Larger tables and remainders
Build facts for 6, 7, 8 and 9 and interpret what is left after equal grouping.
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Rebuild a fact from one you know
Six groups of seven can be five groups and one more: 6 × 7 = 5 × 7 + 7 = 35 + 7 = 42.
Eight groups of seven are twice four groups: 8 × 7 = 2 × 28 = 56. Nine groups of seven are one group fewer than ten: 9 × 7 = 70-7 = 63.
Try a fact before opening the table. If you cannot remember it, rebuild it from a fact you know; the table lets you check the result.
Show the 6 to 9 tables
| Groups | 6 in each | 7 in each | 8 in each | 9 in each |
|---|---|---|---|---|
| 1 | 6 | 7 | 8 | 9 |
| 2 | 12 | 14 | 16 | 18 |
| 3 | 18 | 21 | 24 | 27 |
| 4 | 24 | 28 | 32 | 36 |
| 5 | 30 | 35 | 40 | 45 |
| 6 | 36 | 42 | 48 | 54 |
| 7 | 42 | 49 | 56 | 63 |
| 8 | 48 | 56 | 64 | 72 |
| 9 | 54 | 63 | 72 | 81 |
| 10 | 60 | 70 | 80 | 90 |
Use a fact backwards as well: 8 × 9 = 72 gives 72÷8 = 9 and 72÷9 = 8. In 72÷9 = 8, ask which number multiplied by nine makes 72. A division question can ask for the number of groups or for the amount in each group.
Check your understanding 1 Calculate in both directions
Find 7 × 8 and 72÷9. Check the division with multiplication. Use the table if you need it; this practice has no time limit.
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7 × 8 = 56 and 72÷9 = 8. Checking 8 × 9 = 72 confirms the quotient. The two calculations use different table facts; a multiplication sign asks for the product, while division finds a missing factor.
Sometimes objects are left over
Put 29 counters into groups of four. Seven groups use 28 counters, leaving one. Write 29÷4 = 7 remainder 1.
Check 7 × 4 + 1 = 29. The remainder must be smaller than four; if four were left, you could make another whole group.
Check your understanding 2 Use the whole amount
There are 38 cards. Put five in each full packet. How many cards are left after you fill as many packets as possible?
Show answer
Three cards are left. 7 × 5 + 3 = 38, and three is smaller than a packet of five.
The story decides what the remainder means
Twenty-nine children need cars with four seats for children in each car. Seven cars seat only 28 children. You need eight cars, with one child in the last car.
For 29 beads packed four per full packet, you can sell seven full packets and keep one bead. The division is the same, but the requested answer is different.
Check your understanding 3 Enough containers or full groups?
Fifty-two eggs are packed six per box. How many boxes are needed to hold all the eggs? How many completely full boxes can be made?
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52 = 8 × 6 + 4. Nine boxes hold all the eggs, with four eggs in the last. Only eight boxes are completely full. Explain what is being counted instead of automatically reporting the quotient.
Check a division claim
Lee says 47÷6 = 6 remainder 11. The multiplication check works, but is the grouping finished?
Show answer
No. Eleven can supply one more group of six. The finished result is seven remainder five: 7 × 6 + 5 = 47, with five smaller than six.
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