CLASSROOM ACTIVITY TO DEVELOP STUDENT UNDERSTANDING OF ALGEBRAIC NOTATION

Understanding of algebraic notation is necessary for students to make the transition from arithmetic to algebra. Development of this understanding requires rich classroom opportunities (MacGregor & Stacey, 1997). Algebrafied arithmetic problems were used as a context for engaging the students in dialogue about algebraic notation (Blanton & Kaput, 2003). Students used both shapes and letters to represent unknowns.

To solve the word problem20 Ella supported students to record in a systematic way to emphasise patterns in the equations. For example, during the whole group discussion, Ella

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revoiced Hannah’s explanation scaffolding her to record in a logical manner: Nine plus something equals seventeen…what if the T-shirt was twenty dollars, what would the equation look like if was twenty dollars? In this way, she emphasised the patterns in the equations. She focused the students on what was constant in the equations: Talk to the person next to you about what has stayed the same in all of those equations? Then she facilitated the students to identify the unknowns: What changes in each of those equations?

To introduce students to using algebraic notation to represent a generalised situation Ella asked them to: Come up with a rule or a statement to tell them what you would do no matter what the cost of the T-shirt. Students used informal algebraic notation to represent the situation.

Rachel: [writes □ + 9 = a] We drew a box plus nine equals a.

Heath also shared his group’s alternative strategy z – 9 = x. In subsequent lessons, the students modelled their recording (see Figure 5.2) on Hannah’s explanation which Ella had earlier scaffolded. This enabled them to quickly identify the unknowns and construct algebraic notation to represent the situation. For example, Heath constructed a solution strategy for a problem.

Heath: [writes equation] Triangle divided by five equals spiral.

20 _{If you had $9 in your bank and wanted to buy a t-shirt for $17, how much do you need to save?}

What about if the t-shirt cost $20 or $26 or $40?

Have a go at solving the problem and see what changes and what stays the same.

See if you can find a way to write a number sentence algebraically so someone could use your number sentence to work out how they need to save no matter what the cost of the t-shirt

Figure 5.2 Solution strategy for CD player problem21

Algebrafied word problems provided a context for students to give conceptual explanations. For example, Zhou identified the constant in the equation: The nine because that’s how much you have in the bank. Their explanations of the unknowns also linked to the context of the problem. Gareth said: The amount that you have to save, Sabrina added to the explanation: The cost of the T-shirt. Consistently the students began to link the context of the problem to their explanation. If procedural explanations were used, the students were pressed to provide a more conceptual response. The following vignette illustrates how Ella facilitated the students to collaboratively construct a conceptually focused explanation of their algebraic number sentence and use of notation.

Construction of a conceptually focused explanation of an algebraic number sentence

Students were working in a small group of four to solve the CD player problem.

Bridget He went like this, five. Ella What's the five?

Bridget How much he gets paid. Ella How much he gets paid?

Caitlin/Ruby How much he gets paid every hour. Bridget And then he did times the triangle.

21 _{You would like to buy a CD player that costs $35. You earn $5 an hour at your job. How many hours do}

you need to work?

What about if the CD player costs $45 or $60 or $80?

Have a go at solving the problem and see what changes and what stays the same.

See if you can find a way to write a number sentence algebraically so someone could use your number sentence to work out how many hours they need to work no matter what the cost of the CD player. (adapted from Blanton & Kaput, 2003)

Ella What's the triangle?

Heath It's how much the CD player costs, oh no that's how much you need to earn an hour.

Ella questions this response and the students discuss what the triangle represents. Ruby It's the hours, isn't it?

Caitlin No because that's the end.

Ruby No I think hours isn’t the end, I think it's after.

Heath Five hours…for example you need to times seven…you always times the answer which means that’s how much hours that you need to do it and then the answer is how much the CD player costs.

Ella asks the students to have some reflective thinking time and re-voices Heath’s explanation.

Bridget I agree because that's how much per hour, and how many hours and that's how much the CD player is [points at specific parts of the problem and the equation the group has written].

Caitlin But if you do the divideds, the hours, how much the thing is, is at the start, how much the CD player, and then how much you get per hour and then how much you need, how many hours you need to work before you get that much money, to get the CD player.

(Lesson 6)