Discussing Subtraction Strategies

Explicit Instruction/Teacher modeling

• Explain to students that they will learn or improve their understanding of some vocabulary words related to the lesson. Tell them that they may already know the meaning of some of the terms but by spending time making meaning of them, their understanding will deepen and will help them to become stronger mathematicians.
• Introduce each vocabulary word by displaying the Vocabulary Cards. Leave the cards displayed for the rest of the lesson. Read each word and its definition. On the document camera, show students how to complete a Frayer Model using the word solution. Read aloud the definition, draw a picture or model to further make meaning of the word, and provide examples/non-examples to students.
• Place students into partnerships. Give each pair a blank Frayer Model. Assign one of the remaining vocabulary words to each pair of students. Note: you may have to repeat words. Instruct them to fill out each section of the model collaboratively, meaning they should discuss the image, example, and non-example they add to the model. Tell students that they should copy the definition from the Vocabulary Cards onto their Frayer Model. Make sure that each student participates in the work equally.
• Have each pair of students present their completed model briefly to the whole group. After the presentations are complete, read aloud each vocabulary term and have students show you a thumbs-up if they feel that they have a strong understanding of the word, a thumb to the side if they feel they somewhat understand it, and a thumbs-down if they do not understand the term. Encourage students to be honest in their self-assessment of the vocabulary terms. Note which students may need vocabulary support throughout the lesson.

Guided Practice

• Tell students that today there are two subtraction strategies that they will learn. One is called Expanded Notation and the other is called the Standard Algorithm. Note: students should have some familiarity with these terms after the Frayer Model activity.
• Show students on a piece of chart paper an example of how to solve a three-digit subtraction problem using both strategies (e.g., "There are 382 students in our school. 137 take the bus to school. How many students do not take the bus?"). Read aloud the problem and demonstrate how to solve it.
  • Expanded Notation Subtraction (breaking apart numbers based on their place value and subtracting each part of the number). First, 382 becomes 300 + 80 + 2 and 137 becomes 100 + 30 + 7. Then, we subtract the ones, the tens, followed by the hundreds. Emphasize the regrouping required in the ones section. Finally, we add the three differences we found in each place value to find the total difference.
  • Standard Algorithm Subtraction (lining up the two numbers on top of each other and then subtracting from right to left, with regrouping as necessary): 382 - 137 = 245. First, I subtract the ones, the tens, and then the hundreds.
• Have students turn to their partner and compare the two strategies used to subtract. Ask the following questions and provide the sentence starters as an aid to help them compare.
  • How are the two strategies similar? ("The strategies are similar in that they both...")
  • How are the two strategies different? ("The expanded notation strategy is... while the standard algorithm strategy is...")
• Invite students to share their discussion with the whole class and document their observations on another piece of chart paper.
• Distribute a piece of blank copy paper to each student and have them fold it in half (hamburger style). Inform them that they will solve a subtraction problem using both strategies, one on each side of the paper. Tell students to make sure they label both types of methods on their paper.
• Show students the following problem and read it aloud (define any unfamiliar term as needed): "Samir has been saving all year to buy a new bicycle. He was able to save $447. He decided to buy a bicycle that cost $339. How much money does he still have left after he bought the bicycle?"
• Circulate to make sure students are using both strategies and working together with their partner.
• Ask one pair of students to come up to the document camera to share their two strategies for solving the problem. Provide them with transition words to help them share out ("First, we... Then, we... After that, we... Finally, we...").
• Ask students if anyone has another strategy they think would work well for this problem. Invite students to share any additional subtraction strategies and document their responses (ideas include drawing a base-ten model, breaking the number into friendlier numbers, etc.). Acknowledge other strategies and point out the value of having a tool box with ample methods of solving the same problem so that we become confident and flexible math thinkers.

Group work time

• Facilitate a Number Talk activity for students. Place students into A–B partnerships. Tell students that Partner A will solve the problem using the Expanded Notation strategy while Partner B will solve the same problem using the Standard Algorithm. Read aloud and display this problem: "Michelle collects stamps. She has 563 stamps from around the world. She decides to donate 269 of them to a museum. How many stamps does Michelle have left?"
• Hand out whiteboards and markers to each student. Have them solve the problem using their assigned strategy on the whiteboards.
• Have the pairs share their strategies and answers with each other, and discuss any challenges they encountered. Provide the transition words mentioned earlier. Ask students to share their processes with the whole group and jot down their observations on a piece of chart paper.
• Give students another problem and have them switch strategies (i.e., Partner A now uses the Standard Algorithm and Partner B will use the Expanded Notation strategy): "The orchard has 786 trees. 339 of them are apple trees and the rest are peach trees. How many peach trees are there in the orchard?"
• Allow time for students to complete the subtraction problem on their whiteboards. Repeat the process of sharing and comparing their strategies and answers. Give students access to the questions and sentence starters used in the Guided Practice. Circulate and correct any mistakes. Continue to record students' observations.
• Hold a class discussion about their preferences for the various strategies. Give sentence frames as support:
  • "I prefer the ____ strategy because..."
  • "I dislike the ____ strategy because..."
  • "I find the ____ strategy easiest because..."