Ages 6–8 · 8 little discoveries
Fıstık’s Number Connections
Eight discoveries with tens, unit steps, parts and wholes, addition, subtraction, missing values and picture problems.
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Tens and ones
Represent a two-digit number as whole tens and remaining ones.
A ten rod holds ten equal little units. Beside one rod are four loose units. Count the rod as ten, then four more: fourteen. The rod is not just one loose unit.
Twenty-three has two tens and three ones. The two in twenty-three means two groups of ten, not two loose objects. The position of each digit tells us its place value.
This model has one ten and seven ones: seventeen. Counting the rod as one loose unit would give eight, which does not match its ten cells. Read the ten first, then the seven ones.
Now make the two parts with number cards. One ten and six ones make sixteen. We can check by counting ten units inside the rod, then adding the six loose units.
Unit steps on a number line
Use equally spaced unit marks to locate a value and its immediate neighbours up to twenty.
This is a window of a number line, from eight to sixteen. Every neighbouring mark is one unit apart. Moving one step right increases the number by one; left decreases it by one.
The marked point is at twelve. Find ten, then take two equal steps right: eleven, twelve. Count the gaps we travel, rather than counting the starting mark as the first move.
Immediately left of ten is nine; immediately right is eleven. The same unit spacing makes those neighbours clear. This window does not begin at zero, but its marks still name their actual values.
A new window can show twelve to twenty. At sixteen, one step left is fifteen and one step right is seventeen. Use the labels and equal steps, not the point’s place on the page.
Parts and a whole
Compose a whole from two parts and find a missing part of that same whole.
One group holds three counters and another holds four. Join them without losing or repeating a counter. The whole is seven. The parts tell us how the whole can be split.
If the whole is seven and one part is three, the other part is four. The whole includes both parts. It is not another separate group to add a second time.
A different whole, eight, can split into two and six. Copying the earlier part numbers would change the total. Check the new model and keep every counter in exactly one part.
Now try a whole of nine with one part of four. The missing part must complete nine, not ten. We can count the known four, then count on to find the remaining five.
Make ten to add
Split one addend into a part that completes ten and a remaining part.
Nine and four can be added by making ten first. Nine needs one more to reach ten. Split the four into one and three; all four counters must still be included.
Move the one beside nine to make ten. The three counters remain. Ten and three make thirteen. We rearranged the groups but did not create or remove any counters.
For eight and five, eight needs two to make ten. Split five into two and three. Using all five to fill the gap would overshoot ten; the needed part is only two.
A new example is seven and five. Seven needs three to make ten; two from the five remain. Ten and two make twelve. Keep the split total equal to the original five.
Take away and keep the remainder
Represent taking away as removing a stated part from a starting quantity.
Start with eight counters. Three are crossed out to show they were taken away. The five uncrossed counters remain. Crossed counters still show the starting story, but they are not in the remainder.
Ten counters with four taken away leave six. The starting amount includes both removed and remaining counters. We can check by putting those two parts back together: four and six make ten.
In a new story, eight counters start, two are taken away, and six remain. The order names roles in the story. It is not the same as sorting those numbers from smallest to largest.
Now start with nine and take away two. Find the removed part and the remainder separately. Count only the uncrossed counters for what is left, then join both parts to check nine.
Find a missing value
Determine a missing addend or removed amount from a stated one-step equality.
A question mark can stand for a missing number. In missing plus three equals eight, eight is the whole. Find how much joins three to make eight. That missing part is five.
Six plus a missing part equals ten. Count on from six to ten or split ten into six and four. The question mark belongs to the part, not automatically to the answer after the equals sign.
In twelve minus a missing number equals seven, the gap tells how many were removed. Five were removed, because seven and five make twelve. Seven names what remains, not what was taken away.
Now find the missing part before plus four equals eleven. Keep the known four and the whole eleven in their roles. Seven joins four to make eleven; substitute it to check the equality.
Count in equal jumps
Continue a stated additive sequence with constant steps of two, five or ten.
Counting in twos means adding two each time. From two, the next values are four and six. The jump stays two; it is not simply choosing the next unused number card.
Counting in fives from five gives ten, then fifteen. Check each difference: five more each time. Changing the jump halfway through would make a different rule, not the one in this task.
Counting in tens from ten gives twenty, then thirty. A constant jump can be larger than one. The number cards show the new totals, not how many jumps we have made.
The start can change while the jump stays the same. In twos from four, we get four, six, eight and ten. Listen to the stated start and step before building this new sequence.
One change in a picture story
Model a one-step quantity story using its start, joining or leaving amount and result.
There are six apples in our story and two more join. Each counter stands for one apple. Joining means the total increases: six plus two makes eight. Nothing was taken away in this story.
Another story starts with nine acorns and gives three away. Three leave, so six remain. Match the action as well as the numbers; the same words do not always mean we should add.
Two passengers are on a pretend bus and three join. The story roles are start two, join three, total five. Keeping those roles helps us explain why the final amount is five.
Now seven birds are shown and four fly away. Each counter stands for one bird. Show four leaving, then find the three still in the picture. We use only this stated change, not a second hidden step.