Pin it!
Google Plus

How Many Left?

Number and Operations
Location: Unknown

This lesson encourages the students to explore the familiar set model of subtraction. The students write story problems and find differences using sets, including subtracting all and subtracting zero. They record the differences in a chart.

To help the students become more familiar with the set model for subtraction, ask them to watch as you show six pasta shapes in your hand. Tell them you are going to play "Guess How Many?" Then put your hands behind your back, transferring some, say four, of the pasta shapes to your other hand. Ask a volunteer which hand to show, and then bring that hand forward, opening it so the pasta shapes are visible. Now ask the students to "guess" how many pasta shapes are in your other hand and then count to verify their guess. Then give the students a chance to play this game with one another. [This activity involves a missing addend situation (6 - __ = 4) in concrete form.] Note: This concept involves abstract thinking that may be too challenging for some students. Other students may need to divide manipulatives of their own into two sets to answer the question.

Now place the students in pairs and provide them with pasta shapes and a piece of paper for a work mat. Tell them that they will be telling take-away stories and then using pasta shapes to model the stories. Remind them to record how many they started with, how many they took away, and how many were left using either the vertical or horizontal format practiced in the previous lesson. Note: Students generally find the vertical format easier to use, so you may wish to suggest this format to the students who are more challenged by this material.

To model the process, call on a volunteer to start a story that involves a number that is 10 or less. For example, a set of seven birds were at the feeder. Ask the students to show the corresponding number of pieces of pasta on their paper plate. Then call on another volunteer to continue the story by adding a sentence that involves taking away. For example, a set of three birds flew away. Invite the students to take away the mentioned number of pasta pieces. Then ask, "How many pieces of pasta are on your plate now?" [In the example, four.] Ask the students what that number is called. [The difference.] Now call on a volunteer to record the subtraction in the horizontal format. Ask another volunteer to record the subtraction in the number sentence in vertical format. Repeat with another story until the students are comfortable with the process. Give the pairs time to tell and record at least two stories.

Another example includes the following:

Mary made 5 heart shaped cookies. She ate 2 of them. How many cookies are left?
679 macmath2 

Then tell them they will record subtraction in another way--in chart form. Distribute the Take-Away Activity Sheet, to the pairs.

pdficonTake-Away Activity Sheet 

Ask them to point to the column labeled “Number of Pasta Shapes.” Then ask them to read the titles of the other two columns. Call on a volunteer to make up a story problem. Guide the students in recording values in the first two columns. Ask another student to tell what should be written in the “Number Left” column. Now have the students work together to create new entries for the chart. Encourage them to model each trio of entries with pasta shapes. After they have had time to create several rows on their charts, call the students together and ask them to share one of the rows that they have written.

Number of
Pasta Shapes
Taken Away

As you discuss their entries, you may wish to review the terms “take away” and “difference.” If no student shares a row where zero is taken away, ask the class what would be recorded if you started with five pasta shapes and took zero pieces away. Encourage the students to act this operation out with their pasta shapes and record the result on their charts. Repeat with a model for a difference of 0. Prompt the students to add this entry to their chart as well.

Now ask the students to write a subtraction story problem that uses sets and record the results in two ways. When they have completed this task, encourage them to share their problem with a friend and add it to their portfolios. You may wish to display the charts in the classroom.

679 macmath3

In the example above, a set of 2 was taken away from a set of 6.

Assessment Option 

As you watch the students at work, you will gain valuable insights into the students' present level of understanding. You may find it helpful to add your observations to the Class Notes recording sheet you began earlier. This data may prove helpful as you plan remediation or extension activities.
Move on to the next lesson, Where Will I Land?  

Questions for Students 

1. What is the answer when we take a set of five from a set of eight? Can you show that with these pasta shapes? What is the answer called?

[3; The difference.]

2. Which difference on your chart was the greatest? If we use only eight pasta shapes, do you think we could get a larger difference? How?

[Answers will depend upon the numbers in the students' charts.]

3. What would be the smallest difference we could get with eight pasta shapes? How would you get it? How would you record that in vertical format? On a chart?

[0; 8 - 8 = 0.]

4. Suppose you had nine pasta shapes. What could you do to get a difference of zero? How about a difference of nine? What number sentences will you write to show that? What would that look like on the chart?

[9 - 9 = 0; 9 - 0 = 9.]

Teacher Reflection 

  • Did some students exhibit special strengths? What extension activities are appropriate for these students?
  • Which students did not meet the objectives of this lesson? What caused them particular difficulty?
  • Can most of the students justify the difference when one addend is zero? Can they justify a difference of zero? What additional experiences are needed for those who cannot?
  • What parts of the lesson went smoothly? Which parts will I change the next time that I teach this lesson?
Number and Operations

Recording Two Ways

The students make sets of pasta shapes and count some away, then record the subtraction in vertical and horizontal formats. They draw a set and cross out some shapes, then write in both formats the subtraction that the drawing represents.
Number and Operations

Where Will I Land?

In this lesson, the students find differences using the number line, a continuous model for subtraction. [Number can be considered in two ways: discrete and continuous. The counting and set models use the discrete form of number.] Students are encouraged to predict differences and to compose puzzles involving subtraction.
Number and Operations

What Balances?

This lesson encourages students to explore another meaning of subtraction, the balance. They use subtraction facts to generate related addition facts and explore at the concrete level the idea of subtraction as the inverse of addition.
Number and Operations

Who's in the Fact Family?

In this lesson, the exploration of the relation of addition to subtraction is continued as the students use problem-solving skills to find fact families, including those in which one addend is zero or in which the addends are alike.
Number and Operations

What's the Difference?

During this lesson, students use reasoning to find differences from numbers up to 10, using real and virtual calculators and an addition chart as tools. They also play a concentration game.

Learning Objectives

Students will:

  • Model the set meaning of subtraction.
  • Identify differences.
  • Recognize the effects of subtracting zero and subtracting all.
  • Write story problems involving take-away situations.

NCTM Standards and Expectations

  • Connect number words and numerals to the quantities they represent, using various physical models and representations.
  • Count with understanding and recognize "how many" in sets of objects.
  • Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing 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, Counting & Cardinality

  • CCSS.Math.Content.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).

-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.A.1
    Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

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.A.1
    Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

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.