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Place value with whole numbers and decimals

Keywords:
Achievement Objectives:

Achievement Objective: NA3-4: Know how many tenths, tens, hundreds, and thousands are in whole numbers.
AO elaboration and other teaching resources
Achievement Objective: NA4-6: Know the relative size and place value structure of positive and negative integers and decimals to three places.
AO elaboration and other teaching resources

Purpose: 

The purpose of this unit of sequenced lessons is to build on the students’ understanding of place value with three digit numbers and with one thousand. It supports the students as they generalise their 3 and 4-digit conceptual place value understanding across our numeration system.

Specific Learning Outcomes: 
  • Recognise the importance of zero as a place holder in whole and decimal numbers.
  • Recognise and apply understanding of base ten system repeated naming pattern of hundreds tens and ones.
  • Appreciate and understand the size of one million and beyond.
  • Understand and describe the multiplicative nature of our numeration system.
  • Consolidate understanding of powers of ten and the magnitude of these.
  • Understand what a decimal fraction is and recognise that they arise out of division.
  • Understand that the decimal point is a convention that separates whole units from parts of a unit.
  • Recognise that our base-ten place value system extends indefinitely in two directionss from one,the ‘centre’ of our numeration system.
Description of mathematics: 

In earlier units, students have worked with a range of increasingly abstract place value material representations to develop and consolidate their understanding of grouping in numbers up to and including four digits. Developing the ability of the students to translate between representations, making conceptual connections, has been the focus of earlier work.

Whilst some material representations will be included in these lessons, they will focus on developing a generalised understanding of the base ten system repeated naming pattern of hundreds tens and ones, initially with whole numbers to the left of the decimal point. Students are unable to understand the decimal system unless the positional notation with whole numbers is well understood. These can then be generalised to the fractional parts of numbers to the right of the decimal point.

The key understandings which should be understood as students come to these lessons are that ten in any position makes a single unit in the next larger position and vice versa. The idea of zero as a placeholder has been introduced, but this understanding needs to be further developed and understood. The magnitude of place value shifts to the left, that is that numbers get ten times bigger with each shift in place, is one further key understanding that will be expanded upon here. The factor of ten must be well understood for larger numbers before division by factors of ten will make sense as decimal fractions are explored.

There are many resources, games and activities that complement this series of lessons. However, as having a sound conceptual place value understanding underpins success in mathematics, it makes sound pedagogical sense to focus on developing this understanding in a deliberate, sustained and focused way, rather than through a one-off ‘activities’. This series of lessons seeks to do that.

Links to the Number Framework

Stages 6 and 7

This unit supports the teaching and learning activities in the Student e-ako Place Value 5, 6, 7 and 8 and complements lessons found in Book 5, Teaching Addition, Subtraction and Place Value.

Required Resource Materials: 
MAB (Multibase Arithmetic Blocks), 1, 10, 100, 1000, also known as Dienes blocks
Place value houses (Trendsetter, Thousands and Millions) (MM 4-11)
Arrow cards (MM 4-14)
Decimal arrow cards (MM 7-2)
Interlocking or Unifix cubes
Play dough
Plastic knives
Chart paper
Activity: 

Session 1

SLOs:

  • Consolidate place value understanding of 4-digt numbers.
  • Recognise the importance of zero as a place holder.
  • See and understand the magnitude of place value shifts to the left.
  • Recognise and apply understanding of base ten system repeated naming pattern of hundreds tens and ones.

Activity 1

  1. Begin the lesson by distributing a set of 8 place value variation cards (Attachment 1) to pairs of students. Have them order their set of 4-digit numbers from largest to smallest, laying them out in front of them.
     
  2. Once the sets are ordered, have the students rotate in pairs to an adjacent display of ordered cards and check whether these are ordered correctly.
     
  3. Ask the students to explain what they had to think about as they ordered their cards. Record their place value ideas on a class chart.
     
  4. Distribute sets of arrow cards to the student pairs and have them make one of the numbers in their set:
      (1065). 
    Have them discuss the zero in the number. Then ask them to remove or ‘zap’ another digit in their number and record the value of the digit they are removing. For example: the 6 in the number 1065 is removed. In doing so 60 is being removed or subtracted from the number. This is clearly shown by using arrow cards.
    Ask what they noticed when the 6 (60) was removed. (A zero appeared in its place).
    Discuss zero as a placeholder. Zero adds no value to the number but that it cannot be removed. It has the important job of holding a place. Have the students make another of the 4-digit numbers and repeat this.

Activity 2

  1. Make MAB materials available to the students.
    Have them play Fish for 1000 (Attachment 2) in pairs or groups of four to consolidate their understanding of the composition of 4-digit numbers.
    (Purpose: to work with groupings in numbers to 1000, applying place value understanding).
    They can model numbers with materials if they need to check clues.
     
  2. Discuss the fact that numbers can be composed (and decomposed) in different ways.

Activity 3

  1. Show the students 1,000 dots (Attachment 3) Have students tell you what they see and record all of their ideas using words and numerals. (for example: ten groups of one hundred, 1000, 10 x 100, one hundred groups of ten, 1000, 100 x 10)
     
  2. Ask the students to explain to a buddy what 10 thousand would look like. Listen to their predictions. Have someone attempt to draw this. Show the 10 000 dots image and ask if this is ten thousand. Repeat step 1 above.
    Point out that when we write the number 1000 there is no space after the 1 and before the three zeros, whilst with 10 000 a space is used to separate the ten from the three zeros. Highlight the fact that 10 thousand is ten times bigger, by highlighting one row (1000) and then counting the rows.
     
  3. Write 100 000 on the class chart. Have the students describe what 100 000 might look like. Listen for and record their ideas (ten times bigger). Show one hundred thousand and repeat step 1 above. Again, point out how the number is written with the space between the 100 and the 000.
    Ask the students to explain why.
     
  4. Write a 6-digit number on the class place value houses. Have students read it together, referring to the hundreds, tens and ones headings in each of the houses. Repeat with several 6-digit numbers including several examples with zeros.
    Make sets of Attachment 3 with 9 copies of each representation (9 ones, 9 tens, 9 hundreds, 9 thousands, 9 ten thousands and 9 one hundred thousands.) Have sufficient sets for students to have 1 set for each pair.

Activity 4

  1. Distribute Trendsetter and Thousands place value houses, and dot sets to each pair of students.
     
  2. Have them take turns writing a number with up to 6 digits in the place value house and having their partner read and make the number using the dot materials. Emphasise that it is the number of the dots that is important, not the size of the dots.
     
  3. Introduce the game 11,111 (Attachment 6).

Activity 5

Conclude the lesson by rereading together the points that are recorded on the class chart.
Highlight the hundreds, tens and ones structure of both houses and how this helps when reading bigger numbers.
Emphasise that the number of dots has been ten times more each time the number has an extra zero and as it shifts one more place to the left. Refer again to the dot representations to emphasise this point.

Session 2

SLOs:

  • Appreciate and understand the size of one million.
  • Understand and describe the multiplicative nature of our numeration system.

Activity 1

  1. Begin the lesson by writing a 4, 5 or 6 digit number on place value houses, using an erasable whiteboard marker. Include numbers that have one or more zero as a placeholder. Have students read the number to themselves then read it aloud to a partner.
    Together select a digit to remove and erase this from the chart. Ask students to tell a partner how much was removed (“What was the value of the digit that was removed?”). Leave the empty space and see whether the students tell you that a zero must be written there. If not do this and asks the students why.
     
  2. Introduce the game Zap (Book 4 Teaching Number Knowledge, p26) to the class.
     
  3. Distribute calculators and have students play Zap the digit in pairs. Have them record the numbers made and the amounts removed as they take their turns.

Activity 2

  1. Show the dot image of 10 000 introduced in Session 1.
    Ask, “How many dots are here and how do you know? Repeat with the image of 100 000 dots. Pose to the students, ‘What would 1 million dots look like?’ Have them discuss this in pairs and allow time for them to write down their ideas.
     
  2. Read “How much is a million?” by David M. Schwartz and discuss. You may wish to do this activity linked to the book.
     
  3. Have students share their ideas of how much one million is. Display 10 pages of 100 000 dots, asking, “Is this one million? How do you know?”
     
  4. Draw one dot on the class chart and write beside it 1 x 10 = 10. Draw ten dots. Then list this pattern, having the students complete each equation. 1 x 10 = 10 10 x 10 = (100) 100 x 10 = (1000) 1000 x 10 = (10 000) 10 000 x 10 = (100 000) 100 000 x 10 = (1 000 000)
     
  5. Ask students what they notice about the notation 1 000 000 and record their ideas about “How much is a million?” (Be sure to have the students notice that one million is one thousand lots of one thousand). Consider making a ‘One Million Book’ in which children record their ideas in an ongoing way and in which are pasted copies of the 10 pages of 100 000 dots.
     
  6. Introduce exponent notation, 102, 103, 104, 105, 106 explaining that this is another way (a short hand way) of showing how many times ten is multiplied by itself. For example, write and show 10 x 10 x 10 = 1000, ten multiplied together three times is one thousand.

Activity 3

Conclude the lesson by making available paper, calculators and computers and having the students in pairs choose to complete either of the following:

  1. Plan a “One Million Investigation”. For example investigate, how long their favourite book would be if it had one million words, how old you would be if you lived one million seconds, or how long would the line be if you lined up 1 million one centimetre cubes side by side and from which place to which place this distance might be in the real world.
     
  2. Have students present their findings in a form that can be shared with the class and included in the class ‘One Million book’.
     
  3. Make a poster for the class book, capturing the most interesting facts they have learned about one million. Have students discuss how they could share what they have learned with their parents or whanau.

Session 3

SLOs:

  • Recognise the repeating pattern in numbers beyond one million.
  • Consolidate understanding of powers of ten and the magnitude of these.

Activity 1

  1. Begin the lesson by having the students share their findings from Lesson 2: 3 above. Discuss. Reread ideas recorded so far in the class ‘One Million Book’.
     
  2. Ask a student to write one million in numerals on the class chart. Note the spaces between 3 digits (1 000 000) and ask for an explanation of this. (Listen for recognition that the hundreds, tens and ones naming structure is being repeated).
     
  3. Have students predict what the Millions place value house will look like, then display this.

    Write a range of 7, 8 and 9 digit numbers, emphasizing the hundreds, tens and ones naming in each house, and recognizing how to correctly read numbers which include zeros.
     
  4. Write a 7-digit and have students give some examples of how much that would be in the real world. For example, $3,000,000* might buy a very expensive house or small mansion, 6 000 000 one-centimetre cubes would be 6 cubic metres, in Lotto you choose 6 numbers from 40 and the mathematics of this shows that you have one chance in 3 838 380 of doing this. 
     
  5. Highlight for your students that we use commas to separate the groups of three digits when we are writing amounts of money. However, spaces only are used in other contexts except when writing 1000.

Activity 2

Show the students Attachment 4 and have them write down in pairs five interesting things they notice about this chart. Have them write some of the numbers themselves, making the connection as they do so between the exponent and the number of zeros. Share the discussion points with them.

Activity 3

Have the students play the game Powers or Bust.
(Purpose: to consolidate understanding of the multiplicative nature of our numeration system.)
Each player has a recording sheet (Attachment 5). Pick up cards (also Attachment 5) are placed face down in the centre of the players. Two dice are used. On dice one is written the numbers 1 (once), 2 and 3, twice and on the sixth side, ‘Pick up”. On the other there are three happy faces (increase) two sad faces (decrease) and one exclamation mark (miss a turn).
Each player begins the game with 10 in on their chart and for their first turn rolls the number dice only. The player increases their 10 by the power of the number rolled. For example, if 1 is rolled 10 remains unchanged, if 2 is rolled 10 x10 (102) becomes 100, and if 3 is rolled 10x10x10 (103) becomes 1000. The player records 1000 in the Turn 1 space on their sheet.
Subsequent rolls involve both dice.
At each turn, the number a player has made on the previous round is increased or decreased (happy or sad face) by the power indicated on the number dice and each time the player records the amount.
For example if round one gives a player 1000 and they roll 2 and a happy face on their second turn, this is 1000 x 10 x 10 = 100 000 (105 in total), which is recorded in their Turn 2 space.
If ‘Pick up’ or an ‘Exclamation mark’ (miss a turn) is rolled, the other dice is ignored. If these are rolled at the same time the player can choose which to do. If a player goes bust (is at zero or less) they begin again on their next turn with 10 and a positive roll (as they did at the beginning of the game).
At the end of the game, each player must write out in words their final score. The winner is the player with the highest number at the end of the game.

Activity 4

Challenge students to write, in less than 200 words, a response to the question, “What if we didn’t have a base ten numeration system?” or, “Our base ten numeration system is useful because….” Have the students share these. This should generate some valuable discussion. Use this opportunity to identify and address any place value misconceptions that remain.

Session 4

SLOs:

  • Recognise that one is the ‘centre’ of our numeration system
  • understand what a decimal fraction is and Recognise that they arise out of division
  • understand that the decimal point is a convention that separates whole units from parts of a unit
  • read decimal fractions correctly

Activity 1

  1. Begin the lesson reviewing the place value structure of whole numbers. For example, ask students to explain what is meant by the expression, “powers of ten”.
     
  2. Have student discuss what happens when you divide a big number by ten. (refer to their experiences in playing Powers or Bust).
    Write 1 000 000 on the PV house and discuss dividing this number by 10. It may be helpful to use the ten images of 100 000 dots (1 million) and demonstrate that one tenth of the ten images is one of them (100 000). Model repeated division by ten till one is reached. This can be powerfully demonstrated using the dot images from Attachment 3.
     
  3. Ask the students to discuss in pairs what number our whole number numeration system starts with. (responses may include zero, one, ten). Establish that one is the number with which we begin. We either increase 1 by powers of ten or decrease one by powers of ten.

    Write 1 in the ones place on the place value houses. Have students discuss what happens when we move to the right of one, in other words if we keep on dividing.
    Record the students’ ideas. Listen for important misconceptions, for example: ‘there is a decimal point next then it goes ones, tens, hundreds the other way (an understanding of symmetry around the decimal point) or this may be expressed as oneths, tenths, hundredths. Alternatively no symmetry may be assumed. Some may suggest that the naming pattern continues unchanged hundreds, tens, ones (in fact it reverses to read ones, tens, hundreds)

Activity 2

  1. Distribute a small ball of playdough, a plastic knife, chart paper and pen to student pairs. Explain that the playdough represents one (like the one dot that has been shown on the dot images).
    Set a time limit and challenge the students to work with the materials to show and record what happens when one is divided in a way that reflects the place value pattern that has been discovered with whole numbers.
    Do not mention tenths, hundredths etc. at this point. Also explain that they may have to reshape their play dough ‘one’ to work with it more easily.
     
  2. Observe the students actions. Have them pair share their charts and explanations. Summarize what the students’ work has revealed and discuss this.

Activity 3

  1. On a class model where the class/group can see, model the task again.
    Take the one ball and divide it into ten balls, then take one of those and repeat. Notice how quickly the size of the balls changes and highlight the magnitude of division by ten each time.
    Comment that students may have made a ‘sausage’ with their ball and worked that way. The representation is the same.
     
  2. Use a bright coloured round sticker and place this over the decimal point to emphasize its importance. Have students discuss in pairs and write an explanation to complete this statement:
    This very important dot is...
     
  3. Have students share what they have written.
    If key ideas have not been highlighted by the students, emphasise:
    • the decimal point is a convention that separates a whole unit (one)from parts of a unit (one)
    • the importance of one as the centre of our number system (these are tenths of ONE, and ten and ten times ONE)
    • that the numbers that follow the decimal point are called decimal fractions, which are a special kind of fraction with a tens base
    • decimal fractions are the result of division.

Activity 4

  1. Write on the class/group chart decimal numbers to one or two places in words. Have students volunteer to come forward and write the numeral representation.
     
  2. Reverse the process in step 1 above. Write a decimal number and have students correctly read the decimal number and write this in words. Be aware of misconceptions about reading a decimal number. Highlight correct reading, for example 0.43 is read, zero point four three NOT zero point forty three.

Activity 5

Conclude the lesson by having the students individually write a reflection on what they have learned.
Encourage them to illustrate their reflections with pictures/diagrams.
Display some of the students’ work from this session.

Session 5

SLOs:

  • Correctly read and expand decimal numbers to three places.
  • Understand zero is necessary as a place holder in decimal fractions, just as in whole numbers.
  • Recognise that the grouping pattern extends to the right of the decimal point.
  • Recognise that our base ten place value system extends indefinitely in two directions.

Activity 1

Begin this lesson by having students share their reflections from the end of Session 4. On a class chart summarise key points.

Activity 2

  1. Draw a decimal place value house on the class chart and write several decimal numbers. Distribute decimal arrow cards to the students.
    Have students take turns to make the numbers using the arrow cards and reading them to each other in at least two ways.
    For example:




    4.953 is read as four point nine five three, or, four and nine tenths five hundredths and three thousandths, or, four and nine hundred and fifty three thousandths, or, four and ninety five hundredths and three thousandths 
    3.05 is read as three point zero five, or, three and five hundredths
    7.029 is read as seven point zero two nine, or, seven and two hundredths and nine thousandths, or, seven and twenty nine thousandths.
     
  2. Discuss that thousandths are 1 ÷ 10 ÷ 10 ÷ 10 and that that pattern of dividing by 10 can continue.

Activity 3

  1. Set a time limit and have students work in pairs to brainstorm contexts they can think of in which ONE might be divided into tiny parts (fractions of seconds in Olympic sporting events, measurement of tiny things that can only be seen under a powerful microscope…).
     
  2. As a class/group share ideas and record these on a class chart. Demonstrate how the decimal place value chart continues to the right (go to six places: tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths). Ask, “does it keep going?’ and discuss.
    Challenge students in their spare time to investigate ‘infinity’ and present their own statement or diagram to describe ‘how it works’.

Activity 4

  1. Present students with this statement:
    Zero is worth nothing and it is therefore of no use in decimal numbers.
    In pairs or small groups have them consider and investigate this, then be prepared agree or disagree with it and give evidence to support their position.
     
  2. When the students are ready, have them move to one side of the room or the other, showing their agreement or disagreement.
    Have them present their rationale using illustrations to make their point.
    NB. Zero is an important placeholder, just as in whole numbers. However Recognise that, unlike with whole numbers, if a zero is added to a decimal number it does not change its value. For example 4.5 = 4.50

Activity 5

Conclude the lesson by reviewing and recording key place value ideas that have been developed in this unit.

AttachmentSize
pvWholeNumbersAndDecimalscm1.pdf28.51 KB
pvWholeNumbersAndDecimalscm2.pdf41.46 KB
pvWholeNumbersAndDecimalscm3.pdf1.67 MB
pvWholeNumbersAndDecimalscm4.pdf33.19 KB
pvWholeNumbersAndDecimalscm5.pdf52.22 KB