Ages 9–10 · 8 little discoveries
Fıstık’s Groups and Fractions
Eight discoveries with equal groups, arrays, sharing, remainders, unit and equivalent fractions, comparing fractions and two-step problems.
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Equal groups
Find the total of equal groups from the number of groups and the size of one group.
Fıstık packs acorns into bags. There are 3 bags, and each bag holds 4 acorns. The groups are equal because every bag has the same number. We write 3 groups of 4 as 3 × 4.
To find the total, count on by the group size: 4, 8, 12. Adding the same number again and again is the same as multiplying: 4 + 4 + 4 = 12, so 3 × 4 = 12.
Careful: these bags hold 3, 4 and 3 acorns. The groups are not equal, so one multiplication cannot describe them. Equal groups have the same number in every group.
Now try: 5 plates with 2 pears on each plate. The first number tells how many groups. The second tells how many are in each group. Count by twos: 2, 4, 6, 8, 10. So 5 × 2 = 10.
The same array, turned
Show with a rows-and-columns array that swapping the two numbers of a multiplication keeps the total.
Dots in neat rows and columns make an array. This array has 3 rows, and each row has 4 dots. Three rows of 4 is 3 × 4. Count on: 4, 8, 12.
Now read the same array by its columns. There are 4 columns with 3 dots in each. That is 4 × 3. No dot was added or removed, so the total is still 12: 3 × 4 = 4 × 3.
Swapping works for multiplication, not for every operation. 5 − 2 = 3, but 2 − 5 is not 3, because you cannot take 5 from 2. We use the swap rule only for multiplication.
Try 2 rows of 5 dots: 2 × 5 = 10. Turn the array: 5 rows of 2 dots, 5 × 2 = 10. Know one multiplication, and you know the swapped one too.
Sharing and grouping
Divide a total equally by sharing it among a known number of groups or by making groups of a known size.
Fıstık has 12 acorns for 3 plates. Deal them out one at a time, so every plate gets the same. Each plate ends up with 4. This is sharing: 12 ÷ 3 = 4.
Now Fıstık puts the same 12 acorns into bags of 4. Keep making bags of 4 until none are left. That makes 3 bags. This is grouping: 12 ÷ 4 = 3.
Not every split is a fair share. If the plates get 5, 4 and 3 acorns, all 12 acorns are used, but the shares are not equal. Division needs equal shares.
Division and multiplication are linked. 3 groups of 4 make 12, so 12 ÷ 3 = 4 and 12 ÷ 4 = 3. In 12 ÷ 4, the 4 is the divisor. It tells the size of each group or the number of groups.
The remainder
Divide with a remainder that is smaller than the divisor and say what the remainder means in a story.
Fıstık has 13 acorns and fills bags of 4. 3 bags are full. They use 12 acorns, so 1 acorn is left over. We say 13 ÷ 4 = 3 with remainder 1.
The remainder must be smaller than the divisor. With 14 acorns, 3 bags are full and 2 are left. 2 is smaller than 4, so no new bag can be filled.
Is it finished? Here 3 bags are full but 5 acorns are outside. 5 is bigger than 4, so one more bag can be filled. The answer is 4 bags with remainder 1.
17 children go on boats that carry 5 each. 3 boats are full and 2 children are left. The remainder says 2 children are still waiting. Everyone needs 4 boats to ride.
Unit fractions
Name a unit fraction 1/n as one of n equal parts of a whole and tell it from unequal parts.
A whole bar is cut into 4 equal parts, and 1 part is shaded. That is 1/4, one fourth. The top number, the numerator, tells how many parts we take. The bottom number, the denominator, tells how many equal parts the whole has.
Cut the same whole into 2 equal parts, and one part is 1/2. Into 3 equal parts, one part is 1/3. Into 6 equal parts, one part is 1/6. The more equal parts, the smaller each part.
This bar is cut into 3 parts too, but the parts are not equal. The shaded part is the smallest. It is not 1/3. A unit fraction needs equal parts.
One more try. A whole cut into 5 equal parts with 1 part shaded is 1/5. Every unit fraction has 1 as its numerator. The denominator tells the number of equal parts.
Equal fractions
Show that fractions such as 1/2, 2/4 and 3/6 name the same part of the same whole.
Two bars show the same whole. The first is cut into 2 equal parts with 1 shaded. The second is cut into 4 equal parts with 2 shaded. Both shaded parts are equally long, so 1/2 = 2/4.
Cut the same whole into 6 equal parts and shade 3. The shaded length is still the same. So 1/2 = 2/4 = 3/6. These are equivalent fractions: different names for the same amount.
Careful: changing only one number changes the amount. 1/2 and 1/4 are not equal, because the shaded part gets shorter. To keep the amount, multiply the numerator and the denominator by the same number.
Try 1/3. Cut the whole into 6 equal parts instead of 3. Each third becomes 2 sixths, so 2 sixths are shaded. 1/3 = 2/6. Both numbers were multiplied by 2.
Which is greater?
Compare two fractions of the same whole when they have the same denominator or the same numerator.
Compare 3/8 and 5/8. Both wholes are cut into 8 equal parts, so the parts are the same size. 5 parts are more than 3 parts, so 5/8 is greater than 3/8. Same denominator: the bigger numerator is greater.
Compare 3/4 and 3/8. Both take 3 parts, but fourths are bigger than eighths. So 3/4 is greater than 3/8. Same numerator: the smaller denominator gives the greater fraction.
The wholes must be the same. Here half of a short bar is shaded, and one fourth of a long bar. The shaded lengths are equal even though 1/2 is greater than 1/4. Compare fractions only of the same whole.
Try 2/3 and 2/5 of the same whole. The numerators are the same, so look at the denominators. Thirds are bigger than fifths, so 2/3 is greater than 2/5.
Two-step problems
Solve a two-step quantity problem by finding the amount of the first step and using it in the second step.
Fıstık has 3 boxes with 4 pears in each. Then 5 more pears arrive. Step 1: the pears in the boxes, 3 × 4 = 12. Step 2: add the new ones, 12 + 5 = 17.
Another story: 20 stickers, and Fıstık gives 6 to each of 3 friends. Step 1: what was given away, 3 × 6 = 18. Step 2: 20 − 18 = 2 stickers are left.
A shortcut can go wrong. Taking away 3 and then 6 from 20 gives 11. But the story gives 6 to each of 3 friends: 3 × 6 = 18. Step 1 must match the story.
Try trays: 4 trays hold 3 buns each, and 2 buns are eaten. Step 1: 4 × 3 = 12. Step 2: 12 − 2 = 10 buns are left. Say what each step finds.