Demonstration 1 of 4
The factor decides grow or shrink
When does multiplying a positive length make it shorter?
The top bar stays at 8. The lower bar is the scaled copy. A factor below 1 keeps fewer shares than make up the whole; a factor above 1 keeps more.
\[8 × 1/2 = 4\]
8 is the original length in units of the drawing. The scale factor a/b means: cut the length into b equal shares and take a of them.
Predict first. Choose a factor of 3/2. Will the copy be longer or shorter than 8?
Choose an example
Constructed example: the chapter's 8-unit segment in a half-scale drawing, with other factors.
Calculated values
- Scale factor
- 1/2
- Factor compared with 1
- between 0 and 1
- New length
- 8 x 1/2 = 4
- Change
- shorter by 8 - 4 = 4
8 x 1/2: cut 8 into 2 equal shares of 4 and take 1 of them: 1 x 4 = 4 units. The factor is between 0 and 1, so the length shrinks toward 0.
Use the idea
Before multiplying, compare the factor with 1. That tells you whether the answer should be bigger or smaller, which catches slips.
Where the conclusion applies
A positive starting length and a positive factor. With negative numbers, "bigger" and "smaller" need more care, and a factor of 0 sends any length to 0.
Check your understanding: What is 8 x 5/4, and is it bigger or smaller than 8?
Chapter 11 source: section "A factor tells how the scale changes". Demonstration C11-D01.
Demonstration 2 of 4
A share of a share is an overlap
What is three fourths of two thirds, and why do the bottoms multiply?
The column choice is the first fraction. The row choice takes a share of that. The dark cells belong to both choices, so their count is the numerator and the 12 cells are the denominator.
\[(3/4) × (2/3) = (3 × 2)/(4 × 3) = 6/12 = 1/2\]
\[6 × 1/12 = 6/12 = 1/2\]
Columns cut the whole into thirds; rows cut it into fourths. Together they make 12 equal cells, each 1/12. The overlap counts the cells chosen both ways.
Predict first. Keep 2/3 of the columns and choose 4 of 4 rows. How many overlap cells will there be?
Choose an example
Constructed example: the chapter's three-column, four-row grid, with other row and column choices.
Calculated values
- Cells in the chosen columns
- 2 x 4 = 8
- Overlap cells
- (8 / 4) x 3 = 6
- Product
- (3 x 2)/(4 x 3) = 6/12 = 1/2
- Compared with 2/3
- smaller
(3/4) x (2/3) = (3 x 2)/(4 x 3) = 6/12 = 1/2. The 2 chosen columns hold 8 cells; keep 3 of every 4 rows and (8 / 4) x 3 = 6 cells remain, each 1/12 of the whole. Since 3/4 is less than 1, the result is less than 2/3.
Use the idea
When a recipe or plan says "three fourths of two thirds," draw a quick grid: the overlap is the answer.
Where the conclusion applies
All 12 cells are equal, and both fractions are shares of the same whole. If the rows were unequal, counting cells would not give twelfths.
Check your understanding: Using the same grid, what is (1/4) x (2/3)?
Chapter 11 source: section "Two selections make an overlap". Demonstration C11-D02.
Demonstration 3 of 4
Count the groups that fit
How many groups of one eighth fit inside three fourths?
Rename the available amount in the group's unit, then count. The red line marks the amount; every shaded piece to its left is one group.
\[(3/4) ÷ (1/8) = 6\]
\[6 × 1/8 = 6/8 = 3/4\]
3/4 is the amount available. 1/8 is the size of one group. The answer counts groups, not a share of the whole.
Predict first. Make the groups smaller, 1/16 each. Will more or fewer groups fit inside 3/4?
Choose an example
Constructed example: the chapter's eighths strip, with other group sizes and a half.
Calculated values
- Available amount
- 3/4
- Size of one group
- 1/8
- Renamed
- 3/4 = 6/8
- Groups that fit
- (3/4) / (1/8) = 6
- Check
- 6 x 1/8 = 6/8 = 3/4
Rename in the group's unit: 3/4 = 6/8, which is 6 groups of 1/8. So (3/4) / (1/8) = 6. Reciprocal route: 3/4 x 8 = 6. Check: 6 x 1/8 = 6/8 = 3/4. The count is bigger than 3/4 because each group is small.
Use the idea
To see how many small servings or short lengths fit in an amount, write both in the same small unit and count.
Where the conclusion applies
Each group has the same size, and the group size is not zero. Here every group fits a whole number of times; the next demonstration shows a leftover.
Check your understanding: How many groups of 1/8 fit inside 1/2, and how do you check it?
Chapter 11 source: section "Division counts the unit that fits". Demonstration C11-D03.
Demonstration 4 of 4
A group count can contain a fraction
If five sixths are available and a group needs two thirds, how many groups are there?
Each sixth-sized piece is the same size throughout. Brackets mark full groups. A leftover is named by how much of one group it fills.
\[\frac56\div\frac23=\frac56\div\frac46=5\div4=\frac54\]
5/6 is the amount available, counted in sixth-sized pieces. 2/3 = 4/6 is one group. The answer counts groups, so a leftover is measured against a group.
Predict first. Keep groups of 2/3 and make 2/6 available. Will even one full group fit?
Choose an example
Constructed example: the chapter's five sixths in groups of two thirds, with other amounts.
Calculated values
- Available
- 5/6
- One group
- 2/3 = 4/6
- Groups
- 5 / 4 = 5/4 = 1 1/4
- Leftover pieces
- 1 piece of the 4 a group needs
- Check
- 5/4 x 2/3 = 5/6
(5/6) / (2/3) = (5/6) / (4/6) = 5 / 4 = 5/4 = 1 1/4. 1 complete group fits and 1 piece is left. Measured against a group of 4 pieces, the leftover is 1/4 of a group, not 1/6. Check: 5/4 x 2/3 = 5/6.
Use the idea
When an amount does not divide evenly into servings or containers, say whether a part of a group is allowed, or report full groups and leftovers separately.
Where the conclusion applies
Pieces of equal size, and an amount that can be split, like length or volume. If only full containers count, report 1 group and 1 piece left over instead of 5/4.
Check your understanding: Using groups of 3/4 = 6/8, how many groups are in 3/8?
Chapter 11 source: section "A group count can contain a fraction". Demonstration C11-D04.