Mathematics · 2nd Class

Active learning ideas

Exploring Number Systems: Beyond Base 10

Active learning helps students grasp abstract concepts like place value in different bases by making them concrete. When children manipulate blocks or encode messages, they physically see how grouping changes with the base, which builds deeper understanding than abstract rules alone. These hands-on tasks also encourage collaboration and discussion, reinforcing ideas through peer explanation.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Number - N.1.1

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Stations Rotation
• 30 min · Small Groups

Small Groups: Base 5 Block Builds

Provide blocks or counters limited to 5 per position. Groups build and record base 10 numbers up to 20 as base 5 equivalents, like 14 as 24_5. They explain their builds to another group, noting regrouping steps.

What is the value of each digit in a two- or three-digit number?

Facilitation Tip

During Base 5 Block Builds, circulate and ask groups to explain their bundling process, ensuring they verbalize the shift from ones to fives.

What to look for

Present students with a number in base 5, for example, 32 (base 5). Ask them to draw blocks or use place value charts to show this number and then write its equivalent value in base 10. Observe their use of base 5 place values (fives and ones).

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 02

• Stations Rotation
• 25 min · Pairs

Pairs: Binary Encoding Game

Pairs use 0-1 cards to encode numbers up to 15 in binary, placing cards in powers-of-2 positions (1, 2, 4, 8). Partners decode and verify by counting dots on cards. Switch roles after five numbers.

How can you use hundreds, tens, and units to build any number up to 199?

Facilitation Tip

For the Binary Encoding Game, model how to count digits aloud, emphasizing why binary uses more positions for the same value.

What to look for

Give each student a card with a number in base 10 (e.g., 13). Ask them to convert this number into base 2 and write it down. Then, ask them to write one sentence explaining why the binary representation looks the way it does.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 03

• Stations Rotation
• 35 min · Whole Class

Whole Class: Base Conversion Race

Divide into teams for a relay. Teacher calls a base 10 number under 32; first student writes it in binary or base 5 on a shared chart, next adds the base 10 equivalent. Teams with most correct win.

Can you read, write, and show numbers to 199 using blocks or a place value chart?

Facilitation Tip

In the Base Conversion Race, pause briefly after each round to highlight common errors, such as forgetting to regroup in base 5.

What to look for

Pose the question: 'If we used base 5, how many different digits would we need?' Facilitate a discussion where students explain their reasoning, connecting it to the definition of a base and the digits available in base 5. Prompt them to compare this to base 10.

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Activity 04

• Stations Rotation
• 20 min · Individual

Individual: Personal Base Chart

Each student draws a place value chart for base 5 up to three digits and fills it with given base 10 numbers. They label powers (1, 5, 25) and self-check with a partner.

What is the value of each digit in a two- or three-digit number?

Facilitation Tip

When reviewing Personal Base Charts, ask students to compare their base 5 and base 2 columns to spot patterns in digit placement.

What to look for

Present students with a number in base 5, for example, 32 (base 5). Ask them to draw blocks or use place value charts to show this number and then write its equivalent value in base 10. Observe their use of base 5 place values (fives and ones).

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

A few notes on teaching this unit

Start with base 5 before introducing base 2, as it’s closer to base 10 and easier for children to visualize with physical blocks. Avoid rushing to abstract symbols—instead, let students discover the need for regrouping through guided play. Research shows that alternating between concrete and symbolic representations solidifies understanding, so pair block activities with written conversions. Watch for students who rely too heavily on base 10 comparisons; gently redirect them to use the base-specific materials provided.

Students will confidently represent numbers in base 5 and base 2 using visual models and symbols. They will explain how place values change when the base shifts, using clear language like 'fives and ones' or 'eights and fours.' Missteps in regrouping or digit limits will be corrected through peer feedback and teacher guidance during tasks.

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Watch Out for These Misconceptions

• During Base 5 Block Builds, watch for students who use digits like 6 or 7 in their base 5 representations.

  Prompt them to recount their blocks, asking, 'How many single blocks do you have left?' If they exceed 4, guide them to bundle into a five-block and reset their count.

• During Personal Base Charts, listen for students who assume place values always multiply by 10.

  Have them line up their base 5 and base 2 charts side-by-side, then ask, 'What power of 5 is in the second column? What power of 2?' to redirect their thinking.

• During the Binary Encoding Game, note students who think binary numbers are longer because they represent smaller values.

  Ask them to hold up their binary cards and count the digits aloud, then compare the same number in base 10 to show why binary needs more positions.

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Methods used in this brief

• Stations Rotation