Similarity and congruence – Grade 10 math lesson

Lesson 12 of 16 in the Grade 10 math course

US version. US grades, US spelling and dollars ($). · UK Maths Department

A complete 45-minute lesson for grade 10 (ages 15–16) that runs on this page – at school or at home. Every answer is graded right away, with a worked explanation.

Time: 45 minutes + 20 min homework
Grade: Grade 10 (ages 15–16)
You need: Paper, a pencil and this page
Graded: Instantly, with explanations

For parents: how to use this lesson

You don't need to know this topic yourself. The lesson explains each step on screen, one tidy step at a time. Every answer is graded straight away with a worked explanation, so you never have to grade anything.

Read what your child will learn and the big ideas together.
Press Start the lesson below and sit with your child for the teaching and the worked examples (about 19 minutes).
Let them do the practice on their own. They choose Getting started, On track or Challenge – start on On track and move up after a few right in a row.
Finish with the exit ticket and the talk question: “If you double every length of a box, what happens to its volume?”
Later that day or the next day: the homework (20 minutes), then a practice test.

Watch out for

Matching the wrong sides. Match sides opposite equal angles.
Using the length scale factor for area. Square it for area, cube it for volume.

Tip for home: Compare model cars or toys with the real thing using scale factors.

What your child will learn

Learning goal: use similar and congruent shapes to find missing lengths, areas and volumes.

Success looks like

I can show triangles are congruent.
I can find lengths in similar shapes.
I can use area and volume scale factors.

Key words

congruent: exactly the same size and shape
similar: same shape, different size
scale factor: how many times longer each length is
corresponding: in the same position on both shapes

What to have ready

Paper and a pencil
This page on a computer, tablet or phone
For the hands-on part: Photograph an object at two sizes and check that all lengths changed by the same factor.

The big ideas (for the adult)

Congruent shapes are identical in size and shape. Triangles are congruent by SSS, SAS, ASA or RHS.

Similar shapes have equal angles and lengths in the same ratio. The scale factor is new length ÷ old length.

For similar shapes, areas scale by the square of the length scale factor and volumes by its cube.

The lesson, step by step

45 minutes in 5 parts
Part	Time	How it is checked
Starter: quick recall	5 min	Graded by Classfern
Teaching input	10 min	Not assessed, self-checked
Worked examples and guided practice	9 min	Not assessed, graded by Classfern
Independent practice	15 min	Graded by Classfern
Closure and exit ticket	6 min	Graded by Classfern, self-checked

Starter, "your turn", practice and exit ticket are graded by Classfern. Worked examples and teaching are not assessed. The hands-on task and the reflection are self-assessed. Explanations in talk are judged by the parent and not recorded.

The lesson

It runs here on the page. No sign-up and no names – progress is kept only in this browser.

Homework and practice at home

20 minutes: 10 practice questions, 3 multiple-choice questions and 2 to remember from earlier work, then a short reflection. New questions every time, all graded instantly.

More practice

All Grade 10 math lessons and tests · Math Department

More Grade 10 math lessons

Curriculum

Common Core State Standards HSG.SRT.B.5: “Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.”

Common Core State Standards HSG.CO.B.7: “Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.”

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