Skip it to multiply it

Purpose

This unit explores early multiplication where students are encouraged to skip count to solve story problems, rather than counting all.  For example "John has 3 ponds and there are 2 fish in each pond.  How many fish are there altogether?"  Students will be encouraged to solve this problem by going 2, 4, 6.  "There are 6 fish altogether."

Specific Learning Outcomes
  • Skip count in twos and fives.
  • Skip count to solve simple multiplication problems with a sum of up to 20.
  • Solve simple multiplication problems in various ways and talk about how they found the answer.
Description of Mathematics

This unit develops the skill of skip counting to find the total of several equal sets. At Level One students are expected to use a range of counting strategies such as counting on, counting back, skip counting, etc. Both conceptual understanding and procedural fluency are important to counting. Skip counting involves understanding that a set can be treated as a composite unit. The last number counted tells how many objects are in the set. Composites can be combined whether they are equal or not, but skip counting (e.g. 5, 10, 15, 20) can be used particularly when the sets are equal. The procedural fluency students need to enact skip counting is knowledge of the skip counting sequences. Ideally learning word and numeral sequences, like 2, 4, 6, 8… is learned in conjunction with quantity. That way students realise that the next number is the result of adding two more objects to what is already there.

As well as knowing skip counting sequences, students need fluency in tracking the number of counts. Initially they may use fingers to do that. For example, the fingers in four hands might be skip counted 5, 10, 15, 20, while the number of hands is tracked 1, 2, 3, 4.

Activity

Each of the following sessions is designed to take 20 to 25 minutes. This series of session can be used in whole class or small group situations.  The first session develops students’ ability to skip count in a variety of ways.

Session 1 – Skip Counting

  1. Begin by counting in twos.  Get students to slap their knees and whisper the number 1.  Then get them to push their arms out to the side (or click fingers) and say 2 in a bigger voice.  Continue the sequence 1 (whisper) 2 (big voice) 3 (whisper) 4 (big voice) etc. 
  2. Choose different odd numbers to start from.  Don’t always start from 1. 
    Encourage the students to continue counting while you record the numbers they are saying in a loud voice on the board.  Then stop and talk about the number sequence and the patterns they can see. 
    What can you tell me about the big voice numbers?
    What patterns can you see?
  3. Tie a weighted object to a piece of string and swing it from side to side.  Get the students to count as the pendulum swings. Then omit the odd numbers. This will enable the students to focus on counting in twos.
  4. You might use Animation 1A to connect the spoken words to numerals on the Hundreds Board or Animation 1B to connect numerals to putting counters on a Slavonic Abacus. Animation 1C and 1D deal with skip counting in fives. Links to animations are in the list of required resources.
  5. Challenge students to see if they can count backwards in twos from 10. Animation 1E - 1H use the Hundreds Board and Slavonic Abacus to count backwards by twos and fives.
  6. Counting circles - put students in different sized groups of up to eight students.  Each group sits in a circle.  Students skip count around the circle. One or two students have an individual piece of paper. When the counting reaches them, they record the number they say as a numeral, e.g. 25.  Students continue to count around the circle as far as they can.  Get the students to bring their numbers back to share with the whole class. Place the numbers on the mat.
    What patterns did you see?
    What numbers are missing from the pattern?
  7. Use puppets - have 2, 3, 4 or 5 puppets, students come to the front of the class and count using the puppets.  E.g. Using 3 puppets - puppet 1 says 1, puppet 2 says 2, puppet 3 says 3, puppet 1 says 4, puppet 2 says 5, …
    What do you think puppet 2 is going to say when it his turn next?
    What will she say the next turn?
  8. Encourage students to count up and predict.

Session 2 Horses and Stables

  1. Begin the session by choosing one of the skip-counting activities from Session 1 as a warm up exercise.
  2. Seat students in a circle.  Place four ice cream containers upside down in the middle of the circle.  
    Here are four horse stables.
    Place two horses under each stable. 
    There are two horses in each stable (uncovering the collections and hiding them sequentially). How many horses are there altogether?
  3. Give the students some time to think about the problem.  Encourage students to share their answers and come into the circle to demonstrate what they did.
    Try to record their responses e.g.  Len thought 1, 2, 3, 4, 5, 6, 7, 8; Tom went 1 2, 3 4, 5 6, 7 8; Hemi went 2, 4, 6, 8; and Kelly went 4 + 4 = 8 (note that this is an additive, not counting, response).
  4. Continue to pose several other similar problems.
    For example, “Now there are 5 stables at John’s farm down the road.  He has 5 horses in each stable how many horses are there altogether?” Encourage students to explain how they got their answer. 
  5. Change the bolded numbers in the problem to alter the complexity of the task. Be aware that increasing the number of stables encourages skip counting by making one by one counting inefficient. Changing the number of horses to other multiples such as 3 and 4 greatly increases difficulty, especially if students do not know the skip counting sequence. Using ten animals each stable supports place value development.
  6. Pair up students and give them a series of counters to represent horses and some ice cream containers, or cups to represent stables.  The students take turns to hide the same number of horses under each stable while their partner hides their eyes or turns around.  When the student turns back, their partner says "There are 2 horses under each of these stables.  How many are there altogether?"  Their partner then works out how many horses there are altogether.

Session 3 – Fish in Fishponds

  1. Start the session by choosing one of the skip-counting activities from Session 1 as a warm up exercise. Include the skip counting sequence in fives.
  2. Set up the scenario for this session by seating the students in a large circle. In the middle of the circle make 4 fishponds using four pieces of string joined up to make them look like ponds.  Give each student 1 fish (Copymaster One). 
  3. Ask five students to put their fish in one of the ponds, then another five students to put their fish in another pond.  Continue until all the ponds have five fish. 
    How many fish are there altogether?
    How can we work it out without counting the fish one by one?
  4. Talk about how you might be able to work it out without individually counting each fish.  Give the students some time to work the answer out and then encourage individuals to share their strategies.
    Did anyone do it a different way?
  5. With the skip counting by five sequence students may use additive knowledge, e.g. 5 + 5 = 10, 10 + 10= 20. Thinking like that should be encouraged. Record students’ strategies using numbers and operation signs. Include the multiplication notation 4 x 5 = 20, asking students to explain what the 4, 5, and 20 represent.
  6. Choose a student to be a fisherperson.  Ask them to take a fish out of each pond. 
    Is there a quick way to work out how many fish there are left?
  7. Change the equation to 4 x 4 = □. Do your students use these strategies?
  8. Skip count in twos, i.e., 4, 6, …, 16, 18, 20.
  9. Take away from 4 x 5 = 20, 20 – 4 = 16.
  10. Go back to 4 x 5 = 20. What if someone (student) puts one more fish in each pond? How many fish will there be altogether, then?
  11. Continue to pose similar problems.  Increase the number of ponds and the number of fish put in each pond with awareness of the difficulty level of the problems, and the skip counting knowledge of your students.
  12. Copymaster Two is an activity sheet with further pond and fish problems.  The starting problems are closed but the later problems are open so you or students can add the missing information.

Session 4 – Cartons and Eggs

  1. Begin the session by repeating a skip-counting activity from Session 1.
  2. Discuss how many eggs are usually in a packet. Have dozen and half dozen packets available and larger trays if they are accessible.
  3. Put down three half dozen packets.
    How many eggs are there altogether? (You might have cubes, ping pong balls, or other objects as the make-believe eggs).
  4. Look for students to explain their strategies. Some may count in ones. Others may use skip counting in twos or threes. Some may use addition, such as 6 + 6 = 12 and count on the last six.
    Sixes are hard to count with, so we are going to use smaller packets today.
  5. Set up problems such as “Here are five packets with three eggs each. How many eggs are there altogether?” You might write 5 x 3 = __ to model the equation form. Mask the packets at first but be prepared to uncover them if students need support with image making. Discuss their strategies and the efficiency of count by ones, composites or just knowing facts.
  6. For independent work give small groups of students many egg packets of the same size (Twos, threes, fours and fives) and make-believe eggs.
    I want you to make problems for each other using the same sized packets.
    You might like to record your strategies using numbers. That is up to you.
    Be ready to share your problem with the whole class at the end.
  7. Wander the room. Are students setting problems with equal sets? Are students using non-count-by-ones strategies when they have the available skip counting or addition knowledge.
  8. Gather the class to share their favourite problems. Encourage your students to reflect on what is the same and what is different about the problems (equal sets, different numbers of sets, different set between problems).
  9. Pose problems like this, “Four cartons of five eggs and two extra eggs. How many eggs altogether?” Extras or missing eggs in a packet requires students to adapt skip counting, e.g. 5, 10, 15, 20, 21, 22.

Session 5

  1. Use the context of legs on animals to set problems for the students. Remember to challenge the students to think of quick ways to solve the problems.  Try to encourage them not to count by ones.
  2. Copymaster Three provides some open problems where the number of the legs on each animal is given but the number of animals is left open. Using nursery sticks or iceblock sticks the students can physically model each problem by giving the animals “legs”. They then use whatever strategies they have available to anticipate the number of “legs” that are needed.
  3. You might photocopy and laminate pictures of the ‘legless’ animals to use in problem posing. Students to create their own skip counting problems using outs of the animals’ legs.  Share these with the class.
  4. Finally challenge the class with these problems.
    How many legs would be on five goldfish? Do students still realise that the problem can be written and solved, 5 x 0 = 0? 
    How many legs would be on 10, 50, 100 goldfish?” What is always true?
  5. You might do the same with kiwi with one leg.
    How many legs would be on 10, 50, 100 1-legged kiwi? What is always true?

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