Demonstration 1 of 4
Compare from the left
Two numerals use almost the same digits. Which place decides which one is larger?
The chart lines the digits up under the same place labels. Reading from the top, the first row with different digits settles the comparison, because one unit in that place is worth more than every lower place put together.
\[5{,}084{,}000 = 10 \times 508{,}400\]
Each row is a base-ten place, from millions down to ones. A digit's value depends on its row: one row up means ten times as much. A 0 written in a row holds the other digits in their places.
Predict first. Before you switch to 640,090 and 604,900, guess which row will be outlined as the first place where the digits differ.
Choose an example
Constructed example: the chapter's pair 5,084,000 and 508,400, its 58,400 with the internal zero removed, and its practice pair 640,090 and 604,900.
Calculated values
- First numeral
- 5,084,000
- Second numeral
- 508,400
- First unequal place
- millions (5 against 0)
- Larger
- 5,084,000
- Exactly ten times the second?
- yes: 10 x 508,400 = 5,084,000
Reading from the left, the first unequal row is the millions row, where 5 is more than 0, so 5,084,000 is larger. The subtraction agrees: 5,084,000 - 508,400 = 4,575,600. Every digit also sits one place to the left of its partner, so 10 x 508,400 = 5,084,000 exactly.
Use the idea
When two long numbers look alike, line them up by place before you subtract anything. The first difference from the left tells you which is larger.
Where the conclusion applies
Whole numbers written in base ten and aligned by place. The rule fails if the numerals are lined up by their first digit instead of by place, or if two measurements carry different units.
Check your understanding: Which is larger, 3,402,000 or 3,420,000, and which place decides it?
Chapter 5 source: section "Place determines value". Demonstration C05-D01.
Demonstration 2 of 4
A grouping changes the question
Why do 6 + 4 x 3 and (6 + 4) x 3 give different answers?
Without parentheses, only the 4 is repeated: six items, then three groups of four. With parentheses, the whole sum of ten is repeated three times. Same numbers, different instructions.
\[6+4\times3\]
\[(6+4)\times3\]
The x means multiply. Parentheses mark a group to calculate first. Without them, multiplication is done before addition. Each small square in the picture is one item.
Predict first. With the parentheses added, how many rows of ten will the picture show?
Choose an example
Constructed example: the chapter's worked example 6 + 4 x 3, with one other repeat count.
Calculated values
- Reading
- six plus 3 groups of four
- Rows drawn
- 4
- Total
- 18
- The other grouping gives
- 30
6 + 4 x 3 = 6 + 12 = 18. The expression is read as six plus 3 groups of four. Multiplication comes before addition unless parentheses group the sum first; the other grouping gives 30, a different question with a different answer.
Use the idea
Before calculating a line with several operations, say aloud what each grouping means. The words tell you which part is repeated.
Where the conclusion applies
Ordinary order of operations: groups first, then multiplication, then addition. No exponents appear here. Reading the line strictly left to right would give (6 + 4) x 3 even without parentheses, which is a different instruction.
Check your understanding: What are 50 - 6 x 7 and (50 - 6) x 7?
Chapter 5 source: section "Read grouped arithmetic before calculating it". Demonstration C05-D02.
Demonstration 3 of 4
The example theater
Rounding 18 rows of 28 seats up to 20 rows of 30 gives 600. How far off is it, and which way?
Rounding a positive factor up makes the rectangle bigger, so the product grows. The hatched band shows exactly which seats the rounded model added.
\[18 × (30 - 2) = 504\]
Rows and seats per row are the two factors; their product counts seats. The teal rectangle is the supplied plan. The hatched rectangle is a rounded model.
Predict first. If only the seats per row are rounded, from 28 up to 30, will the estimate still be high?
Choose an example
Constructed example: the chapter's example theater of 18 rows with 28 seats per row.
Calculated values
- Rounded rows x seats per row
- 20 x 30
- Estimate (seats)
- 600
- Exact count (seats)
- 504
- Estimate minus exact
- 96
- Predicted direction
- high
Rounded model: 20 x 30 = 600 seats. Exact count by compensation: 18 x 30 = 540, then 540 - 18 x 2 = 540 - 36 = 504 seats. 600 - 504 = 96. Every rounded factor moved up or stayed, so the estimate was predicted high, and it is 96 seats high. The hatched band is the seats the rounded model added.
Use the idea
When you estimate a total, write the rounded factors and their direction beside the result, so anyone can see why the estimate is high or low.
Where the conclusion applies
Every row has the same number of seats and both factors are positive. If one factor moved up and the other down, the direction would no longer be predictable from the rounding alone.
Check your understanding: A room has 17 rows of 26 chairs. Rounding to 20 x 30, is the estimate high or low, and by how much?
Chapter 5 source: section "An example theater". Demonstration C05-D03.
Demonstration 4 of 4
Which way does the estimate miss?
The tile display has 13 rows of 19. When can you tell, before multiplying, whether an estimate is high or low?
If every move is up or kept, the product can only grow. If every move is down or kept, it can only shrink. One up and one down can pull either way, so only the exact count settles it.
\[13 × 19 = 13 × (20 - 1)\]
Each rounded factor moves up, moves down, or is kept. The gold bar is the estimate; the teal bar is the exact count of 247 tiles.
Predict first. Using 15 rows and 20 tiles per row, will the estimate land above 247 or below it?
Choose an example
Constructed example: the chapter's tile display of 13 rows with 19 tiles per row and its two rounding routes.
Calculated values
- Rounded rows
- 10 (down)
- Rounded tiles per row
- 20 (up)
- Estimate (tiles)
- 200
- Exact count (tiles)
- 247
- Predicted before multiplying
- not settled by the arrows
- Actual
- low by 47
Estimate: 10 x 20 = 200 tiles. Exact: 13 x (20 - 1) = 260 - 13 = 247 tiles. 247 - 200 = 47, so the estimate is 47 tiles low. One factor moved down and one moved up, so the arrows alone could not predict the direction.
Use the idea
Mark an arrow beside each rounded factor. If the arrows agree, say high or low before you multiply exactly; if they disagree, say the direction is not settled.
Where the conclusion applies
Two positive factors and a stated rounding for each. The rule says nothing about how large the miss is, and it fails for mixed directions.
Check your understanding: Estimate 42 boxes of 47 cards as 40 x 50. Can you predict the direction, and what is the miss?
Chapter 5 source: section "Worked example: a tile display". Demonstration C05-D04.