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Looking Back and Moving Forward

Links Away
Grace M. Burton
Location: unknown

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.

With students in their seats, ask them to name the models used during this unit. Prompt them if they forget any of them. Then have them illustrate with links a fact family of their choice. Have a few students share their fact family.

543 flash cardGive students a story problem in which the number left is provided and they have to find the number taken away. For example, on large chart paper, write:

Kim had 8 stickers. She gave some to her friend Sara. Now Kim has 5 stickers. How many stickers did Kim give to Sara?

Provide students access to paper, crayons, links, pan balances, and number lines. Allow them to choose their solution strategies and materials. After students have solved the problem, have them share their strategies and discuss which strategies worked the best.

Next have the children choose five or six facts that they need to learn from their set of flash cards. Put the children into pairs to practice those facts using the cards they selected.

  • Teacher-Generated Subtraction Problem(s)
  • Links or Connecting Cubes (in two or more colors)
  • Paper
  • Crayons
  • Pan Balances
  • Number Lines

Assessment Option 

Give students a subtraction sentence. Have students draw how they would solve this problem. Then have students write the rest of the fact family for this subtraction problem.


Put the children in pairs, and give each pair a die and a supply of paper links. Then tell each child to make a chain of 20 links so they can play "Race From 20." (Instead of using paper links, this game can also be played using the Race From 20 Activity Sheet. However, paper links are preferable, because they allow students to visualize each subtraction.) To play, the children take turns rolling the die and removing the indicated number of links from their chain. The child whose chain disappears first will be the winner for that round. Make sure students understand they can not take away more links than they have. For example, if they have 3 links and they roll a 5, they cannot take 5 away, so play passes to the next person. Ask the children to play several rounds with the links and then challenge them to play without the links, keeping score on a piece of paper.

Questions for Students 

1. How can you subtract 1 from a number?

[Count backward.]

2. How many weights would you need to take away from the right side to balance a scale with 6 links on the right side and 4 on the left side?

[Take away 2 links.]

3. What will you land on if you start at 10 and take 5 hops backward on the number line?

[You will land on 5.]

4. What happens if you subtract 0 from a number?

[The number stays the same.]

5. What are the addition facts and the subtraction facts in one family where the difference is 6?

[One possible answer: 8 – 2 = 6, 8 – 6 = 2, 6 + 2 = 8, 2 + 6 = 8.]

6. What activity did you like most? Which was hardest for you? Why?

[Answers will vary.] 

Teacher Reflection 

  • Can students demonstrate understanding of the terms difference, take away, and equals?
  • What models were the majority of the students most comfortable with?
  • Can students explain how to find differences?
  • Do the students recognize the facts they know and those they have yet to learn?
  • What were the greatest challenges for the students?
  • What other situations would extend their experiences with subtraction?
  • How might I connect the essential ideas of this unit with lessons about similar mathematics content?

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.

Taking Away Sets

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.

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.

Learning Objectives

Students will be able to:

  • Review the models for subtraction.
  • Justify use of specific strategies for solving a subtraction problem.
  • Recall subtraction facts.

NCTM Standards and Expectations

  • Recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another.
  • Use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations.

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.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.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.