Lesson 4
Another Look at the Standard Algorithm
Warmup: Number Talk: Differences (10 minutes)
Narrative
This Number Talk encourages students to think about the baseten 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 multidigit 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
 Record answers and strategy.
 Keep expressions and work displayed.
 Repeat with each expression.
Student Facing
Find the value of each difference mentally.
 \(87  24\)
 \(387  124\)
 \(6,\!387  129\)
 \(6,\!387  4,\!329\)
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
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 multidigit 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 nonzero 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).
Advances: Representing, Conversing
Supports accessibility for: Conceptual Processing, Attention, SocialEmotional 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

Find the value of each difference.

Find the number that makes each expression true.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Advancing Student Thinking
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 twodigit 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 multidigit 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.
 Use both methods to find the difference of 20,000 and 472.

Kiran uses another method. Explain how Kiran found the value of \(20,\!000  472\).
\(\begin{align} 472 + 8 &= 480\\ 480 + 20 &= 500\\ 500 + 500 &= 1,\!000\\ 1,\!000 + 19,\!000 &= 20,\!000\\ \end{align}\)
\(19,\!000 + 500 + 20 + 8 = 19,\!528\)
 Which method do you prefer? Or do you prefer another way? Explain your reasoning.
 Find the value of \(50,\!400  1,\!389\). Show your reasoning.
Student Response
Teachers with a valid work email address can click here to register or sign in for free access to Student Response.
Activity Synthesis
 See lesson synthesis.
Lesson Synthesis
Lesson Synthesis
“Today we used different ways to subtract a number with nonzero 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 multidigit 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.)
Cooldown: A Couple of Differences (5 minutes)
CoolDown
Teachers with a valid work email address can click here to register or sign in for free access to CoolDowns.