Integer Combinations—Postage Stamps Problem (HS Version)

Standard(s) for Mathematical Practice (MP)

MP 1 Make sense of problems and persevere in solving them.
MP 2 Reason abstractly and quantitatively.
MP 3 Construct viable arguments and critique the reasoning of others.
MP 4 Model with mathematics.
MP 8 Look for and express regularity in repeated reasoning.

Grade Level

Grades 8–9

Content Domain

Seeing Structure in Expressions (Algebra Conceptual Category)
Functions

Standard(s) for Mathematical Content

A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see $x^4 – y^4$ as $(x^2)^2 – (y^2)^2$, thus recognizing it as a difference of squares that can be factored as $(x^2 – y^2)(x^2 + y^2)$.
A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★
8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol(★).

Math Topic Keywords

  • equations
  • algebraic expressions
  • greatest common factor
  • relatively prime numbers

The mathematics task is intended to be a problem or question that encourages the use of mathematical practices. The dialogue is meant to show how students might engage in the mathematical practices as they work on the task. Before reading the dialogue, work on the mathematics task. Next reflect on the mathematical practices you engaged in while working on the task. Finally read the student dialogue.

Mathematics Task

Part 1: Suppose the post office sold only five-cent stamps and seven-cent stamps. Some amounts of postage can be made with just those two kinds of stamps. For example, 1 five-cent and 2 seven-cent stamps make 19 cents in postage, and 2 five-cent stamps makes 10 cents in postage. Which amounts of postage is it impossible to make using only five-cent and seven-cent stamps?

Part 2: Suppose the post office only sold six-cent and nine-cent stamps. Which amounts of postage is it impossible to make?

Student Dialogue

Students in this dialogue have been studying the properties of arithmetic and have worked extensively with addition tables. They are now exploring what numbers can and cannot be produced by adding only two types of numbers.

(1) Chris Well, if we want to figure out what postage can’t be made, maybe we should make a list of all the postage that can be made first.
(2) Lee That’s not a bad idea. And we know that all our postage is made using only five-cent and seven-cent stamps so… wouldn’t all the possible postage have to equal $5x+7y$?
(3) Matei That’s right. That expression will give us the possible postage values, but we should probably find the actual numbers that can be made. Let’s plug in different values for $x$ and $y$ to see what postage values we can make. Actually… why don’t we use a table to show the different combinations we can make? We can have the number of five-cent stamps going across and the number of seven-cent stamps going down.
(4) Chris That sounds like a good idea!

[Students take a few minutes and create the following.]

(5) Lee Great! Now let’s see what numbers we have and which ones we don’t.
(6) Chris

Well, we’ve got $5$… I don’t see $6$ but we have $7$ here… no $8$ or $9$… we’ve got 10 but I don’t see 11.

[A few minutes pass in which students are looking to see which values are in the table and which are not.]

No to $23$. I see $24$ here, and $25$, $26$ and $27$, $28$, $29$, $30$, $31$… Hmm, it doesn’t seem like we are skipping over any numbers now. I guess we can make any postage greater than $23$.

(7) Lee That’s strange! Why’s that?
(8) Chris Well…. I’m not sure, but maybe it has something to do with the way I’m moving on the table? To go from one number to the next, I keep going right and up or down and left.
(9) Lee You might be right! Check this out: If you start at $31$, go to the right three squares then up two squares, you get to $32$. [draws green arrows to represent the described path—see figure below] It even works with other numbers. If you start with $38$ and go right three and up two, you’ve got $39$.

(10) Matei I think that has to do with what it means to move on the table. If we moved to the right $3$, that means we are adding $3$ five-cent stamps. And if we are moving up $2$, we are taking away $2$ seven-cent stamps. That’s like saying: $$ \begin{align*} 31+3\cdot5-2\cdot7&=\\ 31+(15-14)&=\\ 31+1&=32 \end{align*} $$
(11) Chris But that doesn’t work for $58$. You’d be going off the table moving to the right and up.
(12) Lee Well, we can just extend the table. I mean, we can ask for more than $7$ five-cent or seven-cent stamps.
(13) Chris You’re right, that makes sense. Oh! But wait, we’ve got a $59$ right here, too! There it is—down three squares from $58$, then to the left four squares.

[draws red arrows to represent the path]

(14) Matei Right, because we’re doing: $$ \begin{align*} 58+3\cdot7-4\cdot5&=\\ 58+(21-20)&=\\ 58+1&=59 \end{align*} $$
(15) Chris So what does all this tell us?.
(16) Lee Well, it looks like we found two ways to add $1$ to a postage amount. So I guess we can build all numbers after $23$.
(17) Matei I’m convinced it works for all numbers after $23$.
(18) Chris Me too. What about the next part, six-cent and nine-cent stamps?
(19) Lee $6$ and $9$ seem really different to me from $5$ and $7$. $5$ and $7$ don’t have any common factors except for $1$.
(20) Matei Right. Their greatest common factor is $3$, not $1$. $6$ is $2\times 3$; $9$ is $3\times3$. That means, any postage amount you can make by combining six- and nine-cent stamps will be some number $m$ times $6$ plus some number $n$ times $9$. [writes $6m+9n$] This is the same as $3(2m+3n)$.
(21) Chris How does that help us?
(22) Matei It says that if we made a $6$ by $9$ table like we did the $5$ by $7$ table, all the numbers on it would be multiples of $3$. So, the final postage amount will always contain a factor of $3$.
(23) Chris Will all multiples of $3$ be on the table?
(24) Lee You mean, except for $3?$
(25) Chris Yeah, except for $3$.
(26) Lee I…think…so.

The teacher reflection questions are intended to prompt thinking about 1) the mathematical practices, 2) the mathematical content and extensions, 3) student thinking, and 4) teaching practices. Begin by reflecting on each of the questions, referring to the student dialogue as needed. After you had some time to do your own reflection, read the possible responses and comment at the bottom of the page with questions or thoughts provoked by the reflection questions and/or responses. Please note some of the mathematics extension tasks are meant for teacher exploration, to prompt engagement in the mathematical practices, and may not be appropriate for student use.

Teacher Reflection Questions

  1. What evidence do you see of students engaging in the Standards for Mathematical Practice?

    Possible Response

  2. What mathematics in this dialogue is most likely to be confusing for students?

    Possible Response

  3. Suppose the post office only sold two-cent and three-cent stamps. After which amount will all postage values be possible? What if they sold three-cent and five-cent stamps? Or four-cent and nine-cent stamps?

    Possible Response

  4. What conjectures do you have about what characterizes two postage denominations, $M$ and $N$, for which all postage values after a certain point can be made? What conjectures do you have about what characterizes two postage denominations, $M$ and $N$, for which there will always be postage values that can’t be made?

    Possible Response

  5. Chris asks a generalizing question at the end. “Will all multiples of $3$ be on the [$6$ by $9$] table?” Will they all be on the table (except for $3$)?

    Possible Response

  6. If you were the teacher listening to this exchange, what might you ask or say at the end of the exchange to help Lee become more certain.

    Possible Response

  7. If only two-cent and three-cent stamps were sold, what movements would take you to the next consecutive number, given a table of postage produced from those denominations similar to the one in the dialogue?

    Possible Response

The mathematical overview is intended to 1) explain how students in the dialogue engaged in the mathematical practices and 2) further explore the mathematical content and its extensions. Read the mathematical overview and comment at the bottom of this page with questions or thoughts it provoked.

Mathematical Overview

Mathematical Practice Evidence

Make sense of problems and persevere in solving them.

The students look for an entry point into the problem by organizing their data in a ($+5$) by ($+7$) table so that they can determine what values of postage they can and cannot make using only five- and seven-cent stamps. When a pattern emerges and the students notice that all postage values after $23$ are possible (line 6), they ask why and persevere in coming up with an adequate explanation based on the movement along the table they’ve constructed.

Reason abstractly and quantitatively.

Matei abstracts from the context of postage stamps to develop numerical procedures that show how to proceed on the table from one postage value to the next greater number (lines 10, 14). Matei’s ability to seamlessly move from the tabular/geometric model to numeric expressions demonstrates that the student can decontextualize or abstract from the scenario in order to explain why the numbers are increasing by $1$ as a result of specific movements on the table.

Construct viable arguments and critique the reasoning of others.

Chris tries to find a counterexample to show that moving three to the right and two up on the table from $58$ is not possible (line 11). This, however, ends up being incorrect and Chris’ argument is critiqued by Lee (line 12). Another example of viable argument occurs in Matei’s two numeric explanations of why the arrow paths yield a $+1$ effect (lines 10, 14). Lastly, students make another viable argument in showing that in the six-cent by nine-cent example, all entries must be multiples of $3$ and so cannot contain all numbers after any point.

Model with mathematics.

Students are able to model the possible postage amounts using a table. Building this model shows the students’ ability to identify the constraints of the problem (they can use only five- and seven-cent stamps) and understand the scenario (using different combinations of the two types of stamps in making postage). Students use the model to draw conclusions (line 6) and provide an explanation for the postage amounts that can be made (lines 9–17).

Look for and express regularity in repeated reasoning.

Chris notices that all numbers greater than $23$ can be found on the table and conjectures that this might have something to do with the regularity of movement on the table (right & up and down & left) that were used to go from one number to its consecutive (lines 6, 8). Lee then sees that the green-arrow and red-arrow paths can be repeated over and over again to guarantee that all postage values after $23$ cents will be on the table (lines 9, 12, 16). In lines 10 and 14, Matei expresses the regular movements Lee found on the table using numerical expressions that are based on the context of the problem and the meaning behind the table. Another example of the students looking for regularity can be seen when Chris asks the generalizing question at the end, “Will all multiples of 3 be on the table?” (line 23).

Commentary on the Mathematics

This problem is most evidently about integer equations of the form $rM+sN=P$, where $r$ and $s$ are integers $\geq0$, and $M$ and $N$ are whole numbers. Also elicited by the problem, as Matei demonstrates a couple of times, are algebraic expressions and equivalence, as can be seen in lines 10 and 14). Matei demonstrates numerically why the geometric paths that Lee and Chris use show a way to go from one number to that number plus $1$. Characterizing integers $M$ and $N$ such that $rM+sN=1$ relates to the Euclidean algorithm, derived from Euclid and one of the most important calculations in arithmetic. For any two integers, the algorithm computes the greatest common factor (GCF). If the two numbers, $r$ and $s$, are relatively prime, then the GCF is $1$, and that fact can be used to show that there are integers $M$ and $N$ such that $rM+sN=1$. This is the relationship Lee and Chris capitalize on in moving between consecutive numbers on the table and Matei capitalizes on in developing numerical expressions that have a $+1$ effect. The postage stamps problem is both rich and deep: younger students can explore number combinations and exercise their algebraic thinking, and students all the way into graduate school can explore generalized aspects of the problem.

Evidence of the Content Standards

In lines 10 and 14, Matei writes numerical expressions for the movements between two consecutive numbers observed in the table. By simplifying and rewriting the expressions with parenthesis (A.SSE.A.2), Matei is able to highlight why those movements produce a $+1$ effect. In line 20 and 22, Matei takes the expression $6m+9n$ and rewrites it to the equivalent expression $3(2m+3n)$ to show why all postage made from six- and nine-cent stamps will be a multiple of $3$ (A.SSE.B.3). Students in this dialogue are also implicitly making use of the fact that $5x+7y$ is a function and that varying the values of $x$ and $y$ (the number of five- and seven-cent stamps respectively) will yield all the possible postage amounts that can be made using those stamps. They understand that for each pair of inputs, only one output will be given (8.F.A.1).

Student discussion questions and related mathematics tasks are supplementary materials intended for optional classroom use with students. If you choose to use this task and dialogue with your students these discussion questions are intended to stimulate discussion and further exploration of the mathematics. Related mathematics tasks are intended to provide students an opportunity to engage in the mathematical practices as they connect to content that is similar to or an extension of that found in the dialogue. Please note, responses are to be read by teachers only in preparation for using the questions and tasks with their class.

Student Materials

Student Discussion Questions

  1. A. Find a combination of five-cent stamps and seven-cent stamps that equals $61$ cents in postage. Find a combination that equals $89$ cents.

    B. Find a combination of four-cent stamps and nine-cent stamps that equals $61$ cents. Find a combination that equals $89$ cents.

    Possible Response

  2. In line 22, Matei claims that every value on a $6$ by $9$ table would be a multiple of $3$. Is this true? Why?

    Possible Response

  3. Assuming $m$ and $n$ can be any integer $\geq0$, what values can $2m+3n$ have?

    Possible Response

  4. Based on your answer to question 3, what values can the expression $3(2m+3n)$, which Matei found in line 20, have? What does this tell you about the postage that can be made using only six-cent and nine-cent stamps?

    Possible Response

  5. A. In the dialogue, students find movements on the table (described as changes in the number of stamps bought of each denomination) that get them to the next consecutive postage value. They then explain the effect of this movement by showing it corresponds to a numerical expression equal to $1$. What is one such movement and its corresponding expression?

    B. If the post office only sold two-cent and three-cent stamps, what movement would cause an increase of $1$ in postage? What is its corresponding expression?

    C. If the post office only sold three-cent and seven-cent stamps, what movement would cause an increase of 1 in postage? What is its corresponding expression?

    D. If the post office only sold five-cent and thirteen-cent stamps, what movement would cause an increase of 1 in postage? What is its corresponding expression?

    Possible Response

Related Mathematics Tasks

  1. If the post office has only two-cent stamps and three-cent stamps, what amounts of postage cannot be made?

    Possible Response

  2. If the post office has only three-cent stamps and five-cent stamps, what amounts of postage cannot be made?

    Possible Response

  3. If the post office has only four-cent stamps and nine-cent stamps, what amounts of postage cannot be made?

    Possible Response

  4. Suppose the post office only sold six-cent and nine-cent stamps. What amount of postage can be made?

    Possible Response

  5. Suppose the post office only sold $m$-cent stamps and $n$-cent stamps. Suppose also that, above some amount of postage, all amounts of postage can be made. What can you say about $m$ and $n$?

    Possible Response

  6. What was the largest impossible amount of postage in each of the questions $1–3$? For $m$-cent stamps and $n$-cent stamps in which all postage after a certain point can be made, what is the largest impossible amount of postage in terms of $m$ and $n$?

    Possible Response

  7. How would the original task from the dialogue change if you could buy a negative number of stamps? (This can also be thought of as buying stamps worth $–5$ or $–7$ cents.) Which amounts of postage are impossible to make using only five-cent and seven-cent stamps? Which amounts of postage are impossible to make using only six-cent and nine-cent stamps?

    Possible Response

 
 
  • MP 1: Make sense of problems and persevere in solving them.

    MP 1: Make sense of problems and persevere in solving them.

    Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

  • MP 2: Reason abstractly and quantitatively.

    MP 2: Reason abstractly and quantitatively.

    Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation  process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

  • MP 3: Construct viable arguments and critique the reasoning of others.

    MP 3: Construct viable arguments and critique the reasoning of others.

    Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their  conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though  they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

  • MP 4: Model with mathematics.

    MP 4: Model with mathematics.

    Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest  depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret  their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

  • MP 8: Look for and express regularity in repeated reasoning.

    MP 8: Look for and express regularity in repeated reasoning.

    Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.