying the scale on the interval.

Exercise 13
Asks students to multiply hundredths in the form of 6 lots of 2/12. This activity introduces students to multiplication of fractions, but uses the more familiar concepts of ‘lots of’. Introductory problems can be modelled using equipment like decipipes or place value blocks before the exercise is introduced. The final problems involve students writing and marking their own problems, including one word problem. This latter should be collected in for marking, to check that students have developed an understanding of the multiplications, and can recognise an instance in which using this process would be of value. Discussing the word problems may be an important follow-up, especially where some students have demonstrated lack of understanding. Modelling what they have created as a problem with materials would be a useful way of correcting misconceptions about which practical problems utilise this skill.

Exercise 14
Asks students to multiply by hundredths in the form of 0.03 x 4. This exercise looks at multiplication of decimals. Note that the standard algorithm will cause some students grief (for example 0.06 x 5 = 0.3, which does not have two decimal places). Again, this topic should be introduced using equipment like decipipes or place value blocks to build an understanding of the process. Using language like 4 lots of three hundredths equals twelve hundredths (and how do we write this) is useful.

Prior knowledge.

• Use the strategy jumping the number line with whole numbers (Book 5 page 33)
• Identify the place value of the tenths hundreds and thousandths columns.
• Make combinations of tenths and hundredths that add to one

Background

In this activity students use additive strategies to solve addition and subtraction problems involving decimals.  Students need to have a good understanding of place value to make sense of the strategies with decimals.

Comments on the Exercises

Exercise 1
Asks students to solve problems by jumping a whole number and then a decimal.  For example, 6 + ? = 8.55, 6 + 2 = 8, 8 + 0.55 = 8.55 so the answer is 2.55.

Exercise 2
Asks students to solve problems by jumping to the nearest whole number, then the next whole number and then make a decimal jump. For example, 4.97 + ? = 8.12, jump 0.03 to 5, then 3 from 5 to 8, then 0.12 to 8.12

Exercises 3 and 4
Asks students to reduce the number of jumps they used in Exercise 2. For example, 3.98 + ? = 9.3, jump 0.02 to 4 then 5.3 to 9.3.

Exercise 5
Asks students to choose their own strategy to solve problems like 11.82 + ? = 38.3.

Exercises 6 and 7
Asks students to solve problems like in Exercise 5 but the numbers are larger, for example 19.88 + ? = 224.52

Written recording

Written recording of mental strategies (that is, how you thought through the problem) is important for developing sound assessment skills as it allows others to follow your reasoning and allows you to have a visual check for accidental errors. It is also something that develops over time, and needs to be discussed regularly with students. Exercises 5, 6 and 7 stress that students should solve the problems mentally, but record enough to show what they have done. Discussing or eliciting different ways of doing this is therefore an important activity, which you may choose to run either before or after setting students to work on this exercise

1. Players are dealt 3 cards each with the remaining cards placed face down on the table.
2. They take turns to draw a card from the pile and place it onto a blank space on their game board.
3. The object of the game is to place the digits to make the largest decimal number. The player with the largest number wins. Once a card is placed it can not be moved.

Extension

Increase the numbers beyond the decimal place. To do this you will need to allow more cards in the pack.

1. Write a decimal number on the board (to two decimal places). 
2. Brainstorm different ways to represent it.
   For example 3.81 could be expressed as;

• Three point eight one
• 3 + 0.8 + 0.01
• (3 x 1) + (8 x 0.1) + (1 x 0.01)
• 381 / 100
• 4 – 0.19
• Three and eighty one hundredths
• 7.62 / 2

3. Give each student a decimal number and ask them to represent in as many ways as they can.  Depending on the level of individual students they could be to one, two, or three decimal places.  Less confident students could work in pairs.
4. Students could present their work on a sheet of A3 paper to be displayed on the wall.
5. As the students work encourage them to use a variety of approaches (not just a whole list of addition sums).

1. Students enter a number into their calculator and push the division sign once. The student keeps pressing the equals sign until 0 is shown (depending on the ability of the student you could allow a one, two, three, or four decimal place number).
2. The student then hands over their calculator to a partner. It is important that no other button is pressed on the calculator as it is handed over.
3. The partner puts in a number and presses =. Again it is imperative that no other number is pressed as this will wipe the memory on the calculator.
4. The number displayed on the calculator screen will tell the partner how close they are to the number entered. A number larger than 1 means the number they entered is bigger than the mystery number and a number less than 1 means the number is smaller than the mystery number.
5. The partner can keep trying numbers, thinking about whether they need a smaller or larger number each time.
6. The challenge is to get 1 displayed on the screen in as few guesses as possible. Once the partner has a 1 displayed on the screen they have guessed the mystery number.

This activity focuses on ordering decimals to two places. Students could model the height of each player using two metre rulers on the floor.

K, R, S, J, and M are the initial letters of the players.
Introduce question 2 carefully. Students may have difficulty fully understanding the question. They need to be clear who is in the original team and who the new players are. Read it through with the students. Highlight the team changes by asking: “Who are the two new players? Who must have been left out if Aroha is now the shortest?”

Game

This game is short and simple but very effective for challenging students’  understanding of the digits involved in decimal numbers.
Have the winning students justify their result by talking about the place value of their digits. Vary the rules to create more learning opportunities. For example, you could ask students to “Make the largest decimal with a two-digit whole number in it. Make the smallest decimal with a one-digit whole number in it. Make the biggest number you can that is smaller than 7 but larger than 3.9.”

Answers to Activity

1. Centre: Minnie (tallest)
Guards: Joe and Stretch (second and third tallest)
Forwards: Ruth and Kevin (shortest)
2. a. Joe and Stretch
b. Joe is a guard, and Stretch is a forward.
c. Ruth and Kevin
Game
A game using decimals