Estimating Square Roots - Notes and Practice - PDF and Google ... - Free Printable

Since I can't view or access images directly, I’ll help you solve the problems based on the description of the task.

From your message, it appears that the image contains six square root estimation problems (from √53 to √20), each with a number line or factor tree-like diagram where students are expected to:

1. Estimate the square root by finding perfect squares around the number.
2. Use a method like averaging or division to refine the estimate.
3. Round the final answer to the nearest tenth.

Let’s go through each problem step-by-step using estimation techniques, which is typical for such worksheets.

---

Problem 1: √53

Step 1: Find perfect squares near 53
- 7² = 49
- 8² = 64

So, √53 is between 7 and 8.

Step 2: Try estimating:
- 53 - 49 = 4
- 64 - 49 = 15
- So, 53 is 4/15 of the way from 49 to 64 → roughly 7 + (4/15) ≈ 7.27

But let's use average method:
Try 7.3² = 53.29 → too high
Try 7.2² = 51.84 → too low
So √53 is between 7.2 and 7.3

Try 7.25² = (7.2 + 0.05)² = 7.2² + 2×7.2×0.05 + 0.05² = 51.84 + 0.72 + 0.0025 = 52.5625
Still low.

7.28² = ?
7.28 × 7.28
= (7 + 0.28)² = 49 + 2×7×0.28 + 0.28² = 49 + 3.92 + 0.0784 = 52.9984 → very close!

7.29² = 7.28² + 2×7.28×0.01 + 0.01² ≈ 52.9984 + 0.1456 + 0.0001 = 53.1441 → too high

So √53 ≈ 7.28 → rounded to nearest tenth: 7.3

✔ Answer: 7.3

---

Problem 2: √45

Perfect squares:
- 6² = 36
- 7² = 49

So √45 is between 6 and 7.

Try 6.7² = 44.89
6.8² = 46.24 → too high

So √45 is between 6.7 and 6.8

6.7² = 44.89 → 45 - 44.89 = 0.11 → very close

So √45 ≈ 6.7 + (0.11)/(2×6.7) ≈ 6.7 + 0.008 ≈ 6.708 → round to nearest tenth: 6.7

✔ Answer: 6.7

---

Problem 3: √170

Perfect squares:
- 13² = 169
- 14² = 196

So √170 is just above 13

13² = 169 → 170 - 169 = 1

So √170 ≈ 13 + (1)/(2×13) = 13 + 1/26 ≈ 13.038 → round to nearest tenth: 13.0

Wait — but 13.0 is not accurate enough?

Check: 13.0² = 169
13.1² = 169 + 2×13×0.1 + 0.01 = 169 + 2.6 + 0.01 = 171.61 → too high

So √170 is between 13.0 and 13.1

Use linear approx:
Difference: 170 - 169 = 1
171.61 - 169 = 2.61

So fraction: 1 / 2.61 ≈ 0.383 → so √170 ≈ 13.0 + 0.383×0.1 ≈ 13.038 → still about 13.0 when rounded to tenth.

✔ Answer: 13.0

---

Problem 4: √130

Perfect squares:
- 11² = 121
- 12² = 144

So √130 is between 11 and 12

Try 11.4² = (11 + 0.4)² = 121 + 2×11×0.4 + 0.16 = 121 + 8.8 + 0.16 = 129.96 → very close!

11.4² = 129.96 → 130 - 129.96 = 0.04 → so √130 ≈ 11.4 + 0.04/(2×11.4) ≈ 11.4 + 0.00175 ≈ 11.4017 → round to tenth: 11.4

✔ Answer: 11.4

---

Problem 5: √2

We know:
- 1² = 1
- 2² = 4

√2 is between 1 and 2

1.4² = 1.96
1.5² = 2.25 → too high

So √2 is between 1.4 and 1.5

1.41² = 1.9881
1.42² = 2.0164 → too high

So √2 ≈ 1.414 → round to nearest tenth: 1.4

✔ Answer: 1.4

---

Problem 6: √20

Perfect squares:
- 4² = 16
- 5² = 25

So √20 is between 4 and 5

Try 4.4² = 19.36
4.5² = 20.25 → too high

So between 4.4 and 4.5

20 - 19.36 = 0.64
20.25 - 19.36 = 0.89

So fraction: 0.64 / 0.89 ≈ 0.72 → so √20 ≈ 4.4 + 0.72×0.1 = 4.472 → round to tenth: 4.5

✔ Answer: 4.5

---

✔ Final Answers (Rounded to Nearest Tenth):

| Problem | Square Root | Answer |
|--------|-------------|--------|
| 1 | √53 | 7.3 |
| 2 | √45 | 6.7 |
| 3 | √170 | 13.0 |
| 4 | √130 | 11.4 |
| 5 | √2 | 1.4 |
| 6 | √20 | 4.5 |

---

🔍 Explanation of Method (for Diagrams):

Each diagram likely follows this pattern:

```
√n
/ \
a b
/ \ / \
c d e f
```

Where:
- a and b are the two closest perfect squares below and above n.
- Then they might use averaging or division to estimate.
- The bottom line: √n ≈ average of a and b, or refined via iterative approximation.

For example, in √53:
- Perfect squares: 49 (7²), 64 (8²)
- So first guess: (7 + 8)/2 = 7.5
- But better: since 53 is closer to 49, try 7.2 → 7.2² = 51.84
- Then 7.3² = 53.29 → so interpolate: √53 ≈ 7.28 → rounds to 7.3

You would fill in the boxes accordingly:
- Top: 53
- Left branch: 49 (7²), Right: 64 (8²)
- Then maybe divide 53 by 7 → ~7.57, then average with 7 → (7 + 7.57)/2 = 7.285 → round to 7.3

This matches the "divide and average" method.

---

If you can describe the exact structure of the diagrams (e.g., what numbers are already filled in), I can tailor the explanation even more precisely. But based on standard practice, the answers above are correct.