Lesson 2 — Comparing, Rounding, and Connecting Decimals to Fractions

Summary

Learners compare and round decimals and connect them to equivalent fractions with denominators of 10 and 100. On a number line they see that 0.5 = 5/10 = 50/100 = 1/2. They compare decimals by lining up place values, and round to the nearest tenth or hundredth using the “look at the next place” rule.

Objectives

• Compare decimals to the hundredths and thousandths and connect them to equivalent fractions with denominators of 10 and 100. (D05.S1.05.01)
• Round decimals to the nearest tenth and hundredth. (D05.S1.05.01)

Connection

Two market stalls sell the same cloth. One writes the price as 0.5, the other as 0.50. Which is cheaper? Trick question — they are the same: 0.5 and 0.50 are equal, just written two ways, like saying “half” or “fifty hundredths.” In a shop, at a farm weighing grain, or measuring rain, we compare decimals to decide what is more or less, and we round long decimals to something easier to think about — a price of 4.376 becomes about 4.38, or 4.4.

Materials

• Decimal number line (0 to 1)
• Decimal comparison cards
• Comparison and rounding page

Preparation

• Print or draw the number line (tenths and hundredths, with fraction labels).
• Copy comparison cards for pairs and the page for each learner.
• Have examples ready: compare 0.45 vs 0.4; round 0.46 to the nearest tenth; round 0.457 to the nearest hundredth.

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) equivalent forms — 0.5, 0.50, 5/10, and 1/2 all name the same amount; (2) comparing by aligning place values (0.45 has 45 hundredths, 0.4 has 40, so 0.45 > 0.4); and (3) rounding by looking at the digit in the next place to the right — 5 or more rounds up. Model each as a worked example on the number line first (Kirschner, Sweller & Clark, 2006), then guide a second before independent practice. Two common slips to watch for: thinking more digits means a larger number (0.4 is not less than 0.45 just because it is shorter), and rounding without looking at the correct place. Reinforce with retrieval: ask “what did we build last lesson?” and have a learner name the places. The equivalence facts here are evidence; the habit of checking our own rounding is a value (docs/philosophy.md §7).

Procedure

1. Gather (5 min). Last time you read and wrote decimals to the thousandths. Quick recall: in 3.456, what does the 5 mean? Today you compare and round those decimals, and see them as fractions.
2. Meet equivalent forms (10 min). Look at the number line. Find 0.5. Notice it is also 5/10, 50/100, and 1/2 — all the same point. Decimals and fractions with denominators of 10 and 100 are two languages for the same amount.
3. Worked example — compare (10 min). Which is more, 0.45 or 0.4? Line them up by place value: 0.45 has 4 tenths and 5 hundredths; 0.4 has 4 tenths and 0 hundredths. 45 hundredths is more than 40 hundredths, so 0.45 > 0.4. Plot both on the number line to check.
4. Worked example — round (10 min). Round 0.46 to the nearest tenth. Look at the next place (hundredths): the digit is 6, which is 5 or more, so round up: 0.46 → 0.5. Round 0.457 to the nearest hundredth: look at the thousandths (7), round up: 0.457 → 0.46.
5. Practice with a partner (10 min). Sort your comparison cards using <, >, or =, and round each decimal to the nearest tenth and hundredth. Check each other: did you look at the right place?
6. Close (5 min). Comparing and rounding are skills you use every day — with money, time, and measures. The trick is the same each time: line up the places, and look at the next place to round.

Differentiation

• Support: Use only tenths and hundredths; round to the nearest tenth only, and keep the number line under each problem. Describe each step aloud and use tactile markers so learners with low vision can follow by touch.
• Extension: Compare and round decimals to the thousandths, and explain a case where two different decimals round to the same number (e.g., 0.43 and 0.44 both round to 0.4).

Assessment

• Formative (observation/self): Can the learner compare two decimals correctly and round a decimal to a given place, explaining why?
• Self-check: The learner asks, “Can I show 0.5 and 0.50 are equal, and can I round 0.457 to the nearest hundredth and explain the rule?”

Home connection

Find two prices at a market or shop, compare them as decimals, and round one to the nearest tenth with someone at home.

Resources

• Decimal equivalence and rounding are standard in elementary mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The claim that checking our own reasoning matters is a value (docs/philosophy.md §7).