Simplifying Ratios

Overview

In this seventh lesson of Mapping Data and Change, students simplify ratios by identifying and dividing by a common factor. They connect factors with equivalent ratios and explain why the comparison remains unchanged. The lesson builds on students’ prior work representing and comparing quantities in ratio form.

Learning intentions

• WALT identify common factors of the numbers in a ratio.
• WALT simplify ratios by dividing both parts by the same common factor.
• WALT represent equivalent ratios in different forms.
• WALT explain why a simplified ratio describes the same comparison.

Success criteria

• I can find a common factor of both parts of a ratio.
• I can divide both parts by the same number to simplify a ratio.
• I can check that my simplified ratio is equivalent to the original.
• I can explain the connection between factors and equivalent ratios.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathematics and Statistics — Mathsteasers / Alignment: challenge connected to familiar mathematical content.
• Mathematics and Statistics — Additional resources for advanced learners: reasoning, explanation and multiple representations.
• Mathematical communication and reasoning: students justify strategies, compare representations and communicate mathematical thinking.

Lesson structure (45 minutes)

1. 0–5 min · Hook and retrieval. Teacher opens with the hook and retrieval slides and displays two groups of counters: 12 red and 8 blue; ask, “How could we describe this comparison in more than one way?” Students discuss with a partner, record a possible ratio and recall factors of 12 and 8.

2. 5–13 min · Explicit teaching. Teacher models the ratio 12:8 using a simple bar or counter diagram, then divides both parts by 2 to make 6:4 and by 4 to make 3:2. Emphasise that both quantities must be divided by the same number and that the comparison, rather than the actual quantities, stays the same. Students use mini-whiteboards to show the common factor used and the resulting ratio.

3. 13–18 min · Build the connection. Teacher uses the worked-example slides to present 15:10, 18:12 and 20:5, asking, “Which common factors can we use?” and “How do we know when a ratio is in simplest form?” Students list common factors, simplify one ratio and explain to a partner why dividing both parts by the same factor preserves equivalence.

4. 18–32 min · Paired practice. Teacher distributes the simplifying-ratios practice worksheet and models the first question only. Students solve ratio questions in pairs, using factor knowledge to simplify ratios such as 8:12, 14:21, 16:24 and 25:15. They represent selected examples with a diagram or multiplication statement, for example 3:2 × 4 = 12:8. Teacher circulates, checking that students divide both terms by the same factor and asking, “Could you simplify it further?” and “How can you prove your answer is equivalent?”

5. 32–39 min · Reasoning challenge. Teacher displays the reasoning-challenge slides with the statement: “6:4 and 9:6 are different ratios because their numbers are different.” Students decide whether the statement is true, use diagrams or factors to justify their response, and then create another pair of equivalent ratios. Invite several students to share different explanations.

6. 39–45 min · Plenary and exit check. Teacher uses the plenary slides to revisit the key idea: “What must happen to both parts of a ratio when simplifying?” Students complete the final worksheet question independently: simplify 24:18 and explain why the answer is equivalent. Take responses as students leave or review them together if time allows.

Resources

• the complete lesson slide deck
• the simplifying-ratios practice worksheet
• Mini-whiteboards and pens
• Two-colour counters or linking cubes
• Pencils and rulers
• Board or display screen
• Prepared ratio examples for teacher modelling

Assessment

• During retrieval and modelling, check whether students identify factors and understand that both parts must be divided by the same number.
• During paired practice, listen for explanations using the terms factor, common factor, equivalent and simplest form. Note students who simplify only one term or stop before reaching simplest form.
• Use the independent exit question to assess accurate simplification and explanation. Collect or photograph responses to inform the next lesson’s grouping and support.

Differentiation

• Support students with counters, bar diagrams, a factor list and sentence starters: “I divided both parts by ___ because ___ is a common factor”; “The ratios are equivalent because ___.”
• Begin with ratios where the greatest common factor is clear, then move to less familiar examples. Pair students strategically so both partners explain rather than one completing the calculations.
• Provide a multiplication check for students who need reassurance: multiply the simplified ratio by the common factor and compare it with the original.
• Extend confident students by asking them to find all equivalent ratios between 1:1 and 30:30, or to create a ratio that can be simplified in two different steps and prove both methods reach the same simplest form.