Demonstration 1 of 4
Find the whole-number product, then place the point
Where does the decimal point in 3.2 x 1.8 come from?
The picture is the book's split 32(10 + 8). The two pieces add to the whole-number product. Two divisions by ten, one for each factor, turn it into the decimal answer.
\[32\times18=32(10+8)=320+256=576\]
3.2 is 32 tenths and 1.8 is 18 tenths. The rectangle is 32 tall and 18 wide, split into a tens piece and a ones piece. Tenths times tenths gives hundredths.
Predict first. Change the second factor to 2.5. What whole-number product appears, and where does the point go?
Choose an example
Constructed example: the chapter's 3.2 x 1.8 calculation, with three other second factors.
Calculated values
- Whole-number product
- 32 x 18 = 320 + 256 = 576
- Decimal product
- 576 / 100 = 5.76
- Estimate band
- between 3.2 and 6.4
- Wrong placements
- 57.6 and 0.576 fail the estimate
32 x 18 = 32(10 + 8) = 320 + 256 = 576. Each factor is one tenth of its whole-number version, so 3.2 x 1.8 = 576 / 100 = 5.76. Estimate check: 1 < 1.8 < 2, so the product lies between 3.2 x 1 = 3.2 and 3.2 x 2 = 6.4. 5.76 fits; 57.6 and 0.576 do not.
Use the idea
Any time a calculator result looks odd, redo the decimal product as whole numbers and count the tenths, then run the quick estimate.
Where the conclusion applies
Both factors have one decimal place. With two decimal places in a factor, the product needs one more division by ten; counting places still works, but the count changes.
Check your understanding: What is 2.4 x 1.3, the warm-up from the start of the chapter?
Chapter 17 source: section "Keep decimal multiplication visible". Demonstration C17-D01.
Demonstration 2 of 4
Convert a side first, then multiply
A mat is 1.5 meters by 80 centimeters. Why do 1.20 and 12,000 both describe its area?
Each side is converted once. Area multiplies two sides, so the factor of 100 is used twice: 100 x 100 = 10,000.
\[1\ \mathrm{m}^2=(100\ \mathrm{cm})(100\ \mathrm{cm})=10{,}000\ \mathrm{cm}^2.\]
m is meters, cm is centimeters, and 1 m = 100 cm. m² and cm² are square meters and square centimeters. Each faint grid square is 10 cm by 10 cm.
Predict first. Change the short side to 50 cm. Will the area in square centimeters still be exactly 10,000 times the area in square meters?
Choose an example
Constructed example: the chapter's 1.5 m by 80 cm mat and practice 17.04, with other side lengths.
Calculated values
- Second side in meters
- 80 / 100 = 0.8 m
- Area in square meters
- 1.5 x 0.8 = 1.20 m²
- Area in square centimeters
- 150 x 80 = 12,000 cm²
- Perimeter
- 2(1.5 + 0.8) = 4.6 m
Convert first: 80 cm = 0.8 m. Area = 1.5 x 0.8 = 1.20 m². Check in centimeters: 150 x 80 = 12,000 cm², and 12,000 / 10,000 = 1.20 m², because one square meter holds 100 x 100 = 10,000 square centimeters. Perimeter = 2(1.5 + 0.8) = 4.6 m.
Use the idea
Before multiplying two lengths, put them in one unit. Then check the area in the other unit and divide by 10,000.
Where the conclusion applies
A flat rectangle with perpendicular sides. Multiplying an area by 100 instead of 10,000 converts only one of its two directions and gives a wrong answer.
Check your understanding: A rectangle is 2 m by 75 cm. What are its area and perimeter?
Chapter 17 source: section "Worked example: a mixed-unit rectangle". Demonstration C17-D02.
Demonstration 3 of 4
The boundary and the interior have different jobs
Can two rooms share the same perimeter and still have different areas?
The dark line is the boundary that P adds up. The hatched region is the interior that A counts in square meters. Reshaping the room keeps P and moves A.
\[A = lw\]
\[P = 2l + 2w\]
l is the length and w is the width, both in meters. A is the area in square meters. P is the perimeter in meters.
Predict first. Every choice here has a 10.0 m boundary. Which one encloses the most interior?
Choose an example
Constructed example: the chapter's 3.2 m by 1.8 m room, plus three other rooms with the same perimeter.
Calculated values
- Perimeter
- 2(3.2) + 2(1.8) = 6.4 + 3.6 = 10.0 m
- Area
- (3.2)(1.8) = 5.76 m²
- Units
- m x m = m² for area; lengths added stay in m
P = 2l + 2w = 2(3.2) + 2(1.8) = 6.4 + 3.6 = 10.0 m. A = lw = 3.2 x 1.8 = 5.76 m². Decomposition check: 3.2 x 1 + 3.2 x 0.8 = 3.2 + 2.56 = 5.76. The two numbers answer different questions, so 10.0 m and 5.76 m² cannot be ranked against each other.
Use the idea
Edging is bought by the meter of boundary; flooring covers square meters of interior. Name the attribute first, then choose the formula.
Where the conclusion applies
A flat Euclidean rectangle with the two values as perpendicular sides in matching units. The comparison stops working across units: 10.0 m and 5.76 m² cannot be ranked.
Check your understanding: A rectangle is 4 m by 3 m. What are its perimeter and its area?
Chapter 17 source: section "The boundary and the interior have different jobs". Demonstration C17-D03.
Demonstration 4 of 4
An exact result and its rounded display
When 5.76 m² is shown as 5.8 m², what has changed and what has not?
The display picks the nearer tenth on the number line. The calculated value does not move; only its presentation does.
\[5.76\ \text{m}^2 \approx 5.8\ \text{m}^2\]
5.76 m² is the exact arithmetic on the stated inputs. The sign ≈ means the display is approximately equal to the calculated value.
Predict first. Choose 5.75. Is it nearer 5.7 or 5.8?
Choose an example
Constructed example: the chapter's 5.76 m² room area, plus three other calculated values.
Calculated values
- Calculated (exact)
- 5.76 m²
- Nearest-tenth display
- ≈5.8 m²
- Distances to neighbors
- 5.76 - 5.7 = 0.06; 5.8 - 5.76 = 0.04
5.8 - 5.76 = 0.04 and 5.76 - 5.7 = 0.06, so 5.76 is nearer 5.8. The exact result stays 5.76 m²; the display ≈5.8 m² is a presentation choice, not a new measurement.
Use the idea
Keep the exact value for later work and label the rounded one where it is shown, so the display never quietly replaces the calculation.
Where the conclusion applies
Display rounding only. It says nothing about how a real length was measured or how precise an instrument is. A value exactly halfway needs a stated tie rule.
Check your understanding: What are the nearest-hundredth and nearest-tenth displays of 8.346 m²?
Chapter 17 source: section "Exact arithmetic and a rounded display". Demonstration C17-D04.