standard_description: Know and apply the Binomial Theorem for the expansion of (x + y)^(n )in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)
problem_activity: Narrative
This optional activity gives students additional opportunities to compare multi-digit numbers by reasoning about the value of the digits in different places. It also prompts students to generalize the relative size of two numbers based on their understanding of place value. Students practice constructing logical arguments (MP3) as they explain whether it is possible for
\(4 \underline{\phantom{5}},\!300\)
to be less than
\(3 \underline{\phantom{5}},\!400\)
, or for
\( \underline{\phantom{5}}4,\!300\)
to be less than
\( \underline{\phantom{5}}3,\!400\)
.
Required Materials
Materials to Gather
Materials from a previous activity
Required Preparation
Each group of 2 needs a set of cards from the previous activity.
Launch
Groups of 2
Give each group a card with a digit between 0 and 9 for each group
Activity
“Use the digit on the card to complete the comparison statements in the first problem and decide if they are true.”
4–5 minutes: group work time
Pause to collect responses from all the groups.
“Is the first comparison statement true for all digits?” (Yes, because if the digits are the same, we can just compare 999 to 500.)
Repeat the question with all statements. Record responses in a chart such as shown:
{"table": [{"_attributes": {"border": "1", "class": ["c-table-spread"]}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "statement"}, {"_attributes": {"scope": "col"}, "_value": "true"}, {"_attributes": {"scope": "col"}, "_value": "false"}]}]}], "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\( \\underline{\\phantom{00}} \\ , 999 > \\underline{\\phantom{00}} \\ , 500\\)"}]}, {"_value": "all digits"}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(15,2 \\underline{\\phantom{00}}0 > 15, \\underline{\\phantom{00}}02\\)"}]}, {"_value": "0, 1, 2"}, {"br": [{}], "_values": ["3, 4, 5, 6,", "7, 8, 9"]}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(4 \\underline{\\phantom{00}},700 < 7 \\underline{\\phantom{00}} ,400 \\)"}]}, {"_value": "all digits"}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(1 \\underline{\\phantom{00}}5,000 > 5 \\underline{\\phantom{00}}1,000\\)"}]}, {}, {"_value": "all digits"}]}]}]}]}
“Why are the statements in parts a and c true no matter what digit is used?” (Sample response:
In part a, both numbers are missing the first digit, in the thousands place. Since the missing digit is the same, we’re comparing the hundreds, and 999 is always greater than 500.
In part c, the missing digit is in the thousands place of both numbers, but the digits in the ten-thousands place need to be compared first, and 4 ten-thousand is always less than 7 ten-thousand.)
“Why is the statement in part d false no matter what digit is used?” (Ten-thousand is never greater than 50 thousand.)
“Take a few quiet minutes to work on the last two problems independently.”
5 minutes: independent work time
Student Facing
Each of the following pairs of numbers is missing the same digit but in different places.
Your teacher will assign a digit to you. Use it as the missing digit and decide if each comparison statement is true.
\(\large \boxed{\phantom{0}} \ , \boxed{9} \ \boxed{9} \ \boxed{9} > \boxed{\phantom{0}} \ , \boxed{5} \ \boxed{0} \ \boxed{0}\)
\(\large \boxed{1} \ \boxed{5} \ , \boxed{2} \ \boxed{\phantom{0}} \ \boxed{0} > \boxed{1} \ \boxed{5} \ , \boxed{\phantom{0}} \ \boxed{0} \ \boxed{2}\)
\(\large \boxed{4} \ \boxed{\phantom{0}} \ , \boxed{7} \ \boxed{0} \ \boxed{0} < \boxed{7} \ \boxed{\phantom{0}} \ , \boxed{4} \ \boxed{0} \ \boxed{0}\)
\(\large \boxed{1} \ \boxed{\phantom{0}} \ \boxed{5} \ , \boxed{0} \ \boxed{0} \ \boxed{0} > \boxed{5} \ \boxed{\phantom{0}} \ \boxed{1} \ , \boxed{0} \ \boxed{0} \ \boxed{0}\)
Here are two numbers, each with the same missing digit.
\(\large \boxed{4} \ \boxed{\phantom{0}} \ , \boxed{3} \ \boxed{0} \ \boxed{0} \qquad \qquad \boxed{3} \ \boxed{\phantom{0}} \ , \boxed{4} \ \boxed{0} \ \boxed{0}\)
Choose a digit to complete the numbers and show where they would be on the number line.
[IMG]
Is it possible to fill in the two blanks with the same digit to make each statement true? If you think so, give at least one example of what the digits could be. If not, explain why it is not possible.
\(\large \boxed{4} \ \boxed{\phantom{0}} \ , \boxed{3} \ \boxed{0} \ \boxed{0}\)
is less than
\(\large {\boxed{3}} \ \boxed{\phantom{0}} \ , \boxed{4} \ \boxed{0} \ \boxed{0}\)
.
\(\large \boxed{\phantom{0}} \ \boxed{4} \ , \boxed{3} \ \boxed{0} \ \boxed{0}\)
is less than
\(\large \boxed{\phantom{0}} \ \boxed{3} \ , \boxed{4} \ \boxed{0} \ \boxed{0}\)
.
Student Response
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Activity Synthesis
Select students to share their responses to the last two problems.
Highlight explanations that make it clear that:
\(4 \underline{\phantom{5}},\!300\)
will always be greater than
\(3 \underline{\phantom{5}},\!400\)
because 4 ten-thousand is always greater than 3 ten-thousands.
\( \underline{\phantom{5}}4,\!300\)
will always be greater than
\( \underline{\phantom{5}}3,\!400\)
because, given the first digit is the same, the digits to compare are the thousands, and 4 thousands is always greater than 3 thousands.


aligns: no