Lesson 4

Another Look at the Standard Algorithm

Warm-up: Number Talk: Differences (10 minutes)

Narrative

This Number Talk encourages students to think about the base-ten structure of whole numbers and properties of operations to mentally solve subtraction problems. The reasoning elicited here will be helpful later in the lesson when students find differences of multi-digit numbers.

Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”

Activity

• 1 minute: quiet think time
• Keep expressions and work displayed.
• Repeat with each expression.

Student Facing

Find the value of each difference mentally.

• $$87 - 24$$
• $$387 - 124$$
• $$6,\!387 - 129$$
• $$6,\!387 - 4,\!329$$

Activity Synthesis

• “How is each expression related to the one before it?”
• “How might the first expression help us find the value of the last expression?” (Sample response: Knowing $$87 - 24$$ can help us find $$87 - 29$$. The latter is 5 less than the former. Then we can just find $$6,\!300 - 4,\!300$$.)

Activity 1: Lots of Zeros (15 minutes)

Narrative

In this activity, students subtract multi-digit numbers. They do so in two ways: by using the standard algorithm for subtraction and by finding unknown addends. Students find the value of a string of related differences that encourage them to look for and express regularity in repeated reasoning (MP8).

Students may notice that when a subtraction problem requires them to decompose multiple units to subtract in one place when using the standard algorithm—as is the case when the minuend has multiple zeros and the subtrahend has mostly non-zero digits—the standard algorithm for subtraction may not be the most practical. Students use their work in the lesson activity to discuss alternatives to the standard algorithm in these cases, including methods based on the relationship between addition and subtraction and reasoning about sums and differences that are easier to calculate (MP7).

MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “What did the approaches have in common?”, “How were they different?”, or “Did anyone solve the problem the same way, but would explain it differently?”
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge while activating background knowledge. Provide access to materials that students found helpful in Unit 4, such as base-ten blocks, $$10 \times 10$$ grids, digit cards, and colored pencils. As the value of the numbers increase, invite students to consider how they might use mental pictures of these materials instead of the materials themselves.
Supports accessibility for: Conceptual Processing, Attention, Social-Emotional Functioning

Required Materials

Materials to Gather

Launch

• Groups of 2
• Display the first set of equations.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• Share responses.

Activity

• 6–8 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor students who:
• use the standard algorithm and record how they decompose each place
• think flexibly about how to decompose a larger place one time (for example, for $$7,\!000 - 16$$, rather than thinking about decomposing by starting with 7 thousands, they think of decomposing 700 tens and record to show 7,000 as 699 tens and 10 ones)
• use the results of solved problems to help solve new problems (by using known differences or by making use of a pattern)
• use additive reasoning (for example, thinking about what number to add to 16 to find 100 and adding on to 100 to reach the total)

Student Facing

1. Find the value of each difference.

2. Find the number that makes each expression true.

Student Response

If students appear to subtract or add to find each expression in a string without making connections between expressions, consider asking:

• “How did you find the unknown value in the previous expression?”
• “What do you notice about the value you found in each expression?”
• “Before you start in any steps on the next expression, what do you think the unknown value will be?”

Activity Synthesis

• Select previously identified students to share their strategies they used to find the value of $$7,\!000 - 16$$.
• Ask students who didn’t use the standard algorithm to subtract why they chose another way.
• “How was finding the value of the differences in the first set of equations like finding the unknown addend in the second set of equations?” (They both had totals that kept adding a new place. The two-digit number in each did not have any zeros, but the total number in each had many zeros. There was a pattern in each. You could think about adding on and adding to a number like 100 first for each set.)

Activity 2: Ways of Finding Differences (20 minutes)

Narrative

Previously, students used the standard algorithm for subtraction to find the differences involving minuends with multiple zeros. They also used the standard algorithm of addition to find a missing addend that gives a sum with multiple zeros for its digits. In this activity, students make connections between these two ways of reasoning about differences. They also analyze another way to find differences between multi-digit numbers.

Launch

• “Look at Priya and Han’s calculations. How are their setups alike? How are they different?”
• 1 minute: quiet think time
• 1 minute: share responses
• Highlight that Priya is subtracting 472 from 20,000, while Han is finding a number to add to 472 to get 20,000, but both are finding the same missing number.

Activity

• 6–8 minutes: independent work time on the first three questions
• Pause for a discussion before the last question.
• Select students to share their calculations, or display:

Kiran

\begin{align} 472 + 8 &= 480\\ 480 + 20 &= 500\\ 500 + 500 &= 1,\!000\\ 1,\!000 + 19,\!000 &= 20,\!000\\ \end{align}
$$\displaystyle 19,\!000 + 500 + 20 + 8 = 19,\!528$$

• Ask a student to explain what is happening in each calculation.
• “Did anyone think of another way to find the difference?” (See sample student response.)
• Poll the class on their preferred strategy. Ask a student from each camp to explain their reasoning.
• 3 minutes: independent work time on the last question
• Monitor for students who try Priya’s way, Han’s way, Kiran’s way, or a different way for sharing in the lesson synthesis.

Student Facing

To find the value of $$20,\!000 - 472$$, Priya and Han set up their calculations differently.

1. Use both methods to find the difference of 20,000 and 472.
2. Kiran uses another method. Explain how Kiran found the value of $$20,\!000 - 472$$.

\begin{align} 472 + 8 &= 480\\ 480 + 20 &= 500\\ 500 + 500 &= 1,\!000\\ 1,\!000 + 19,\!000 &= 20,\!000\\ \end{align}

$$19,\!000 + 500 + 20 + 8 = 19,\!528$$

3. Which method do you prefer? Or do you prefer another way? Explain your reasoning.

4. Find the value of $$50,\!400 - 1,\!389$$. Show your reasoning.

Activity Synthesis

• See lesson synthesis.

Lesson Synthesis

Lesson Synthesis

“Today we used different ways to subtract a number with non-zero digits from a number with zeros.”

“What strategy did you use to find the difference between 50,400 and 1,389?” (Display the strategies used in the last question. Or, display the three ways here and any additional methods. Ask students to explain each method.)

Using Kiran’s method:

\begin{align} 1,\!389 + 11 &= 1,\!400\\ 1,\!400 + 49,\!000 &= 50,\!400\\ 49,\!000+11 &= 49,\!011\\ \end{align}

Using Han’s method:

Using Priya’s method
(standard algorithm):

“When might it be convenient to use the standard algorithm to subtract two multi-digit numbers?” (When most digits in the second number are smaller than those in the same place in the first number.)

“When might it be inconvenient to use the standard algorithm to subtract?” (When most digits in the second number are greater than those in the same place in the first number, making it necessary to do multiple rounds of regrouping.)