Learning to count: Five-based grouping

Purpose

This unit develops students’ understanding of, and proficiency in, using five-based grouping.

Achievement Objectives
NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
Specific Learning Outcomes
  • Identify five-based groupings.
  • Represent five-based groupings in a variety of ways.
Description of Mathematics

Gelman and Gallistel (1978) provided five principles that young students need to generalise when learning to count. These principles are:

  1. The one-to-one principle
    Just like in reading when one spoken word is matched to one written word, counting involves one-to-one correspondence. One item in a collection is matched to one spoken or written word in the whole number counting sequence.
  2. The stable order principle
    The spoken and written names that are said and read have a fixed order. If that order is altered, e.g. “One, two, four, five,…”, the count will not work.
  3. The cardinal principle
    Assuming the one-to-one and stable order principles are applied then the last number in a count tells how many items are in the whole collection.

The first three principles are about how to count. The final two principles are about what can be counted.

  1. The abstraction principle
    Items to count can be tangible, like physical objects or pictures, or they can be imaginary, like words, sounds, or ideas, e.g. Five types of animal.
  2. The order irrelevance principle
    The order in which the items are counted does not alter the cardinality of the collection. This is particularly challenging for students who think that counting is about assigning number names to the items, e.g. “This counter is number three.”
Opportunities for Adaptation and Differentiation

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

  • Providing extended opportunities to use equipment to explore five based groupings.
  • Ensuring students are confident with groupings of 5, before moving onto groupings with 5.
  • Support students to use counting strategies to identify grouping patterns, if needed.

The contexts for activities can be adapted to suit the interests and experiences of your students. For example, use Māori counting numbers.

Required Resource Materials
Activity

Use a variety of equipment to explore five based grouping: a slavonic abacus, five-based tens frames, unifix (or similar) linking cubes, and students’ hands. Using a variety of representations is a powerful way to develop grouping knowledge. These tasks illustrate grouping with numbers from 5 – 10 but can be used with smaller and larger quantities.

Session 1: Finger pattern pairs

An important connection is between the parts that make ten. If a student knows that in 7 + ? = 10 the missing number is three, then they may transfer that fact to the answer the problem 3 + ? = 10. For example:
Show me seven fingers.
How many more fingers make ten? How many fingers are you holding down?
Write 7 + 3 =10 and say, “Seven plus three equals ten.”
Show me three fingers.
How many more fingers make ten? How many fingers are you holding down?

Variations
Students work in pairs. One student makes a number up to ten with their fingers. The other says the number and writes the numeral for it big in the air.
Find different ways to make a number to ten, for example, seven can be 5 + 2, 3 + 4, 1 + 6 and 0 + 7. Finding ways to make numbers between 10 and 20 is possible in pairs or threes.

Session 2: Slavonic abacus and finger patterns

The Slavonic abacus is five based. The purpose of the colouring is to enable instant recognition of a quantity without counting. Try not to use counting to confirm a quantity as that is counter-productive to the intention of either knowing the quantity or working it out from known facts.
For example:
Make a number between 5 and 10 on the top row. Shift the quantity in one move, not one counter at a time.
Show me that many fingers. Note that this gives all students time to work out an answer and it also provides a way for you to see what each student is thinking.
How many beads are there?
How did you know there were eight?

Encourage grouping-based strategies, such as “I can see five and three” and “There are two missing from ten, so I held two fingers down.”

Variations
Ask the students to convince a partner how many beads have moved.
Ask the students to write the number for the beads on the palm of their hand in invisible ink then show you.
Move to “ten and” groupings such as ten and four to develop teen number knowledge. Students work in pairs to show that many fingers or write the number on the palm of their hand.

Session 3: Five based tens frames

Hold up a single tens frame, such as nine, for no longer than one or two seconds. The aim is for students to image the five-based patterns rather than count the dots one by one.
How many dots did you see?
Show me that number on your fingers.
Write that number big in the air for me.

Discuss the structure that students saw. “I saw five and four.” “I saw one missing from ten.” I saw three threes.”

Variations
Play tens frame flash in pairs or threes. Players take turns to be the ‘flasher’ and show the tens frames, with the other students stating the number of dots on each tens frame as quickly as possible.
Instead of writing the number, talk to a partner about what you saw.
Write what is found with symbols like, 8 + 2 = 10, 10 – 2 = 8.
Progress to two tens frames being shown. Start with numbers less than five, e.g. four and three. Move to ten and another tens frame for teen numbers, e.g. ten and six. Try ‘close to ten’ frames, like nine and eight, with another tens frame, e.g. nine and five.

Session 4: Cube stack

  1. Begin with a stack of ten cubes made from five cubes of two colours. Like the slavonic abacus the colours are used to support non-counting methods to establish quantity.
  2. Like the tens frame and abacus activities students can match a quantity you hold up, using their fingers or writing the number. However, finding a missing part encourages part-part-whole knowledge. Show the students a stack of cubes with some missing (put into your pocket). Show the stack for a second or two then hide it.
  3. Ask: How many cubes did you see? How do you know? How many did I put in my pocket?
  4. Praise risk-taking even if the answers are incorrect and try to offer knowledge that might be helpful. For example:
    I think there are eight because I saw five and two.
    Good work. This would be eight (showing five and three). Can you fix it?
  5. After the students have found the missing part reveal it from your pocket to check.


Variations
Students play in pairs with one being the hider and the other the estimator.
Start with a number other than ten, e.g. eight stacks made of five and three.
Students match stacks to finger patterns to help them find the number of missing cubes.
Students write equations for the stacks problems, e.g. 7 + ? = 10.
Progress to taking some cubes from each end. Progress to using two stacks of ten to start, depending on the number knowledge of the students.

Session 5: Ordering fitness fun

  1. Organise students into 5 or 6 relay lines.  Each relay line has an ice cream container with the numeral cards 1-10 or 10-20 in them.  At the other end of the court place the empty number line.
  2. When the teacher says go. The first student in the line closes their eyes and picks a numeral card out of the container.  They run to the other end of the court and place the number in the correct place on the number line.  Then they run back and tag the next person in line.  This continues until all the numbers are placed on the number line. 
  3. When all the numbers have been removed from the container, the whole team runs in a line to the end of the court to check that the sequence is in the right order.  When they are satisfied it’s correct they sit around the number line in a circle.
  4. Conclude the session with students by talking about how they knew that they had put the number in the correct place.
    How can the shading help to work out where number 6 would go, without having to count?
    For example, I knew that 5 went there because it’s at the end of the shaded box.
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Level One