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Year 7 activity - Exploring the need for a convention when rounding decimal numbers
In this activity, students are supported to develop an understanding of why and how we round numbers to a given number of decimal places.
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Key objectives: Numbers and the number system
Place value, ordering and rounding
Round positive whole numbers to the nearest 10, 100 or 1000 and decimals to the nearest whole number or one decimal place.
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Preparation:
Display the Number line software using a data projector (and interactive whiteboard if available).
If laptops are available, students could use the software to explore the effects of rounding for themselves.
Number line setup: zero to 1 with 5 scale marks and select 1 decimal place. Display the decimals only.
Additional resources: A3 laminated number lines and pens, A4 paper copies of the number line on which students could record their work.
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Activity
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Questions to ask students
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| With the value of n hidden, move the point n to about 0.14
Invite students to move n to different positions on the number line and follow a similar line of questioning.
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Look carefully at the number line… What do you estimate the value of n to be?
If we were going to write the value of n to only 1 decimal place, how could we decide what to write?
Does the value of n look closer to 0.1 or 0.2?
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| Move n to a point such as 0.35 to discuss with the mathematical convention of 'rounding-up'. |
How are we going to decide what to write if we are rounding a number that is exactly halfway between, say 0.3 and 0.4 to 1 d.p.?
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Give students the opportunity to choose some numbers of their own and round them to one decimal place, recording their work on paper.
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| Reveal the value of n and invite a student to slowly drag the point n along the line.
Use this feature to support students to check their previous solutions.
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Look carefully at the value of n that is being displayed. What do you notice?
Where does its value change?
Why?
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Year 8 activity - Exploring equivalent algebraic relationships
In this activity, students are supported to make conjectures about equivalent algebraic expressions that involve brackets.
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Key objectives: Algebra
Equations, formulae and identities
Begin to distinguish between the different roles played by letter symbols in equations formulae and functions. Simplify or transform linear expressions by collecting like terms; multiply a single term over a bracket.
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Preparation:
Display the Number line software using a data projector (and interactive whiteboard if available). If laptops are available, students could use the software to explore the effects of rounding for themselves.
Number line setup: -10 to 10 with no scale marks and select 2 decimal places. Display the decimals only. Away from the pupils view, define a function a=2n+2.
Additional resources: Resource sheet (see example)
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Activity
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Questions to ask students
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| Display the Number line tool with the relationship a=2n+2 already defined.
Take several suggestions and record them for everyone to see.
Choose one of the suggestions, for example, a = n+2.
Note: Avoid the correct one at this stage! Invite a student to move n so that n=1. Discuss with students why, although the rule a = n+2 was true when n = 0, as it is no longer true when n = 1.
Drag n to various positions on the line and build a table of values of n and a. For more able year 8 groups, include decimal and negative numbers. Support students to establish that the actual relationship is a = 2n+2 and reveal the calculator to them to how the relationship was created.
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Would anyone like to suggest a relationship that might connect (or link) the values of n and a?
If the rule a = n+2 was true, what would you expect the value of a to become if we move n so that it equals one?
Do we think that the rule a = n+2 is true now?
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| Set the task for students, which is to substitute different vales of n into expressions to determine which appear to give the same result, always, sometimes and never.
Choose the expressions to provoke discussion amongst students.
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What do you notice about your results?
Which expressions do you think could be equivalent?
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| Use the Number line tool to test the students’ conjectures.
Invite students to suggest pairs of expressions.
Define the first expression as a and the second expression as b.
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If the expressions are equivalent, what will happen as we drag n along the line?
How can we check that they are equal if we can only see one of the points?
(Either reveal the values of a and b to show they are equal or switch off a or b from theOptions menu)
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| Use the Number line tool to confront and discuss the common misconception that 2n and n2 are equivalent. |
How could we use the Number line tool to prove that n2 and 2n are not equivalent, although they are equal when n = 0 and n = 2?
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Student Task Sheet
Exploring equivalent expressions
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2(n+1) |
n2 |
2n |
2n+2 |
3+5n–3n-1 |
| n = 2 |
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| n = 3 |
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| n = 0 |
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| n = 0.5 |
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| n = |
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