PAPER · REASONING · PRACTICE

Real numbers: classification, bounds and approximation

Use exact definitions and square comparisons; keep an approximation distinct from an equality.

40 minutes · Year 7, lower volume · text pp. 35–38 · textbook lessons 18–19

Textbook reference: Year 7 lower · Chapter 6 Real numbers · text lessons 18–19

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

18 minutes in the roots lab with target area 2: predict whether 1.4 is exact (3 min). Hold the target fixed while changing precision from 0 to 1 to 2 decimal places; square trial sides 1.4,1.41,1.42 and record exact lower/upper comparisons (8 min). Change only the target to 9 and explain why its bounds collapse to one exact value (3 min). Write a strict bound for √2 and explain why neither a narrow interval nor a finite display proves the entire decimal expansion or irrationality (4 min).

Organise the key idea

Use exact definitions and square comparisons; keep an approximation distinct from an equality.

Open the original chapter Open the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

Use $\frac{7}{5}$ and $\frac{3}{2}$ to give strict bounds for $\sqrt{2}$ . Justify by squaring.

Square each proposed bound.
Compare those exact squares with the radicand.
Both endpoints are nonnegative; use strict inequalities when neither square equals the target.

Both bounds are nonnegative, where squaring preserves order.
Their squares are $\frac{49}{25}$ and $\frac{9}{4}$ , with $\frac{49}{25}<2<\frac{9}{4}$ .
Therefore the square root lies strictly between the two bounds.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

Given $\sqrt{5}\approx2.2361$ , estimate $4-\sqrt{5}$ to two decimal places.

Substitute the supplied decimal: $4-\sqrt{5}\approx4-2.2361$ .
Before rounding, $4-2.2361=\square$ .
The third decimal digit is ____; to two places the estimate is ____.

Use 2.2361 only as an approximation to √5.
Subtract from 4 before rounding; retain all four supplied decimal places in that subtraction.
Use the third decimal digit to round to two places, and retain the approximation sign.

Use the supplied approximation: 4−2.2361=1.7639.
Round to two decimal places: approximately 1.76, not an exact equality.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

Classify $\sqrt{9}$ .

Rational.
Irrational.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

√9=3, an integer.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

Classify $0.125$ .

Irrational.
Rational.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

0.125=1/8, a ratio of integers.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Classify $0.\overline{12}$ . The bar means the block repeats forever.

Rational.
Irrational.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The block 12 repeats forever; its value is 12/99=4/33.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

Compare $-\sqrt{5}$ and $-\sqrt{6}$ . Explain the effect of the negative signs.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

5<6 gives √5<√6 for nonnegative principal roots.
Taking opposites reverses order.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

Find $|-\sqrt{7}|$ and the opposite of $-\sqrt{7}$ .

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

−√7 is negative; its distance from zero is √7.
Its opposite changes sign, also giving √7.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Use $\frac{173}{100}$ and $\frac{174}{100}$ to give strict bounds for $\sqrt{3}$ . Justify by squaring.

Square each proposed bound.
Compare those exact squares with the radicand.
Both endpoints are nonnegative; use strict inequalities when neither square equals the target.

Both bounds are nonnegative, where squaring preserves order.
Their squares are $\frac{29929}{10000}$ and $\frac{30276}{10000}$ , with $\frac{29929}{10000}<3<\frac{30276}{10000}$ .
Therefore the square root lies strictly between the two bounds.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

Which statement is justified by 1.41²<2<1.42²?

$\sqrt{2}=1.41$
These two calculations alone prove every digit of √2.
$1.41<\sqrt{2}<1.42$

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Both endpoints are positive and their squares bracket 2.
This proves a bound, not equality or an entire decimal expansion.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

Classify $\sqrt{2}$ .

Rational.
Irrational.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

2 is not a perfect square; use the taught irrationality result for √2.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Classify $-\sqrt{16}$ .

Rational.
Irrational.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

−√16=−4.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

Classify $\sqrt[3]{27}$ .

Irrational.
Rational.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

The cube root of 27 is 3.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

Simplify $3\sqrt{2}-\sqrt{2}$ exactly, without replacing √2 by a decimal.

3\sqrt{2}-\sqrt{2}

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Factor the common real number √2: (3−1)√2.
The exact result is 2√2.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Use $2$ and $3$ to give strict bounds for $\sqrt{5}$ . Justify by squaring.

Square each proposed bound.
Compare those exact squares with the radicand.
Both endpoints are nonnegative; use strict inequalities when neither square equals the target.

Both bounds are nonnegative, where squaring preserves order.
Their squares are $\frac{4}{1}$ and $\frac{9}{1}$ , with $\frac{4}{1}<5<\frac{9}{1}$ .
Therefore the square root lies strictly between the two bounds.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

A screen shows 0.333333. Can that finite display alone decide whether the intended number is 1/3 or a terminating decimal? Explain.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

A rounded display omits information about later digits.
Use exact notation or a specified repeating rule before classifying.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

Use $\frac{223}{100}$ and $\frac{224}{100}$ to give strict bounds for $\sqrt{5}$ . Justify by squaring.

Square each proposed bound.
Compare those exact squares with the radicand.
Both endpoints are nonnegative; use strict inequalities when neither square equals the target.

Both bounds are nonnegative, where squaring preserves order.
Their squares are $\frac{49729}{10000}$ and $\frac{50176}{10000}$ , with $\frac{49729}{10000}<5<\frac{50176}{10000}$ .
Therefore the square root lies strictly between the two bounds.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Preparation and assessment notes

Prerequisites

Distinguish principal square roots from two square roots; compare nonnegative squares and round decimals.

Look for in students’ work

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original 40-minute core consolidation after the mapped textbook lessons; not a replacement for every first-teaching period. Only the supplied text has been mapped.

大湾区初中数学三维课堂（七年级下册）正文.pdf

Year 7, lower volume · text pp. 35–38 · textbook lessons 18–19

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Real numbers: classification, bounds and approximation

Year 7, lower volume · text pp. 35–38 · textbook lessons 18–19 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Use $\frac{7}{5}$ and $\frac{3}{2}$ to give strict bounds for $\sqrt{2}$ . Justify by squaring.

Guided attempt2

Given $\sqrt{5}\approx2.2361$ , estimate $4-\sqrt{5}$ to two decimal places.

Substitute the supplied decimal: $4-\sqrt{5}\approx4-2.2361$ .
Before rounding, $4-2.2361=\square$ .
The third decimal digit is ____; to two places the estimate is ____.

Prerequisite3

Classify $\sqrt{9}$ .

Rational.
Irrational.

Prerequisite4

Classify $0.125$ .

Irrational.
Rational.

Real numbers: classification, bounds and approximation

Year 7, lower volume · text pp. 35–38 · textbook lessons 18–19 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Classify $0.\overline{12}$ . The bar means the block repeats forever.

Rational.
Irrational.

Independent practice6

Compare $-\sqrt{5}$ and $-\sqrt{6}$ . Explain the effect of the negative signs.

Independent practice7

Find $|-\sqrt{7}|$ and the opposite of $-\sqrt{7}$ .

Independent practice8

Use $\frac{173}{100}$ and $\frac{174}{100}$ to give strict bounds for $\sqrt{3}$ . Justify by squaring.

Exit question9

Which statement is justified by 1.41²<2<1.42²?

$\sqrt{2}=1.41$
These two calculations alone prove every digit of √2.
$1.41<\sqrt{2}<1.42$

Exit question10

Classify $\sqrt{2}$ .

Rational.
Irrational.

Real numbers: classification, bounds and approximation — Further practice

Year 7, lower volume · text pp. 35–38 · textbook lessons 18–19

Further practice11

Classify $-\sqrt{16}$ .

Rational.
Irrational.

Further practice12

Classify $\sqrt[3]{27}$ .

Irrational.
Rational.

Further practice13

Simplify $3\sqrt{2}-\sqrt{2}$ exactly, without replacing √2 by a decimal.

3\sqrt{2}-\sqrt{2}

Further practice14

Use $2$ and $3$ to give strict bounds for $\sqrt{5}$ . Justify by squaring.

Extension15

A screen shows 0.333333. Can that finite display alone decide whether the intended number is 1/3 or a terminating decimal? Explain.

Extension16

Use $\frac{223}{100}$ and $\frac{224}{100}$ to give strict bounds for $\sqrt{5}$ . Justify by squaring.

Real numbers: classification, bounds and approximation — Teacher answers

Year 7, lower volume · text pp. 35–38 · textbook lessons 18–19

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Distinguish principal square roots from two square roots; compare nonnegative squares and round decimals.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

Inquiry and discussion: 18 minutes in the roots lab with target area 2: predict whether 1.4 is exact (3 min). Hold the target fixed while changing precision from 0 to 1 to 2 decimal places; square trial sides 1.4,1.41,1.42 and record exact lower/upper comparisons (8 min). Change only the target to 9 and explain why its bounds collapse to one exact value (3 min). Write a strict bound for √2 and explain why neither a narrow interval nor a finite display proves the entire decimal expansion or irrationality (4 min).

Worked example1

Use $\frac{7}{5}$ and $\frac{3}{2}$ to give strict bounds for $\sqrt{2}$ . Justify by squaring.

Reference answer

$\frac{7}{5}<\sqrt{2}<\frac{3}{2}$ .

\frac{7}{5}<\sqrt{2}<\frac{3}{2}

Both bounds are nonnegative, where squaring preserves order.
Their squares are $\frac{49}{25}$ and $\frac{9}{4}$ , with $\frac{49}{25}<2<\frac{9}{4}$ .
Therefore the square root lies strictly between the two bounds.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

Given $\sqrt{5}\approx2.2361$ , estimate $4-\sqrt{5}$ to two decimal places.

Substitute the supplied decimal: $4-\sqrt{5}\approx4-2.2361$ .
Before rounding, $4-2.2361=\square$ .
The third decimal digit is ____; to two places the estimate is ____.

Reference answer

$1.76$ .

4-\sqrt{5}\approx1.76

Use the supplied approximation: 4−2.2361=1.7639.
Round to two decimal places: approximately 1.76, not an exact equality.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

Classify $\sqrt{9}$ .

Rational.
Irrational.

Reference answer

Rational.

√9=3, an integer.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

Classify $0.125$ .

Irrational.
Rational.

Reference answer

Rational.

0.125=1/8, a ratio of integers.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Classify $0.\overline{12}$ . The bar means the block repeats forever.

Rational.
Irrational.

Reference answer

Rational.

The block 12 repeats forever; its value is 12/99=4/33.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

Compare $-\sqrt{5}$ and $-\sqrt{6}$ . Explain the effect of the negative signs.

Reference answer

$-\sqrt{5}>-\sqrt{6}$ .

-\sqrt{5}>-\sqrt{6}

5<6 gives √5<√6 for nonnegative principal roots.
Taking opposites reverses order.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

Find $|-\sqrt{7}|$ and the opposite of $-\sqrt{7}$ .

Reference answer

Both are $\sqrt{7}$ .

|-\sqrt{7}|=\sqrt{7}

−√7 is negative; its distance from zero is √7.
Its opposite changes sign, also giving √7.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Use $\frac{173}{100}$ and $\frac{174}{100}$ to give strict bounds for $\sqrt{3}$ . Justify by squaring.

Reference answer

$\frac{173}{100}<\sqrt{3}<\frac{174}{100}$ .

\frac{173}{100}<\sqrt{3}<\frac{174}{100}

Both bounds are nonnegative, where squaring preserves order.
Their squares are $\frac{29929}{10000}$ and $\frac{30276}{10000}$ , with $\frac{29929}{10000}<3<\frac{30276}{10000}$ .
Therefore the square root lies strictly between the two bounds.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

Which statement is justified by 1.41²<2<1.42²?

$\sqrt{2}=1.41$
These two calculations alone prove every digit of √2.
$1.41<\sqrt{2}<1.42$

Reference answer

$1.41<\sqrt{2}<1.42$

Both endpoints are positive and their squares bracket 2.
This proves a bound, not equality or an entire decimal expansion.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

Classify $\sqrt{2}$ .

Rational.
Irrational.

Reference answer

Irrational.

2 is not a perfect square; use the taught irrationality result for √2.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Classify $-\sqrt{16}$ .

Rational.
Irrational.

Reference answer

Rational.

−√16=−4.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

Classify $\sqrt[3]{27}$ .

Irrational.
Rational.

Reference answer

Rational.

The cube root of 27 is 3.
Use an exact value or the stated repeating/nonrepeating definition, not a finite calculator display.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

Simplify $3\sqrt{2}-\sqrt{2}$ exactly, without replacing √2 by a decimal.

3\sqrt{2}-\sqrt{2}

Reference answer

$2\sqrt{2}$ .

2\sqrt{2}

Factor the common real number √2: (3−1)√2.
The exact result is 2√2.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Use $2$ and $3$ to give strict bounds for $\sqrt{5}$ . Justify by squaring.

Reference answer

$2<\sqrt{5}<3$ .

2<\sqrt{5}<3

Both bounds are nonnegative, where squaring preserves order.
Their squares are $\frac{4}{1}$ and $\frac{9}{1}$ , with $\frac{4}{1}<5<\frac{9}{1}$ .
Therefore the square root lies strictly between the two bounds.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

A screen shows 0.333333. Can that finite display alone decide whether the intended number is 1/3 or a terminating decimal? Explain.

Reference answer

No. Both 1/3 and 333333/1000000 can give that rounded display; the intended definition matters.

A rounded display omits information about later digits.
Use exact notation or a specified repeating rule before classifying.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

Use $\frac{223}{100}$ and $\frac{224}{100}$ to give strict bounds for $\sqrt{5}$ . Justify by squaring.

Reference answer

$\frac{223}{100}<\sqrt{5}<\frac{224}{100}$ .

\frac{223}{100}<\sqrt{5}<\frac{224}{100}

Both bounds are nonnegative, where squaring preserves order.
Their squares are $\frac{49729}{10000}$ and $\frac{50176}{10000}$ , with $\frac{49729}{10000}<5<\frac{50176}{10000}$ .
Therefore the square root lies strictly between the two bounds.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.