Demonstration 1 of 4
An array shows the roles
In 6 x 8 = 48, which number counts rows, which counts tiles per row, and what happens if one doubles?
Each row is one equal group, so the total is rows times tiles per row. Doubling the rows doubles the total. Turning the array around swaps the roles but not the count.
\[6 × 8 = 48\]
The first factor counts rows. The second counts tiles in each row. The product counts tiles. Every row has the same length.
Predict first. Keep 8 tiles per row and go from 6 rows to 12. How many tiles will there be?
Choose an example
Constructed example: the chapter's six-by-eight tile array, its 12-row scaling, and its turned-around version.
Calculated values
- Rows
- 6
- Tiles per row
- 8
- Tiles
- 48
- Compared with 6 x 8 = 48
- same as 48
6 x 8 = 48. As repeated addition: 8 + 8 + 8 + 8 + 8 + 8 = 48. One factor counts rows, the other counts tiles in each row.
Use the idea
When you multiply, say what each factor counts, with its unit. The unit of the product then comes for free.
Where the conclusion applies
Every row has the same number of tiles, and both factors are positive whole numbers. A ragged arrangement may hold 48 tiles but no longer shows 6 equal groups of 8.
Check your understanding: An array has 7 rows of 9 tiles. How many tiles, and how many with 14 rows of 9?
Chapter 7 source: section "An array makes the roles visible". Demonstration C07-D01.
Demonstration 2 of 4
Signs in a product
Why is (-5)(6) negative while (-5)(-6) is positive?
Multiply the magnitudes, then choose the sign: different signs give a negative product, equal signs a positive one. A factor and its opposite give products that cancel, because 6 + (-6) = 0.
\[(-5)(6)=-30\]
\[(-5)(-6)=30\]
Two numbers in parentheses side by side are multiplied. The magnitude is the size without the sign. The dashed arrow shows the product with the opposite second factor.
Predict first. Change the second factor from 6 to -6. Which way will the arrow point?
Choose an example
Constructed example: the chapter's worked example comparing (-5)(6), (-5)(-6) and (-5)(0), plus the same products with 5.
Calculated values
- Magnitudes
- 5 x 6 = 30
- Signs
- negative
- Product
- -30
- Pair check
- -30 + 30 = 0
Magnitudes first: 5 x 6 = 30. Then the sign: the signs differ, so the product is negative. So (-5) x 6 = -30. With the opposite factor, (-5) x (-6) = 30, and the pair adds to zero: -30 + 30 = 0, just as (-5) x (6 + (-6)) = 0.
Use the idea
When a sign is uncertain, write the magnitudes and the signs separately, then check that the opposite factor gives the opposite product.
Where the conclusion applies
Signed whole-number factors and the distributive rule. Zero has no sign, so a product with a zero factor is simply 0.
Check your understanding: What are (-4)(3) and (-4)(-3), and why must they add to zero?
Chapter 7 source: section "Signed multiplication keeps direction visible". Demonstration C07-D02.
Demonstration 3 of 4
Sharing or grouping?
48 tiles, boxes, and the numbers 6 and 8. Which division question is being asked?
Both questions come from one multiplication fact: boxes x tiles per box = tiles. Whichever factor is missing, division finds it and multiplication checks it.
\[48 ÷ 6 = 8\]
\[48 ÷ 8 = 6\]
Sharing division knows the number of boxes and finds tiles per box. Grouping division knows tiles per box and finds the number of boxes. The ÷ sign in the equations and the / sign in the calculations both mean divide.
Predict first. Switch from sharing among 6 boxes to packing 8 tiles per box. Will the 48 tiles fill 6 boxes or 8?
Choose an example
Constructed example: the chapter's 48 tiles shared among 6 boxes or packed 8 per box, plus a 24-tile version.
Calculated values
- Kind of division
- sharing (group size unknown)
- Known
- 6 boxes
- Quotient
- 8 tiles per box
- Multiplication check
- 6 x 8 = 48
48 tiles / 6 boxes = 8 tiles per box. Check by multiplication: 6 x 8 = 48. The same multiplication fact answers both questions; the units say which number the quotient is.
Use the idea
Before dividing, write the missing-factor sentence with a blank and its unit. The blank tells you whether the answer is a group size or a number of groups.
Where the conclusion applies
Every box holds the same number of tiles and the division comes out even. If it does not, a remainder appears and the question must say what to do with it.
Check your understanding: 72 labels go equally on 9 sheets. How many labels per sheet, and is that sharing or grouping?
Chapter 7 source: section "One product, two division questions". Demonstration C07-D03.
Demonstration 4 of 4
When something remains
53 cards go into groups of 8. Is the answer 6 or 7?
Full rows of 8 are the complete groups; the short row is the remainder. Counting complete groups ignores the short row. Placing every card needs one more group for it.
\[53 = 6 × 8 + 5\]
The quotient counts complete groups. The remainder is what is left after making as many full groups as possible, and it is always less than 8 here.
Predict first. Keep 53 cards and switch to asking how many groups hold every card. Will the answer be 6 or 7?
Choose an example
Constructed example: the chapter's 53 cards in groups of 8, with other card totals.
Calculated values
- Quotient and remainder
- 53 / 8 = 6 remainder 5
- Rebuild check
- 6 x 8 + 5 = 53
- Answer to this question
- 6 complete groups
6 x 8 = 48 cards, and 53 - 48 = 5 cards remain, which is less than 8. Check: 6 x 8 + 5 = 53. Only full groups count, so the answer is 6, with 5 cards left over.
Use the idea
When a division leaves a remainder, reread the question before rounding: does it count full groups, or does every item need a place?
Where the conclusion applies
Whole cards and a final group that may be incomplete. Some settings forbid an incomplete group or split the leftovers another way; the setting decides.
Check your understanding: 61 cards go into groups of 8. How many complete groups, and how many groups to hold every card?
Chapter 7 source: section "When something remains". Demonstration C07-D04.