Try This First
Try this short activity before reading the lesson.
> Tip a handful of buttons or blocks onto the table, no more than about twelve. Touch each one and say a number as you touch it. Say the last number again, loudly. Now spread them out and ask: how many now? You do not have to count again. It is still that number.
The Big Idea
Counting is touching things and saying number words, one word for each thing. Touch, say. Touch, say. If you touch one twice, or skip one, the count goes wrong. So move each thing to a new pile as you count it.
The last word you say is the answer. If you count "one, two, three, four, five" and stop, then there are five. You do not need to count again to know how many.
Moving the things around does not change how many there are. Five blocks in a row and five blocks in a heap are both five. The number belongs to the group, not to the shape it makes.
For the grown-up: Counting is the first mathematics a child owns: say one number word for each object, and the last word said is how many there are. Students count sets to 20, match a count to a written numeral, and count out a given number of objects from a larger pile. Every number idea that follows, from place value to fractions, rests on this one-to-one count and the idea that the count does not change when the objects are moved.
Words to Know
- count: To touch each thing once and say the next number word each time.
- how many: The last number you said when you counted. Five blocks: the last word was five, so there are five.
- numeral: The written number. The numeral for seven is $7$.
Worked Examples and Your Turn
Study each example, then try the paired problem.
Example 1: Count a small group
Problem: There are some spoons on the table. How many spoons?
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 220 100" width="220" height="100" role="img" aria-label="Seven spoons: a row of five and a row of two." style="display:block;margin:12px auto"><rect x="28.4" y="28" width="3.2" height="20" fill="#ffffff" stroke="#333" stroke-width="1"/><ellipse cx="30" cy="21" rx="6.4" ry="9" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="68.4" y="28" width="3.2" height="20" fill="#ffffff" stroke="#333" stroke-width="1"/><ellipse cx="70" cy="21" rx="6.4" ry="9" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="108.4" y="28" width="3.2" height="20" fill="#ffffff" stroke="#333" stroke-width="1"/><ellipse cx="110" cy="21" rx="6.4" ry="9" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="148.4" y="28" width="3.2" height="20" fill="#ffffff" stroke="#333" stroke-width="1"/><ellipse cx="150" cy="21" rx="6.4" ry="9" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="188.4" y="28" width="3.2" height="20" fill="#ffffff" stroke="#333" stroke-width="1"/><ellipse cx="190" cy="21" rx="6.4" ry="9" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="28.4" y="68" width="3.2" height="20" fill="#ffffff" stroke="#333" stroke-width="1"/><ellipse cx="30" cy="61" rx="6.4" ry="9" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="68.4" y="68" width="3.2" height="20" fill="#ffffff" stroke="#333" stroke-width="1"/><ellipse cx="70" cy="61" rx="6.4" ry="9" fill="#ffffff" stroke="#333" stroke-width="1.5"/></svg>
Solution:
- Touch the first spoon and say "one". Slide it to the side. Why: a spoon that has moved has been counted, so it cannot be counted twice.
- Touch the next and say "two". Slide it over. Keep going: three, four, five, six, seven.
- There are no spoons left in the first pile. The last word was seven. So there are $7$ spoons.
Answer: $7$ spoons
> Notice: Slide each thing as you count it. Then the counted pile and the not-yet pile can never mix.
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 300 130" width="300" height="130" role="img" aria-label="Nine forks lying in a messy heap." style="display:block;margin:12px auto"><rect x="30.7" y="37.7" width="2.9" height="18" fill="#ffffff" stroke="#333" stroke-width="1"/><path d="M26.8 23.3 V34.1 Q32.2 39.5 37.6 34.1 V23.3 M32.2 23.3 V36.8" fill="none" stroke="#333" stroke-width="1.8" stroke-linejoin="round" stroke-linecap="round"/><rect x="173.6" y="83.4" width="2.9" height="18" fill="#ffffff" stroke="#333" stroke-width="1"/><path d="M169.6 69 V79.8 Q175 85.2 180.4 79.8 V69 M175 69 V82.5" fill="none" stroke="#333" stroke-width="1.8" stroke-linejoin="round" stroke-linecap="round"/><rect x="128.6" y="94.2" width="2.9" height="18" fill="#ffffff" stroke="#333" stroke-width="1"/><path d="M124.7 79.8 V90.6 Q130.1 96 135.5 90.6 V79.8 M130.1 79.8 V93.3" fill="none" stroke="#333" stroke-width="1.8" stroke-linejoin="round" stroke-linecap="round"/><rect x="221.7" y="30.6" width="2.9" height="18" fill="#ffffff" stroke="#333" stroke-width="1"/><path d="M217.7 16.2 V27 Q223.1 32.4 228.5 27 V16.2 M223.1 16.2 V29.7" fill="none" stroke="#333" stroke-width="1.8" stroke-linejoin="round" stroke-linecap="round"/><rect x="76.3" y="33" width="2.9" height="18" fill="#ffffff" stroke="#333" stroke-width="1"/><path d="M72.3 18.6 V29.4 Q77.7 34.8 83.1 29.4 V18.6 M77.7 18.6 V32.1" fill="none" stroke="#333" stroke-width="1.8" stroke-linejoin="round" stroke-linecap="round"/><rect x="81.5" y="92.6" width="2.9" height="18" fill="#ffffff" stroke="#333" stroke-width="1"/><path d="M77.5 78.2 V89 Q82.9 94.4 88.3 89 V78.2 M82.9 78.2 V91.7" fill="none" stroke="#333" stroke-width="1.8" stroke-linejoin="round" stroke-linecap="round"/><rect x="34.5" y="88.4" width="2.9" height="18" fill="#ffffff" stroke="#333" stroke-width="1"/><path d="M30.5 74 V84.8 Q35.9 90.2 41.3 84.8 V74 M35.9 74 V87.5" fill="none" stroke="#333" stroke-width="1.8" stroke-linejoin="round" stroke-linecap="round"/><rect x="263.7" y="94.1" width="2.9" height="18" fill="#ffffff" stroke="#333" stroke-width="1"/><path d="M259.7 79.7 V90.5 Q265.1 95.9 270.5 90.5 V79.7 M265.1 79.7 V93.2" fill="none" stroke="#333" stroke-width="1.8" stroke-linejoin="round" stroke-linecap="round"/><rect x="261.8" y="39.2" width="2.9" height="18" fill="#ffffff" stroke="#333" stroke-width="1"/><path d="M257.9 24.8 V35.6 Q263.3 41 268.7 35.6 V24.8 M263.3 24.8 V38.3" fill="none" stroke="#333" stroke-width="1.8" stroke-linejoin="round" stroke-linecap="round"/></svg>
Your Turn 1: Put out $9$ forks in a messy heap and count them by sliding each one across.
Example 2: Count out some from a pile
Problem: There is a big pile of $20$ blocks. Count out $12$ blocks for a tower.
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 200 182" width="200" height="182" role="img" aria-label="Twenty blocks in four rows of five. The first twelve are colored yellow: they are counted out. The last eight are white: they are left behind." style="display:block;margin:12px auto"><rect x="15" y="15" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="51" y="15" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="87" y="15" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="123" y="15" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="159" y="15" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="15" y="51" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="51" y="51" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="87" y="51" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="123" y="51" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="159" y="51" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="15" y="87" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="51" y="87" width="25.9" height="25.9" rx="2" fill="#f2c14e" stroke="#333" stroke-width="1.5"/><rect x="87" y="87" width="25.9" height="25.9" rx="2" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="123" y="87" width="25.9" height="25.9" rx="2" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="159" y="87" width="25.9" height="25.9" rx="2" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="15" y="123" width="25.9" height="25.9" rx="2" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="51" y="123" width="25.9" height="25.9" rx="2" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="87" y="123" width="25.9" height="25.9" rx="2" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="123" y="123" width="25.9" height="25.9" rx="2" fill="#ffffff" stroke="#333" stroke-width="1.5"/><rect x="159" y="123" width="25.9" height="25.9" rx="2" fill="#ffffff" stroke="#333" stroke-width="1.5"/><text x="100" y="170" font-family="Source Sans 3" font-size="14" text-anchor="middle" fill="#333">12 counted out, 8 left</text></svg>
Solution:
- Say the target first: "I need twelve." Why: knowing where to stop is the whole job here.
- Take one block at a time into a new pile, saying the number as it lands: one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve.
- Stop at twelve. Leave the rest. Check by counting the new pile again: twelve.
Answer: A pile of $12$ blocks, with $8$ left behind.
> Verification & Check: Count the new pile a second time. If the last word is twelve again, it is right.
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 200 164" width="200" height="164" role="img" aria-label="A pile of twenty blocks in four rows of five." style="display:block;margin:12px auto"><rect x="15" y="15" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="51" y="15" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="87" y="15" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="123" y="15" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="159" y="15" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="15" y="51" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="51" y="51" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="87" y="51" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="123" y="51" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="159" y="51" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="15" y="87" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="51" y="87" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="87" y="87" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="123" y="87" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="159" y="87" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="15" y="123" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="51" y="123" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="87" y="123" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="123" y="123" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><rect x="159" y="123" width="25.9" height="25.9" rx="2" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/></svg>
Your Turn 2: From the same pile, count out $15$ blocks.
Example 3: Write the numeral
Problem: Count the dots. Write the number.
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 220 140" width="220" height="140" role="img" aria-label="Eleven dots: two rows of five and one more." style="display:block;margin:12px auto"><circle cx="30" cy="30" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><circle cx="70" cy="30" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><circle cx="110" cy="30" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><circle cx="150" cy="30" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><circle cx="190" cy="30" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><circle cx="30" cy="70" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><circle cx="70" cy="70" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><circle cx="110" cy="70" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><circle cx="150" cy="70" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><circle cx="190" cy="70" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/><circle cx="30" cy="110" r="14.4" fill="#6fa8dc" stroke="#333" stroke-width="1.5"/></svg>
Solution:
- Count, sliding a finger under each dot: one to eleven. The last word is eleven.
- Eleven is written with two digits: a $1$ and then another $1$. Write $11$.
Answer: $11$
> Notice: Numbers past nine need two digits. Ten is $10$, eleven is $11$, twelve is $12$. Say the word, then write the numeral, so the two stay glued together.
<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 220 180" width="220" height="180" role="img" aria-label="Stars to count, in rows of five."
The chapter continues in the book.
