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Numbers Many Ways

Number and Operations
Grace M. Burton
Location: unknown

Students work with subtraction at the intuitive level as they explore number families and ways to decompose numbers to 10. They will also identify members of fact families. (A fact family is a set of three [or two] numbers that can be related by addition and subtraction, for example: 7 = 4 + 3, 7 = 3 + 4, 7 - 4 = 3, and 7 - 3 = 4. When the number is a double, there are only two members of the fact family. An example would be 10 - 5 = 5, and 5 + 5 = 10.)

Provide pairs of students with two number cubes, buttons, and the red and blue strip of paper from the previous lesson. If these strips are not available, give the students a strip of paper and ask them to color one half of the paper red and the other half blue. Next ask them to count out seven buttons and write that number in purple on a separate sheet of paper. Then ask them to find as many ways as they can to separate the seven buttons into two sets, putting some (or no) buttons on the red side of the paper and some (or no) on the blue side. Ask them to record each way using red and blue numerals to match what they have done. Some of the possible sorts are shown below. There will be five more sorts when the set is complete.

293 Red-Blue Strip 

When they have decomposed seven in several ways, help them enter two or three of their findings on a class record chart. Once all their sorts made have been recorded, ask the students whether there are any sorts missing. [There are eight possible sorts of seven objects. Students may need to be reminded to use 0 on each of the sides.]

Call on a volunteer to read one row of the chart. Now demonstrate how to rewrite that entry on the chart as a pair of addition sentences. Then challenge the students to write the same row as a pair of subtraction sentences. [You may wish to model how this is done if the students have not previously encountered the subtraction sign.]

Call on volunteers to read their subtraction sentences aloud. Then write the sum (7) and the four sentences that can be derived from a pair of its addends (say, 5 and 2.) [It will help the students to see the relationships regardless of whether they continue the convention of writing the addends in blue and red and the sum in purple.] Tell the students that this is called a fact family.

The fact family for 7, 5 and 2 is:

5 + 2 = 7
2 + 5 = 7
7 – 5 = 2
7 – 2 = 5

Now ask the students to count out eight buttons on the red side of their strip and repeat the activity, recording all the possible two-addend combinations for eight. [This time there will be nine ways to sort the buttons. You may wish to encourage them to look for a pattern as they make the sets.]

Repeat with other numbers. [The number of ways to sort the buttons will be one more than the number of buttons used. To be sure all the sorts are found, the students might be encouraged to start with all the buttons on one side of the strip, then move them one at a time to the other side of the strip, recording each addend pair as it is displayed.]

pdficonSums to 10 Chart 

Finally, ask the students to choose a number from zero to 10 and write one fact family that has that sum. When the students are ready, ask several volunteers to demonstrate the fact family using buttons and a red and blue strip of paper. Then ask them to find the addends and sum for that fact family on their Sums to 10 Chart.

You may wish to ask the students to select two addends and their sum to record for their portfolio by drawing a picture illustrating that fact family.

Assessment Options

  1. At this stage of the unit, it is important for students to know how to:
    • represent numbers in flexible ways
    • connect numerals to the quantities they represent
    • identify the addition and subtraction sentences related to a specific sum and pair of addends
  2. To assess students’ attainment of these abilities, give them chances to display their understanding using objects, words, and pictures.
Move on to the next lesson, Lost Buttons.

Questions for Students 

  1. How many buttons are on the red side of this sheet? On the blue side? How many in all? What addition sentences could you write to show that? What subtraction sentences? 
  2. What was one sum that we used? What pairs of addends had that sum? 
  3. What pairs of addends make a sum of four? Of one? 
  4. Would you get the same sum if you had two buttons on the blue side and three on the red side as you would if you had three on the blue side and two on the red side? Can you write the addition sentences that show that? What are the related subtraction facts for this family?

Teacher Reflection 

  • Were all students able to find pairs of addends for a given sum?
  • Could they write the numeral for a given set to 10? Which caused difficulty?
  • Could they use the sum and the addend pair in an addition sentence?
  • Are all students able to explain in their own words what a fact family is?
  • Can they demonstrate a fact family using buttons?
  • Which students were able to use the addend pair to create two subtraction sentences?
  • Did some students exhibit reluctance to participate? Why?
  • Which students met all the objectives of this lesson? What extension activities are appropriate for these students?
  • Which students did not meet the objectives of this lesson? What misconceptions did they demonstrate?
  • What parts of the lesson went smoothly? Which parts would I change the next time that I teach this lesson?
ButtonTrains ICON
Number and Operations

Button Trains

In this lesson, students describe order by using vocabulary such as before, after, and between. They also review and use both cardinal and ordinal numbers.
Number and Operations

Many Sets of Buttons

Students classify buttons and make disjoint and overlapping Venn diagrams. In an extension, they make and record linear patterns.
HowManyButtons ICON
Number and Operations

How Many Buttons?

In this lesson, students review classification, make sets of a given number, explore relationships between numbers, and find numbers that are one more and one less than a given number. They apply their knowledge of classification as they play a game similar to bingo.
MoreAndMoreButtons ICON
Number and Operations

More and More Buttons

Students use buttons to create, model, and record addition sentences. They also explore commutativity in addition contexts.
LostButtons ICON
Number and Operations

Lost Buttons

In this lesson and the following one, students investigate subtraction more directly, beginning with the easier “take away” mode. They model “take away” subtraction with buttons and write subtraction sentences. They also work with the additive identity (0) as an addend and as a difference and find missing addends.
Number and Operations

Shirts Full of Buttons

Students explore subtraction in the comparative mode by answering questions of “How many more?” and “How many less?” as they match sets of buttons. They also make and discuss bar graphs based on the number of buttons they are wearing.
Number and Operations

Looking Back and Moving Forward

This final lesson of the unit reviews the work of the previous lessons through a variety of activity stations, one of which involves using an interactive graphing tool. Students model with buttons and record addition and subtraction.

Learning Objectives

Students will:

  • Represent numbers in flexible ways.
  • Connect numerals to the quantities they represent.
  • Identify the addition and subtraction sentences related to a specific sum and pair of addends.

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.
  • 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, 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.3
    Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

-Kindergarten, Algebraic Thinking

  • CCSS.Math.Content.K.OA.A.4
    For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

-Kindergarten, Algebraic Thinking

  • CCSS.Math.Content.K.OA.A.5
    Fluently add and subtract within 5.

-Kindergarten, Number & Operations

  • CCSS.Math.Content.K.NBT.A.1
    Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

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 1, Number & Operations

  • CCSS.Math.Content.1.NBT.C.4
    Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, 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 and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

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.