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Taking Away Sets

Links Away
Grace M. Burton
Location: unknown

This lesson encourages students to explore another model for subtraction, the familiar set model. Reading one of the many books that feature subtraction set the stage for this lesson in which the students write story problems, find differences using sets, and present results in a table. In the discussion of the table, they focus on the effects of subtracting all and subtracting 0.

To set the stage for this lesson, you may wish to read another of the counting books.

pdficonCounting Books

Appropriate books include Ten, Nine, Eight, How Many Snails?, and Mean Machine. Now ask students to create subtraction story problems that use sets bigger than 1. For example, for 6 – 2, a student could ask, "Jose had 6 marbles and lost 2 of them. How many does he have now?"

529 marbles

Encourage a few volunteers to share their problems with the class. Discuss whether these problems can be solved. If they provide too little or too much information, solicit student help to revise the story problems.

pdficonFind the Difference Activity Sheet

Then post a large piece of chart paper displaying a Find the Difference chart where all the students can see it. Repeat one of the student’s story problems. Demonstrate how to fill in the columns labeled "Number of Objects" and "Number Taken Away" with the information from the story. Then explain to the students that the column labeled "Number Left" is the difference and that you will find the difference together using real objects. Display a chain that matches the story problem. For the example above, you will make a chain of 6 links and take away 2 links. Place the two links where they are separate from the chain but still visible to students. Ask students to tell you the difference. Enter this information in the chart. Repeat this process with the other student-generated story problems.

Give students the opportunity to practice writing and solving their own subtraction story problems individually or in pairs. After each student or pair of students writes a problem, have them record the information from their problem in a Find the Difference chart. Provide links for the students to solve their story problems.

When they are ready, call students together to share their story problems and enter their findings on the class chart. Afterward, review the terms take away and difference. Then ask the students what would be recorded if you started with 7 links and took 7 away. Repeat with a model for 7 – 0. Prompt them to add entries to the chart. At this point, you may choose to encourage children to also notice rows in which the first column ("Number of Objects") shows the same number.

At the end of the lesson, ask students to choose one of the rows from the chart and draw a picture illustrating that number fact. You may allow students to display these in the classroom or in a more public place.

Assessment Options

  1. Use the Questions for Students above to assist you in determining your students’ level of understanding. Other questions may suggest themselves as you talk with your children as well. Record your observations on the Class Notes teacher resource sheet you began earlier in this unit.
  2. Have students re-use their subtraction problem (or create a new subtraction problem) using a theme such as "At The Pond." Provide students with appropriate stamps to illustrate their problems. Assess whether students are able to model subtraction in written and pictorial form.
Move on to the next lesson, Hopping Backward on the Number Line

Questions for Students 

1. What happens when we subtract?

[We take something away, and the group usually gets smaller. (The group will not get smaller if the number we take away is 0.)]

2. Which difference on our chart was the greatest?

[Answers will vary.]

3. If we start with 10 links, what is the greatest difference we can get? How do you know?

[The biggest difference is 10. We get the biggest difference when we subtract 0.]

4. What would be the smallest difference we could get with 10 links? How would you get it?

[The smallest difference would be 0. We get 0 if we take all 10 links away.]

Teacher Reflection 

  • Can students explain the terms difference and take away?
  • Were students able to create story problems for subtraction? If not, what activities can you use to give them additional experience?
  • Which students need assistance to record what they know and need to find out from story problems?
  • Can most of the children justify the difference when 0 is taken away? Can they justify a difference of 0?
  • What other books would you use in this lesson?

Counting Back and Counting On

This lesson, which focuses on the counting model for subtraction, begins with reading a counting book. The students model the numbers as the book is read. Then they make a chain of links and write in vertical and horizontal format the differences suggested by adding and subtracting one link at a time from their chains. Finally, they draw a chain showing one link being taken away and write in two formats the difference it represents.

Hopping Backward on the Number Line

In this lesson, students generate differences using a number line model. Because this model highlights the measurement aspect of subtraction, it is a distinctly different representation from the models presented in the previous lessons of this unit. The order property for subtraction is investigated. At the end of the lesson, children are encouraged to predict differences and solve puzzles involving subtraction.

Finding the Balance

Pre-K-2, 6-8
This lesson encourages students to explore another model of subtraction, the balance. Students will use real and virtual balances. Students also explore recording the modeled subtraction facts in equation form.

Finding Fact Families

In this lesson, students explore the relation of addition to subtraction with books and links. Then the children search for related addition and subtraction facts for a given number. They also investigate fact families, including those where one addend is 0 and where the addends are alike.

Practice Makes Perfect

During this lesson, students use what they know about fact families to play a concentration game. They will also identify subtraction facts they need to learn.

Looking Back and Moving Forward

This final lesson reviews the work of the previous lessons and suggests a framework for summative assessment. Students will self-select a solution strategy for subtraction from the models introduced in this unit. An extension activity is suggested in which students use the mathematical knowledge and skills developed in the previous lessons to demonstrate understanding and ability to apply that knowledge to playing a new game.

Learning Objectives

Students will be able to:

  • Create subtraction story problems.
  • Explore the results of subtracting sets.
  • Define the term difference.
  • Explore the effects of subtracting 0 and subtracting all.
  • Construct a table showing differences.

NCTM Standards and Expectations

  • Use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations.

Common Core State Standards – Mathematics

-Kindergarten, Counting & Cardinality

  • CCSS.Math.Content.K.CC.A.2
    Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

-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.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.5
    Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

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.