Comparing two numbers to 10
Two numeral cards lie on the table with nothing to count beside them. How can a child tell which number is greater and which is less?
Where you are 23 in Counting and cardinality
When you reach for it
Once a child can compare two groups of things and can read the numerals 1 to 10, the next step is to compare two written numbers with no objects in sight: 9 is greater than 3, and 3 is less than 9. The child can always check by making each number with fingers and matching them, and slowly learns the faster way: the number said later when counting from one is the greater number.
Each day below takes about ten to fifteen minutes. Use the words “greater than”, “less than” and “equal” from the first day, and always say the sentence both ways.
Day 1: Finger match
You need: numeral cards 1 to 10, written large, with a short line under the 6 and the 9 so they cannot be read upside down.
Sit facing the child. Each of you takes a card and shows that many fingers. Then press your fingertips together, one of yours against one of theirs, until someone runs out. Ask, “Who has fingers left over? Whose number is greater?” Then say both sentences together: “5 is greater than 3. 3 is less than 5.”
The fingers are only a way to check. Keep pointing back to the cards: “So the card that says 5 is greater than the card that says 3.” The goal is to know it from the numerals alone.
Practice. Show the fingers, match them, then say both sentences.
- The cards in the picture, 3 and 5. Answer: 2 fingers are left over on the 5 side, so 5 is greater than 3 and 3 is less than 5.
- Cards 9 and 6. Answer: 3 fingers are left over on the 9 side, so 9 is greater than 6 and 6 is less than 9.
- Cards 2 and 4. Answer: 2 fingers are left over on the 4 side, so 4 is greater than 2 and 2 is less than 4.
- Cards 7 and 7. Answer: no finger is left over, so the numbers are equal: 7 is equal to 7.
Day 2: The number path
You need: masking tape or sidewalk chalk to make ten squares in a row on the floor, and sticky notes numbered 1 to 10, one in each square in counting order.
Ask the child to stand on 1 and walk the path, saying each number aloud. Point out that every step forward is one more. Now call out two numbers. One child (or the child and a stuffed toy) stands on each. Ask, “Who is further along? Whose number is greater?” The number further along is the one said later in counting, so it is greater.
Ask the “less” question too: “Whose number is less? How do you know?” A good answer sounds like “3 comes before 8 when I count.”
Practice. Stand on both numbers, then answer.
- The picture, 7 and 10. Which is greater? Answer: 10, because it is 3 hops further along; 10 is greater than 7.
- 3 and 8. Which is greater? Answer: 8, because it is 5 hops further along; 8 is greater than 3.
- 1 and 6. Which is less? Answer: 1, because it comes 5 hops sooner; 1 is less than 6.
- 4 and 9. Which is less? Answer: 4, because it comes 5 hops sooner; 4 is less than 9.
Day 3: Flip and compare
You need: two sets of numeral cards 1 to 10, shuffled together into one face-down pile.
Each of you flips one card. Whoever has the greater number says the sentence first (“6 is greater than 5”), and the other says it the other way (“5 is less than 6”). If both cards show the same number, say together, “They are equal.” When the child is unsure, say the counting words aloud from one and listen for which number comes later, or match fingers as on Day 1.
Numbers next to each other, like 7 and 8, are the hardest pairs, because nothing about the cards looks different. Slow down on these: “Which one do we say right after the other?”
Practice. Read each pair of cards and say both sentences.
- Pair A, 6 and 5. Answer: 6 is greater than 5, and 5 is less than 6. Six comes right after five.
- Pair B, 7 and 8. Answer: 8 is greater than 7, and 7 is less than 8. Eight comes right after seven.
- Pair C, 10 and 2. Answer: 10 is greater than 2, and 2 is less than 10. A child should not be fooled by the 1 in 10 looking small.
- Pair D, 1 and 2. Answer: 2 is greater than 1, and 1 is less than 2.
Day 4: Guess my number
You need: numeral cards 1 to 10, laid face up in a row in counting order.
Say, “I am thinking of a number. It is greater than 6.” Ask the child to turn face down every card that cannot be it: 1 through 6. Then give the second clue: “It is less than 8.” Now 8, 9 and 10 go face down, and one card is left. Turning the cards over shows the child that “greater” means everything further along the row and “less” means everything before.
Then swap roles: the child picks a secret number and gives you the clues.
Practice. Start each round with every card face up.
- Greater than 6 and less than 8. Answer: 7.
- Less than 2. Answer: 1, the only card before 2.
- Greater than 9. Answer: 10, the only card after 9.
- Greater than 3 and less than 5. Answer: 4.
- Say both sentences for 5 and 9. Answer: “9 is greater than 5. 5 is less than 9.”
Exercises
- Show the two cards in the picture. Which number is greater, and which is less? Ask the child to say both sentences.
- Say aloud, with no cards and no fingers: “Which is less, 10 or 5? How do you know?”
Answers
- 8 is greater than 4, and 4 is less than 8. A child who has it answers from the numerals, then can check by matching fingers: 4 fingers meet 4 of the 8, and 4 are left over on the 8 side.
- 5 is less than 10. A child who has it gives a reason such as “I say 5 before 10 when I count” or “5 is one hand and 10 is two hands.”
Where this goes next
The words “greater than”, “less than” and “equal” get their written signs in Comparing numbers and the symbols >, < and =, where 9 is greater than 3 is written 9 > 3. Knowing that each number on the path is one more than the one before it is the idea behind One more and one less. Putting several counted groups in order, from most to fewest, is the work of ordering groups by how many.