2907
/www/nrich/html/content/id/2907/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Mrs Trimmer's class had been drawing different shapes with straight sides. On Tuesday the sun was shining and Mrs Trimmer took all twenty-four of the children out into the playground. She also took some long loops of string.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt=" a loop of string" height="146" src="string1.gif" width="227"></mdo:image></div>
<br></br>
<br></br>
Mrs Trimmer held up one of the string loops. "How many of you will we need to make a triangle?" she asked.<br></br>
<br></br>
<div class="toggle">They said "Three!"</div>
<br></br>
<br></br>
She chose Ellie, Winston and Andy. They held the string taut and so made a beautiful triangle.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="3 children holding string to make triangle" height="327" src="string2.gif" width="347"></mdo:image></div>
<br></br>
<br></br>
Then other children made triangles, squares, rectangles, pentagons and hexagons. Some made regular polygons and others made more irregular shapes.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="irregular hexagon out of loop of string" height="57" src="string3.gif" width="249"></mdo:image></div>
<br></br>
<br></br>
<br></br>
Six of the class made a shape. Nick pointed at it. "That's nearly a triangle!" he laughed. Mrs Trimmer came up. "It's still a hexagon," she explained, "It's got six sides and six people holding the corners."<br></br>
After a while Mrs Trimmer called all the twenty-four children together. "Now we are all going to make triangles," she said, "So get into threes." They made lots of different ones. Some looked like these:<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="different triangles" height="48" src="string4.gif" width="268"></mdo:image></div>
<br></br>
<br></br>
If all the children were making a triangle, how many triangles did they make altogether?<br></br>
<br></br>
Then the children made four-sided shapes.<br></br>
What different shapes could they have made?<br></br>
Can you draw some of them?<br></br>
How many four-sided shapes did the class make altogether if all the children were involved?<br></br>
<br></br>
Then the children made hexagons and then octagons.<br></br>
<br></br>
How many hexagons and how many octagons could the class make?<br></br>
<br></br>
"We haven't made pentagons yet, Mrs Trimmer," complained Nick.<br></br>
How do you think they managed to make five good pentagons?<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p class="editorial">This was an engaging problem. It encouraged
you to think about factors and multiples and showed how these
principles may be applied to real-life situations! It also prompted
you to think about different kinds of shapes.</p>
<p class="editorial">Mrs. Heffernan's students from Paton School,
Shrewsbury, Massachusetts, USA targeted this problem using the
"chunking" method of division. There are twenty four students, and
one group of three students makes a triangle. So, to find out the
number of triangles they can make, we need to know how many groups
of three there are.</p>
<p class="editorial">Caitlin, from Deer Hill School worked out the
correct answer, also noting that no-one should be left out. Nadia
from Wimbledon High School and Ryan from Rhu Primary both sent
excellent solutions to this problem. Nadia says:</p>
<div>Well for the triangle one the answer is eight because there
are $24$ children and you need three children for each triangle so
you divide $24$ by $3$ and you get $8$.</div>
<br></br>
Then for the question that asks you what different shapes could be
made the answer is a square, a rectangle, a diamond and many
more.<br></br>
<p><span class="editorial">It might be better not to use the word
"diamond" in maths - can you think of the mathematical names of
shapes that we might think look like a diamond? Ryan suggests that
parallelograms are possible too. What other four-sided shapes could
we make?</span></p>
<p class="editorial">Carolyn, from Pigeon Mountain Primary School
suggested some four-sided shapes, including parallelograms, and
rhombuses. Priya and Ashleigh, from Penrhos College also mentioned
a trapezium for another four-sided shape. Mrs. Heffernan's students
pointed out that a rectangle is actually a type of parallelogram;
it is a special version, because all of the angles are
$90^{\circ}$. Similarly a square is a special type of rhombus, with
right angles.</p>
<p class="editorial">The problem then asks about the numbers of
other shapes that the class can make. Again, Mrs. Heffernan's class
used the same method ("chunking") to find the answers. Nadia, and
Ryan also submitted correct solutions. Ryan explains:</p>
<div>They can make six four-sided shapes ($24\div 4=6$) , four
hexagons ($24\div 6=4$) and three octagons ($24\div 8=3$). To make
five pentagons Mrs. Trimmer can help.</div>
<p class="editorial">Like Ryan, Nadia also thought that Mrs Trimmer
could join in - that's a good idea. Sarah from Greenlands Secondary
School also suggested this, as did Matthew from Stambridge.
Caitlin, and Margaret from Deer Hill School, and Ryan and Trystan
also sent in correct solutions, with great
reasoning.</p>
<p class="editorial">Kieran from Newman Primary School, and Ebony
from Gordon Primary suggested that one child could hold two corners
of a pentagon, which would be another good way around the
difficulty. Priya and Ashleigh suggested that four children could
be left out so that four pentagons could be made.</p>
<p class="editorial">Can you think of any other ways that the class
could make shapes with the string? What if some people (or even all
of them!) held two pieces of string, one in each hand? What shapes
could they make? How many?</p>
<p class="editorial">Well done, and thank you for
your solutions.</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Mrs Trimmer's String</h2>
<br></br>
Mrs Trimmer's class had been drawing different shapes with straight sides. On Tuesday the sun was shining and Mrs Trimmer took all twenty-four of the children out into the playground. She also took some long loops of string.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt=" a loop of string" height="146" src="string1.gif" width="227"></mdo:image></div>
<br></br>
<br></br>
Mrs Trimmer held up one of the string loops. "How many of you will we need to make a triangle?" she asked. She chose Ellie, Winston and Andy. They held the string taut and so made a beautiful triangle.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="3 children holding string to make triangle" height="327" src="string2.gif" width="347"></mdo:image></div>
<br></br>
<br></br>
Then other children made triangles, squares, rectangles, pentagons and hexagons. Some made regular polygons and others made more irregular shapes.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="irregular hexagon out of loop of string" height="57" src="string3.gif" width="249"></mdo:image></div>
<br></br>
<br></br>
<br></br>
Six of the class made a shape. Nick pointed at it. "That's nearly a triangle!" he laughed. Mrs Trimmer came up. "It's still a hexagon," she explained, "It's got six sides and six people holding the corners."<br></br>
After a while Mrs Trimmer called all the twenty-four children together. "Now we are all going to make triangles," she said, "So get into threes." They made lots of different ones. Some looked like these:<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="different triangles" height="48" src="string4.gif" width="268"></mdo:image></div>
<br></br>
<br></br>
If all the children were making a triangle, how many triangles did they make altogether?<br></br>
<br></br>
Then the children made four-sided shapes.<br></br>
What different shapes could they have made?<br></br>
Can you draw some of them?<br></br>
How many four-sided shapes did the class make altogether if all the children were involved?<br></br>
<br></br>
Then the children made hexagons and then octagons.<br></br>
<br></br>
How many hexagons and how many octagons could the class make?<br></br>
<br></br>
"We haven't made pentagons yet, Mrs Trimmer," complained Nick.<br></br>
How do you think they managed to make five good pentagons?</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=2907&part=index">This problem</a> gives children opportunities to explore the properties of $2$D shapes but also to apply their knowledge of dividing into equal groups.<br></br>
<br></br>
<h3>Possible approach</h3>
<div>The activity could be introduced by involving the whole class in the story of Mrs Trimmer and her children. The questions in the story could be raised in a simliar order. You will need plenty of room and a large loop of rope or string for the whole class. After that the children could be practically engaged in small groups of three to six children each with a loop of string about $3$ metres
in length and consider the questions themselves as well as ways of recording their answers. You could pose the questions in the problem orally for them to investigate using loops of string, encouraging them to make conjectures and justify them. You could give each group some paper to record their shapes, or you may want to take photographs of the children as they experiment.</div>
<br></br>
<div>You could then return to the classroom to discuss their findings. Their answers will depend on the number of children who held the string to make the shapes and this can lead into discussions about the underlying mathematics of factors by considering how many triangles/quadrilaterals could be made with this number of children. Encourage the children to use accurate descriptive language to
explain what they see so the vocabulary of sides, corners (vertices) and angles can be introduced.</div>
<br></br>
<div>(<a href="/content/id/2907/MrsTrimmer.pdf">This sheet</a>, which contains all the questions asked but has a shortened introductory part and no illustrations, may be useful if you wish children to have paper copies of the problem as written.)</div>
<br></br>
<h3>Key questions</h3>
<div>How many children are needed for one triangle? Then how many children would be needed to make two triangles? How many triangles can we make at the same time with our class? What is the same/different about these two triangles/quadrilaterals/shapes?</div>
<div>Why don't you use counters to help?</div>
Can you think of the names of any other shapes with four sides?<br></br>
<div>How many sides has a pentagon got? How many can we make with our class if everyone holds one corner each? Can you think of a way we could make five pentagons?</div>
<br></br>
<h3>Possible extension</h3>
Learners could try one of these Stage 2 problems, <a href="http://nrich.maths.org/public/viewer.php?obj_id=79&part=index">Bracelets</a> or <a href="http://nrich.maths.org/public/viewer.php?obj_id=1058&part=index">Where Are They?</a>.<br></br>
<br></br>
<h3>Possible support</h3>
If it is not possible to work on the problem practically using string, some children might find it useful to make the shapes with some apparatus, such as geostrips. Alternatively, they could draw the triangles and number the corners to show the children or use counters to group together to represent the children or to represent the corners of the shapes.<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
How many children are needed for one triangle? How many children
would be needed to make two triangles then?<br></br>
So if there are twenty-four children, how many triangles can they
make at the same time?<br></br>
You could try drawing the triangles and numbering the corners to
show the children.<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p class="editorial">This was a fun problem. It encouraged you to
think about factors and multiples, and showed hoe these principles
may be applied to real-life situations! It also prompted you to
think about the different kinds of shapes there are.</p>
<p class="editorial">Mrs. Heffernan's Students from Paton School
Shrewsbury Massachusetts USA targeted this problem using the
"chunking" method of division. There are 24 students, and one group
of three students makes a triangle. So, to find out the number of
triangles they can make, we need to know how many groups of three
there are.</p>
<p class="editorial">Caitlin, from Deer Hill School worked out the
correct answer, also noting that on-one should be left out. Nadia
from Wimbledon High School and Ryan from Rhu Primary both sent
excellent solutions to this problem. Nadia says:</p>
<div>Well for the triangle one the answer is eight because there
are $24$ children and you need three children for each triangle so
you divide $24$ by $3$ and you get $8$.</div>
<br></br>
Then for the question that asks you what different shapes could be
made the answer is a square, a rectangle, a diamond and many
more.<br></br>
<p><span class="editorial">It might be better not to use the word
"diamond" in maths - can you think of the mathematical names of
shapes that we might think look like a diamond? Ryan suggests that
parallelograms are possible too. What other four-sided shapes could
we make?</span></p>
<p class="editorial">Carolyn, from Pigeon mountain primary school
suggested some four sided shapes, including parrallelograms, and
rhombuses. Priya and Ashleigh, from Penrhos College also mentioned
a trapezium for another four-sided shape. Mrs. Heffernan's Students
pointed out that a rectangle is actually a type of parrallelogram;
it is a special version, because all of the angles are
$90^{\circ}$. Similarly a square is a special type of rhombus, with
right angles.</p>
<p class="editorial">The problem then asks about the numbers of
other shapes that the class can make. Again, Mrs. Heffernan's class
used the same method ("chunking") to find the answers. Nadia, and
Ryan also submitted correct solutions. Ryan explains:</p>
<div>They can make six four-sided shapes ($24\div 4=6$) , four
hexagons ($24\div 6=4$) and three octagons ($24\div 8=3$). To make
5 pentagons Mrs. Trimmer can help.</div>
<p class="editorial">Like Ryan, Nadia also thought that Mrs Trimmer
could join in - that's a good idea. Sarah from Greenlands Secondary
School also suggested this, as did Matthew from Stambridge.
Caitlin, and Margaret from Deer Hill School, and Ryan and Trystan
also submitted correct solutions, with great reasoning.</p>
<p class="editorial">Kieran from Newman Primary School, and Ebony
from Gordon Primary suggested that one child could hold two corners
of a pentagon, which would be another good way around the
difficulty. Priya and Ashleigh suggested that four children could
be left out so that four pentagons could be made.</p>
<p class="editorial">Can you think of any other ways that the class
could make shapes with the string? What if some people (or even all
of them!) held two pieces of string, one in each hand? What shapes
could they make? How many?</p>
<p class="editorial"></p>
<p class="editorial">Well done, and thank you for the submitted
solutions!</p>
<br></br></mdoxml>
2
4
1
0
0
0
0
Mrs Trimmer's String
Can you help the children in Mrs Trimmer's class make different
shapes out of a loop of string?
Numbers and the Number System
Factors and multiples
2D Geometry, Shape and Space
Octagons
2D Geometry, Shape and Space
Hexagons
2D Geometry, Shape and Space
Pentagons
2D Geometry, Shape and Space
Quadrilaterals
2D Geometry, Shape and Space
Mixed triangles