The purpose of this unit is to support students to represent number problems as equations, and to strategically choose the best operation to solve problems in context.
Linear equations are equations which, when plotted on a graph, create a straight line.
In these types of equations, an unknown amount or number can be represented with a symbol (e.g. the letter “x”). The process of finding these unknown values is foundational knowledge for a good understanding of algebra at Level 5 of the New Zealand Curriculum. Key to this knowledge is the understanding that the equals sign communicates equivalence between two amounts. Often, students perceive the equals sign as meaning “the answer is”. It is important to build on this understanding using written equations which show that the amount on the left hand side of the equals sign is the same as the amount on the right hand side. Single-step finding of unknowns can be classified in this helpful way:
Operation | Result unknown | Change unknown | Start unknown |
Addition | 4 + 6 = [ ] | 4 + [ ] = 10 | [ ] + 6 = 10 |
Subtraction | 10 - 6 = [ ] | 10 - [ ] = 6 | [ ] - 6 = 4 |
Multiplication | 3 × 5 = [ ] | [ ] × 5 = 15 | 3 × [ ] = 15 |
Division | 15 ÷ 3 = [ ] | 15 ÷ [ ] = 5 | [ ] ÷ 3 = 5 |
To solve these equations, knowledge of inverse operations is needed. Addition and subtraction are inverse operations, and so are multiplication and division. Students may find it helpful, in this context, to think of inverse operations as “cancelling” each other out.
To engage students in the context of linear equations, it is important they have opportunities to solve problems that reflect relevant real-life situations.
For example, 4 + [ ] = 10 might be framed as “Sid has 4 apples. He picks some more apples and now he has 10 apples. How many apples has he picked?” The problem could be solved as subtraction, 10 - [ ] = 4, but the context is one of joining sets of apples.
Varying the location of the unknown substantially changes the difficulty of the problem, assuming the numbers are similar. To scaffold your students in their understanding of this, you might refer to “change” unknown, “result” unknown, and “start” unknown problems. Result unknown is significantly easier than change unknown and start unknown. Generally students are more familiar with result unknown so it is important to ensure they are meeting many examples of the latter two.
Change unknown problems, such Sid’s original apple picking problem, involve considering the possibilities for change. They can be expressed as a + ? = b
"Result" unknown problems involve using given values to find a total amount. They can be expressed as a + b = ? .
"Start" unknown problems require inverse thinking since there is no beginning state. For example, “Sid has some apples. He picks six apples and now he has 10 apples. How many apples did Sid have to start with?” This "start" unknown problem changes the operation to subtraction. This can be represented as ? = 10 - 6. This involves reconceptualising the role of the whole and parts.
To support knowledge of these different types of equations, students should experience opportunities for estimating values and explaining their reasoning. Calculators and materials (e.g. laminated number lines, counters) may be a useful tool to use within this stage. When using a calculator, it is important to emphasise that the use of correct operations is essential.
For some students, algebra may be a source of anxiety. However, algebra should be celebrated as an important means of communicating mathematical statements. To support students in their learning of the concepts in this unit, you may adapt the learning opportunities provided, by removing or adding support for students and by varying the task requirements. Students need to focus on the decision-making process of which operation/s to use. While the development of calculation strategies can be facilitated through solving the problems, that is not the primary purpose. Ways to support students include:
The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. A menu from a school cafeteria, or tuck shop, and a fruit and vegetable shop provide the contexts of the unit. You may need to show students a video clip of lunch schemes in larger schools or overseas to help them appreciate the usefulness of the context. If the food-based contexts are culturally inappropriate to your students, change the items to those that match the everyday situations students regularly encounter, e.g. prices at the school fair, the cost of new equipment for the sports-shed, the cost of food for a community hui (meeting), the cost of new native plants for a school garden. When utilising money as a context for learning, it is also important to recognise that individuals have different experiences of, and perspectives towards money.
Te reo Māori vocabulary terms such as tāpiri (add, addition), tango (subtract, subtraction), huatango (difference), whakarea (multiply), whakawehe (divide, division) and ōrite (equal, same) could be introduced in this unit and used throughout other mathematical learning.
In this session students use the clues provided to find out the prices of items on a menu. The clues involve applying addition and subtraction with unknowns.
In this session students explore "change" and "start" unknown addition and subtraction problems. They should record their solution strategies as equations with specific unknowns, and should recognise when either addition or subtraction can be used to solve a problem.
Answers to Copymaster questions:
In this session students represent change and start unknown problems with multiplication and division. They do so in the context of the Kiwi School Cafeteria used in the previous two lessons. The focus in this session is more on making sensible decisions about the operations to perform than on strategies for calculation. ‘Unfriendly’ amounts are used for the prices and students are required to choose appropriate operations to solve the problems.
In this session students explore situations in which more than one operation is involved, and they learn to discriminate between additive and multiplicative situations. To allow for more flexibility the context is changed from a canteen to a fruit and vegetable store. At this stage, the context could be changed to further reflect the cultural diversity of your students. For example, students could remake Copymaster 5 with foods of their choice, and use the provided questions to plan a menu for a shared celebration involving kai. An investigation into traditional hākari (feasts) could ignite interest in this learning. In te ao Māori, different times of the year (e.g. the kūmara harvest in March) were celebrated with different hākari. To extend more knowledgeable students, hardcopy (or online) supermarket pamphlets could be used to provide more complex numbers for the different food items.
Dear parents and whānau,
In mathematics this week, our focus has been on writing and solving equations. Students are applying their understanding that an equation represents a balance, or sameness. The equals sign represents that two amounts on each side of it are the same, though the operations may look different and represent different operations.
Have your child explain their common sense understanding of these two problems, how they would solve the equation, and justify why their strategy works. Using a calculator is fine, provided your student can explain why they chose the operation they did.
Find the number that goes in the box to make each equation correct. |
[ ] - 27 = 51 |
7 × [ ] = 161 |
300 ÷ [ ] = 12 |
Listen to your student’s explanations, and ask questions if the strategies don’t make sense to you. Compliment them when their explanation is clear, and their solution is efficient and correct.
Thank you
Printed from https://nzmaths.co.nz/resource/food-thought-using-equations at 2:00am on the 25th April 2024