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Recording Two Ways

Number and Operations
Location: Unknown

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.

Display the Electronic Abacus and demonstrate how to create a set of up to 10 counters on the top line (by clicking on the beads on the left).

appicon Electronic Abacus 

Then select a student to count one bead away from the set that you made. (By clicking on the beads, they will move from the left side of the abacus to the right side.) Call on a volunteer to write on the board the number sentence that shows what happened. (Example: 5 – 1 = 4.) Repeat and have other students take turns counting away one bead and writing the number sentence. Then repeat counting back more than one.

Give each student 10 to 12 pasta shapes and a paper plate. Then tell a story that involves counting back one such as, there were six pieces of pasta on the plate and Jerry took one. How many are left? Ask the students to model with pasta shapes what is happening as they listen to the story. Then ask them to write the numeral for the whole set, for what was taken away, and for what was left. Repeat with other stories. Check to see that all the students are able to write the numerals that correspond to each set.

Next, hold up an index card displaying a number (for example, 4), and ask the students to make a set with that many pasta shapes, place them on the plate, and record the number of pasta shapes on the plate. Then ask the students to count three pieces of pasta from the plate and record the number remaining on the plate. Now write the appropriate equation in horizontal format, and ask them to copy it. Now show another number and ask the students to make a set with that many pasta shapes, then call on a volunteer to tell how many pieces should be taken away this time. As the students model the subtraction, introduce the vertical subtraction format and ask them to also record the subtraction in this way. Ask them to compare the two formats.

677 mac math 

Now put the students into pairs and give each pair a number cube. Assign one student in each group to roll the number cube twice, recording both numbers that come up. Tell the other student to model a subtraction by counting away the smaller number from the larger number. Tell the first student to record the subtraction in two different ways. Have them reverse their roles and repeat several times.

When the students are ready, ask each pair to read one of their subtraction examples and show the two ways they recorded it. Then ask the students to draw one subtraction situation of their choosing and record it in both vertical and horizontal formats.

  • Pasta
  • Paper plates
  • Paper
  • Crayons
  • Index cards with numerals 0 to 10 written on them
  • Number cubes
  • Electronic Abacus Tool 

Assessment Options 

  1. As you observe the students completing the activities in this lesson, you will be able to gather data about their growth in understanding. To help you document information on the individual needs and strengths of your students, a recording format, Class Notes recording sheet, is provided. The information you record at the end of each lesson may be useful when you discuss the students' progress toward learning targets with the students themselves and with their parents.
  2. Collect the subtraction situations created by the students at the conclusion of the lesson. Reviewing these situations will help you assess the students' level of understanding thus far.
Move on to the next lesson, How Many Left? 

Questions for Students 

1. How many pasta shapes are in this set? (Show a set with eight pasta shapes.) Can you write that number? Now count three pieces of pasta away. Can you write that number? How many are left? Can you write that number?

[8; 3; 5.]

2. What number sentence would show that you started with 10 pasta shapes and took 3 away? How could you show that in another way?

[10 - 3 = 7; Students should be able to show this vertically.]

3. What is alike between the two ways we recorded the differences? What was different?

[Both ways showed the same subtraction equation; One format was vertical, while the other format was horizontal.]

4. How could you help a younger student model 7 - 2?

[Student responses may vary.]

5. Suppose you started with eight pasta shapes and took four pieces of pasta away. What would the number sentence look like? What would a recording in the vertical format look like?

[8 - 4 = 4; Students should be able to write this in a vertical format.]

6. If the same number came up on both tosses of the number cube, what is the answer to the subtraction equation?


7. What two symbols did you use to mean "equals"?

[The equal (=) symbol; the bar in the vertical format functions as an equals sign.]

Teacher Reflection 

  • Which students were able to recognize sets up to 10 and write the numeral for the set?
  • Which students could record subtraction in the horizontal format? Which could record it in the vertical format?
  • What mathematical ideas need clarification?
  • What adjustments will I make the next time that I teach this lesson?
Number and Operations

How Many Left?

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.
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 numbers to 10.
  • Identify sets to 10 and write the numeral for the set.
  • Find differences from numbers to 10 by counting back.
  • Record differences in vertical and in horizontal format.

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.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.