Printable guide · ages 9–10

Fıstık’s Groups and Fractions

Use exact groups, arrays and fraction bars to link a picture, a spoken explanation and a number sentence. Each lesson has four explanations, three activities and a new showing task. A correct card is evidence for that task only. Words such as divisor, remainder, numerator and denominator are named and then explained with a picture.

Materials: The on-screen models are enough. Optional: paper plates, drawn circles, paper strips of the same length and counters chosen by an adult. No money, purchase, account, timer or personal performance profile is used.

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What to notice

Notice whether the child checks that groups and parts are equal, reads an array by rows and by columns, says what a remainder means in a story and names the whole when comparing fractions. Accept speaking, pointing or placing cards; allow a pause and another try.

Ask “How does your picture show that?” Completion stars are not a timed test, a diagnosis or proof of mastery. Let a child go back to an earlier picture.

Lesson 1

Equal groups

Goal: Find the total of equal groups from the number of groups and the size of one group.

Use drawn circles or paper plates with the same number of dots in each. Ask the child to say “groups of” before writing ×. Counting on, pointing and drawing are all good ways to explain. The goal is the idea of equal groups, not only a memorised fact.

Starting point: Draw 3 circles with 4 dots in each. Count by fours together.
A helpful phrase: “How many groups? How many are in each group?”
A common mix-up: Adding the two numbers (3 + 4 = 7) instead of repeating the group size once for every group.
Notice the transfer: Does the child check that every group has the same number? Try a new picture with groups of different sizes.

Show: New picture: apples in baskets. Show the number of groups, the size of each group and the total with cards.

Lesson 2

The same array, turned

Goal: Show with a rows-and-columns array that swapping the two numbers of a multiplication keeps the total.

Show an array on paper and turn the paper a quarter turn. The dots do not change, only the way we read them. Ask the child to say the rows and the columns before writing a multiplication. Do not suggest that swapping works for subtraction or division.

Starting point: Draw 3 rows of 4 dots. Turn the paper and read the columns as rows.
A helpful phrase: “What stayed the same? What changed?”
A common mix-up: Thinking the swap changes the total, or swapping the numbers of a subtraction too.
Notice the transfer: Does the child name the rows and the columns? Offer a new array such as 5 rows of 3 dots.

Show: New array: 5 rows with 3 dots in each row. Turn it, then show the turned array and its total with cards.

Lesson 3

Sharing and grouping

Goal: Divide a total equally by sharing it among a known number of groups or by making groups of a known size.

Use real or drawn counters and plates. Let the child try both ways: dealing out one at a time (sharing) and making groups of a set size (grouping). Keep the two stories side by side so that 12 ÷ 3 and 12 ÷ 4 are not mixed up. No timing and no memorised table is needed.

Starting point: Deal 12 paper counters onto 3 plates, one at a time.
A helpful phrase: “Does every plate have the same? How many are on one plate?”
A common mix-up: Giving the divisor as the answer, or subtracting (12 − 3) instead of sharing.
Notice the transfer: Does the child say whether the story gives the number of groups or the size of a group? Try 15 stickers for 3 friends.

Show: 15 stickers are shared equally among 3 friends. Put the numbers in order: all the stickers, the friends, the stickers for each friend.

Lesson 4

The remainder

Goal: Divide with a remainder that is smaller than the divisor and say what the remainder means in a story.

Use a small number of real or drawn objects. Make as many equal groups as possible, then say how many are left. Ask what the left-over objects mean in the story: still waiting, extra, or needing one more group. A remainder is never as big as the divisor.

Starting point: Draw bags of 4 and put 13 counters into them. Count the full bags first.
A helpful phrase: “Could the left-over ones fill another bag?”
A common mix-up: Stopping early with a remainder as big as the divisor, or giving the remainder as the number of groups.
Notice the transfer: Does the child explain what the remainder means in the story? Try 23 pears in baskets of 6.

Show: 23 pears go into baskets of 6. How many baskets are full, and how many pears are left over?

Lesson 5

Unit fractions

Goal: Name a unit fraction 1/n as one of n equal parts of a whole and tell it from unequal parts.

Use paper strips of the same length folded into equal parts. Name the parts only when they are equal. Let the child say “one of four equal parts” before using the name fourth. Compare a folded strip with an unequal cut to show why equal parts matter.

Starting point: Fold a paper strip into 4 equal parts and colour one part.
A helpful phrase: “How many equal parts? How many do we take?”
A common mix-up: Counting parts without checking that they are equal, or reading 1/4 as the numbers 1 and 4.
Notice the transfer: Does the child say why 3 unequal parts are not thirds? Try a strip folded into 8 equal parts.

Show: A new whole is cut into 8 equal parts, and 1 part is shaded. Which fraction names the shaded part?

Lesson 6

Equal fractions

Goal: Show that fractions such as 1/2, 2/4 and 3/6 name the same part of the same whole.

Use paper strips of the same length. Fold one strip in halves and another in fourths, then lay them side by side. Name equivalent fractions only when the wholes are equally long. The rule of multiplying both numbers should follow the pictures, not replace them.

Starting point: Fold two same-length strips, one in halves and one in fourths. Colour the matching lengths.
A helpful phrase: “Where does the shaded part end on each strip?”
A common mix-up: Changing only the numerator or only the denominator, or comparing strips of different lengths.
Notice the transfer: Does the child check the lengths before using numbers? Try 1/4 and 2/8.

Show: A new fraction: touch every fraction that is equal to 1/4.

Lesson 7

Which is greater?

Goal: Compare two fractions of the same whole when they have the same denominator or the same numerator.

Always ask “Is it the same whole?” first. Let the child compare with strips or drawings before using the rules. The rules here cover only fractions with the same denominator or the same numerator; do not say they cover every pair. Avoid the shortcut “bigger numbers mean a bigger fraction”.

Starting point: Lay two same-length strips folded into eighths. Colour 3 parts on one and 5 on the other.
A helpful phrase: “Is it the same whole? Are the parts the same size?”
A common mix-up: Thinking that a bigger denominator makes a bigger fraction, so 1/8 would be greater than 1/4.
Notice the transfer: Does the child say which rule applies, same denominator or same numerator? Try 2/6 and 5/6.

Show: Same numerator again. From these three fractions, choose the greatest and the smallest.

Lesson 8

Two-step problems

Goal: Solve a two-step quantity problem by finding the amount of the first step and using it in the second step.

Ask the child to name what each step finds before calculating: first the groups, then the rest. Drawing the two steps is welcome. Check that the second step uses the answer of the first. Keep the numbers small; the stories are made up and use one multiplication or division plus one addition or subtraction.

Starting point: Read the story aloud and point to the groups in the picture.
A helpful phrase: “What does the first step find? What do we do with it?”
A common mix-up: Using all the numbers in one step, or adding when the story takes away.
Notice the transfer: Does the child say what each answer means? Try a new story with a different second step.

Show: 12 apples are shared equally among 3 friends. Then each friend eats 1 apple. Show how many apples each friend gets first and how many each has now.

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