Thank you for your interest in NCTM’s Illuminations. Beginning in mid-April, all Illuminations content will be moving to Interactives will remain openly available and NCTM members will have access to all Illuminations lessons with new filtering and search options. We hope you will continue to utilize and enjoy these resources on

Pin it!
Google Plus

How Many More?

Number and Operations
Grace M. Burton
Location: unknown

Students write subtraction problems, model them with sets of fish-shaped crackers, and communicate their findings in words and pictures. They record differences in words and in symbols. The additive identity is reviewed in the context of comparing equal sets.

Call seven students to the front of the room, and then roll a number cube to decide how many more will come to form a second group. Next, ask each group to form a line so that the lines are parallel and the last person in each line is standing against the board. Then tell each student to hold hands with the student across from him or her. Have a volunteer say how many more students were in the larger group, and then record the subtraction equation where all the students can see it.

Continue the lesson by reading a book that features fish, such as Swimmy, by Leo Lionni. To help the students become more familiar with the set meaning for comparison subtraction, tell the students you are going to compare sets of crackers. Show a plate of fish-shaped crackers. Then roll a number cube and ask how many there will be in a plate with that many more fish-shaped crackers. Make a second plate with that many crackers to verify the students' responses. Then encourage them to write the subtraction equation that would be used to compare the two sets.

Repeat this procedure several times. Then, if necessary, review the terms "addend," "compare," and "difference." Ask what the addends and difference would be if one plate has four crackers and the other has six? [4, 6, 2] Ask what the addends and difference would be if both plates had seven fish-shaped crackers [7, 7, 0]. Prompt the students to create other such entries.

Next, ask the students to watch as you solve a subtraction problem in which two sets are compared. For example, you could say that Jen’s plate has five crackers and Sally’s plate has three crackers. Then create two sets where everyone can see them, surrounding one set with red yarn and the other with blue yarn. Ask the students to imagine the yarn loops represent plates. (If you prefer, you can use red and blue plates for the demonstration and for the student activity.)

Then ask questions such as the following:

  • What comparing questions can we ask and answer about the plates?
  • How many more crackers are on (student's name) plate?
  • How many fewer crackers are on (another student's name) plate?

Now give each student two lengths of yarn and some crackers. Have the students pose comparison situations, model them, and answer similar questions.

When the students are ready, present a subtraction story problem in which a set of three and a set of four are compared. Demonstrate how to make a horizontal bar graph that will allow the students to compare the data. Next, guide the students through the solution of another problem, this one showing the comparison of a plate of two crackers with a plate of three crackers. (For example, Meg had three fish and Pete had two fish. How many more fish did Meg have?) Then ask the students to record the sets in a bar graph.

pdficonGraph Paper

435 Plate chart

When they are ready, call on the students to share their problems and the graph. You may wish to suggest that they record a comparison in pictures, as a bar graph, and in an equation for their portfolios.

  •  Number cubes 
  • Fish-shaped crackers in resealable bags  
  • Paper plates 
  • Yarn in two colors 
  • Paper and crayons 
  • Graph paper 
  • Book of your choice about fish 

Assessment Option

You may find it helpful to add to your recordings on the Class Notes recording sheet you began earlier in this unit. This data may be helpful as you plan strategies for regrouping students.
Move on to the next lesson, Hopping Backwards to Solve Problems.

Questions for Students 

1. What do we find when we compare two sets?

[The difference.]

2. Make two plates of crackers. Can you show how to compare the two plates?

[Student responses may vary, but they should be able to identify the major concepts covered in today's lesson.]

3. If we compare a set of five and a set of eight, what will the difference be? How would you verify that?

[3; 8 - 5 = 3; students may use a bar graph, subtract, draw two plates, etc.]

4. What would be the smallest difference we could get between two plates if one plate has four fish-shaped crackers? How will I get that difference?

[0; Put four crackers on each pate.]

5. Suppose you had a plate of five fish-shaped crackers. How many crackers would you put on a second plate so that there would be a difference of 4 with that plate? Is there another way?

[There are two ways--a plate of one and a plate of nine.]

6. Look at one of the bar charts. How could we act it out with lines of students? With sets of crackers?

Teacher Reflection 

  • Which students were able to pose appropriate comparison problems?
  • Which students were able to model problems with objects? Which could record the comparison with an equation?
  • Which students could identify addends and differences?
  • Can most of the students justify the difference when one addend is 0? Can they justify a difference of 0?
  • Which students met all the objectives of this lesson? What extension activities are appropriate for these students?
  • Which students did not meet all the objectives of this lesson? What caused them particular difficulty?
Number and Operations

Counting Back

Students count back to compare plates of fish-shaped crackers, and then they record the comparison in vertical and horizontal format. They apply their skills of reasoning and problem solving during this lesson in several ways. [Because students have associated the word "more" with addition, the comparative approach to subtraction is typically more challenging for the students to understand.]
Number and Operations

Hopping Backward to Solve Problems

In this lesson, students determine differences using the number line to compare lengths. Because this meaning is based on linear measurement, it is a distinctly different representation from the meanings presented in Lessons One and Two. At the end of the lesson, the students use reasoning and problem solving to predict differences and to answer puzzles involving subtraction.
Number and Operations

Balancing Equations

This lesson encourages the students to explore another meaning for operations of subtraction, the balance. This meaning leads naturally into recording with equations. The students will imitate the action of a pan balance and record the modeled subtraction facts in equation form.
Number and Operations

Fact Family Fun

In this lesson, the relation of addition to subtraction is explored with fish-shaped crackers. The students search for related addition and subtraction facts for a given number and also investigate fact families when one addend or the difference is 0.
Number and Operations

Wrapping Up the Unit

During this final lesson in the unit, the students use the mathematical knowledge and skills developed in the previous lessons as they visit five stations to review comparative subtraction.

Learning Objectives

Students will:

  • Find differences by comparing sets.
  • Review the terms "addend" and "difference."
  • Explore the effects of subtracting 0 and subtracting all.
  • Record differences in pictures, words, and symbols.

NCTM Standards and Expectations

  • Count with understanding and recognize "how many" in sets of objects.
  • Understand the effects of adding and subtracting whole numbers.
  • Understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations.
  • Develop and use strategies for whole-number computations, with a focus on addition and subtraction.
  • Develop fluency with basic number combinations for addition and subtraction.
  • Use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators.

Common Core State Standards – Mathematics

-Kindergarten, Algebraic Thinking

  • CCSS.Math.Content.K.OA.A.1
    Represent addition and subtraction with objects, fingers, mental images, drawings1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

-Kindergarten, Algebraic Thinking

  • CCSS.Math.Content.K.OA.A.2
    Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

-Kindergarten, Algebraic Thinking

  • CCSS.Math.Content.K.OA.A.5
    Fluently add and subtract within 5.

Grade 1, Algebraic Thinking

  • CCSS.Math.Content.1.OA.B.4
    Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.

Grade 1, Algebraic Thinking

  • CCSS.Math.Content.1.OA.C.6
    Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Grade 2, Algebraic Thinking

  • CCSS.Math.Content.2.OA.B.2
    Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Grade 2, Number & Operations

  • CCSS.Math.Content.2.NBT.B.7
    Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP6
    Attend to precision.