In How estimation helps with ‘more’ and ‘less’, we explored how estimation and the Compare structure, along with finding the math main idea, can help students make sense of word problems.

Estimation and the Compare structure are not just about finding answers; they’re about uncovering relationships between numbers and understanding the context behind those relationships. Here are some practical ways to challenge your students while keeping the focus on building their reasoning skills through estimation.

1. Working with Numbers Closer Together

We often start with clear differences, like comparing 3 and 14, because they’re easier to grasp. But once students are comfortable, try shifting to numbers closer together, such as 8 and 10. This subtle change requires greater precision and encourages students to think more critically about the concept of difference.

Here are some guiding questions to spark discussion:

• How can you tell which value is larger when the numbers are so close?
• What does the difference between these numbers look like on a Compare model?*

By working with smaller differences, students refine their ability to focus on the details of the comparison, enhancing both their conceptual understanding and visual modeling skills.

*Remember: Not all your students will draw their visual models to scale, and that’s okay. Your models should be representationally accurate and can serve as references during extension discussions.

2. Replacing ‘More’ with ‘Less’

The concept of ‘less’ often trips students up, but it’s just as important as understanding ‘more.’ Flip the narrative to focus on comparisons involving ‘less’ and watch how it challenges their thinking. For example:

• Hudson has 14 Pokemon cards. Luke has 3. How many less cards does Luke have than Hudson?

This adjustment shifts the focus:

• The line of equality stays the same, but now we’re examining the difference next to the shorter bar.

Encourage students to estimate: How far is Luke’s value from Hudson’s?

This small tweak helps students see that the Compare structure applies universally, whether the focus is on ‘more’ or ‘less.’

Notice that I didn’t draw a ring around the area that represents less. That’s because the structure should reflect the situation as described in the story. Since Luke has fewer cards, the difference isn’t physically present—it’s implied by the comparison.

3. Using Estimation as a Warm-Up Activity

Estimation can be an excellent warm-up to engage students and set the tone for deeper thinking. Here’s a simple way to incorporate it:

• Display a Compare model without context, such as two bars labeled 12 and 9.
• Ask: What could the difference be? What’s a number that’s too small? Too big?

Then take it a step further:

• What’s a number story could match this model?
• What would be the math main idea of a number story that matched this model?
• How does estimation help you decide on a reasonable answer?

This approach encourages students to think critically and primes them for more complex problem-solving.

4. Gradually Increasing Complexity

When your students are ready, increase the challenge level:

• Use numbers that are harder to estimate, like 123 and 118.
• Ask: Which comparisons are easier or harder? Why?
• Combine estimation with other Structures of Equality to demonstrate how different models connect.

These changes build flexibility and deepen students’ understanding of the relationships of the given amounts, or values.

5. Reinforcing the Connection Between Model and Context

One of the most critical aspects of estimation is linking it back to the story. If students only focus on numbers without context, they miss out on understanding the relationships those numbers represent. To reinforce this, ask:

• How does the story help you decide if your estimate makes sense?
• What does the line of equality tell us about the relationship between these numbers?

These questions encourage students to see estimation as part of a broader problem-solving process, not just an isolated skill.

Why It Matters

The beauty of these modifications is that they’re not just about adding variety—they’re about building foundational skills. By tackling different scenarios, students learn to adapt, think critically, and connect their learning to broader mathematical concepts. Estimation becomes a tool for reasoning, not just finding answers.

Wrapping Up

By challenging your students while keeping the focus on building their reasoning skills through estimations, you equip your students to see beyond the immediate problem and into the relationships and reasoning that underpin it. The goal isn’t just to solve problems but to help students grow into confident, curious thinkers. Keep experimenting, adapting, and watch your students thrive as mathematical thinkers.