Using Diagrams to Represent Addition and Subtraction

5 min

Teacher Prep
Setup
Students in groups of 2. 1–2 minutes of quiet think time, followed by partner and whole-class discussions.

Narrative

The purpose of this Warm-up is for students to review place value when working with decimals. There are many ways students might find the numbers represented by the large rectangle and large square. However, the goal is to recognize that each digit in a base-ten number represents a unit that is 10 times larger than the digit immediately to its right. This observation can be made by looking for structure in diagrams and in the numbers represented by each small shape and the larger shape it composes (MP7).

Launch

Give students 1–2 minutes of quiet think time. Encourage students to look for patterns as they work. Select students with correct responses, and ask them to share during the whole-class discussion.

Students may benefit from reviewing place-value names for decimals. Consider displaying a place-value chart for reference, or inviting students to name each number in the Student Task Statement before they answer the questions.

Student Task

  1. Here is a rectangle.

    A rectangle divided vertically into 10 equal squares.

    What number does the rectangle represent if each small square represents:

    1. 1

    2. 0.1

    3. 0.01

    4. 0.001

  2. Here is a square.

    A square divided horizontally into 10 equal rectangles.

    What number does the square represent if each small rectangle represents:

    1. 10

    2. 0.1

    3. 0.00001

Sample Response

    1. 10
    2. 1
    3. 0.1
    4. 0.01
    1. 100
    2. 1
    3. 0.0001
Activity Synthesis (Teacher Notes)

Ask previously selected students to share their responses. Record each set of answers in a table, aligning the decimal points vertically, as shown:

value of a small square value of the rectangle
1.0001\phantom{.000} 10.0010\phantom{.00}
0.1000.1\phantom{00} 01.00\phantom{0}1\phantom{.00}
0.0100.01\phantom{0} 00.10\phantom{0}0.1\phantom{0}
0.0010.001 00.01\phantom{0}0.01
value of a small rectangle value of the large square
10.0000110\phantom{.00001} 100.0001100\phantom{.0001}
00.10001\phantom{0}0.1\phantom{0001} 101.0001\phantom{10}1\phantom{.0001}
00.00001\phantom{0}0.00001 100.0001\phantom{10}0.0001

Ask students:

  • “What relationship do you see between the values in the two columns of each table?” (The values in the right column are 10 times the values in the left column.)
  • “What do you notice about the position of the 1 in the two numbers in each row?” (It moves one place to the left, for example, from being in the tens place to being in the hundreds place.)
  • “How does the value of the 1 change as it shifts one place to the left?” (Its value is 10 times as much.)
  • “Why might these representations be called ‘base-ten diagrams’?” (Each larger shape is made of 10 smaller shapes and has a value that is 10 times that of the smaller shape.)
Anticipated Misconceptions

Some students may continually use skip-counting (by 10, by 0.1, and so on) to find the value of the rectangle and the square, rather than making connections to place value. Ask these students if they see a pattern in their skip-counting (for example, in the number of times they skip-counted to answer each question), or if they see a relationship between the value of each of the smaller units to that of the larger unit they compose.

Standards
Building On
  • 5.NBT.1·Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
  • 5.NBT.A.1·Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

15 min

15 min