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This True or False prompts students to look carefully at the sum and difference of the digits in each place and to remember to compose units when needed. It also prompts students to make use of structure (MP7). For example, the last expression would be cumbersome to calculate with the standard algorithm. Recognizing that 99,999 is 1 less than 100,000 would enable students to find the difference much more quickly. Likewise, students who notice that is only 1 away from 400,000 would know that 311,111 is far too high and cannot be the difference between 400,000 and 99,999.
Decide if each statement is true or false. Explain your reasoning.
In this activity, students perform multi-digit addition and subtraction to solve problems in context and assess the reasonableness of answers. The situation can be approached in many different ways, such as:
Students reason abstractly and quantitatively when they make sense of the situation and decide what operations to perform with the given numbers (MP2).
This activity uses MLR6 Three Reads. Advances: reading, listening, representing
MLR6 Three Reads
A school’s track teams raised \$41,560 from fundraisers and concession sales.
In the fall, the teams paid \$3,180 for uniforms, \$1,425 in entry fees for track meets, and \$18,790 in travel costs.
In the spring, the teams paid \$10,475 in equipment replacement, \$1,160 for competition expenses, and \$912 for awards and trophies.
Is the amount collected enough to cover all the payments? Explain how you know.
In this activity, students practice adding and subtracting multi-digit numbers through a game. Students draw several cards containing single-digit numbers, arrange the cards to form two numbers that would give the greatest and the least sums and differences. To meet these criteria, students look for and make use of structure in base-ten numbers (MP7).
Your teacher will give you and your partner a set of 10 cards, each with a number between 0 and 9. Place the cards facedown in a stack.
Draw 3 cards. Use all 3 cards to form 2 different numbers that would give:
the greatest possible sum
the least possible sum
the greatest possible difference
the least possible difference
Shuffle the cards and draw 4 cards. Use them to form 2 different numbers that would give:
the greatest possible sum
the least possible sum
the greatest possible difference
the least possible difference
Display:
732
3,005
8,401
12,475
218,699
“In this lesson, you added and subtracted lots of large numbers to solve problems. Suppose we’re working with these large numbers.”
“What are some ways to estimate the sum or difference of a list of numbers without adding them?” (Round each number to make them easier to add or subtract. Look at the digits and the place values of the numbers involved to get a sense of the sizes of the numbers.)
“If careful calculations are needed, what are some ways to organize the numbers and add or subtract them efficiently?” (Start with the largest numbers first. Start with the numbers with more zeros and fewer non-zeros. Start with computations that would result in multiples of 10, 100, 1,000, and so on, which would make other calculations easier. For example, in the given list of numbers, we could add 218,699 and 8,401 first because it’d give 227,100. Then add 12,475 and 3,005 because both numbers end in 5 and would add up to 15,480.)
“How might we find two numbers that give the greatest sum or greatest difference without trying to find the sum and difference of every pair of numbers?” (Pay attention to the size of each number, based on the number of digits and their place values.)
We used our understanding of place value and expanded form to add and subtract large numbers using the standard algorithm.
We learned how to use the standard algorithm to keep track of addition of digits that result in a number greater than 9.
When there are 10 in a unit, we make a new unit. Then record the new unit at the top of the column of numbers in the next place to the left.
When we subtract numbers, it may be necessary to decompose tens, hundreds, thousands or ten thousands before subtracting.
Finally, we learned if the digit we are subtracting is a zero, we may need to decompose one unit of the digit in the next place to the left.
Sometimes it is necessary to look two or more places to the left to find a unit to decompose. For example, here is one way to decompose a ten and a thousand to find .