standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: Narrative
The purpose of this activity is to examine subtraction cases in which non-zero digits are subtracted from zero digits. In some cases, students could simply look at the digit to the left of a 0 and decompose 1 unit of that number. But in other cases, the digit to the left is another 0 (or more than one 0), which means looking further to the left until reaching a non-zero digit. Students learn to decompose that unit first, and then move to the right, decomposing units of smaller place values until reaching the original digits being subtracted. The problems are sequenced from fewer zeros to more zeros to allow students to see how to successively decompose units.
Recording all of the decompositions can be challenging. For the last problem, two sample responses are given to show two different ways of recording the decompositions. The important point to understand is that because there are no tens, hundreds, or thousands to decompose, a ten-thousand must be decomposed to make 10 thousands. Then, one of the thousands is decomposed to make 10 hundreds, and so on, until reaching the ones place. Those successive decompositions can be lined up horizontally, but this can make it hard to see what happened first. A second way shows more clearly the order in which the decompositions happen, but it may be challenging to see which place the successive decomposed units are in.
To add movement to this activity, the second problem could be done as a gallery walk where each group completes one problem and then walk around the room to look for similarities and differences in others’ posters.
MLR8 Discussion Supports.
Display sentence frames to support partner discussion: “I noticed _____ so I . . .”, and “I wonder if . . . .”
Advances: Conversing, Representing
Representation: Develop Language and Symbols.
Maintain a visible display to record language that students can use to explain their work. Include important vocabulary, such as the name of each place value category, decompose, and exchange. Also include sentence frames, such as: “I looked at the _____ place and saw that I did not have enough ____ to subtract.” and “I exchanged one ______ for 10 _____.”
Supports accessibility for: Conceptual Processing, Language, Organization
Launch
Groups of 2–4
5 minute: independent think time
3 minute: partner share
Activity
“Take some time to think about the first problem and then discuss it with your partner.”
2 minutes: quiet think time
3 minutes: partner discussion
Pause for clarifying questions, as needed.
“Now, work with your partner to complete the next 2 problems.”
5 minutes: partner work time
Student Facing
Here are some numbers you saw earlier. Each number has at least one 0. From each number, 1,436 is being subtracted.
Make sense of the problems and explain to a partner.
a
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b
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Use the approach in the first problem to find these two differences:
a
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b
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Find the value of each difference. Be prepared to explain your reasoning. If you get stuck, try subtracting using the expanded form.
a
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b
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c
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d
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Student Response
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Advancing Student Thinking
Students may lose track of the units when multiple rounds of decompositions are required. Offer base-ten blocks and prompt students to show multiple decompositions when finding the value of
\(100 - 1\)
using a large square base-ten block that represents 100. Consider asking:
“How can you show subtraction of 1?” (Exchange it for 100 ones.)
“What if we could only exchange 1 unit for 10 units at a time?” (Exchange the hundred for 10 tens, then exchange 1 ten for 10 ones, so that we’d have 9 tens and 10 ones. Then, we can remove 1.)
“How would you show the subtraction of 1 from 100 using the standard algorithm?”
Activity Synthesis
Ask students to share responses and to demonstrate consecutive decomposing when multiple zeros are involved.
If needed, use expanded form to represent the decompositions students are explaining.


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