# How Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem. Will is w years old.

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1 How Old Are They? This problem gives you the chance to: form expressions form and solve an equation to solve an age problem Will is w years old. Ben is 3 years older. 1. Write an expression, in terms of w, for Ben s age. Jan is twice as old as Ben. 2. Write an expression, in terms of w, for Jan s age. If you add together the ages of Will, Ben and Jan the total comes to 41 years. 3. Form an equation and solve it to work out how old Will, Ben, and Jan are. Will is Ben is Jan is years old years old years old Show your work. Copyright 2007 by Mathematics Assessment Page 75 How Old Are They? Test 9

2 4. In how many years will Jan be twice as old as Will? years Explain how you figured it out. 7 Copyright 2007 by Mathematics Assessment Page 76 How Old Are They? Test 9

3 Task 4: How Old Are They? Rubric The core elements of performance required by this task are: form expressions form and solve an equation Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives a correct expression: w Gives a correct expression: 2(w + 3) Gives correct answers: Will is 8 years old Ben is 11 and Jan is 22 years old Shows correct work such as: w + w (w + 3) (allow follow through) 4w + 9 = 41 4w = Gives a correct answer: in 6 years time 1 1 1ft. 1 3 Gives a correct explanation such as: Will is 14 years younger than Jan so when Will is 14 Jan will be = 6. Accept guess and check with correct calculations. 2 Solves correct equation. Total Points 7 Copyright 2007 by Mathematics Assessment Page 77 How Old Are They? Test 9

5 Looking at Student Work on How Old Are They? Student A is able to set up expressions to represent the ages of the children. Notice that when writing the equation to find the ages of children each expression is defined. The student is able to think about the difference in ages and uses mathematical reasoning to find the time when Jan will be twice as old as will. Student A Algebra

6 Student A, part 2 Student B is able to represent the situation using algebra to quantify the relationships of the ages and then apply the constraints to set up and solve an equation. Student B is also able to think about growing older as a variable and set up an equation to find out when the Jan will be twice as old as Will. The student is able to use the symbolic language of algebra to represent the problem. Algebra

7 Student B Algebra

8 Student B, part 2 Student C adds a new variable to help express Jan s age in part 2. However the student realizes, when pushed to find the age of all the children, that the new variable won t help solve the problem and makes adjustments to the expression for Jan s age. The student uses a table to find the time when Jan will be twice as old as Will, but does not interpret the elapsed time correctly. Student C Algebra

9 Student C, part 2 Student D uses w to mean different things. In part 2, w is used much the same way as student C as a new variable, without maintaining the referent to Will. Now w is used to refer to Ben instead of Will. When pushed to do further thinking, the student is able to correctly describe all the relationships with a single variable. The student struggles with the idea of growing older and only ages Will. Jan does not grow older in this model. What do you think some of the issues are for students making sense of this idea of elapsed time? Is it an equality problem: whatever you do to one side of the equation you do to the other side? Is it a question of describing the situation first in your own words? What ideas might help this student? Algebra

10 Student D Algebra

11 Student E is able to use algebra to solve parts 1,2, and 3. However when thinking about the relationship of Jan being twice Will s age, the student misinterprets the meaning of doubling and struggles with the idea of elapsed time for both children. How do we help students to develop the habit of mind of checking their work for sense-making? Is it reasonable to have a negative amount of time in this situation? Do students in your class have enough opportunities to make sense of answers in context? Student E Student F is confusing variable with a fixed solution. In part one Ben is given an exact age. In part 2 the student is trying to find a number sentence for twice Ben s age or 6, but does compute correctly with the negative number. In part 3 the student ignores all the previous work and the relationships between the students ages and just finds a numerical solution to the total ages equals 41. The student either doesn t understand or ignores the prompt in part 4. How do we help develop the big idea of variable early in the curriculum? How do help students distinguish between an unknown and a variable? What would be your next steps with this student? Algebra

12 Student F Algebra

13 Student G seems to understand the relationships of the students ages in terms of the variable w, but then seems uncomfortable with a simple expression and adds on Will s age to each part. The student doesn t use the relationships to find the ages in part 3 and the student doesn t use the constraint of having the ages add to 41. The student leaves part 4 blank. The thinking of Student H may show how the student found the original 17 1/2. Student G Student H confuses doubling with using an exponent to square a number. However when using numbers instead of symbols, the student knows to multiply by 2. Do we give students adequate opportunities to relate symbolic notations to numerical expressions? What is involved in developing fluency with mathematical notation and how can we foster that in our classrooms? The student tries to work backwards from the doubling by doing a series of dividing by two. Notice that the totals add to 31 instead of 41. Student H also gives the most popular incorrect answer for part 4. How do we help students develop a sense of tinkering with an idea, before making a conclusion? Algebra

14 Student H Algebra

15 Student I uses an incorrect operation to define Ben s age. The student uses values that match the relationships described in part 1 and 2, but ignores the constraint that the ages should add to 41. In part 4 the student feels that all the numbers in part 3 were doubles, but ignores that constraint that it is Jan who should be twice as old as Will and doesn t take into account the idea of both students growing older. Student I Algebra

16 Student J is unable to write an expression for Jan s age, even though the guess and check work shows that the student understands the mathematical relationships. Again, there is a disconnect between the mathematical ideas and how to express it symbolically. The student has the nice habit of mind to be persistent in trying to find the time when Will will be half of Jan s age, however the student seems to only consider Will aging in groups of eight years. What do you think is causing the confusion? Student J Algebra

17 Student J, part 2 Algebra

18 Algebra Task 4 How Old Are They? Student Task Core Idea 3 Algebraic Properties and Representations Form expressions and solve an equations to solve an age problem. Represent and analyze mathematical situations and structures using algebraic symbols. Use symbolic expressions to represent relationships arising from various contexts. Judge the meaning, utility, and reasonableness of results of symbolic manipulations. Mathematics in this task: Write algebraic expressions to represent the relationships between students ages Understand how to use symbolic notation to express distributive property for a multiplicative relationship Solve an equation Reason about elapsed time to find when Jan s age is double Will s age Identify constraints and use them to set up an equation Based on teacher observations this is what algebra students know and are able to do: Write an algebraic expression for an additive relationship Find the ages for the three children Areas of difficulty for algebra students: Writing a multiplicative expression that involves distributive property Writing and using an equation in a practical setting Understanding elapsed time and expressing elapsed time numerically or algebraically Algebra

19 The maximum score available for this task is 7 points. The minimum score for a level 3 response, meeting standards, is 5 points. Most students, 84%, were able to write an algebraic expression for an additive relationship. More than half the students, 66%, could write an additive expression and find the ages for the 3 students which met all the constraints of the problem. Almost half the students, 45%, could write an expression for an additive relationship, find the ages of the 3 students, and find a strategy to correctly calculate the time when Jan will be twice as old as Will. Almost 21% of the students could meet all the demands of the task including writing a multiplicative relationship involving distributive property and writing and solving an equation to find the ages of the 3 children. Almost 16% of the students scored no points on this task. 69% of the students with this score attempted the task. Algebra

22 The second issue is the idea of aging. Again look at student work, to help pick examples of students thinking to help pose the question. For example: I overheard a table in a different class discussing part 4, in how many years will Jan be twice as old as Will. Lily thinks that Jan is already twice as old as will. Eugene thinks in 3 years Will will be twice as old as Jan, because 8+ 3= 11 and 11 x 2 = 22. Cody says I think you are both wrong. They both need to get older. But I don t know how to write that. What advice could you offer the group? Who do you think is right? How can you convince the others that they have made a mistake? These types of discussions give students the opportunity to hone in their logical reasoning skills, clarify mathematical ideas, and practice using academic language. What questions would you pose? Why? Algebra

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