Number representations: An evidence-based math strategy

Many students start kindergarten already knowing how to count. They can say the numbers “one, two, three…” and so on up to 10. But they don’t always understand that each number they’re saying is a quantity.

On top of that, they may be confused by the fact that the same number can be represented in different ways — by its number name (“one”) and by its numeral (“1”). Confusion about different number representations can make it hard for students to learn addition and subtraction. 

Practicing the different ways to show a number can help students understand each number as a quantity. With this strategy, you’ll use explicit instruction to teach three different representations for the numbers 1–10: the number name, the numeral, and a picture or set of objects showing the quantity.

Students will move through a series of collaborative practice activities with these representations. The strategy ends with an assessment of learning as well as options for continued practice.

Download: Printable number puzzles

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Read: How to use this number representation strategy 

Objective: Students will identify a number by its written numeral, its number name, and a picture of repeated objects showing the quantity.

Grade levels (with standards): 

  • K (Common Core K.CC.A.3: Write numbers from 0 to 20. Represent a number of objects with a written numeral 0–20, with 0 representing a count of no objects.)

  • K (Common Core K.CC.B.4.A: When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.)

  • K (Common Core K.CC.B.4.B: Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.)

  • K (Common Core Math Practice MP2: Reason abstractly and quantitatively.) 

Best used for instruction with: 

  • Whole class

  • Small groups

How to prepare:

Gather materials. Print a number puzzle for each group of three to five students. Consider cutting the puzzle pieces out ahead of time. You can keep them clipped together or bundled in small bags. Printing on card stock or laminating the puzzle pieces can make it easier for students to work with them.

Prepare a set of index cards for the numerals 1–10. Write one numeral on each card. Prepare a matching set for the number names. 

Organize small objects like straws, blocks, erasers, or counters into groups of one to 10. You can group the objects in small plastic bags or cups, like five straws in one bag or six erasers in a cup.

Teaching tip: Make a word wall to support English language learners (ELLs) and other students who benefit from visual supports. Having the number names alongside the written numerals will help students learn them as sight words. If a word wall is not an option, make an anchor chart large enough for all students to see.

How to teach:

1. Warm up by reviewing number identification. Write a number between 1 and 10 on the board, or point to a number on a word wall or anchor chart. Say the number aloud. Ask students to repeat the number. Then ask students to show and count the number of fingers to match the number. Model the correct answer. For example, for 3, you would model the answer by holding up three fingers and saying, “That’s right. This is 3.” Then count each finger aloud. ”One, two, three.” 

After several rounds, explain to the students that there are many ways to represent numbers, and using fingers is just one of them. Ask, “What is another way we can show the number 3?” Take different responses from students. Model the responses by writing, drawing, and using objects depending on what students say.

2. Model representing numbers in different ways. Tell students the goal of today’s lesson. “Today we’re going to learn three different ways you can show a number. You can use a numeral. You can use a number name, which is the word for the number. Or you can use a picture or a set of objects.” Have students repeat the concept. Say, “So, you can use a numeral. Say numeral with me. You can use a number name. Say number name with me. Or you can use a picture or set of objects. Say picture or set of objects with me.”

Show them what these representations look like. For example, hold up a card with the numeral 5 on it and another with the number name “five.” Then say, “The number 5 can be written as a numeral like this. But it can also be written as the word five. F-i-v-e. Five. Five is the number name. And it can be shown by five of the same object.” Draw five squares on the board counting each one until you get to five. “One, two, three, four, five. Five squares. All of these mean the same thing: 5 (point to the written numeral), 5 (point to the number name), and 5 (point to the squares).”

“Let’s try it again.” Repeat the steps with another number from the set of 1–10.

Use objects around the room to represent the quantity. You can use blocks, counters, bean bags, or anything large and bright enough for students to see from their seats. 

3. Practice matching numerals and number names to counted objects. Before moving on to working with all the numbers 1–10 and their representations, have students practice with just a few numbers. Place a set of objects in front of each group of students, varying the number of objects from group to group. These objects can be counters, blocks, straws, etc. One student should be in charge of displaying the objects for the other members to see.

Next, give students four index cards: two with numerals and two with matching number names. For example, if a group has six objects in front of them, they might get the following four cards: 1, 6, one, six. As a group, have the students decide which index cards (one numeral and one number name) go with the set of objects. Let the students discuss it and come to an agreement. Check in with each group and offer support as needed. Once all the groups have made their decisions, have them share out. As they hold up the cards, prompt them to say the number and to spell the number name: “s-i-x...six.”

Rotate the set of objects from one group to the next and pass out a new set of cards to each group. Repeat the same steps. After two rounds, if each group has had a successful match, move onto the number puzzles activity. 

4. Practice with number puzzles. Pass out a set of number puzzles to each group (pre-cut or with scissors if students are making the sets). Give the directions: “Now we’ll use number puzzles to practice with the different ways we can show numbers. Sort all the puzzle pieces into three piles: numerals, number names, and pictures.” 

Once students have made three piles, have them shuffle each pile so its pieces are in no particular order. Say, “Now each student will take a turn selecting one piece from the numeral pile. Tell your group members the number you have selected. Together the group will search for a matching number name and a matching picture card from the other two piles to go with the number. Continue taking turns and matching your pieces together. When all of the pieces have been matched into groups of three (numeral, number name, and picture), raise your hands to have your work checked.” Model how to make a matching set before students begin. 

5. Assessment of learning. Have students meet with you one-on-one at a table with objects on it. Give each student a numeral (an index card or from the puzzle piece set). Ask students to show you the matching quantity using objects on the table. Then, ask them to pick the matching number name from the set of index cards or number puzzle pieces. Have students count out the objects and say the number name to show their understanding that all three represent the same quantity. Students who have difficulty with this task should be given additional small group instruction focused on a limited set of numbers like 1–5 and then 6–10. 

6. Continued practice. Give students opportunities to keep practicing representing numbers. Try these as follow-up activities to reinforce learning: 

  • Make a set of puzzles into necklaces by punching a hole on either side of each piece and adding string. Give each student a necklace. Ask them to get up and find the other two classmates who are wearing the same “number” but in a different form.

  • Use a word wall as a way to check in with small groups of students or individuals. Have them identify a number, spell out the number name, and count the objects.

  • Show a number in any form that you choose. Give students a moment to think about what number it is. Ask them to find examples of that number around the room. Then, ask students where else they might see forms of this number in their lives. 

Understand: Why this strategy works

Students who struggle with math in the early elementary grades, especially with number sense, often have a hard time switching between different number representations. They have trouble understanding that the quantity is the same. By directly teaching the three representations of the same quantity, you can help students avoid this confusion. 

Using physical objects and pictures to represent quantities helps students practice and apply the concepts of stable order and one-to-one correspondence. These concepts are essential to understanding addition, subtraction, and comparison of quantities.

The cooperative learning activities in this number puzzle strategy give students practice with showing and communicating their understanding. Cooperative learning is especially beneficial for ELLs because they may find peers more accessible and easier to understand than teachers. Students who may not be ready to complete the task independently also benefit from working with peers. At times peers may offer corrective feedback that other students can learn from. 

Give suggestions for ways that families can practice number representation at home. For example, families can give students a number name written on a small piece of paper and ask them to find that quantity of objects. Or families can ask students to label a set of items in the house with a numeral or number name on a piece of paper to show the quantity. 

Playing games is also a great way to practice numbers at home. Board games, like Five Little Monkeys and Hi Ho Cherry-O, represent quantities in more than one way. 

Research behind this strategy

“Better Learning Through Structured Teaching: A Framework for the Gradual Release of Responsibility (2nd ed.),” by Douglas Fisher and Nancy Frey

“Experimental evaluation of the effects of cooperative learning on kindergarten children's mathematics ability,” from International Journal of Educational Research

“The transition from informal to formal mathematical knowledge: Mediation by numeral knowledge,” from Journal of Educational Psychology


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