Skip it to multiply it

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Purpose

This unit explores early multiplication where ākonga are encouraged to skip count to solve story problems, rather than counting all.

Achievement Objectives
NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
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.
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, ākonga are expected to use a range of counting strategies such as counting on, counting back and skip counting. 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 ākonga need to enact skip counting is knowledge of the skip counting sequences. Ideally, learning word and numeral sequences, like 2, 4, 6, 8… are learned in conjunction with quantity. That way, ākonga 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, ākonga 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 as 5, 10, 15, 20, while the number of hands is tracked as 1, 2, 3, 4.

The learning opportunities in this unit can be differentiated by providing or removing support to ākonga and by varying the task requirements. Ways to differentiate the tasks include:

• extending the number of skip counts in the pattern. For example, ākonga could extend the pattern to 10 or more skips
• reducing the difficulty by focusing on skips of 2 until ākonga are secure in their understanding, and before moving to skips of 5
• providing opportunities for ākonga to skip count forwards and backwards, and from different starting points (e.g. 55, 20).

The contexts for skip patterns used in this unit can be adapted to suit the interests and experiences of your ākonga. For example:

• Ask ākonga to suggest the contexts for different For example, cockles in groups of 5 or 10, poi per tamariki, tuĪ in kōwhai trees, and as an extension counting sheep in the two way sort gate (many farmers count in threes). Consider how links can be made between the items you are counting and area of ākonga interest, or relevant learning from other curriculum areas (e.g. learning about Minibeasts in science).

Te reo Māori vocabulary terms such as tatau māwhiti-rua (skip count in twos), tatau mawhiti-rima (skip count in fives), whakamua (forwards), and whakamuri (backwards) could be introduced in this unit and used throughout other mathematical learning.

Required Resource Materials
Activity

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

Session 1 – Skip Counting

1. Begin by skip counting in twos. Get ākonga to tap their knees and whisper the number 1. Then get them to clap their hands 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. Try backwards skip counting in twos as well.
Encourage ākonga 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 ākonga to count as the pendulum swings. Then omit the odd numbers. This will enable ākonga to focus on counting in twos. Try backwards skip counting in twos as well.
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 ākonga 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 ākonga in different sized groups of up to eight ākonga.  Each group sits in a circle.  Ākonga skip count forward and backwards in fives around the circle. One or two ākonga have an individual piece of paper (a post-it note would work well). When the counting reaches them, they record the number they say as a numeral, e.g. 25.  Ākonga continue to count around the circle as far as they can. If a mistake is made or there is a hesitation, the group can start again. The person who hesitated or made the mistake can choose the starting number and whether they will go forward or backward (the teacher may need to support ākonga with the restarts). After a period of time, ask ākonga 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?

Session 2 – TūĪ in Kōwhai trees

1. Begin the session by choosing one of the skip counting activities from Session 1 as a warm up exercise.
2. Seat ākonga in a circle.  Place containers (for example, ice cream containers) upside down in the middle of the circle. You could print out pictures of kōwhai trees and tūī to use in this session or you could pretend by using other objects from around your classroom.
Here are four kōwhai trees.
Place two tūī in (under) each kōwhai tree.
There are two tūī in each kōwhai tree (uncovering the collections and hiding them sequentially). How many tūī are there altogether?
3. Give ākonga some time to think about the problem.  Encourage ākonga 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; Elinda 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, “There are 6 kōwhai trees on High Street. There are 5 tūī in each kōwhai tree. How many tūī are there altogether?” Encourage ākonga 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 kōwhai trees encourages skip counting by making one by one counting inefficient. Changing the number of tūī to other multiples such as 3 and 4 greatly increases difficulty, especially if ākonga do not know the skip counting sequence. Using ten tūī in each kōwhai tree supports place value development.
6. Pair up ākonga and give them counters to represent tūī and some containers or cups to represent kōwhai trees. A tuakana/teina model could work well here. Ākonga take turns to hide the same number of tūī under each kōwhai tree while their partner hides their eyes or turns around.  When the ākonga turns back, their partner says "There are 2 tūī under each of these kōwhai trees.  How many are there altogether?" Their partner then works out how many tūī 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 ākonga 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 (or use chalk). Give each ākonga 1 fish (Copymaster 1).
3. Ask five ākonga to put their fish in one of the ponds, then another five ākonga 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 ākonga 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, ākonga may use additive knowledge, e.g. 5 + 5 = 10, 10 + 10 = 20. Thinking like that should be encouraged. Record ākonga strategies using numbers and operation signs. Include the multiplication notation 4 x 5 = 20, asking ākonga to explain what the 4, 5, and 20 represent, as well as considering what the x and = symbols mean.
6. Choose one ākonga to go fishing.  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 ākonga use these strategies?
• Skip count in twos, i.e., 4, 6, 8, 10, 12, 14, 16.
• Take away one set of four from 4 x 5 = 20, 20 – 4 = 16.
8. Go back to 4 x 5 = 20. What if someone puts one more fish in each pond? How many fish will there be altogether, then?
9. 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 ākonga.
10. Copymaster 2 is an activity sheet with further pond and fish problems.  The starting problems are closed but the later problems are open so you or your ākonga 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 carton. Have a dozen and half dozen cartons available and larger trays if they are accessible.
3. Put down three half dozen cartons.
How many eggs are there altogether? (You might have cubes, ping pong balls, or other objects as the make-believe eggs).
4. Ask ākonga 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 cartons today.
5. Set up problems such as “Here are five cartons with three eggs each. How many eggs are there altogether?” You might write 5 x 3 = __ to model the equation form. Mask the cartons at first but be prepared to uncover them if ākonga need support with image making. Discuss their strategies and the efficiency of counting by ones, composites or just known facts.
6. Ākonga can then work in small groups to reinforce this learning - a tuakana/teina could work well here. Provide each group with many egg cartons 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 cartons.
You can record your strategies using numbers.
Be ready to share one of your problems with the whole class at the end.
7. Roam and check if ākonga are setting problems with equal sets. Also support ākonga 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 ākonga to reflect on what is the same and what is different about the problems (equal sets, different numbers of sets, different sets 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 carton require ākonga to adapt skip counting, e.g. 5, 10, 15, 20, then 21, 22.

Session 5: Legs on animals

1. Use the context of legs on animals to set problems for ākonga. Remember to challenge ākonga to think of efficient ways to solve the problems.  Try to encourage them not to count by ones.
2. Copymaster 3 provides some open problems where the number of the legs on each animal is given, but the number of animals is left open. Using toothpicks or bits of paper so that some ākonga 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. Ākonga can create their own skip counting problems using different animals or providing ākonga with 'legless' animal printouts such as Copymaster 3. Ākonga can share these with the class.
4. Challenge the class with these problems.
How many legs would be on five goldfish? Do ākonga still realise 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?