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Counting on Counting

Keywords:
Achievement Objectives:

Achievement Objective: NA1-5: Generalise that the next counting number gives the result of adding one object to a set and that counting the number of objects in a set tells how many.
AO elaboration and other teaching resources
Achievement Objective: NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
AO elaboration and other teaching resources
Achievement Objective: NA1-2: Know the forward and backward counting sequences of whole numbers to 100.
AO elaboration and other teaching resources

Purpose: 

The purpose of this unit is to explore and develop understanding of cardinality and one-to-one matching, conservation, counting patterns and simple addition and subtraction of sets.

Specific Learning Outcomes: 
  • recognise that counting tells how many objects are in the set irrespective of how they are arranged or the order in which they are counted.
  • solve problems involving one more or less to a given set using their knowledge of the forward and backward number sequences.
  • skip count in 2s
Description of mathematics: 

Kamii suggests that young students need to understand both the inclusive (cardinal) nature of counting and the hierarchal (ordinal) nature of counting. This unit focuses on students generalising their idea about what it means to count the objects in a set. This generalisation should mean students understand that the count gives the number of objects irrespective of the order they are counted, how they are arranged, or whether they are directly sensory like physical objects, or need to be “captured”, like sounds or touches.

Number Framework

Stage 4, Advanced Counting

Required Resource Materials: 
number strips (1-20)
transparent coloured counters
attribute blocks
plastic cups or yoghurt containers
scissors
glue
paper
dice
Copymaster 1: Farmyard cards
Copymaster 2: Hundreds board (Material Master 4 -4)
Copymaster 3: Operation cards
Copymaster 4: Storyboard pictures
Copymaster 5: Pattern cards
Copymaster 6: Digit cards (Material Master 4 -1)
Key Vocabulary: 

 how many? E hia? difference, more, same, order, enough, each, compare

E tahi,... rua, toru, wha, rima, ono, whitu, waru, iwa, tekau....tekau ma tahi, tekau ma rua.... rua tekau, toru tekau....

tasi, lua, tolu, fa, lima, ono, fitu, valu, iva, sefulu... sefulu-tasi, sefulu-lua, ..........lua- sefulu, tolu-sefulu

Activity: 

Session 1

  1. Ask the students to count a set of five counters, but making sure that they use at least three different colours. Place your own set of five counters on the strip, one colour at a time, until the last counter is placed. Get the students to do the same with their collection, starting with a given colour.
    diagram
     
  2. Tell them to take the five counters off their strip. Ask:          
    How many counters will you have if you start by counting the green counter first?
    Invite their ideas. Get them to check their predictions by replacing the counters starting with the green counter.
    diagram
    Ask:
    Why do you think there were five counters both times?
    Do you think that counting the blue counters first will make any difference?
    Get them to check by replacing the counters on their strip.
     
  3. Do the same thing with larger collections of counters, and different mixes of colours. After three different     collections students are likely to say that the order of counting the colours make no difference, “the answer   (count) is always the same.” Be aware that they may be saying this because it has occurred three times in     sequence rather than because they recognise the cardinal (measuring) nature of counting.
     
  4. Pose problems that involve adding to or removing from given collections. For example:
    counters
    How many counters do I have here?
    I’m going to count the red ones first, then the blues, .. Will that make any difference?

    diagram
     
  5. Tell the students that you are going to add one blue counter to the collection. Ask:
    How many counters there will be? How they know?
    Will it make any difference whether the new counter is put on the end or placed with the other blues and the greens and yellows moved along one space?
    Note that it is important that students realise that the result of adding one or removing one from a collection   is given by the next number after or before in the number sequence.
     
  6. Put the students in pairs to pose similar problems, one student making a collection of counters with mixed     colours and the other student getting them to predict the result of adding or removing up to 5 counters of       one colour. The students’ predictions can be checked, if necessary, by placing counters on a number strip.
     
  7. Use an overhead projector to create shadow collections with attribute blocks. It is very important to vary the size and shapes of the blocks used as these variables are distracters in the counting process. Make two           identical collections, left and right, separated by a line (see below). Arrange the collections differently and   cover them with a book.
    shape collections
     
  8. Uncover the collections one at a time briefly (about 3 seconds), and ask the students to draw what they         remember seeing.
    Can you guess which has more, left or right?
  9. Uncover both collections and discuss why the collections have the same number of objects (they have the     same number of hexagons, triangles, circles and squares).

    Show collections that have the same number of objects, but the shapes are different. Also compare               collections that are different by one or two blocks For example:
    shape collection example
                                  Same blocks                                                     Two blocks different
     
  10. Students might like to draw their own “different or same” collections by tracing around attribute blocks and colouring them in. A book of problems could be assembled with one collection on the front of a page and its comparison collection on the back. In this way students will encouraged to visualise the patterns, and hold their count of the first collection while counting the second collection.
     
  11. Discuss with the students what they have learned about counting today. Record this as a learning statement, e.g. “The number of objects in a collection stays the same no matter what order I count them in.”

Session 2

  1. Draw a farm with two paddocks on a sheet of paper or the whiteboard. Use counters to represent animals that live on the farm. Ask a student to put ten animals on the farm. diagram
     
  2. Tell them to watch as you move one or two animals across the “bridge” to a different paddock.
    Ask, “How many animals are on the farm now?” Some students are likely to recount while some may accept that the total number has not changed. Repeat, moving animals until the students accept that the total number of animals has not changed.
  3. Move two or three animals to different paddocks, then be very obvious as you invite the students to watch you putting another animal on the farm.
    diagram 
  4. Ask, “How many animals are on the farm now?” Students who trust their previous counting are likely to realise that, in effect, one more animal has been added so the next number, eleven, gives the new number of animals.
  5. Repeat this by moving some animals around then removing two animals.
  6. Send the students away in pairs with a Farmyard card (see: Copymaster 1), a number strip, and some counters to act as animals. Then begin by putting ten animals on the farm and take turns to rearrange the existing animals, then add or remove up to three animals. The partner must then work out how many animals are now on the farm and point to the number on the number strip.
  7. After students have in pairs explored shifting the animals, encourage them to image the splitting of collections. Begin with a given number of animals on the farm. Tell the students to close their eyes. Move some animals around and then place a plastic cup (barn) over the animals in one paddock. Tell the students the same animals are on the farm but some are in the barn. Ask them how many animals they think are in the barn. Discuss their strategies for working this out.
  8. Get them to play the barn game in pairs. Students could be encouraged to record their answers as equations, e.g. 4 + 6 = 10.
  9. Discuss with the students what they have learned about counting today. Record this as a learning statement, e.g. “The number of objects in a collection stays the same no matter how I move them around.”

Session 3

  1. Begin the session by rehearsing counting using counters and student copies of the hundreds board (Copymaster 2). Roll a dice or select numeral cards from a deck to generate numbers of counters to add on or take off. For example, to play the adding games, roll the dice, and students collect that many counters, and build on to their existing collection from the last number. Encourage them to predict how many counters they will have on their board before they place the new ones.
  2. To play a subtracting game, get the students to place a collection of counters on their board (say 20), roll the dice to find how many counters to remove, students predict how many will be left when they have taken the dice number away.
  3. To combine adding and subtracting, the students start with ten counters on the hundreds board, roll the dice, and add or subtract the number shown in turns. Play finishes when either 0 or 20 are reached.
  4. Use a plastic ice-cream or yoghurt container with a slot cut in its base. Turn it upside- down on an overhead projector and feed counters into it in sets of one, two, or three. Each time a new set is added ask the students to predict how many counters will be in the container in total. Link this to the process of adding counters to a hundreds board.
  5. Similarly start with a collection already in the container and either remove or add one, two, or three counters each time.
  6. Students can practise this game with a partner to consolidate their understanding. An extension of the activity is to use the put in or take off cards (see Copymaster 3) to generate instructions. Students start with a number of counters in their container. They choose two cards and follow both of the instructions on the card. If they can successfully work out how many objects are left they get a point. Discuss the most efficient way to solve these problems. For example, getting “take away 3” and “put in 4” has the same effect as adding one.
  7. Develop a learning statement with the students, e.g. “ When I put one more thing in a collection the next number tells me how many, when I take one thing out the number before tells me how many.”

Session 4

Pose the following problem to the students using the pictures from Copymaster 4 to make a storyboard. The pictures can be fastened onto a whiteboard using blue-tack, or magnetic tape.

 Here are some puppies and some bones. Are there enough bones to give each puppy a bone?   

  PuppiePuppiePuppiePuppiePuppie     

bonebonebonebonebonebone

Discuss how the problem might be solved. Students may suggest matching the puppies and bones in one-to-one correspondence while others may suggest counting both collections. Either method is valid but the counting method is more generalisable. Vary the puppies and bones problems by altering the numbers and the length of arrangements. For example, Are there enough bones for the puppies to have one each?

PuppiePuppiePuppiePuppiePuppiePuppiePuppiePuppiePuppie              bonebonebonebonebonebonebonebone

Provide the students with copies of the pictures from Copymaster 4. Using these pictures they are to make up their own comparison problems.  After students have completed their problems put them into groups of four to share. Bring the class together to discuss strategies for solving the problems.

Pose more comparison problems that involve finding the difference, for example:“How many more garages would you need so there was one for each car?”   

      GarageGarageGarageGarageGarageGarageGarage       

      carcarcarcarcarcarcarcarcarcarcar

“How many garages are left over once the one car has been put in each garage?”

        garagegaragegaragegaragegaragegaragegaragegaragegarage

carcarcarcarcar   

Promote the use of counting to compare the sets by masking the lines of garages/cars. Uncover one collection at a time and record how many objects are in the collection on top of the masking strip. Invite students to find the difference without being able to see the collections. Encourage the students to use the result of counting to solve the problems.

Develop a learning statement, e.g. “You can compare how many things are in two sets by counting and comparing the numbers.”

Session 5

Play the game, “Messages”, with the students in pairs. The students sit back to back with one student facing the teacher. The teacher flashes a pattern card (Copymaster 5) at one student in the pair. That student turns around and gently “pokes” the number of dots they saw on their partner’s back. The partner shows how many dots they felt by holding up that many fingers. Both players turn around to check to see if the number of fingers matches the dot pattern.

Similarly, a digit card (Copymaster 6) can be held up and the student taps their partner that many times on the back.

Use the pattern cards and digit cards to play the following game. Put the students in groups of three or four. Each group needs a set of pattern cards and a set of digit cards.

The pattern cards are spread out individually, face down on the mat. The digit cards are shuffled and put in a pack, face down, in the centre. Players take turns to turn over the top digit card then use their memory to turn over one of the pattern cards. If the cards match the player keeps both cards and has another turn. Once all of the cards are matched the player with the most pairs wins.

After the game, discuss with the class how they recognise some patterns instantly. Look for students to use combinations of smaller groupings, e.g. “I know it’s seven because four and three are seven.”

Use transparent counters to pose problems that involve building on or taking away from instantly recognised patterns. For example, form a pattern of six counters on the projector. Cover it with a book. Uncover it briefly and allow the students to tell you how many there were. Build on or take off from the pattern. With each change show the students the revised pattern for only two seconds and get them to work out how many counters are in the pattern.

For example:counters

This is six counters                    How many is this?                     How many is this?

Extend this further by combining known patterns. For example:

counters

How many is this? (5 and 3)                  How many is this? (4 and 5)

Session Six

  1. Begin the lesson with some skip counting using body percussion. For example, count in twos by touching knees then clapping on every second number. Progress to the students miming the ones counting, “like goldfish”, and saying the second count aloud.

  2. Ask the students to recall which numbers they slapped their knees on. Use the hundreds board to show the numbers and ask the students if they can see any patterns in the numbers. Some may suggest that the numbers go down the board in vertical lines.

  3. Ask students to predict whether larger numbers you say will be multiples of two (knee-slap numbers). For example, “Will we slap our knees on the number 28?”

  4. Ask each student in the group to get two counters of one colour. Put each pair of counters onto the number strip progressively, encouraging the students to predict how many counters will be on the strip in total.

  5. Have the students work in pairs. One partner takes a small handful of counters and the other partner counts the counters in pairs. Be aware that students may count an odd numbered collection incorrectly. The last counter may be counted as a two count. Get students to place their counters on the number strip to check if this occurs.

  6. Use the Slavonic abacus to move across several rows of two beads and ask the students to tell you how many beads have moved. Change this slightly to include rows of two with an extra bead to develop the idea of remainders for the skip count.

  7. Put the students in pairs with a calculator. One partner keys in + 2 and gives their partner an even target, below ten (The hundreds board can be used as a reference). The other partner pushes the equals key, without looking, until they believe the target number is in the window. Note that the calculator will add two to the window number each time the equals key is pressed, until the AC button is pushed.

  8. Extend the skip counts to include threes running through the same sequence of body counting, counting sets, abacus counting and calculator targets.

 

AttachmentSize
CountingCM1.pdf130 KB
CountingCM2.pdf67.86 KB
CountingCM3.pdf62.47 KB
CountingCM4.pdf116.93 KB
CountingCM5.pdf231.41 KB
CountingCM6.pdf72.43 KB