Odd Numbers Left Over

Overview

This is the second of two consecutive Year 4 lessons, building on students’ understanding that even numbers can be grouped into pairs with no remainder. Students investigate odd numbers as quantities that leave one left over when paired, identify odd numbers to 1000, compare odd and even numbers, and connect odd numbers to multiplication by 2 plus 1.

Learning intentions

Students will:

• understand that an odd number has one left over when arranged in pairs
• identify and classify odd and even numbers to 1000
• use the ones digit to describe the pattern of odd and even numbers
• connect odd numbers to multiplication by 2 plus 1
• justify their classification using materials, drawings or number sentences

Success criteria

• I can explain why a number is odd or even.
• I can identify odd numbers to 1000 by checking the ones digit.
• I can show an odd number as 2 groups of equal size plus 1.
• I can use words, drawings or a number sentence to justify my answer.

Vocabulary

Odd, even, pair, group, remainder, left over, ones digit, multiple, multiplication, number sentence, justify.

Curriculum links

• Multiplicative relations: represent and use the structure of multiplication and division to solve problems.
• Multiplicative relations: complete number sentences involving multiplication and division by finding missing values.
• Place value: use the ones digit and the role of zero to represent and classify numbers.
• Additive and multiplicative reasoning: explain and justify strategies using mathematical language and representations.

Lesson structure (60 minutes)

1. 0–8 min · Retrieve and connect. Open with the retrieval and hook slides. Display 12 counters and ask students to make pairs, then recall: “What makes a number even?” Students complete a quick think-pair-share and explain that even numbers have no counters left over when paired. Show 7 counters and ask, “What is different?” Introduce the term odd: one counter is left over.

2. 8–20 min · Concrete investigation. Give each pair a tub of counters and number cards from 0–20. Using the odd and even modelling slides, students build each number, arrange counters into pairs and record whether there is no remainder or one left over. Model 9 as 4 pairs and 1: 9 = 2 × 4 + 1. Emphasise that the single counter cannot make a pair. Students justify two classifications to another pair using the sentence stem, “___ is odd/even because ___.”

3. 20–30 min · Pictorial and abstract pattern. Draw paired counters and an open number line, then connect the representations to number sentences: 2 × 3 = 6 (even) and 2 × 3 + 1 = 7 (odd). Students use the odd and even number investigation worksheet to colour or mark numbers from 0–30 and circle the ones digit. Guide students to notice that odd numbers end in 1, 3, 5, 7 or 9, while even numbers end in 0, 2, 4, 6 or 8. Clarify that this pattern works for numbers to 1000 because the ones digit determines whether one is left over.

4. 30–43 min · Pair challenge: Odd or even detective. In pairs, students choose or receive three-digit numbers and classify them without building every counter. They must prove each answer in one of three ways: identify the ones digit, draw pairs with one left over, or write 2 × ___ + 1 for an odd number. Partners take turns asking, “How do you know?” Include numbers such as 408, 517, 620, 735, 842 and 999. Circulate and ask, “What changes if I add 1?” and “What happens if I add 10?”

5. 43–53 min · Group sorting and reasoning. Groups of four sort a teacher-prepared set of number cards into odd and even columns, using the sorting instructions and reasoning prompts. Each group selects one number from each column and prepares a justification. Challenge groups to find an odd number greater than 500, an even number with a 0 in the ones place, and a pair of consecutive numbers where one is odd and one is even. Groups share one explanation; classmates agree or respectfully challenge it.

6. 53–60 min · Plenary and exit ticket. Revisit the learning intention on the plenary slides. Students complete an exit ticket: a) Is 684 odd or even? Explain. b) Is 927 odd or even? Explain. c) Complete: 2 × 6 + 1 = __; is the answer odd or even? Collect responses as students leave and invite two students to explain different justifications.

Resources

• the odd numbers teaching and activity deck
• the odd and even number investigation worksheet
• Counters or linking cubes, approximately 15 per pair
• Number cards from 0–1000
• Mini-whiteboards and markers
• Document camera or interactive whiteboard
• Pencils, coloured pencils and highlighters
• Prepared odd/even sorting headings
• Exit-ticket slips or small pieces of paper

Assessment

• Listen during retrieval and concrete work for the distinction between “one left over” and “no remainder”.
• Check whether students use the ones digit accurately and can justify rather than merely state a classification.
• Use the exit ticket to identify students who confuse the tens digit with the ones digit or cannot connect 2 × n + 1 with an odd number.

Differentiation

• Support students with counters, a visible ones-digit reference chart, pre-selected numbers below 100 and the sentence stem, “___ is odd because it makes ___ pairs and has ___ left over.”
• Provide a place-value chart for students who need help locating the ones digit; allow oral explanations, drawing or practical modelling instead of written explanations.
• Challenge confident students to find all odd numbers between 200 and 230, explain why adding 2 preserves oddness, or complete 2 × __ + 1 = 19.
• Pair students strategically, read number cards aloud for EAL/D learners, and use consistent gestures for “pairs” and “one left over”. Avoid relying on colour alone when sorting.

Teacher notes

• Include 0 in discussion: it is even because it can be arranged into zero pairs with none left over.
• Avoid saying that odd numbers are “not divisible” without explanation; focus on equal pairs and the remainder of 1.
• Reinforce that the ones digit, not the number of digits, determines odd or even.
• During group work, require every answer to include “How do you know?” so reasoning remains central.