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Looking Back and Moving Forward

Number and Operations
Grace M. Burton
Location: unknown

This final lesson of the unit reviews the work of the previous lessons and suggests a framework for summative assessment. During this lesson, students use the mathematical knowledge and skills developed in the previous lessons to demonstrate understanding and the ability to apply that knowledge to playing subtraction games.

To begin the class, ask students to name the models for subtraction (counting backward, set, number line, balance, and inverse of addition) used during this unit. Then name a difference and have the children use pairs of connecting cube trains to represent that difference.

Next, ask the students to write two numbers that are less than 10 (for example, 5 and 7) on each of four index cards. Collect and shuffle the cards, and call on individuals to select two cards and find the difference between the numbers using any of the models studied in the unit.

Now assign students to work at one of the following stations that you have set up in advance. Encourage students to visit at least three of the stations during the remainder of the class time.

Station 1

338 Station 1 

Materials: cards with one digit numbers on them, paper 

Using the cards generated by the students at the beginning of the lesson, have the children distribute the cards equally among four players. Each child displays one of the cards and the children find all the differences indicated. The children who displayed the card with the greatest difference make a tally mark. If more than one player displayed a card with the greatest difference, each player makes a tally mark. Play continues until one child has eight tally marks.

Station 2

338 Station 2 

Materials: connecting cubes, paper, number cubes  

Provide each player with 2 number cubes and 12 connecting cubes. The players roll the number cubes, make a train that has as many cubes as the sum of the numbers thrown, and then compare the trains. The player with the longest train makes a tally mark on a piece of paper. After 10 rounds, the players compare their tallies; the one with the most tallies wins the game.

Station 3

338 Station 3 

Materials: 21 pennies per player, paper  

Give each pair of players a bag containing 21 pennies. Assign one player in each team to count heads, and the other to count tails. Have the children toss the pennies and count how many of their assigned sides came up. The child with more sides announces how many more heads or tails were counted and records that amount on a score sheet. The first child to reach or pass 25 wins the round.

Station 4

338 Station 4 

Materials: 20 counters per team, brightly colored paper, coin  

Distribute to each team a coin, 20 small counters, and a brightly colored sheet of paper. Have players choose “heads” or “tails” and flip a coin to decide who will count those counters that land on the paper. The other child will count those that land off the paper. The children take turns dropping the counters, with the child who won the coin toss dropping first. After each drop, the players count their designated counters and compare the numbers. The player whose group is greater records the difference between the groups as a score for that round. The play continues for 10 rounds, and the child with the highest score wins the game.

Station 5

338 Station 5 

Materials: Bag of 20 connecting cubes with 5 each of 4 colors, crayons to match the 4 colors, Grid Paper for a bar graph  

Prepare a bag of cubes with 5 each of 4 colors of cubes. Give the bag, some crayons, and grid paper on which they can make a bar graph to the players. Assign each child one of the colors in the bag and have the children pull a cube from the bag, color an appropriate square on a grid, and replace the cube. Tell them to make 20 draws in all and then compare the bars on the graph. The player whose color has been drawn the most often wins the round.

Station 6

338 Station 6 

Materials: Race to Zero Activity Sheet, Number cubes 

Tell students they are to take turns rolling a number cube and subtracting the number they rolled each time from 20. The first child to reach 0 wins the round. They should record the results on the Race to Zero Activity Sheet.

You may wish to document specific models of subtraction that students understand and apply.

After the students have had time at the stations, call them together and ask them to record in their journals what happened when they played one of the games.

As another summative assessment activity, display two trains of connecting cubes and have the children compare them, recording the results in both vertical and horizontal notation. Then write a number less than 5 and have the students create two trains showing that difference.

Assessment Option

The guiding questions help students focus on the mathematics they have studied in the lessons of this unit and help you gather summative assessment data. The documentation you have collected about students’ understanding and skills throughout the unit will help you plan appropriate remediation and enrichment opportunities.

Questions for Students 

  1. What addends less than 10 have differences of 2? Of 5?
  2. What subtraction sentence shows that we have compared a set of 7 red pencils with a set of 5 blue pencils?
  3. How many weights would you need to add to the right side of a balance with 3 connecting cubes on the right side and 5 on the left side? What number sentence would show what you did?
  4. How would you compare a train of 8 cubes with a train of 5 cubes on the number line?
  5. If you subtract 0 from a number, what happens?
  6. What are the addition facts and the subtraction facts in one fact family where the sum is 6? Where the sum is 8?
  7. What activity did you like the most? Which activity was hardest for you? Why?
  8. Suppose that you could do some of your work over again. What would you choose to do?

Teacher Reflection 

  • Which children met all the objectives of this unit? What extension activities are appropriate for those children?
  • Which children are still having difficulty with the objectives of this unit? What additional instructional experiences do they need?
  • Are children able to recognize the facts that they know by heart and those that they still need to learn? How can you provide more practice on these facts?
  • What would you do differently the next time that you teach this unit?
  • With which models of subtraction were most of the students the most comfortable?
  • Did all students display understanding of the subtraction models?
  • Can students explain how to compare to find differences?
  • What were the greatest challenges for the students?
  • How can I help students continue to focus on the important ideas in this set of lessons?
  • What other learning situations would extend the students’ experiences with comparison subtraction?
  • How might I connect the essential ideas of this unit with lessons about related mathematics content? Measurement and data are two areas that might be extensions from this unit.
  • What learning experiences would help students not yet comfortable with these concepts become more agile with them?
  • Which activities would help students continue toward mastery of the subtraction facts?
Number and Operations

Counting Back and Counting On

In this lesson, students model subtraction with connecting cubes while the teacher reads to them from counting books. Then children make a train of connecting cubes and write in vertical and horizontal format the differences suggested by adding to and subtracting from the train one cube at a time. Finally, they record, in written form, a train showing one cube being taken away and record the difference in vertical and horizontal format.
Number and Operations

Comparing Sets

A children’s book sets the stage for this lesson which encourages students to review counting back. In this lesson, children write subtraction problems and model them with cubes. They compare sets and record differences in the form of a table. The additive identity is reviewed in the context of comparing equal sets.
Number and Operations

Using the Number Line to Compare

In this lesson, students determine differences using the number line to compare lengths. Because this model is based on linear measurement, it is a distinctly different representation from the models presented in the previous two lessons. At the end of this lesson, children are encouraged to predict differences and answer puzzles involving subtraction.
Number and Operations


This lesson encourages students to explore another model of subtraction, the balance. This model leads naturally to recording with equations. Students use actual and virtual pan balances in their explorations and record the modeled subtraction facts and the related addition facts in equation form.
Number and Operations

Fact Families

In this lesson, the relationship of addition to subtraction is explored with books and with connecting cubes. Students search for related addition and subtraction facts for a given number using a virtual or actual calculator to find differences. They also investigate fact families when one addend is 0 as well as when the addends are the same.

Learning Objectives

Students will:

  • Review the models for subtraction.
  • Practice subtraction in a variety of formats.

NCTM Standards and Expectations

  • 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.
  • Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections.
  • Use multiple models to develop initial understandings of place value and the base-ten number system
  • 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.

Common Core State Standards – Mathematics

-Kindergarten, Counting & Cardinality

  • CCSS.Math.Content.K.CC.C.6
    Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.

-Kindergarten, Counting & Cardinality

  • CCSS.Math.Content.K.CC.C.7
    Compare two numbers between 1 and 10 presented as written numerals.

-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 1, Number & Operations

  • CCSS.Math.Content.1.NBT.B.3
    Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

Grade 1, Measurement & Data

  • CCSS.Math.Content.1.MD.C.4
    Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

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.A.4
    Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

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.

Grade 2, Measurement & Data

  • CCSS.Math.Content.2.MD.D.10
    Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

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.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP6
    Attend to precision.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.
  • CCSS.Math.Practice.MP8
    Look for and express regularity in repeated reasoning.