The unit involves students in solving problems that can be modeled with algebraic equations or expressions. Students are required to describe patterns and relationships using letters to represent variables.
Algebraic thinking is about generalisation, describing something that consistently occurs in a particular situation. In algebra, letters are used to express generalisations. Einstein’s famous equation e=mc2 expressed a relationship about the amount of energy, the mass of matter, and the speed of light.
Letters may be used to express an important number. In Einstein’s equation c represents the speed of light, a number. Letters can also refer to a specific unknown. For example, “Mere gathers some pipi. Rawiri gives her seven more pipi. Now she has 18 pipi in her bucket.” x + 7 = 18 expresses the unknown number of pipi she first collected.
Letters are used more powerfully to express generalised relationships as Einstein did. The equation v=d/t expressed velocity as a function of distance divided by time. All of the letters refer to variables, quantities that can change relative to each other.
Algebra is loaded with conventions about how unknown variables are expressed. For example, 5x refers to x multiplied by five, t2 refers to t multiplied by itself, and n/m refers to the division of n by m. Decoding algebra is similar to learning to read. Students need lots of experience linking symbols with the meaning of those symbols.
Students can be supported through the learning opportunities in this unit by differentiating the nature and complexity of the tasks, and by adapting the contexts. Ways to support students include:
Tasks can be varied in many ways including:
This unit is focussed on learning how to use algebraic expressions and equations, and the teaching sessions are not set in a real world context. You may wish to explore real world applications of algebra in the teaching sessions following the unit, for example bags of ingredients going into a manufacturing machine, servings of food at a hāngī, or ticket prices for an event.
Make up a set of three containers by trimming down 1.5 litre plastic soft drink bottles. It is preferable that these containers are of the same type. Straight-sided bottles are better than curved ones as they make it easier for the students to predict the relationships. The three cut down containers should hold different amounts of water when full. Make their capacity no more than 500 ml and ensure that these capacities are not multiples of each other, e.g. 120 ml, 230 ml, and 400 ml would be better than 100 ml, 200 ml, and 400 ml. Label the containers p, q, and r respectively.
Students could make up their own set of containers, label them with algebraic letters, and develop challenges for each other. Alternatively, the class containers could be left as a station for students to use independently.
In this session students look for patterns within each equation set and use these patterns to predict further equations in the set. They may do this using recursion, that is finding a relation between consecutive equations, rather than by looking for relationships within the equations across the equals sign. Highlight relationships that might be found between the numbers in each set of equations and encourage the students to look for ways to describe these relationships. It is important that students find the unknowns using mental calculation rather than with calculators, as their attention needs to be on the relative size of numbers.
Some learners will need support from physical materials to notice and describe the patterns. Connecting cubes in stacks make a useful representation.
Below are some suitable equation patterns:
Students solve “What’s my Number?” problems and record how they found the final answer. At this stage trial and improvement are legitimate strategies though the problems encourage students to attend to structure, and apply their understanding of inverse operations.
Flowcharts can support students to apply inverse operations. Consider the problem; “Take my number, multiply it by three then subtract 4. The answer is 20. What is my number?” The problem is equivalent to 3x - 4 = 20.
A flowchart looks like this:
Applying inverse operations gives:
In this session students work out the functional rule for given input and output numbers. The functions increase in complexity as the week progresses. Each example offers the students three input/output pairs. to the right of these three pairs are three other pairs (typed in bold) that could be used if needed. Copymaster 1 can be used to make stimulus cards by cutting and folding the output of the three additional pairs behind so they can not be seen unless needed.
Discuss how a rule might be found.
If the output numbers are always more than the input numbers what types of rules do you try first?
Students are likely to suggest addition or multiplication. That strategy is useful but needs to be qualified. Multiplying by a fraction less than one or an integer will result in smaller output numbers.
How can you check if only addition is involved?
The difference between input and output numbers must be constant. That generalisation is true of subtraction as well.
How can we check if only multiplication is involved?
The differences between input and output numbers change. That will not work as a strategy. Students will need to estimate what the multiplier is, possibly from two pairs, then check to see if the multiplier applies to the other pairs.
If division is in the rule, how can you tell?
Students usually say that division, like subtraction, makes the output number smaller. That generalisation is naïve since division with a fraction between zero and one makes the output number larger. However, provided the divisor is larger than one the claim that “division makes smaller” holds. As with multiplication, the divisor will need to be estimated first then tried on the input-output pairs.
Challenge students to come up with their own tables of values. You might set up a spreadsheet of tables as a template, with hidden rules. Alternatively create a paper template as seen in Copymaster 1.
These activities involve students in working out the number of counters or cubes that are in each cup of a given colour. Several clues are provided, and students must combine these clues to find a solution. Use opaque coloured plastic cups, that are readily available in supermarkets, warehouses, and dollar stores. For each problem all students should solve it, then record their reasoning, before the solution is “revealed.”
Encourage students to check their solutions by putting the values they assign to each cup back into the original clues. Do all the clues work? If not, try again!
Examples of cups and counters problems follow. The first sentence is the instruction for the teacher and gives the answer. It does not appear on (Copymaster 2) as students use the clues to work out how many cubes are in each coloured cup.
Dear parents and whānau,
This week we have been looking at equations with unknowns. We are learning to express relationships using the language of algebra, particularly using letters to represent amounts that we do not know.
Try the following with your child:
Write 8 on a small piece of paper and 4 on another piece. Hold the numbers in different hands so that your student can’t see them. Say, “One number is twice the other, and both numbers add to 12. What numbers are in my hands.”
Get your child to make a problem like this for you. He or she might share the problems they are solving in class.
Enjoy challenging each other!
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/balancing-acts at 8:41pm on the 26th February 2024