Facts for seven, eight and nine

When you reach for it

By now a child knows the twos, fives and tens from Facts for two, five and ten, and the threes, fours and sixes from Facts for three, four and six. Zero groups give 0 and one group gives the other factor, so the zeros and ones are known too. The commutative property lets each fact count for both orders, so 7 \times 4 is already known as 4 \times 7. Once all of that is in place, only six facts of the table are still new:

These are the largest products in the table and the ones that take longest to settle. This lesson builds each of them from a fact that is already known, with a move that fits the factor:

• 9 is one less than 10, so a nine fact is a ten fact take away one group;
• 8 is 2 \times 2 \times 2, so an eight fact is the other factor doubled three times;
• 7 is 5 + 2, so a seven fact is a five fact and a two fact added.

Each move rests on a property. The first and third are the The distributive property, which lets a factor be split into parts. The second is the associative property: in a product of three or more numbers, which pair is multiplied first does not change the answer.

Reach for these moves whenever one of the six facts is half remembered. The setting is a scout troop and the relay race it runs at its summer field day.

The lesson takes four days, one part each. The aim of the fourth day is the whole table: every product from 1 \times 1 to 10 \times 10 comes without rebuilding it, with the moves kept as the way to check.

The procedure

Each part gives one family, says which property makes the move correct, and shows it as an array: rows of equal groups, as in Arrays.

Part 1: nines are tens take away one group

At the relay race, Isaure’s troop fields 9 teams with 7 runners on each team. The number of runners is 9 \times 7. The tens are easy: 10 teams of 7 would be 10 \times 7 = 70. Nine teams is one team fewer, so take one group of 7 away.

Draw it as an array of ten rows of 7 and strike out the last row.

The distributive property works with take-away as well as with adding: the 9 was written as 10 - 1, the 7 multiplied each part, and the second product was taken from the first. In plain words, a nine fact is a ten fact take away one group. The group taken away is a whole group of the other factor, here a whole row of 7.

The subtraction is always a ten fact minus a one-digit number, which can be done by counting back or by going down to the ten below. For 90 - 9: take away 10 to reach 80, then give back the 1 that was taken too many, 81.

By the commutative property each row is also a column: 7 \times 9 = 9 \times 7 = 63. The nines products also have a digit pattern of their own, which Patterns in the multiplication table takes up.

Part 2: eights are doubles, three times over

Malin packs tent bags for the troop, 8 bags with 9 tent stakes in each. The count is 8 \times 9. Eight is 2 \times 2 \times 2, so the count can be reached by doubling.

• Two bags hold 2 \times 9 = 18.
• Four bags are two sets of two bags: 18 + 18 = 36.
• Eight bags are two sets of four bags: 36 + 36 = 72.

Why doubling three times gives eight groups: the associative property lets the factors be grouped in any way, so

Each “2 \times” is one doubling. An eight fact is the other factor doubled three times. The second doubling gives the four fact, so an eight fact is also a four fact doubled: 8 \times 9 is double 4 \times 9.

The last doubling is the one that needs care, because it adds two numbers in the twenties or thirties. Add the tens and the ones apart: 36 + 36 is 60 and 12, which is 72.

Every eight fact ends in a doubling, so every product of 8 is even. Part 1 gives a second route to the same products: 9 \times 8 is 80 - 8 = 72, the same as 8 \times 9.

Part 3: sevens are fives and twos

Lennart sets out cones for relay practice in 7 rows of 7. The count is 7 \times 7. Seven does not split into equal parts, but it does split as 5 + 2, and both of those are known. Cut the array after the fifth column.

This is the distributive property again, with the second factor split this time. A seven fact is a five fact and a two fact added. The cut can run either way: cutting after the fifth row instead splits the first 7, and gives the same 35 + 14.

Of the six facts, 7 \times 7 is the only one with no eight or nine in it, so it is the only one that has to come from the sevens. The other two seven facts are already reached from Parts 1 and 2 by the commutative property: 7 \times 8 = 8 \times 7 = 56 and 7 \times 9 = 9 \times 7 = 63. The sevens move then gives each of them a second route, and a second route is a check:

Part 4: the whole table

Here are the seven, eight and nine rows in full, from 1 to 10:

Most of these cells were known before this lesson: the columns for 1, 2, 5 and 10 from the easy facts, and the columns for 3, 4 and 6 from the middle block. Only the cells under \times 7, \times 8 and \times 9 are new. Those nine cells hold the six new facts: 7 \times 8, 7 \times 9 and 8 \times 9 appear twice, once in each order, and 7 \times 7, 8 \times 8 and 9 \times 9 once each. With them, every fact from 1 \times 1 to 10 \times 10 is known.

Fluency means the product comes to mind without rebuilding it. It grows from using the facts. The moves are what make that safe: a fact half remembered can always be rebuilt and checked by a second route. Three habits help on this last day.

Choose the move that fits. A nine is closest to a ten, an eight is a chain of doubles, a seven splits as 5 + 2. When both factors are big, either one can lead. For 8 \times 9, doubling 9 three times and taking 8 from 80 both work, and the two together are a check.

One group more or less. Neighbors in a row differ by one group. If 8 \times 8 = 64 is known, then 8 \times 9 is one more eight, 64 + 8 = 72, and 8 \times 7 is one less, 64 - 8 = 56. This is the distributive property with the factor split as 8 + 1 or written as 8 - 1.

Run it backward. Every fact answers a missing-factor question. If 72 granola bars are shared by 8 patrols, each patrol’s share is the number in 8 \times \square = 72. Searching the eights for 72 finds 8 \times 9. This is division, and Division facts takes it up for the whole table.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.

Boundaries

• The group taken away is a whole group of the other factor. 9 \times 7 is 70 - 7, one row of 7. Taking away 1, or taking away 9, does not remove a row.
• Ten take away one gives nine, and only nine. The move 10 \times n - n gives 9 \times n. Other near-ten factors take away more groups: 8 \times n is 10 \times n - 2 \times n, two groups.
• Doubling three times gives eight because 2 \times 2 \times 2 = 8. Doubling once gives two groups, twice gives four, three times gives eight. A fourth doubling gives sixteen groups, which is past the facts to know by heart.
• The parts of a split must add up to the factor. 7 = 5 + 2, so the pieces of the array fill it exactly. Parts that add to 6 count only six columns.
• A backward question needs an exact fit. \square \times 8 = 70 has no whole number answer, because 8 \times 8 = 64 and 8 \times 9 = 72 jump over 70.
• Whole groups and factors from 0 to 10. The moves still hold at zero (9 \times 0 = 0 - 0 = 0) and past ten, but the facts to know by heart stop at 10 \times 10.

Where this goes next

With this lesson the table is complete, and every fact also answers a division question: 8 \times \square = 72 is 72 \div 8, and Division facts builds division on that link. Patterns in the multiplication table looks at the finished table as a whole: the digit pattern down the nines, rows that are doubles of other rows, and the squares along the diagonal. A fact like 9 \times 7 grows into 9 \times 70 in Multiplying by tens and hundreds. Later, changing a product by doubling one factor and halving the other becomes a strategy of its own in Doubling, halving and quick products. All of these sit under Multiplication facts.

Exercises

Exercises 1 to 5 go with Part 1, 6 to 9 with Part 2, 10 to 13 with Part 3, and 14 to 21 with Part 4. The reasoning group draws on all four.

Part 1: nines

1. Find 9 \times 4 by taking one group of 4 away from 10 \times 4.
2. Fill in the boxes: 9 \times 6 = (10 \times 6) - \square, which is 60 - \square, which is \square.
3. A relay race has 9 teams with 3 runners on each team. How many runners are there?
4. Find 9 \times 5 two ways: as a ten fact take away one group, and as a five fact.
5. Find 9 \times 2 two ways: as 20 take away one group of 2, and by doubling 9.

Part 2: eights

6. Find 8 \times 6 by doubling 6 three times.
7. Eight relay teams have 3 runners each. Find the number of runners by doubling.
8. Find 8 \times 5 by doubling 5 three times, then check with a five fact.
9. Fill in the boxes: 8 \times 4 = 2 \times (4 \times 4) = 2 \times \square = \square.

Part 3: sevens

10. Find 7 \times 6 by splitting the 7 as 5 + 2.
11. Malin’s troop has 7 patrols with 4 scouts in each. How many scouts is that?
12. Fill in the boxes: 7 \times 10 = (5 \times 10) + (2 \times 10) = \square + \square = \square.
13. Find 7 \times 5 by splitting the 7 as 5 + 2, then check with a five fact.

Part 4: the whole table

14. Fill in the missing factors: 9 \times \square = 27, 8 \times \square = 32 and 7 \times \square = 42.
15. Lennart puts 48 runners into teams of 8. How many teams are there?
16. Isaure sews 54 badges onto sashes, 9 badges on each sash. How many sashes does she use?
17. The troop has 7 patrols of 8 scouts and 9 leaders. How many people is that in all?
18. Which is larger, 8 \times 6 or 7 \times 7, and by how much?
19. Write the nine row, 9 \times 1 to 9 \times 10.
20. A relay race has 9 teams of 4 runners. Malin has 80 bibs. How many bibs are left over after every runner has one?
21. The troop has 70 scouts. Can they be split into patrols of 7 with none left over? Of 8? Of 9? Give the number of patrols where it works.

Reasoning

22. Explain, with an array, why 9 \times 8 is the same as 80 - 8. Name the property that makes it true.
23. Explain why doubling a number three times gives eight groups of that number. Name the property that makes it true.
24. Explain why, once the facts for 0, 1, 2, 3, 4, 5, 6 and 10 are known, exactly six facts of the table are left.
25. Is every product with a factor of 8 also a product with a factor of 4? Explain, and give an example.